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Determination of Risk Aversion and Moment-Preferences: A Comparison of Econometric models

DISSERTATION der Universit¨ at St.Gallen, Hochschule f¨ ur Wirtschafts-, Rechts- und Sozialwissenschaften (HSG) zur Erlangung der W¨ urde eines Doktors der Wirtschaftswissenschaften

vorgelegt von

Fabian Wenner von Murten (Freiburg) und aus Deutschland

Genehmigt auf Antrag der Herren Prof. Dr. Klaus Spremann und Prof. Dr. Alex Keel

Dissertation Nr. 2606 Difo-Druck GmbH, Bamberg 2002

Die Universit¨at St. Gallen, Hochschule f¨ ur Wirtschafts-, Rechts- und Sozialwissenschaften (HSG) gestattet hiermit die Drucklegung der vorliegenden Dissertation, ohne damit zu den darin ausgesprochenen Anschauungen Stellung zu nehmen.

St. Gallen, den

Der Rektor:

Prof. Dr. Peter Gomez

To my parents Gerhard Wenner and Cornelia Wenner-Delos´ea and my wife Beatrice Zellweger

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Acknowledgements I would like to thank Professor Takeshi Amemiya and Professor Klaus Spremann for their patient supervision and their ingenious advice. Without their encouragement, their guidance, confidence and their academic example this thesis could not have been written. Profound gratitude goes to my coexaminer Professor Alex Keel who made me familiar with the fundamentals of statistics during my studies and spontaneously and genuinely agreed to support and supervise my thesis. Professor Spremann has been very supportive since the very beginning of my studies. His academic gatherings in V¨attis and the accompanying discussions provided fertile ground for the development of new ideas - one of their products being the thesis on hand. I am also much indebted to Prof. Takeshi Amemiya for giving me the opportunity of a stimulating and very hospitable stay at the Economics department at Stanford University. It was an academic experience that will accompany me for the rest of my life. Inexpressible is my gratitude for the help and support, the love and care of my parents. My mother’s and sister’s love for languages not only urged me to write this thesis abroad, it also cut them out for the ungrateful job of correcting this manuscript. Many thanks on that score, especially to my Mum. For valuable and helpful discussions I am indebted to Kenneth Arrow, Andreas Dische, Luigi Pistaferri, Clemens Sialm, Ed Vytlacil, Alexandre Ziegler. Valuable and ingenious input during a presentation of my paper also came from Whitney Newey and James Powell. I’d also like to thank Stephanie Winhart for helping me with evaluating a survey on risk taking in the early i

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stage of this thesis. All errors are mine alone. Financial support from the Swiss National Fund for my one-year stay at Stanford University is gratefully acknowledged. It considerably helped to focus on academic research in Silicon Valley in a time when its real estate ‘bubble’ was at its peak. Many thanks also go to Camilla and Antonio Cozzio who welcomed us heartily in our new environment in Palo Alto and made us a part of their family during our stay in the bay. Last but definitely not least I would like to thank the person most essential for the stepwise development of this thesis: Beatrice has walked along with the ups and downs of its growth path, patient but always urging on, cheering on the way, listening to partly obscure thoughts and making the time working on it worth living. Zurich, Summer 2001. Fabian Wenner

Table of Abbreviations Abbreviation

Meaning

AIC αa , αAP αi αr appe ARA BNL CAPM CD CE CLM CLT CPPI CRRA DCM ε ER EU EVD FS GBM Govnmt IID

Akaike Information Criterion Pratt-Arrow measure of absolute risk aversion Measure of relative risk aversion for investor i Pratt-Arrow measure of relative risk aversion Average per period exposure Absolute risk aversion Binomial Logit Model Capital Asset Pricing Model Certificate of Deposit Certainty Equivalent Conditional Logit Model Central Limit Theorem Constant proportion portfolio insurance Constant relative risk aversion Discrete Choice Model Error term Expected Return Expected Utility Extreme Value Distribution Fisher skewness = m3 /σ 3 Geometric Brownian Motion Government Independently and Identically Distributed continued on next page iii

iv

Abbreviation

Meaning

IRA j k KEOGH

Individual Retirement Account Subscript representing the risk category Subscript representing the independent factor Tax-deferred pension account for employees of unincorporated businesses or for persons who are self-employed. Likelihood function Log-Likelihood function Linear Probability Model Likelihood Ratio Test Non-normalized third central moment of the return distribution Multinomial Logit Model Subscript representing the observation of the sample Negative Exponential Utility (Null hypothesis) cannot be refuted Ordinary Least Squares Probability Probability Distribution Function Density function of the standard normal variable Distribution function of the standard normal variable Risk premium Skewness preference of investor i, defined as W 2 · U  /U  Dividend yield of index option Simple return Riskfree rate Risk Aversion (Null hypothesis) can be refuted Skewness or non-normalized third central moment Standard Deviation of a random variable Total dollar value Time invariant portfolio protection Total market value continued on next page

L logL LPM LRT m3 MNL n NEU n.r. OLS P PDF φ Φ π ψi q R rf RAV ref. S σ tdv TIPP tmv

Chapter 0. Table of Abbreviations

Abbreviation

Meaning

U W WT WLS yi

Utility Wealth End-of-period Wealth Weighted Least Squares Continuous dependent variable representing the stock ratio of investor i’s portfolio Discrete dependent variable representing the stock ratio j of investor i’s portfolio

Yi (j = 1)

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Contents Acknowledgements

i

Table of Abbreviations

iii

Abstract

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1 Introduction 1.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Earlier assessment methods: overview and critique . . . . 1.2.1 Intuitive approaches . . . . . . . . . . . . . . . . . 1.2.2 Gambles . . . . . . . . . . . . . . . . . . . . . . . . 1.2.3 Derivation of risk aversion in asset pricing models 1.3 Research idea . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.1 Portfolio theory’s old and new facts . . . . . . . . 1.3.2 Overview of models . . . . . . . . . . . . . . . . . 1.3.3 Structuring the Asset Allocation Decision . . . . . 1.3.4 Outline of thesis . . . . . . . . . . . . . . . . . . .

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Two-Moment Risk Preference

2 The CAPM and two moment risk preference 2.1 Assumptions of mean-variance portfolio selection . 2.2 The concept of two-moment risk aversion . . . . . 2.2.1 The Markowitz Premium . . . . . . . . . . 2.2.2 The Pratt-Arrow measure of Risk Aversion vii

1 2 3 3 4 7 10 10 13 14 16

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21 23 25 28 28

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CONTENTS

2.3

2.4 2.5 2.6

Relevance of assessing the correct asset allocation . . . . . 2.3.1 Calculating Expected Utility loss . . . . . . . . . . 2.3.2 All stocks half the time or half stocks all the time? Switching and the time horizon controversy . . . . . . . . Risk aversion and changes in wealth . . . . . . . . . . . . Bounded rationality and goal of methodology . . . . . . . 2.6.1 Fundamentals of bounded rationality . . . . . . . . 2.6.2 Limits of Expected Utility Maximization . . . . . . 2.6.3 Presentation determines investment choice . . . . . 2.6.4 Goal of questionnaire . . . . . . . . . . . . . . . . 2.6.5 General problem of the approach . . . . . . . . . .

3 Empirical Analysis 3.1 The Data set . . . . . . . . . . . . . . . . . . . . . . 3.2 Selection of factors and hypotheses . . . . . . . . . . 3.3 Testing for bounded rationality . . . . . . . . . . . . 3.3.1 Putting things in perspective . . . . . . . . . 3.3.2 ‘Correctly’ assigned observations . . . . . . . 3.3.3 ‘Wrongly’ assigned observations . . . . . . . . 3.3.4 Stated and observed preferences . . . . . . . 3.4 Structure of Analysis and Nests . . . . . . . . . . . . 3.5 Characterization of econometric models . . . . . . . 3.5.1 The Objective Function . . . . . . . . . . . . 3.5.2 Ordinary and Weighted Least Squares Model 3.5.3 The Tobit Model . . . . . . . . . . . . . . . . 3.5.4 The Ordered Logit Model . . . . . . . . . . . 3.5.5 The Binomial- and Multinomial Logit Model 3.5.6 The Conditional Logit Model (CLM) . . . . . 3.5.7 The Nested Logit Model . . . . . . . . . . . . 3.6 Goodness of fit and hypotheses testing . . . . . . . . 3.6.1 Classification Tables and Error Distance . . . 3.6.2 Akaike Information Criterion (AIC) . . . . . 3.6.3 Likelihood ratio tests . . . . . . . . . . . . . . 3.7 Results of regressions for Setting 1 - “All Categories” 3.7.1 Coefficients . . . . . . . . . . . . . . . . . . .

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51 51 55 58 60 62 65 67 68 70 70 73 74 75 77 78 80 81 81 83 83 85 85

CONTENTS

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3.7.2

Results of Classification Tables . . . . . . . . . . . . . . 88

3.8

3.9

Results of regressions for Setting 2 - Two-step estimation . . . 89 3.8.1

Setting 2a - “Assetholder or Non-asset holder” . . . . . 89

3.8.2

Setting 2b - “Assetholders only” . . . . . . . . . . . . . 92

Results of regressions for setting 3 - Three-step estimation . . . 93 3.9.1

Setting 3b - “Stock- or Non-stock holder” . . . . . . . . 94

3.9.2

Setting 3c - “Stockholders only” . . . . . . . . . . . . . 95

3.10 Conclusion of the two-moment setting . . . . . . . . . . . . . . 96 3.10.1 The factors and their explanative power . . . . . . . . . 96 3.10.2 Performance of the econometric models . . . . . . . . . 97 3.10.3 Transferability of results . . . . . . . . . . . . . . . . . . 98 4 Joint estimation by gambles and observed stock ratio

II

101

4.1

Determining Two-Moment Risk Aversion by Gambles . . . . . 102

4.2

Joint estimation of econometric choice and gamble . . . . . . . 104

Three-moment risk preference

107

5 Shortcomings of two-moment asset pricing

111

5.1

5.2

Critique of the mean-variance approach . . . . . . . . . . . . . 111 5.1.1

Quadratic utility . . . . . . . . . . . . . . . . . . . . . . 112

5.1.2

Normal distribution . . . . . . . . . . . . . . . . . . . . 113

5.1.3

Performance measurement of optioned portfolios . . . . 115

Summary of critique . . . . . . . . . . . . . . . . . . . . . . . . 116

6 Skewness

117

6.1

Higher moment preferences . . . . . . . . . . . . . . . . . . . . 118

6.2

Prospect theory and adaptive aspiration . . . . . . . . . . . . . 120

6.3

Gambling and insurance habits . . . . . . . . . . . . . . . . . . 123 6.3.1

Cubic utility and skewness

6.3.2

Implications of the cubic utility . . . . . . . . . . . . . . 123

. . . . . . . . . . . . . . . . 123

6.3.3

Insurance and Gambling . . . . . . . . . . . . . . . . . . 125

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CONTENTS

7 Determining skewness preference through gambles 7.1 Risk premia for three moment approximation . . . . 7.2 Creating gambles for skewness preference . . . . . . 7.2.1 Interpretation of three-moment preferences . 7.2.2 Example for the assessment of risk aversion .

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127 . 128 . 130 . 135 . 136

8 Creating portfolio skewness with options 8.1 Portfolio distribution . . . . . . . . . . . . . . . . . 8.2 Truncating the distribution’s lower end with Puts . 8.3 Enhancing the distribution’s upper end with Calls 8.4 Implementation and Limits of Options strategies .

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141 142 143 153 154

9 Joint estimation of three moment preference

157

10 Summary and Conclusion

159

A Asset Allocation and Higher Moment Models A.1 Optimal Asset Allocation in the 2-Moment Model A.1.1 Objective Function . . . . . . . . . . . . . . A.1.2 Solution of the Objective Function . . . . . A.2 Optimal Asset Allocation in the 3-Moment Model

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163 163 163 166 168

B Deriving the MNL model from Utility maximization

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C Likelihood function of the Nested Logit Model

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D Structure of Estimations

177

E Empirical Results of Regressions 179 E.1 In-sample estimation SCF1998 . . . . . . . . . . . . . . . . . . 179 E.2 Out-of-sample estimation SCF 1995 in SCF1998 . . . . . . . . 215 E.3 Likelihood Ratio Tests . . . . . . . . . . . . . . . . . . . . . . . 247 F Independent Factors

251

G Various Riskrulers 261 G.1 Risk Quota by Fidelity Investments . . . . . . . . . . . . . . . . 261 G.2 Allianz Anleger Analyse . . . . . . . . . . . . . . . . . . . . . . 261

CONTENTS

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G.3 Union Investment . . . . . . . . . . . . . . . . . . . . . . . . . . 261 H Practical implementation of a Risk Ruler H.1 Proceeding when developing a Risk Ruler H.2 Example of an interactive Risk Ruler . . . H.2.1 Small sample Internet survey . . . H.2.2 Calculating the predicted choice . I

Empirical performance of risk classes

J Survey of Consumer Finances: Details

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267 . 267 . 268 . 275 . 275 283 287

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CONTENTS

List of Figures 1.1 1.2

Structuring the asset allocation decision . . . . . . . . . . . . . 15 Structure and outline of thesis . . . . . . . . . . . . . . . . . . 18

2.1 2.2 2.3

Dividing the CML into 6 funds . . . . . . . . . . . . . . . . . . 22 Typical utility function of a risk averter . . . . . . . . . . . . . 26 Skewed gambles and their Pratt-Arrow risk premia . . . . . . . 31

3.1 3.2 3.3 3.4

The data sets: SCF 1995 and 1998 . . . . . . . . Factor coefficients of different risk classes, setting Factor coefficients of different risk classes, setting Factor coefficients of different risk classes, setting

5.1

Distorted performance of option strategies in the CAPM . . . . 116

8.1 8.2 8.3

Lognormal portfolio hedged with puts, floor 85% . . . . . . . . 144 Lognormal portfolio hedged with puts of different strikes . . . . 145 Moments of hedged portfolios . . . . . . . . . . . . . . . . . . . 149

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C.1 Simple 3 response nested logit model . . . . . . . . . . . . . . . 175 D.1 Structure and nests of empirical analyses . . . . . . . . . . . . . 178 G.1 G.2 G.3 G.4

‘Risk Quota by Fidelity Investments’ . ‘Allianz Anleger Analyse’ . . . . . . . ‘Union Investment’ - Answering Sheet ‘Union Investment’ - Evaluation Sheet xiii

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LIST OF FIGURES

H.1 Internet questionnaire: Small Sample Data . . . . . . . . . . . 276 H.2 Internet Survey: Evaluation Sheet . . . . . . . . . . . . . . . . 281 I.1 I.2

Return distribution of portfolios with low stock ratios . . . . . 285 Return distribution of portfolios with high stock ratios . . . . . 286

Abstract The assessment of an individual investor’s risk aversion is an obligatory part of professional investment advice. It is also essential for the financial planning process in general. The assessment approaches of banks, funds and insurances however, usually lack structure, scientific methods and empirical testing. An investor’s total wealth can be conceptually divided into tied assets which are budgeted for specific short-or mid-term projects - and into free assets - whose purpose is yet undefined. While the shortfall risk approach seems best suited for the first quantity, allocation according to individual risk aversion is adequate for the latter part of wealth. The thesis at hand shows how to determine individual risk aversion with different discrete choice models, with gambles and jointly with both methods. The methods developed thus aim at allocating the investor’s free part of wealth. In the first part empirical estimation of selected socioeconomic factor coefficients that determine risk preference is carried out by OLS, WLS, Tobit, Ordered Logit and Multinomial Logit models. The Survey of Consumer Finances 1995 and 1998 proves to be the ideal data sample for this purpose. The preliminary analysis of the first part is carried out assuming the traditional two-moment mean-variance framework. While demographic factors such as age, gender and marital status will prove insignificant, different saving reasons, the investment horizon and expectations about the economic development - among others - allow a stepwise out-of-sample assignment precision of up to 90%. In the second part the traditional asset pricing model analysis is extended to incorporate the third moment, skewness. A joint estimation employing obxv

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served and stated preferences assists in determining an individual investor’s preference pattern regarding the trade-offs between the three moments - mean, variance and skewness. Investors favor positive skewness and some will accept a lower mean or higher return variance to obtain higher positive skewness. The study briefly shows how to translate these preference patterns into option strategies that are added to the portfolio. For the skewness analysis the traditional Pratt-Arrow measure of risk aversion is extended to a three moment risk premium. The study exemplifies how to proceed when determining risk aversion on an individual basis. It shows how to create a questionnaire based on hypotheses, it examines what model to use for what data basis in order to achieve the highest level of prediction and it illustrates how to include higher moment preferences within this framework.

Chapter 1

Introduction “...and they see only their own shadows, or the shadows of one another, which the fire throws on the opposite wall of the cave [...]” – Plato, The Republic, VII.

While the theoretical knowledge about diversification, risk-return tradeoff and portfolio-separation has long been at our disposal, the overwhelming majority of both investors and investment advisors has not implemented the concepts introduced by Markowitz (1952), Sharpe (1965), Ross (1976), Tobin (1958) and others: Puzzlingly stock-ownership especially in Europe is still limited to a minority among savers, and financial institutions are extremely lax about profoundly educating their clients and show little effort determining their investor’s risk aversion. Banks and brokers usually content themselves with informing investors about the risk-return trade-off and advise them to adapt their portfolio to changing needs. For profound investment advice, however, the conceptual division of one’s financial assets into reserved funds and free wealth is of paramount importance to the asset allocation decision. Sound assessment of the investor’s risk aversion is its second most relevant step. One reason for the lack of more serious analysis of risk aversion and the development of a profound quantitative assessment method in practice is obviously the difficulty to determine risk aversion (RAV) with confidence. This 1

2

1.1. Motivation

is rather puzzling, as it is such a fundamentally important variable in most financial and economic models. While some studies have chosen a cross-sectional approach to quantify it on an aggregate basis, others have taken interest in the individual investor and have chosen a more direct approach using gambles or questionnaires whose significance was not tested empirically. Whatever method used, RAV has been determined only within a certain range and often not without paradoxical results. The latter arise due to well-known behavioral anomalies described in articles by Tversky, Kahneman, Thaler and others. The study at hand aims at developing a methodological framework for assessing investors’ risk preferences for their free part of wealth. The analysis of risk aversion is carried out by discrete choice models1 and at a later stage by gambles. The framework lays out the structure of the investor’s asset allocation and examines the patterns underlying the choice of his risk level. The gathered information can be utilized to set up a questionnaire that predicts an individual’s investment choice by weighing those factors found significant in explaining the choice of the investment risk level. The goal is to determine significant behavioral factors that capture as many aspects of investment decision making as possible and integrate them into an empirical model.

1.1

Motivation

Several studies as the one by Kritzman (1992) but also many behavioral finance studies2 , prove that single investors are typically not conversant with the mathematical and theoretical underpinnnings of asset allocation. A necessary condition for good advice is therefore to present the investment choice and asset allocation in a way that appeals to an individual’s intuition and that lets the investor easily understand and relate to the basic differences of concurring investment choices. The study will develop a framework of questions and gambles for individuals whose evaluation yields an estimate for the 1 In the year 2000, the year this thesis was written, D. McFadden and J. Heckman were awarded the Nobel prize in economics for the development of the models and methods used in this thesis’ empirical analysis. 2 See for example Shleifer (1999).

Chapter 1. Introduction

3

asset allocation of the free part of wealth. Its major benefit lies in the fact that the answering of the questionnaire does not require the investor to have much financial expertise. Another motivation for the development of more profound assessment methods is the legal urge for European banks to develop sound practice in advising investors3 . Sound practice necessitates quantitative methods for assessing risk aversion. Despite the large body of knowledge related to individual risk-taking and investment allocation, there still remain many open questions. As public policy makers consider changes to public and private pension regulation, it is increasingly important that there be a clear understanding of the factors that influence individual investment choices. Financial behavior of individuals has also increasingly awakened academic interest, as it is either explicitly or implicitly at the core of traditional predictive models of asset pricing and aggregate behavior. As a by-product the analysis of factors which explain investment behavior and the assessment of investment preferences proves valuable to product development in asset management. Surveys of financial behavior help decide what type of funds to offer to whom.

1.2

1.2.1

Earlier assessment methods: overview and critique Intuitive approaches

Most risk assessment methods used in practice lack empirical foundation. One example is the ‘Risk Quota’ (RQ) Questionnaire of 1991 by Fidelity Investments as described in Luenberger (1998)4 . Fidelity’s RQ consists of 20 factors that are supposed to be most relevant to the asset allocation decision. All questions are derived from normative theory and thus concerned with how people should invest. The answers given result in an overall score that assigns 3 The

insight that the average individual has insufficient understanding of the financial markets has led most European countries as well as the U.S. to enact laws about the general duties of care for the protection of individual investors: Art.31, Abs. 2 & 3 WpHG in Germany, Art. 11 BEHG in Switzerland, Financial Services Act of 1998 (H.R. 10) in the U.S. 4 please see Appendix G.1

4

1.2. Earlier assessment methods: overview and critique

the investor to one of five investment funds that have distinctly different stocks ratios or risk levels. The calculation of the factor weights and the overall score occurs by rule of thumb. These assessment attempts can in general be identified as intuitive methods that do not analyze the influence and weight of each independent factor empirically - let alone their interaction effects. Instead, the weights are assigned arbitrarily or they are derived from separate normative theories. The most severe drawback of these assessment attempts, however, is to mix questions about future projects with questions about one’s personality. The two concepts relating to these issues are to be treated separately, as will be discussed later. Many other Risk Rulers used in practice are based on psychological theory and were implemented without having to pass empirical testing. Examples are the ‘Allianz Anleger Analyse’5 , ‘Bœrsencoach’ by aixigo6 , ‘riskquestionnaire’ by promistar7 , ‘Psychotrainer’ by Bœrse Online/Dialego8 and Schwab’s ‘Risk Ruler’9 . Major banks provide rather simplified versions of their Risk Rulers online, as they are afraid of imitation. The ‘Riskruler’ by UBS and the ‘Risk Analyser’ by CreditSuisse10 fit this description. Closely related to Risk Rulers are so-called Asset-Allocation Tools provided by most banks, insurance groups and Online brokers: Accutrade’s ‘Asset Allocation Worksheet’11 , Smartmoney’s ‘Asset Allocator’12 and ‘Allocator’ by Investsearch.

1.2.2

Gambles

Gambles are based on the Bernoulli principle13 which assumes that individuals maximize expected utility when given the choice between different payoffs. These payoffs are typically described by two or more outcomes equalling monetary amounts that will arise with a given probability. Each gamble and choice refers to a different point on the investor’s utility curve. By having the 5 please

see Appendix G.2

6 http://www.boersen-coach.com/frames.php3?start=fb

7 http://www.promistar.com/LTCRiskProfileQuestionnaire.pdf 8 http://www.dialego.de/bo11/

9 http://www.schwab.com/SchwabNOW/navigation/mainFrameSet/0,4528,552—829,00.html

10 http://investmentmanager.cspb.com/en/riskanalyzer/index.html 11 http://www.accutrade.com/fhtml/assetallocation.fhtml 12 http://www.smartmoney.com/oneasset/ 13 See

the translation of Bernoulli’s original text of 1738 in Sommer (1954).

Chapter 1. Introduction

5

investor choose among gambles several times, an approximate outline of his utility function can be derived14 . The form of the utility function contains the investor’s degree of risk aversion. More demanding and less accurate gambles as the ones by Mosteller and Nogee (1951) present risky and riskless choices and let the investor quantify his certainty-, outcome- or probability- equivalent. As Schoemaker and Hershey (1992) proved, this assessment method differs even intra-personally depending on the method chosen and thus yields very unreliable results. Examples of the avenue of investigating risk taking behavior as outlined above are: Gertner (1993), Metrick (1995), Altaf (1993) and Levy (1994). Evidence on those studies is mixed. Most find that people’s behavior is inconsistent with the predictions of expected utility theory. Another issue that comes up with gambles is whether there is any down-side risk presented. If there is none then the results may not be generalizable to behavior in the real world. A serious drawback is often the often small sample size of most of these experiments. Lastly, for average investors gambles generally seem to be too difficult to understand. This might also explain intrapersonal inconsistencies and paradoxical outcomes. Gambles and the rigorous belief in expected utility maximization as the general decision making rule during the 1950s have been criticized for their ambiguous and paradoxical results. Three critiques seem worthwhile mentioning: 1. Maurice Allais (1953) opposed the American school for using the Bernoulli principle as its major working hypotheses at the time. He remarked that the experimental observation of the behavior of men who are considered rational by public opinion invalidates it. Four facts account for its failure in reality: • Very prudent people behave non-according when gambling small sums. • The choice of risks bordering on certainty contradicts the independence principle of Savage. • The choice of risks bordering on certainty contradicts the substitutability principle of Samuelson. 14 This

was shown by Meyer and Pratt (1968)

6

1.2. Earlier assessment methods: overview and critique • The behavior of entrepreneurs when great losses are possible deviates from expected utility maximization. For Allais the dispersion as well as the general properties of the form of the probability distribution of psychological values are essential to the theory of risk. They must be taken into account by every theory of risk if it is to be realistic. The Bernoulli principle as a description of rationality has no interest per se. Rationality must be defined either in the abstract by referring to a general criterion of internal consistency15 or it must be defined experimentally by observing the actions of people who can be regarded as acting in a rational manner. Allais viewed all the fundamental postulates leading to the Bernoulli principle as based on false evidence.

2. Machina (1982) showed that the outcome of individuals’ decision making differs significantly from what Expected Utility Theory would suggest. 3. Kahneman and Tversky (1979) coined the term ‘bounded rationality’16 and proved that individuals do not interpret probabilities according to expected utility. Tversky and Kahneman (1986) show that preferences are quite insensitive to small changes of wealth, but highly sensitive to corresponding changes in the reference income level. Losses and gains must therefore be considered with respect to this reference income level. Also, individuals exhibit loss aversion, i.e. they are much more responsive to losses than to gains. A certain income decrease thus results in a higher loss of utility than the utility gain associated with the same income increase. Furthermore, the utility function becomes convex in the loss region which implies local risk loving behavior. In other words, there exists a range of income levels below the reference income where bounded rationality reduces risk aversion.17 15 Criterion

implying the coherence of desired goals and the use of appropriate means for attaining them. 16 See Section 2.6. 17 This revives the discussion of the shape of the utility function. Confer in this context Friedman and Savage (1948) as well as Kahneman and Tversky (1979) among others.

Chapter 1. Introduction

7

As an alternative to expected utility theory the concept of bounded rationality demonstrates better compatibility with empirical findings18 . As Brunnermeier (1996) shows, it can provide theoretical explanation for inconsistent behavior of individuals towards income lotteries and other anomalies like loss aversion and status quo bias. In contrast to rational individuals, boundedly rational decision makers are restricted both by the availability of information and their ability to learn. However, their persistent search for ‘better’ heuristics leads to a rational thinking process eliminating systematic errors.

1.2.3

Derivation of risk aversion in asset pricing models

Various empirical studies employing intertemporal equilibrium models derive risk aversion from option prices and realized asset returns19 . Studies on the equity-premium puzzle20 and its implications are one example. All of them choose a cross-sectional approach to determine risk aversion on average. They infer statements about properties of absolute and relative RAV and the change of RAV over time which are relevant for asset pricing. As shares are generally riskier than bonds and as the average investor can be said to show risk averse behavior, stocks earn a premium on the capital markets. Over the last 20 years this risk premium was about 8% higher than the one for bonds - a difference inexplicable by the traditional asset pricing models alone. In fact a difference so significant that relating theories in asset pricing are rendered incompatible. Given the above equity premium, the measure of absolute risk aversion implied by the traditional CAPM would come to about 18. This is significantly higher than the coefficient of proportional risk aversion which Blume and Friend (1975) estimated to be in excess of two. Either our asset pricing models do not explain reality convincingly or the empirically estimated measure of risk aversion is in fact drastically higher than we have assumed so 18 Non-expected

utility and non-utility models as proposed by Fishburn (1988), Machina (1989) as well as Epstein and Zin (1990) also offer feasible alternative solutions. 19 Cf. Ait-Sahalia and Lo (1999), Jackwerth and Rubinstein (1996) and Jackwerth (1998). 20 Cf. Mehra and Prescott (1985). They state that standard asset pricing models fail to explain the high average excess return on the US stock market. Their study is based on a consumption-based asset pricing model with power utility function which causes the elasticity of intertemporal substitution to be the reciprocal of the coefficient of relative risk aversion. This relation is an undesirable feature and avoidable if another utility function is assumed, as done by Epstein and Zin (1991) and Weil (1989).

8

1.2. Earlier assessment methods: overview and critique

far. Arrow (1971) found constant relative risk aversion to be around 1. The proposed log utility function, however, does not seem to be an appropriate assumption for the average investor. Tobin and Dolde (1971) suggest a constant relative risk aversion (CRRA) of 1,5. They supplement their observations by the theory of life cycle patterns and changing risk aversion. Mehra and Prescott (1985) quote several microeconometric estimates that bound the risk aversion coefficient from above three. Pindyck (1988) in his research of the US stock market found relative risk aversion to equal approximately 3 - 4. Mankiw and Zeldes (1991) continue Mehra and Prescott’s thoughts arguing that an individual with a coefficient of relative risk aversion above ten would be willing to pay unrealistically large amounts to avoid bets. Empirically they find constant relative risk aversion to be about 26, provided the rational expected utility maximization models hold. Kandel and Stambaugh (1991) show that even values of relative risk aversion as high as 30 imply quite reasonable behavior when the bet involves a maximal potential loss of around one percent of the gambler’s wealth. Blake (1996) contradicts these views by arguing for a CRRA of about 2. Other works about risk aversion were carried out by Aggarwal (1990) who showed that distributions with low variance, positive skewness and negative kurtosis are preferred by investors. Cohn, Lewellen, and Schlarbaum (1975), King and Leape (1984) as well as Morin and Suarez (1983) reject the CRRAthesis and instead suggest decreasing risk aversion for increasing levels of wealth. Kandel and Stambaugh, Mankiw and Zeldes and others, believe it possible that there is no equity premium puzzle. Individuals might be more risk averse than we thought and this high degree of risk aversion is reflected in the spread between stocks and bonds. Benartzi and Thaler (1995) and Bonomo and Garcia (1993) give a different explanation by showing that loss aversion combined with a short horizon can rationalize investors’ unwillingness to hold stocks even in the face of a large equity premium. Supporters of the young field of behavioral finance - Haugen (1995), Shefrin and Statman (1999) among others - find it unlikely that the observed patterns of excess return predictability can be explained purely by time-varying risk premia generated by highly risk averse agents in a complete markets economy.

Chapter 1. Introduction

9

Studies about habit-formation by Constantinides (1990) and by Campbell and Cochrane (1998) disagree and present evidence by explaining a good part of the excessive stock returns.21 While the first study is able to explain the intertemporal dynamics of returns, it lacks to account for the differences in average returns across assets. The latter study by Campbell and Cochrane suggests that investors seem to fear stocks primarily because they do poorly in occasional serious recessions - in the times of low surplus consumption ratios - unrelated to the risks of long-run average consumption growth. Empirical observations do suggest that equity risk premia are higher at business cycle troughs than they are at peaks. Risk premia shrink significantly in up-markets as investors become eager to profit from rising stock prices and increase their risk exposure. The latter fact sometimes gives rise to speculative bubbles, as documented by Kindleberger (1996) and Shiller (2000). Excess returns on common stocks over treasury bills seem forecastable, and many of the variables that predict excess returns are correlated with or predict business cycles. While models of habit formation are able to explain the equity premium puzzle, their validity seems incompatible with the structure of stock-ownership as was pointed out by Benartzi and Thaler (1995): most of the assets on the American stock market and even more on the European stock markets are owned by three groups of investors: pension funds, endowments and very wealthy individuals. It is indeed “hard to understand why habit formation should apply to these investors”. Based on Kahneman and Tversky (1979), Benartzi and Thaler (1995) propose a model of myopic loss averse investors that demonstrate a high sensitivity to losses with a prudent tendency to frequently monitor one’s wealth. The domain of the utility function is thus shifted from consumption to returns whose variability is accepted only in exchange for a large premium. Their results suggest that behavioral anomalies deserve consideration, but that single phenomena cannot explain the riddle. 21 Habit-models

are an important area of consumption-based pricing models. The Habit formation describes a positive effect of today’s consumptions on tomorrow’s marginal utility of consumption. Utility today can be presented as a constant elasticity function of current consumption and future utility. In this function, the degree of risk aversion of the consumer as well as the elasticity of intertemporal substitution is governed by different parameters. Disentangling risk aversion and intertemporal substitution helps explaining various aspects of asset pricing behavior that appear anomalous in the context of the preferences.

10

1.3. Research idea

Thus, the equity premium puzzle still remains unsolved.

1.3

Research idea

We assume that common behavioral factors relating to risk preferences account for people choosing the same equity ratio in their portfolio. Significant independent factors can be used for predicting the choice of a particular investor. They can assist in advising customers who do not have sufficient financial knowledge to make a choice themselves. The following two subsections briefly summarize the status quo of portfolio theory and draft the theoretical framework of financial assumptions relevant for this study. The first subsection revises some old facts about portfolio theory while the second reviews the categories of asset pricing models. These two subsections lay the foundation for the structure of the asset allocation decision that is employed in this study and presented in the third subsection.

1.3.1

Portfolio theory’s old and new facts

In traditional asset pricing models markets were, to a good approximation, assumed to be informationally efficient.22 While the CAPM is per se irrefutible, the evidence on the predictability of long-term returns reported in empirical studies is overwhelming. Also, there are strategies with which one can reliably outperform simple indexes and passive portfolio strategies. The assets they consist of produce high average returns without large betas implying that beta as the sole risk measure does not suffice and in fact does not do justice to reality. Multifactor models such as the Arbitrage Pricing Theory have much improved our knowledge about the risk factors that earn a premium on the market. They also enabled us to better adjust for risk when measuring investment performance. Several studies showed that stock and bond returns are predictable at long horizons. Their negative serial correlation implies that asset returns follow a mean reverting process23 that seems to be associated with business cycles 22 The

statements in this subsection are partly based upon Cochrane (1999, 2000) reversion implies stationarity and thus constant unconditional moments. Most important, however, is the fact that under mean reversion return variance grows less than 23 Mean

Chapter 1. Introduction

11

and financial distress.24 However, annual returns prove much less predictable and all the more so daily, weekly and monthly returns. Their process comes closest in resembling a random walk.25 The value-added of most actively managed funds seems questionable. Their returns prove to be slightly predictable: Past winning funds perform better than average in the future while past losing funds do worse. Most of this seemingly persistence however can be explained by fairly mechanical investment styles derived by multifactor models and not by persistent skill at stock selection. All of these findings suggest that one can earn a premium for holding macroeconomic risks associated with the business cycle and for holding assets that do poorly in times of financial distress, in addition to the risks represented by overall market movements. Cochrane (2000) thus advises the investor to hold - in addition to the market26 and the risk-free asset - a number of passively managed ‘style’ funds that capture the broad risks common to the majority of investors. In addition to the overall level of risk aversion, an investor must determine his aversion to some additional risk factors: Each Survey of Consumer Finances reconfirms that the major proportion of US stockholders largely own stocks in the company they work for. Clearly, such an allocation runs counter to diversification of one’s financial risk exposure. In the case of the company’s distress the investor is in danger of losing both his job, his income and a large part of his savings. A wealthy investor, on the other hand, who is less dependent on his income from work, can afford to invest in value stocks and similar assets that seem to offer a premium in return for potentially poor performance in times of financial distress. It is the purpose of the stock market to help transfer risks from those unable or unwilling to bear them to those who for a risk premium proportionally with time. 24 Cochrane (1999) 25 While these rather short-term returns are still strikingly unpredictable, it is the returns at five-year horizons and more that seem highly predictable, confer Fama and French (1988). A substantial amount of their variation can be explained by variables like the dividend/price ratio and the term premium. 26 Where the market is a blend of styles (risk factors) identified by an adequate multifactor model.

12

1.3. Research idea

can and will afford to take them. Since returns are somewhat predictable, investors can enhance their average returns by moving their assets around among broad categories or ‘styles’ of investments. The question what styles to choose depends on one’s risk tolerance and other specific circumstances. An investor desiring more return and willing to take more risk than the market portfolio is best advised to borrow to invest in the market rather than to compose a portfolio of riskier stocks. An alternative for adjusting the risk-return profile is to purchase call or put options on the market portfolio. The basic decision for the investor is to what proportion he wants to hold the market portfolio. The empirical evidence of market efficiency and the poor performance of professional managers of active funds relative to passive indexation, strongly suggests that stock-selection and timing attempts will not pay off for most investors. Thus, the standard advice is to hold passively managed funds that concentrate on minimizing transaction costs and fees. The two-fund separation theorem leaves open the possibility that the investor’s time horizon matters as well as his risk aversion. An important question concerns the intertemporal allocation of assets: To what extent does the time horizon of the investment impact the allocation decision? - The customary advice that a long-term investor can afford to sit out all the market’s short-term volatility needs to be qualified. It is true that a short-term investor should avoid stocks, as he may have to sell at the bottom rather than wait for the inevitable recovery after a price drop. However, for the long-term investor the time diversification fallacy lies in the ‘inevitable’ recovery implied by the customary advise. If returns are close to independent over time, and prices are close to a random walk, a price drop makes it no more likely that prices will rise more in the future. It may take a long time until the former, all-time-high value of the portfolio is reached again.27 27 This

implies that stocks are not safer in the long run and the stock/bond allocation should be independent of the investment horizon. If returns are iid over time then the mean and variance of continuously compounded returns rises in proportion to the horizon: the mean and variance of ten-year returns are ten times those of one-year-returns, so the ratio of mean to variance is the same at all horizons. Merton (1969) and Samuelson (1969) show that an investor with a utility exhibiting constant relative risk aversion who can continually rebalance his portfolio between stocks and bonds will always choose the same stock/bond proportion regardless of investment horizon, when returns are independent over time.

Chapter 1. Introduction

13

Only if returns mean-revert, if they exhibit negative serial correlation, prices are more likely to come back after a shock. The evidence, in fact, suggests that stock prices do tend to mean-revert. Return variances at horizons of five years are about 50-60% larger than short-horizon variances. If long-term returns are predictable, the mean and variance no longer change proportionally with the time horizon. With negative serial correlation or mean reversion the variance of long-horizon returns is proportionally lower than the variance of single-period returns. In this case, stocks are more attractive in the long run. The most relevant conclusion of the above exposition for this study materializes in the fact that the one input essential to optimal portfolio advice is risk tolerance. Providers of investment services have at last started to think about how to measure risk tolerance using a series of questionnaires. This is the trickiest part of the conventional advice, partly because conventional measures of risk tolerance often seem incompatible with aggregate risk aversion displayed in asset markets. The basic pragmatic question is whether one is more or less risk tolerant than the average investor. This question will be conceptualized and answered in quantitative terms in the study.

1.3.2

Overview of models

The traditional discrete-time, single-period Capital Asset Pricing Model (CAPM) by Sharpe (1964), Lintner (1965), Mossin (1966) has been criticized extensively by Roll (1977)28 , Ross (1976) and Elton and Gruber (1987)29 . The 28 Roll’s

CAPM critique comprised:

• Because of various actual trading restrictions such as unlimited riskless borrowing and short-selling, investors may not be able to hold MVE frontier portfolios. • Due to the fact that security returns are not normally distributed, investors may have skewness preferences and end up holding inefficient portfolios. • Since transaction costs and taxes affect security returns, investors who face different costs may take inefficient portfolio positions gross of cost. • Investors may suboptimize their portfolio positions when holding indivisible assets such as their own human capital (i.e., the present value of their future earnings). 29 Elton and Gruber (1987) showed that with homogeneous expectations and meanvariance efficiency, portfolio selection reduces to simple after-tax optimization. Their study

14

1.3. Research idea

CAPM will be briefly dealt with in Chapter 2. Otherwise its fundamental assumptions, statements and results are presupposed in this study and won’t be discussed here. The continuous time models exhibit a normative advantage, as they are more realistic even though their assumption of continuous rebalancing is not unproblematic. Continuous rebalancing (and mean reversion) leads to a geometric Brownian motion (GBM) of stock prices (with drift). If continuously compounded returns are normally distributed as implied by the GBM, then in the discrete-time case single period returns are lognormally distributed. Investors are thus faced with positively skewed returns due to which they choose portfolios not efficient in terms of the mean-variance framework. Skewness cannot be portrayed in the traditional two-moment model. When there are more than two moments, diversity in tastes e.g. moment preference will ensure that unidentical portfolios result. It is for this reason that Rubinstein wrote his paper on the Taylor expansion of the utility in 1973. Moment preference models incorporating skewness can approximate optimal strategies more accurately. Periodic lognormally distributed returns imply continuous normally distributed returns.30 If one wishes to use the properties of the normal distribution to estimate an investment’s likelihood of loss, one must convert the investment’s periodic returns to continuous returns. It is then feasible to use the mean and standard deviation of these values (continuous returns), along with the normal distribution, to estimate the likelihood of loss.

1.3.3

Structuring the Asset Allocation Decision

According to modern portfolio theory all investors are to allocate their assets between the riskfree asset and the market portfolio. They thus decide for a specific point on the capital market line to represent their investment choice. Which point an investor exactly chooses depends on his individual risk aversion. Risk aversion itself is a concept derived from decision making also relates to the “clientele effect” first reported by Lewellen, Lease, and Schlarbaum (1977). This effect claims that investors of different income tax brackets will prefer stocks that differ in their dividend policy. 30 Kritzman (2000).

Chapter 1. Introduction

15

theory and closely related to expected utility theory. Unfortunately behavior as predicted by this theory strongly deviates from that observable in reality. In order to be able to determine risk preference in practice, the problem of allocating free and reserved wealth must be treated separately31 . The distinction is crucial to isolate the effect of individual risk aversion on the allocation decision. Factors such as the investment horizon, the proximity to retirement, in short all questions concerning risk taking capacity are thereby taken into consideration (see Figure 1.1). The usual pitfalls can be avoided Figure 1.1: Structuring the Asset Allocation Decision: Distinguishing between free and tied assets: This thesis will show how an individual investor’s risk aversion can be determined. The resulting estimation applies only to that part of the investor’s wealth that has not yet been budgeted for specific projects.

by structuring the decision in this way, as the two methods for allocating the two parts of wealth differ significantly.32 For reserved wealth only the shortfall-approach is adequate while for free wealth the only relevant criterion is individual risk aversion. The shortfall-approach has the single goal to minimize the probability of not meeting a predefined minimum target return that depends on the investment goal. This target return usually represents the yearly return that is necessary to meet a planned project. Roy (1952) initially phrased this 31 Cf.

Spremann (2000), 303ff . wealth conceptually in this way contributes to solving the problem techni-

32 Separating

cally.

16

1.3. Research idea

concept “safety-first”. It later led to the development of shortfall-risk and more generally to the implementation of lower partial moments into portfolio management and financial planning33 . The study at hand concentrates on the assessment of risk aversion and thus on the question how to allocate free wealth. It is that part of wealth that has not been reserved for any project at the time of survey - neither in terms of time nor in terms of purpose. The investment horizon or the maturity for this part of wealth is thus undetermined.34 In the analysis of the SCF data this distinction between free and reserved wealth could unfortunately not be considered. However, the data was ideal for portraying how a survey assessing risk preferences must be designed and how the analysis must be conducted. All questions in the SCF concern the respondent’s total financial assets. Thus, questions of the investment horizon, possible budgeted projects and similar issues interfered with the object of analysis, distorting both the choice of the stock ratio and the independent factors used for explaining it. Therefore, the independent factors will not have as much explanative power as if the investors had been asked to answer only in the light of their free wealth.

1.3.4

Outline of thesis

In a first empirical analysis, discrete predefined intervals of the stock ratio which serves as a proxy for the risk level - are regressed onto different factor sets. Out of these the one with the highest overall significance was then chosen. The resulting estimates and different models are compared by their predictive power for in-sample assignment of the observations. As a result a single factor set was created from those sets. The final group of factors consists almost entirely of individual variables that are based on different hypotheses of financial theory. For the proceeding of the analysis it is convenient to have the same factors available in both the SCF 1995 and 1998. The SCF guarantees this fact, as all factors relate to financial behavior, one could argue that, in fact, we test for consistency of the overall financial decisions of an 33 Other

authors that dealt with this subject in detail are: Telser (1955), Kataoka (1963), Leibowitz and Henriksson (1989) and Van Harlow (1991). 34 For a detailed discussion of this topic, see Spremann (2000), Ch. 8

Chapter 1. Introduction

17

investor. It has been stated before that single investors might not be conversant with fundamentals of financial markets, with its statistics and the quantitative measures of risk, return and skewness. The proceeding of the method, however, implicitly assumes that at least on average the individual decides correctly. Otherwise the regressions wouldn’t yield significant results for factors that are projected to be relevant. As such, the investors are assumed to be boundedly rational. Their search for their optimal investment risk level leads to asset allocations approaching the latent optimal result. Surprisingly, the majority of individual investors have been shown to reconsider the risklevel of their portfolios seldomly even though they check their values frequently.35 It must be noted that the emphasis of this study is not on factor analysis or the derivation and selection of regressors that have the greatest explanative power for the two specific samples that were employed. The goal is to describe the procedure, methodology and testing possibilities for developing a tool to determine investment risk preference. We will proceed as follows: After structuring the asset allocation decision, the theoretical part of the paper will outline the basic two-moment model, its assumptions and the hypotheses motivating the choice of the factor sets. The two-moment model lays the theoretical justification for using the stock ratio as a proxy of risk aversion. The second part will present the empirical analysis and determine the classification power of different regression models for the final set of individual factors for in-sample and out-of-sample prediction of the SCF 1995 and 1998. The last part of the study offers an outlook on the extension of the model and research idea to three moments. Figure 1.2 visualizes and sums up the outline of this study.

35 Cf.

Lee (2001), Bertaut (1998), De Bondt (1998) and Roth and Mueller (1998).

18

1.3. Research idea

Figure 1.2: Structure and outline of thesis: The first part comprises the determination of risk preferences in two moments - mean and variance. An empirical analysis of the Survey of Consumer Finances (SCF) helps to identify factors that are significant in explaining different degrees of risk aversion. The determination of one’s certainty equivalent for specific gambles is an alternative way of assessing one’s risk aversion. At the end of part one a joint estimation procedure is presented. It combines both methods to increase accuracy. The second part extends the first by incorporating the third moment as an essential attribute and criterion of investment preference. A new risk aversion measure is derived to replace the one by Pratt-Arrow for three-moment setting. The assessment through gambles is extended by the third moment to yield the investor’s skewness preference which in practice can be implemented by option strategies. The second part concludes with a description of how to jointly estimate the regression model and the gamble to achieve higher predictive precision in the investment choice.

Part I

Two-Moment Risk Preference

19

Chapter 2

The CAPM and two moment risk preference “The one input to the optimal portfolio advice is risk tolerance, and many providers of investment services have started to think about how to measure risk tolerance using a series of questionnaires. This is the trickiest part of the conventional advice...” – Cochrane (2000).

The CAPM (one factor model) is still one of the most widely used asset pricing model - maybe less in academic research than in practice. Its most relevant statements are: 1. there are only two criteria for the pricing of an asset j: its expected return µj and its return volatility σj . 2. the Investment Opportunity Set (IOS) integrates all risky assets in its boundaries. Different correlations between the assets give rise to the Efficient Frontier that comprises those combinations of risky assets that are highest in its return/volatility ratio. 3. adding a riskless asset to this setting and combining it with the efficient frontier results in the Capital Market Line (CML) that is tangential to the Efficient Frontier in the Market Portfolio (M). 21

22

4. as the Market Portfolio is the single optimal risky asset to hold, Tobin’s Two-Fund-Separation Theorem tells us that the asset allocation decision can be divided into the delegated management of M and the individual choice for a point on the CML. For the investor the last statement raises the single most important question about the degree of his risk aversion. The risk aversion determines the percentage of risky assets, that is, the proportion of M in the investor’s portfolio. To simplify the analysis this study assumes that the market portfolio consists solely of stocks. Bonds are assumed to be held unto maturity and thus considered riskless. The asset allocation decision thus reduces to merely choosing the individually desired stock ratio. Figure 2.1: Dividing the Capital Market Line (CML) into 6 funds representing six different risk categories. The individual investor will be assigned to one of these by the discrete choice model.

The Capital Market Line is divided as illustrated above, yielding six funds of distinctly different risk levels. This division is purely arbitrary and can be modified easily to yield finer or cruder results of the risk aversion. The decision

Chapter 2. The CAPM and two moment risk preference

23

for five different stock ratios was motivated by the different risk profiles of equity funds. For a derivation of the risk aversion within the CAPM setting please refer to Appendix A.

2.1

Assumptions of mean-variance portfolio selection

The relevance of the investor’s risk taking capacity for the asset allocation decision is ruled out by the before-mentioned conceptual division of assets into free and reserved wealth. All foreseeable future short-time to mid-term expenses must be provided for by the shortfall approach. As frequent rebalancing of the portfolio is costly and yields below-average returns1 , the study focuses on determining a risk level that suits the investor in the long run. The study maintains the view that for an individual investor a buy-and-hold strategy seems most profitable for longer investment horizons. This view is supported by studies of Barber and Odean (1999), Siegel (1998)2 Schlarbaum, Lewellen, and Lease (1978) as well as Schlarbaum, Lewellen, and Lease (1979). It is thereby refuted that markets are fully efficient. Systematic out-performance of major market indices by price- and earnings-momentum strategies has been proven by Jegadeesh and Titman (1993), La Porta (1996) and Rouwenhorst (1998) among others. However, such strategies are researchintensive and extremely time-consuming and thus infeasible to pursue for an individual investor. An investor could of course invest in funds exploiting market inefficiencies, such as overreaction and other behavioral phenomena. Such a strategy is well advised and supported, but it does not influence the 1 Cf.

the following in-text references as well as Malkiel (1999). sources cite stocks as the superior investment vehicle when judged from a longer time horizon. Siegel (1998) states that over the last 200 years stocks have beaten T-bonds as measured by the average yearly return. Prerequisite is a well-diversified portfolio of at least 12 stock titles (preferably blue chips) that is to be held for at least 15 years. He warns against attempts to time the market and empirically finds that most timing and stockpicking strategies of individual investors end in a disaster. Even the selection of proven growth stocks turns out to be difficult, as the development of the “Nifty Fifty” from 1972 - 1997 clearly illustrates. The technology shares within the nifty-fifty had high P/E-ratios and underperformed the S&P500, while stocks of consumer goods companies contributed the largest part of the growth. 2 Many

24

2.1. Assumptions of mean-variance portfolio selection

decision over what stock ratio the investor should choose. This decision must be made solely on the basis of one’s risk aversion. The rationale behind the importance of RAV lies in the fact that both continuous rebalancing and active trading strategies cannot be implemented by an individual investor. They are simply too costly in terms of time and money for individuals have earn a living. The same holds true for any attempt to time the market. Price decreases and increases happen much too quickly for an individual investor to anticipate. Any delay in rebalancing and timing is punished severely: The price increases that accounted for 80% magnitude of the index increases in the last 25 years happened in about 20-30 days. Everyone not invested in those days was never able to catch up with those who had followed a buy-and-hold strategy. Therefore, stock picking, markettiming and switching strategies are considered futile attempts to beat the market and just involve brokerage commissions, higher taxes and losses in money and time.3 A critic could argue that by investing in actively managed mutual funds one could delegate or evade the asset allocation decision. Such a view implies the belief that fund-managers have the ability to time the market and could rule out the relevance of deciding for a specific static stock ratio. However, the skill of fund managers as compared to a benchmark of passive style indices seems very questionable. Studies by Fisher and Statman (1997) and Gruber (1996) have proven that in past crashes more than 80% of the actively managed mutual funds have declined at least as much in value as the market index. The advice to the investors is thus to allocate their assets according to their risk aversion in a fund that tracks one or several broad global indices - depending on what kinds of risks the investor is willing to bear. - In this context it is needless to repeat that they should not adjust their risk exposure every time the market moves. Volatility for such a long-term, buy-and-hold strategy is still important, as the average investor checks his portfolio value frequently and worries if it falls below a certain boundary. Studies about individual trading4 have revealed that in times of market downturns investors frequently react in a panic and 3 For

a more detailed exposition and reasoning, see Malkiel (1999), 350ff. and Odean (1999) among others.

4 Barber

Chapter 2. The CAPM and two moment risk preference

25

sell their assets, thereby materializing the former virtual losses. This kind of behavior is in fact the greatest investment risk of all to the individual investor. An initial adequate risk level might help to prevent it. As we will see later, the adjustment of the skewness of expected returns might also prove effective in preventing the investor from realizing losses or from changing a chosen risk exposure. Option strategies can enable an investor to take on more risk when investing in stocks. They are therefore integrated into the analysis in the second part of the study. As a summary and a set of assumptions for the first part, it can be stated that the investor will hold the market as suggested by the two fund-separation. The riskless asset is represented by money market accounts, while the market portfolio will be a major market index such as the S&P500 total return index. The Pratt-Arrow measure of risk aversion is the quantity relevant to this study’s first part’s problem set. As Appendix B proves, in the two-moment model this measure is closely tied to the ratio of risky assets within the investor’s portfolio. For simplification, bonds possessed by the investor were counted to the riskfree assets. It is understood that bonds generally constitute a part of the risky market portfolio, as they bear significant risks when not held to maturity. The character of the analysis is static. Though risk aversion is generally assumed to be a quantity relatively stable over short time-horizons, it is bound to change with age, wealth and varying circumstances in the investor’s life. The assessment thus has to be repeated as those changes occur.

2.2

The concept of two-moment risk aversion

In the economics of uncertainty, the term risk aversion in its more general meaning stands for a cautious attitude in the context of reasonable decision making. A risk averse individual shows a diminishing willingness to accept more and more risk, even if taking more risk is related to more positive mean return. Thus “risk aversion refers to preferences with (i) decreasing rates of substitution between risk and mean return, and (ii) signifies a behavior, where diversification [...] pays.”5 5 Spremann and Kotz (1981). In contrast to risk-averse behavior a risk-neutral investor attaches no importance to diversification. He aims at maximizing his return and accordingly

26

2.2. The concept of two-moment risk aversion

Figure 2.2: Typical concave utility function of a risk averse investor. At the current wealth level W = 50 the depicted gamble pays off 60 with 40% probability and causes a loss of 40 with 60% probability. The investor’s certainty equivalent of the gamble is CE = U −1 [E(U (W ))]. The gamble’s risk premium equals π = E(W ) − CE. A person who prefers the gamble to the certain payoff is a risk lover; one who is indifferent between taking the gamble and accepting the actuarial amount is risk-neutral; and one who avoids the gamble choosing the actuarial value is risk-averse.

The higher the risk aversion the greater the utility gain implied by diversification. Deviations from the market portfolio should be the exception. Under risk aversion the condition that an uncertain wealth gain is not preferred to the expected value of the gain (w ˜ ≺ E w) ˜ holds. This condition is equivalent to the concavity of U . Risk-aversion means that an investor when given the choice, will avoid to take fair gambles. A gamble is called ‘fair’ when it has a zero risk premium chooses the asset with the highest return disregarding its risk level. Risk neutral investors judge risky prospects solely by their expected rates of return. The level of risk is irrelevant. This type of investor will even consider taking out a credit in order to increase the expected value of his investment. Such behavior patterns are known as ‘Plunging’. A behavior that will be mentioned again in Section A.2 after Equation A.18.

Chapter 2. The CAPM and two moment risk preference

27

or in other words when its expected payoff is zero. Risk-averse investors are willing to consider only risk-free or speculative prospects. The reason is that the potential loss represents an amount of “displeasure” that is greater than the amount of “pleasure” associated with the potential gain. Formally speaking, a person is risk averse, if the utility of expected wealth is greater than the expected utility of wealth: U [E(W )] > E[U (W )]. The additional assumption of insatiation, common in economics, together with risk-aversion, causes indifference curves to be positively sloped and convex. Even though all investors are said to be risk-averse, they do not have identical degrees of risk aversion. Different investors will have different maps of indifference curves. The more risk-averse an investor, the steeper his indifference curves in the mean-variance-diagram. In the classical sense, risk aversion measures attitudes towards pure wealth bets. It is therefore conventionally captured by the second partial derivative of the value function with respect to individual wealth. The value function depends on individual wealth W . The investor’s personal trade-off between portfolio risk and expected return is expressed by his utility function. The latter assumes that investors can assign a welfare or a score to any investment portfolio depending on its risk and return. Because we can compare utility values to the rate offered on risk-free investments when choosing between a risky portfolio and a safe one, we may interpret a portfolio’s utility value as its “certainty equivalent” rate of return to an investor. The certainty equivalent rate of a portfolio is the rate that risk-free investments would need to offer with certainty to be considered as equally attractive to the risky portfolio. A portfolio is desirable only if its certainty equivalent return exceeds that of the risk-free alternative. In general, the risk premium must be added to the gamble in order to make the investor indifferent between the gamble and its actuarial value. She’d be willing to pay the premium in order to receive the expected value right away instead of playing the gamble. However, the risk premium in itself does not express whether the gamble is favorable or not. For that purpose the cost of the gamble has to be calculated: C = CurrentW ealth − CE. If it is positive then one would pay to avoid the gamble; if it is negative, one would pay to take the gamble. The overwhelming majority of financial models assumes individuals to be

28

2.2. The concept of two-moment risk aversion

risk averse. A reasonable utility function belonging to the class of HARA utility functions is the following isoelastic utility which exhibits decreasing absolute risk aversion and constant relative risk aversion6 : U (W ) =

1 γ W γ

(2.1)

where 0 = γ < 1

2.2.1

The Markowitz Premium

The degree of the utility function’s curvature is measured by the difference between the expected value of a gamble and the investor’s certainty equivalent. The measure can also be interpreted as the amount that an investor would pay to insure against a particular risk. Given the numbers of the gamble in Figure 2.2, assuming initial wealth of W = 50 and the isoelastic utility of Equation 2.1 with γ = 0.3, the Markowitz Risk Premium is  E(W ) − CE(W )

=

50 − 0.3 ·

=

17.72



1  0.3 0.6 0.3 0.4 0.3 100 + 10 · 0.3 0.3

as the Certainty Equivalent, CE = U −1 [E[U (W )]] and U −1 = γW 1/γ . In other words, the investor is willing to pay up to $17.72 to insure himself against the gamble and the risk of losing $40.

2.2.2

The Pratt-Arrow measure of Risk Aversion

The most widely used measure of risk aversion is based on the insight that the degree of curvature in a function is related to its second derivative. Pratt (1964) and Arrow (1965) derived the risk premium by constructing an actuarially neutral gamble of Z dollars, equating the utility of expected wealth and the expected utility of wealth. By using a two-term Taylor approximation of the utility function they disregard third and higher order terms or, in other words, skewness and higher moments of the distribution. As a result, the methodology is numerically 6 Thus allowing for a much more realistic setting of the analysis than both the quadratic and the exponential utility function.

Chapter 2. The CAPM and two moment risk preference

29

exact only for infinitesimal risks.7 The postponement of the truncation of higher order terms as depicted in Chapter 7 permits the extension of the analysis to larger risks. For deriving his two-moment risk premium, Pratt (1964) uses the Taylor ˜ and series and equates the expected utility of a (random variable) gamble X the utility of its expected value: ˜ E[U (W + X)]

˜ − π(W, X)] ˜ = U [W + E(X)

(2.2)

˜ = 0: As it is a fair gamble, E(X) ˜ E[U (W + X)]

˜ = U [W − π(W, X)]

Expanding the left-hand side using Taylor series and disregarding third-order and higher terms: ˜2 X U  (W ) + ... 2! ˜2 ˜  (W ) + E[X ] U  (W ) = E[U (W )] + E[X]U 2! σ 2  = U (W ) + U (W ) 2 Expanding the right-hand side using Taylor series and disregarding secondorder and higher terms in { }-brackets:  2  π  ˜ U (W ) − ... U [W − π(W, X)] = U (W ) − πU  (W ) + 2! ˜ E[U (W + X)]

˜  (W ) + = E[U (W ) + XU

Putting both sides together σ 2  U (W ) = U (W ) − πU  (W ) 2 Solving for the Pratt-Arrow measure of a local risk premium π we get   1 2 U  (W ) π = σZ −  (2.3) 2 U (W ) U (W ) +

7 For large risks, Pratt/Arrow state that if, at every wealth level, a utility function U 1 displays greater local risk aversion than U2 , the risk premium associated with any risk irrespective of its size, will be greater for U1 than for U2 . Thus, the cardinal measure of risk aversion in the small corresponds to an ordinal measure in the large, provided that the risk aversion functions do not intersect. However the measure does not permit inferences about risk aversion in the large for all utility functions.

30

2.2. The concept of two-moment risk aversion

where σ 2 represents the variance of the random variable Z˜ which indicates the outcome of the gamble in dollar terms. Whether the variance is an adequate measure of portfolio risk will be discussed later, here it is just noted that the extent to which variance lowers utility depends on α. Where α is the investor’s degree of absolute risk aversion ARA, expressed as   U  (W ) (2.4) αA = ARA = −  U (W ) the Pratt-Arrow measure risk aversion. Multiplying the measure of absolute risk aversion by the level of wealth W we obtain a measure of relative risk aversion (RRA):   U  (W ) αR = RRA = W −  (2.5) U (W ) The RRA can be interpreted as an elasticity measuring the percentage change in marginal utility given a percentage change in the individual’s wealth. ARA on the opposite, measures the percentage change in marginal utility, given an absolute change in wealth. Each of these two measures indicates to what extent the odds have to be better than fair in order to induce an individual to accept a bet of a certain size, measured either as an absolute amount or as a percentage of income. For the gamble in Figure 2.2 above which is actuarially neutral, the variance of the asymmetric risk is: 0.4(110 − 50)2 + 0.6(10 − 50)2 = 2400. The Pratt-Arrow risk premium thus comes to   −0.7(50)−1.7 1 2400 − = 16.8 π = 2 50−0.7 Thus the risk averse individual will be indifferent between winning $60 and losing $40 when the actuarial value is $16.8. The difference for the example between the Markowitz and Pratt/Arrow premium is only small. It results from the fact that relative to the assumed reference wealth level of $50 the risk is not small. The gamble is also skewed. Consider a gamble that is even more skewed: On the right side in Figure 2.3 the gamble pays off $150 with a chance of 20%, while the gambler loses $37.5 with 80% probability. Initial wealth is again $50. Calculating the Pratt-Arrow measure for this gamble yields a premium of $39.4.

Chapter 2. The CAPM and two moment risk preference

31

Figure 2.3: Two differently skewed gambles and their PrattArrow risk premia

The investor’s isoelastic utility function with γ = 0.3 and its derivatives are 0.3−1 W 0.3

U (W )

=

U  (W )

= W −0.7

U  (W )

= −0.7W −1.7

U  (W )

=

1.19W −2.7

From these derivatives the moments of the two gambles in Figure 2.3 and the corresponding premia can be calculated: σ2 Skewness Pratt-Arrow πA Markowitz

Gamble1 2’400 48’000 17 18

Gamble2 5’625 632’812 39 23

Due to a higher variance, the premium of Gamble2 is higher than that of the less skewed Gamble1. In both cases the expected value of the gamble was 0, both were actuarially neutral. With the mean being constant, an increase in variance (from 2’400 to 5’625) and skewness (Fisher skewness grew from 0.41 to 1.50) results in a deterioration of the two-moment expected utility and the local rejection of positive skewness-variance trade-offs. Judging from the characteristics of the Pratt-Arrow risk measure, investors will never accept a higher variance for an increase in skewness regardless of

32

2.3. Relevance of assessing the correct asset allocation

the size of the trade-off. Intuitively, this result is not satisfactory. Various studies have proven that investors value positive skewness. The Markowitz and Pratt/Arrow risk premium measures only consider the variance as a factor of risk. Excluding negative skewness as a risk source seems only acceptable when the source itself is symmetrically distributed. Explicit consideration of the gamble’s third moment in an alternative risk premium measure might prove valuable for weighing larger risks that have skewed distributions. Such proceeding will be portrayed in Part II. In general, it can be stated that the Pratt-Arrow definition of risk aversion provides useful insight into the properties of ARA and RRA, but it assumes that risks are small and actuarially neutral. The Markowitz concept, which simply compares E[U (W )] with U [E(W )] is not limited by these assumptions. The difference between these two measures becomes most accentuated when the risk in question is large and very asymmetric (skewed). In such a case the Pratt-Arrow measure tends to underestimate the risk. It tells us nothing about the preferences of investors when return distributions are skewed. The Markowitz measure of a risk premium is thus superior for large or asymmetric risks.8

2.3 2.3.1

Relevance of assessing the correct asset allocation Calculating Expected Utility loss

The expected utility approach assigns utiles to monetary values and thus allows to measure the utility loss caused by a wrong asset allocation. The magnitude of the calculated loss indicates whether the accurate assessment of risk preferences is of any significant relevance for the investor.9 With regard to the return differences between different stock ratios - as depicted in Formula A.14 in Appendix A.1 - one might argue that the monetary loss in expected utility for an assignment to a wrong risk class (stock ratio) might be neglectable10 . As shown below, the precise assessment of individual 8 Cf.

Copeland and Weston (1992). (2000), 328. 10 Even though it is believed that the expected utility approach is not a realistic reflection 9 Spremann

Chapter 2. The CAPM and two moment risk preference

33

preferences is indeed trivial when considering a short investment horizon of one to three years, as the monetary disutility of a wrong assignment is very small. However, when looking at a longer investment horizon, the loss in utility and the potential absolute monetary loss can be significant.

Time horizon of 1 year We assume the following values for the parameters of the formula in Appendix A.14: • financial wealth, W = 100 (in ’000 $) • optimal asset allocation or stock ratio, b = 80%, where b =

µM −rf 2 αW σM

• return of riskfree asset, rf = 5% p.a. • expected return of risky asset, µM = 10.5% p.a. • annual volatility of risky asset, σM = 22% p.a. With the above values, we obtain the investor’s risk aversion: α=

.105 − .05 = 0.0142 0.8 · 100 · 0.222

The certainty equivalent, CE, of the investor’s end-of-period wealth, WT , for his optimal asset allocation of 80% stocks amounts to: α Var[WT ] 2 0.0142 309.76 = 107.20 109.4 − 2

CE[WT ] = E[WT ] − CE[WT ] =

where WT = W0 (1 + µw ) and µw = rf + b · (µM − rf ) Let’s assume the investor whose optimal asset allocation is 80% stocks is assigned to a mutual fund with a stock ratio of only 20%. The certainty of the average investor’s decision making criterion, we will utilize it here to analyze the relevance of developing an accurate risk assessment method.

34

2.3. Relevance of assessing the correct asset allocation

equivalent of the investor’s end-of-period wealth, WT , for a stockratio of 20% is: 0.0142 19.36 = 105.96 2   The relative utility loss thus equals 1 − 107.20 105.96 = 0.012 or approximately 1%. A utility loss of such magnitude is in fact neglectable. CE[WT ] = 106.1 −

Time horizon of 10 years By prolonging the investment horizon, the utility loss increases as expected: • return of riskfree asset, rf = 10 · 5% = 50% • expected return of risky asset, µM = 10 · 10.5% = 105% √ • annual volatility of risky asset, σM = 22% 10 • the investor’s risk aversion remains the same as for one-year horizon:

α=

1.05 − 0.5 = 0.0142 0.8 · 100 · 0.69572

The certainty equivalent, CE, of the investor’s end-of-period wealth, WT , for his optimal asset allocation of 80% stocks amounts to: α Var[WT ] 2 0.0142 3097.6 = 172.01 194 − 2

CE[WT ] = E[WT ] − CE[WT ] =

where WT = W0 · (1 + µw ) and µw = rf + b · (µM − rf ). Let’s assume the investor whose optimal asset allocation is 80% stocks is assigned to a mutual fund with a stock ratio of only 20%. The certainty equivalent of the investor’s end-of-period wealth, WT , for a stockratio of 20% is: CE[WT ] = 161 −

0.0142 193.6 = 159.6 2

Chapter 2. The CAPM and two moment risk preference

35

  The relative utility loss thus equals 1 − 172.01 159.6 = 0.078 or approximately 8%. It increases almost proportionally with the investment horizon. In general, the loss is more pronounced the larger the equity premium and the greater the difference between the optimal and the assigned stock ratio. The most considerable disutility, however, comes in the form of unplanned portfolio rebalancing in unfavorable market situations when these rebalancings are triggered by an inadequate risk level e.g. a misspecified measuring of risk aversion.

2.3.2

All stocks half the time or half stocks all the time?

When discussing the optimal asset allocation strategy for a given investor, it is often argued that the answer depends on the time horizon, the dynamics of the market or the timing abilities of the investor. Fortunately, this claim is a misconception provided that there is a premium for risky assets. It is solely the investor’s utility function, his risk preferences that determine the optimal investment strategy. This section will show why. The preceding analysis in Sections 1.3.1-1.3.3 has already raised a more general question essential to asset allocation: Is it better to follow a constant balanced strategy or to shift between extreme allocations by trying to time the market?11 For the following brief exposition it will be assumed that the average return on stocks is higher than the riskless rate and that stock returns follow a random walk. Furthermore, transaction costs and taxes will be neglected. The balanced strategy allocates 50% of W in stocks and the rest in the riskless asset. The switching strategy invests 100% of W in stocks for half of the investment horizon, the rest of the time it invests in the riskless asset. Intuitively, one would presume that both strategies yield the same results. And even though both strategies will yield the same expected return and the same terminal wealth in the limit12 , the balanced strategy has a higher average per period exposure (appe) than the switching strategy. As a higher 11 Among

others, Clarke and de Silva (1998) and Kritzman (2000) analyzed this problem. fact, the expected cumulative wealth is not to be expected, as due to continuous compounding the pdf of terminal wealth is positively skewed. The mean is thus greater than both the median and the modus. The geometric average return would be a better measure for what is to be expected even if it does not mirror the figure that will result on average over many repetitions. 12 In

36

2.3. Relevance of assessing the correct asset allocation

exposure implies a better performance, the balanced strategy dominates the switching strategy: appe balanced strategy

=

appe switching strategy

=

  N −1 1 1 Wn − W0 1+ N n=0 2 W0  K−1  Wk − W0 1 1+ N W0

(2.6)

(2.7)

k=0

r˜S stands for the random return on the stock investment, rf depicts the riskless rate, W0 symbolizes the total wealth in the period of the first stock exposure of the corresponding strategy, n = 0, 1, 2, ..., N represents the investment period and k = 0, 1, 2, ...K is the index of the periods in which the switching strategy is invested purely in stocks. By definition K = N/2. As an example, the two strategies are compared for a time horizon of N = 20 periods (therefore K = 10). Stocks are assumed to yield either r˜S = +20% or r˜S = −6% alternately, while the riskfree rate returns a steady rf = 6.2073%. The expected return for both strategies in the long run is: rbs = 0.5 · 7% + 0.5 · 6.21% = 6.60%. For alternating risky returns the growth paths of wealth for the balanced (B) and for the switching (S) strategy are: t 0 1 2 3 4 5 6

B 1 1.13 1.13 1.28 1.28 1.45 1.45

S 1 1.2 1.13 1.35 1.27 1.53 1.44

t 7 8 9 10 11 12 13

B 1.64 1.64 1.86 1.86 2.1 2.11 2.38

S 1.72 1.62 1.94 1.83 1.94 2.06 2.19

t 14 15 16 17 18 19 20

B 2.38 2.7 2.7 3.05 3.06 3.46 3.46

S 2.32 2.47 2.62 2.78 2.96 3.14 3.33

An initial investment of 1 in the balanced strategy grows to 3.46 at the end of period 20. The switching strategy yields a lower total capital of 3.33 which stems from the lower average per period exposure (appe): appe balanced strategy

=

19 1 1 Wn = 0.9919 20 n=0 2

appe switching strategy

=

9 1 Wk = 0.9013 20 k=0

The expected cumulative wealth for both strategies is 1 · (1.066)20 = 3.59 and thus higher than the two results produced by the series above. The sequence

Chapter 2. The CAPM and two moment risk preference

37

of switching is thereby essentially irrelevant if long time horizons and thus infinite repetitions of the investment strategies are considered. Apart from higher performance, the balanced strategy has another advantage compared with the switching strategy: it bears less unsystematic risk as measured by the return’s volatility13 and is thus more efficient. While the balanced strategy’s volatility is 0.5 · σ 2 , the switching strategy’s volatility14

equals 0.5 · σ 2 + 0.52 · (˜ rs − rf )2 ). Thus, even if an investor is a skillful market-timer, he needs to achieve a substantial excess return over the balanced strategy in order to outperform it on a risk-adjusted basis and even more so when considering transaction costs over longer investment periods. At this point, one might argue that deciding whether switching pays solely by the criterion of terminal wealth might be too one-dimensional. What if an investor derives utility not only from terminal wealth, but also from the interim realizations of wealth along its growth path? Samuelson (1994) proves that even in such a case the balanced strategy is still to be preferred provided the investor exhibits constant relative risk aversion. Then, the sum of expected utilities over all periods is greater for the balanced strategy. This is not surprising, as the switching strategy bears higher unsystematic risk. For every point in time it thus has higher potential deviations than the balanced strategy. The assumption of constant relative risk aversion ascertains that the valuation of the volatility does not change over the growth path of wealth. The specific form of the underlying utility function does not influence this finding provided rf is the certainty equivalent of r˜s .15 The above subsection tried to indicate why a steady, balanced strategy beats a switching or timing strategy on average. This line of reasoning was necessary as the balanced strategy serves as the default assumption in the following chapters.

13 As

opposed to the before-mentioned average per period exposure Clarke and de Silva (1998), 63. 15 U [W (1 + r )] = 0.5 · U [W (1 + r sup )] + 0.5 · U [W (1 + rsdown )] must hold f 14 See

38

2.4

2.4. Switching and the time horizon controversy

Switching and the time horizon controversy

The preceding chapter showed why a strategy of continuous dynamic adjustments16 to the asset allocation is on average inferior to a balanced buy-andhold strategy. This section briefly reviews the relevance of the investment horizon for any asset allocation advice.17 In his 1963 paper Samuelson analyzes the behavior of a colleague who turns down a single play of a bet that has a positive expected value though exhibiting the possibility of a loss. Puzzlingly, the same colleague is willing to accept multiple plays of the same bet.18 Samuelson’s famous theorem about the ‘fallacy of large numbers’ states that this kind of behavior is inconsistent. He accuses his colleague of erroneously believing that the variance of outcomes decreases as the number of trials increases. In fact, the variance of terminal wealth increases proportionally with an increasing time horizon. It is solely the variance of the average annualized simple return that decreases with time. The above line of reasoning indicates that the behavioral inconsistency is of more than just academic interest.19 Repeated plays of independent gambles (addition problems) essentially portray investments over time where returns are compounded (multiplication problems). Refusing a single gamble while accepting a series of them hints at the hope of realizing the positive expected value rather than one extreme. It also relates to the idea that there is greater probability of meeting a given 16 This

refers to adjustments that are not motivated by variations in underlying market fundamentals that change the composition of the market portfolio. 17 In the study at hand the question of the investment horizon has conceptually been uncoupled from the link between asset allocation and risk aversion by the separation between the tied and the free part of investors’ wealth. See Subsection 1.3.3 18 A finding also confirmed with experiments by Keren and Wagenaar (1987). Subjects who were shown the explicit multi-year distribution were willing to accept more risks than when faced with the characteristics of the one-year return distribution. Related studies about the change of behavior induced by the increase of gambles played come from Lopes (1984), Montgomery and Adelbratt (1982), Keren (1991) and Redelmeier and Tversky (1992). 19 The role played by the investment horizon in optimal portfolio selection has been of recurring interest in the financial literature. Under the term “Time diversification” this topic received great attention by: Merton (1969), Samuelson (1989, 1990, 1994), Benartzi and Thaler (1995), Bodie (1995), Thorley (1995), Merrill and Thorley (1996) and Bierman (1998) among others.

Chapter 2. The CAPM and two moment risk preference

39

return objective by investing in high-risk/high-return assets than there is by investing in low-risk/low-return assets and that the tendency for the high-risk portfolio to dominate the low-risk portfolio increases with the length of the investment horizon. This tendency has been called the “time diversification effect”. Merton (1969) and Samuelson (1969) showed that when returns follow a random walk and utility functions exhibit constant relative risk aversion there is no such effect. Asset allocation is then independent of the time horizon of the investor.20 Samuelson’s contributions to this question can be separated into two paths: first he assumed iid returns meaning that successive returns are independent of each other and not autocorrelated. He thus assumed a random walk and concluded that an investor’s optimal choice of single-period portfolios will remain constant over the investor’s life cycle. Later he examined the same question allowing for autocorrelation of returns and assuming a mean-reversion process (rebound effect). Again he concluded that there is no argument for the long-horizon investor to hold a riskier portfolio. Still later, he developed the conventional wisdom to hold under both regimes (random walk and mean reversion) - all under the assumption that the investor is anxious not to fall under a subsistence level of terminal wealth.21 Empirically the “colleague’s” behavior could not be confirmed unambiguously. Evidence about time induced financial behavior is mixed. While Kahneman and Tversky (1979) justified the ‘colleague’s’ inconsistent decision by myopic loss aversion22 , several studies proved that most subjects seemed to 20 These results have always been controversial and in reality investors were told to do differently. Investment advisors explicitly recommend changing the asset mix as retirement comes closer. Eventually the necessity to reduce portfolio risk due to retirement has to be answered in a detailed financial plan and is not an issue of this analysis. However, even if an investor’s risk taking capacity is considerable, it is conceivable that he feels very uncomfortable with every day value changes of his portfolio. Though knowing that these daily changes balance each other out in the long run if prices mean revert, such an investor could be advised to buy portfolio insurance occasionally and in times of obvious ‘irrational exuberance’ in order to hedge the portfolio’s value. 21 A similar argument was made by Roy (1952) and was later shown by Zelney to characterize the way investors perceive risk. Roy coined the well-known “safety-first” approach that later led to the development of the concept of shortfall-risk and more general to the implementation of lower partial moments in portfolio management and financial planning. 22 Loss aversion is an important property of the prospect theory value function introduced

40

2.4. Switching and the time horizon controversy

suffer from a fallacy of small rather than large numbers.23 Less well experienced subjects overestimate the variance of a multiple-play gamble rather than underestimate it. As a consequence less risky investments are made. This finding might explain why such a low proportion of the population holds stocks and why the average relative stockholdings are so small. If Samuelson’s colleague really mis-estimated the variance of the manybet-portfolio then he would presumably change his mind when shown the correct distribution of final wealth resulting from multiple gambles. However, experiments proved that this is not the case. It became clear that subjects in fact favor repeated plays of a positive expected value gamble rather than refrain from accepting them when shown the payoff distribution.24 Table 2.1: The relevance of the time horizon for the asset allocation depicted in a matrix subdivided into return process and form of utility function. The Bernoulli investor with U = ln(W ) exhibiting constant relative risk aversion (crra) serves as a benchmark for all conceivable utility functions. Return More Risk Averse Bernoulli (Log-W) Less Risk Averse ∗ † process than Log-W Investor (crra) than Log-W Mean reversion

Random Walk

Relatively more wealth is allocated to the risky asset

Time horizon has no impact on the asset allocation decision

Longer time horizon means that less wealth is allocated to the risky asset Asset allocation is independent of time. It depends solely on individual risk aversion. The optimal strategy is fully myopic.

∗ An

investor with a utility function logarithmic in wealth generally, any investor exhibiting constant relative risk aversion. Any isoelastic investor can be assumed: U (W ) = 1/γW γ .

† More

When assuming iid returns and consequently a constant and known investment opportunity set (IOS), then the time horizon is irrelevant for an by them in 1979. It states that reductions in wealth, relative to the current reference point, are weighted about twice as much gains. 23 See study by Benartzi and Thaler (1995). 24 Benartzi and Thaler (1995) show that by holding the distribution of final outcomes constant they can predict which repeated gambles people will find attractive.

Chapter 2. The CAPM and two moment risk preference

41

isoelastic investor, as an isoelastic utility implies constant relative and decreasing absolute risk aversion. According to his degree of risk aversion the investor allocates a constant proportion of his wealth in risky assets regardless of the length of the period considered. This equals Samuelson’s original proposition in the ‘fallacy of large numbers’-debate in Samuelson (1963a). The above result remains almost unchanged for the investor exhibiting crra when returns are assumed to either mean-revert or to be partially predictable. However, the investors who are more risk averse than log-wealth experience an increase in expected utility under mean reversion as their horizon increases, as Winhart (1999) and Kritzman (2000) point out. The reason is that mean reversion25 causes that part of wealth allocated to the risky asset to disperse at a “sufficiently slow rate so that conservative investors can tolerate greater exposure over longer horizons.” It can thus be concluded that there is no theoretical or empirical evidence for the so-called “time-diversification”. If risky returns follow a random walk then variance increases proportionally with time and the investor’s expected utility is invariable to the investment horizon provided the investor has constant relative risk aversion. Thus, under iid returns investors will wish to hold their asset allocation constant as their time horizon increases. Apart from the specific form of the utility function, for the asset allocation decision it is also crucial to define the correct objective function. Kr¨ anzlein (2000) points out that maximizing expected terminal wealth might not be the right goal given the fact that positive skewness of the terminal wealth’s pdf increases with the time horizon.26 For a positively or right skewed pdf the expected value overestimates the actual return that results in the end. Modus or median would provide better target measures for the objective function. A closely related and equally important factor concerning intertemporal asset allocation is the retirement-induced reduction in portfolio risk.27 This 25 The data history is too short to enable an unambiguous statement about the specific form of the return process. Evidence for mean reverting stock returns find Poterba and Summers (1988) as well as Fama and French (1988). 26 Kahneman and Tversky (1979) have shown that investors care about the (interim) change in wealth and not about the final asset value. That is after all why they are myopic loss averters. 27 Other factors that play a role in life-time utility maximization and influence intertemporal asset allocation are: retirement postponement, the inflation rate, taxes, changing

42

2.5. Risk aversion and changes in wealth

effect is often extended to mean that investors become generally more risk averse as they age. However, while there is considerable evidence of different degrees of risk aversion within each age cohort, there is little evidence on tolerances for risk bearing among different age cohorts. There are also no reported long-term studies which show conclusively how attitudes toward risk change as individuals age. Marshall (1994) presents the so-called draw-down criterion as a plausible reason.28 With this criterion one aims at maximizing terminal portfolio return subject to not falling below a prespecified threshold. Even if an investor has the same risk aversion all his life, he will choose progressively less risky portfolios as his investment horizon shortens due to the rising relevance of the return variance of a high-risk portfolio. An investor’s perception of the riskiness of any given single-period portfolio changes as his horizon grows shorter. It is thus not necessary for an investor to become more risk averse to observe a gradual shift from more to less risky portfolios with the passage of time.

2.5

Risk aversion and changes in wealth

The previous section has pointed out that the relevance of the time horizon depends largely on the course and the change of risk aversion with wealth. In other words it depends on whether relative risk aversion is constant or not. The following paragraphs will briefly examine what form of relative risk aversion is best compatible with reality. Intuitively, one would expect the coefficient of absolute risk aversion for a given individual to decrease with wealth, while the coefficient of relative risk aversion might be roughly constant across different wealth levels.29 That is, consumption preferences, consumption opportunity sets and changing investment opportunity sets. These have been considered by Bodie and Crane (1997), Chen and Moore (1985), Khaksari, Kamath, and Grieves (1989) and Bierman (1997, 1998) among others. 28 A drawdown is defined as a percentage of initial investment capital (similar to shortfall risk), and as the lower bound of a confidence interval when measured over a continuum of investment horizons. 29 Cf. Taggart (1996). Within the Consumption based asset pricing model with power utility (CCAPM) the coefficient of relative risk aversion γ is typically estimated by 1 = Et [(1 + Ri,t+1 )δ

Ct+1 −γ ] Ct

Chapter 2. The CAPM and two moment risk preference

43

we would expect an individual starting with little wealth to be more cautious about entering a bet in which he could win or lose the amount of initial endowment than the same individual who starts with a fortune. On the other hand, there are reasons which support the assumption of at least constant (very high) absolute risk aversion. Factors such as social status might play an important role. Some people might dread the thought of losing 10% of their wealth through plunges in stock market prices, especially if the decreases necessitate restrictions in the standard of living. The characteristics of an individual’s absolute risk aversion (ARA) allow us to determine whether he treats a risky asset as a normal good when choosing between a single risky and a riskless asset.30 Arrow (1971) showed that decreasing ARA over the entire domain of −U  /U  implies that the risky asset is a normal good - that is, the dollar demand for the risky asset increases as the individual’s wealth increases. Increasing absolute risk aversion as in the quadratic utility function suggested by the CAPM implies that the risky asset is an inferior good, and constant absolute risk aversion implies that the individual’s demand for the risky asset is invariant with respect to his initial wealth. Under increasing relative risk aversion the wealth elasticity of the individual’s demand for the risky asset is strictly less than unity. In other words the proportion of the individual’s initial wealth invested in the risky asset will decline as his wealth increases. For short time periods the question of whether the investor’s utility function exhibits constant absolute or constant relative risk aversion or even decreasing relative risk aversion has little relevance for the optimal asset allocation.31 It is longer investment horizons of ten years or more that call for a more precise analysis of the higher moments of the utility-function. The longer the investment horizon the more distinct the distribution’s skewness.32 that can be derived from U  (Ct ) = Ct−γ . Ri,t+1 is the return on asset i at time t + 1, δ is the time discount factor and Ct is aggregate consumption in period t. Asset returns and aggregate consumption are assumed to be jointly homoskedastic and lognormal. Also see Hansen and Singleton (1983). 30 Huang and Litzenberger (1988) 31 Cf. Spremann (2000), 331. 32 It must also be noted that the longer the horizon considered the more probable that the investor’s preferences have changed. Time-varying preferences necessitate a periodic

44

2.6. Bounded rationality and goal of methodology

The assessment of moment preferences thus must aim at determining what form of return distribution is desired by the investor. Some will want to hold risky assets and buy portfolio insurance due to their high risk aversion. Others might be able to afford to sell insurance or cap their upside potential by employing covered call strategies. In part two this study will show how to determine such different preference patterns.

2.6

Bounded rationality and goal of methodology

The following subsections will briefly present some basic concepts that the formal analysis rests on. One of them - the assumption that individuals are boundedly rational - is a crucial prerequisite for the usefulness of advising investors with an empirically calibrated questionnaire. The following section will show why and discuss what criteria are appropriate for rating the performance of the econometric models. Bounded rational decision makers are not only restricted by the availability of information but also in their ability to learn. In order to save information processing costs and time, boundedly rational decision makers apply simplified thinking and calculation procedures. The irrational decision maker, too, is restricted in his learning abilities but in contrast to the bounded rational decision maker he does not take constraints suhc as processing information cost or time into account. He just randomizes without reasoning and has thus infinitely high information costs.33

2.6.1

Fundamentals of bounded rationality

In his Clarendon lectures on inefficient markets Shleifer (1999) names several examples for behavioral mistakes of individual investors34 : Investors’ reasonre-assessment of the investor’s risk aversion. 33 For further details see Brunnermeier (1996) 34 “They trade on noise rather than information, follow the advice of financial gurus, fail to diversify, actively trade stocks and churn their portfolios, sell winning stocks and hold on to losing stocks increasing their tax liabilities, buy and sell actively and expensively

Chapter 2. The CAPM and two moment risk preference

45

ing errors are systematic, as they make use of ‘heuristics’ or rules of thumb which fail to accommodate the full logic of a decision.35 The three main deviations from standard decision making theory36 are: First, investors do not assess risky gambles according to the von NeumannMorgenstern principle. They weight gains and losses by a reference point rather than by their prospective final wealth37 . Secondly, individuals systematically violate Bayes’ rule38 , as they predict future uncertain events by extrapolating a short history of data. Lastly, subjects make different investment choices depending on the framing of problems39 . Perhaps the most relevant point of behavioral finance for this study is that boundedly rational investors do not trade randomly, but rather tend to make the same mistakes. Therefore, their deviations from traditional decision making theory are expected to be systematic, synchronous and significant for the pricing of securities even if experiments seem to prove that experienced subjects move toward rational expectations. As it is difficult to determine whether the investment level of a whole sample is biased due to ‘herding’ or other situational circumstances, the data sample in 3.3 will at least be examined and tested for systematic biases in the independent factors.

2.6.2

Limits of Expected Utility Maximization

The Von Neumann and Morgenstern (1947) utility function for money income assigns utilities to sums of money in such a way that the individual chooses the action with the highest expected value of utilities.40 Unfortunately, the axiomatic system cannot be directly subjected to empirical tests. In fact, extensive experimental research indicates that the principle is widely violated and that people often have motives inconsistent with the maximization of expected utility. Certain observations created specific difficulties with the system. One of them is the popularity of gambling. As the expected value of managed mutual funds, follow stock price patterns.” 35 See Conlisk (1996) 36 See Kahneman and Riepe (1998) 37 Kahneman and Tversky (1979) 38 Cf. Kahneman and Tversky (1973). 39 See Benartzi and Thaler (1995). 40 Cf. Niehans (1990)

46

2.6. Bounded rationality and goal of methodology

most gambles is negative, they will only be accepted if the utility gain from gaining amount x is regarded as much larger than the utility loss from losing amount x.41 This implies that the function must be convex (rising marginal utility) in that region. Such a risk-prone behavior - implying a convex utility function - is hardly compatible with the widely observable buying of insurance, a risk-averse behavior that implies a concave utility function (decreasing marginal utility).42 Friedman and Savage (1948) pursued this line of research attempting to reconcile these two obviously contradicting behavioral patterns. Another central problem concerns the nature of the postulated probabilities. In Von Neumann and Morgenstern’s theory they are portrayed as objective frequencies. Experiments, however, suggest that human decisions are rather guided by subjective judgments which are modified perceptions of objective probabilities. These were later axiomatized together with utility in an approach developed by Savage (1954). The above mentioned violations of Expected Utility Maximization made way for psychological models of preferences.43 One of the best-known is the prospect theory by Kahneman and Tversky (1979) that defines preferences over gains and losses relative to some benchmark outcome rather than over consumption as the traditional models. Losses cause greater disutility than gains utility and each possible outcome can be weighted in a nonlinear fashion by its probability through the expectations operator.44 Some preferences models that have been applied to asset pricing include Hogarth and Reder (1987) as well as Kreps (1988). 41 According

to the Bernoulli hypothesis. style of representation is one of the great disadvantages of expected utility theory, as it leaves no room for the pleasure or pain of risk itself. What appears as risk aversion is rationalized as an implication of the diminishing marginal utility of income. 43 Campbell, Lo, and MacKinlay (1997) identify three main components that are altered by psychological preference models: the period utility function, the geometric discounting with discount factor delta and the mathemetical expectations operator Et : 42 This

 Et 

+∞

 δ U (Ct+j ) j

j=1 44 Some examples for general models of subjective expectations are Barberis, Shleifer, and Vishny (1998), DeLong, Shleifer, Summers, and Waldmann (1990) and Froot (1989).

Chapter 2. The CAPM and two moment risk preference

2.6.3

47

Presentation determines investment choice

Not only probabilities are perceived subjectively, the choices themselves generally seem to be judged on the basis of their presentation. Benartzi and Thaler (1999) studied how decision makers choose when faced with multiple plays of a gamble or investment. When faced with simple mixed gambles they are concerned about the amount they can lose on a single trial (holding the distribution of returns for the portfolio constant). They display “myopic loss aversion”. Many subjects who decline multiple plays of such a gamble will accept it when shown the resulting distribution as opposed to the single gamble. Benartzi and Thaler apply this analysis to the problem of retirement investing and show that workers invest more of their retirement savings if they are shown long-term (rather than one-year) rates of return. This in turn indicates that people cannot infer the probability distribution underlying multiple plays of gambles.

2.6.4

Goal of questionnaire

At the outset of the study, the goal of analysis needs to be defined. One important objective is the identification of the most significant factors explaining the dependent variable’s variance. Equally important is to identify that model and method which - with the given factors - achieves the highest rate of correctly predicted observations. However, it cannot be the goal to design an econometric model that classifies all respondents correctly to the choice they made. Apart from its impracticability, it would amount to asking directly each new out-of-sample individual what stock ratio he wishes. The value added of the questionnaire method (regression models) must be founded in its ability to derive and filter out underlying normative patterns of investment choice. These patterns that are portrayed in the independent factors must represent universally applicable correlations to the true latent factors that determine investment choice. If they do not represent these correlations, new investors are mislead by the questionnaire’s result. Reasons for such inconsistencies between actual and latent factors are multifarious.

48

2.6.5

2.6. Bounded rationality and goal of methodology

General problem of the approach

The main threat of the questionnaire-method arises from the fact that the respondents’ answers are implicitly viewed as normatively correct, as they are weighted to calibrate the final questionnaire. The method uses a sample set of ‘ordinary’ respondents representative of the whole population. However, among these respondents there are naturally some whose asset allocation due to bounded rationality does not match their own latent optimal allocation. In other words, some respondents’ stock ratio is way too low or way too high for them. In the process of trial and error they haven’t yet arrived at their own optimal investment strategy. The question arises whether it would be wrong to assume unbounded rationality. - If people were fully rational, they would not need advice on how to invest, as their self-assessment yielded the asset allocation optimal for them. It thus seems straightforward for a study that aims at advising individual investors to exclude the assumption of unbounded rationality and rational expectations. The fact that individuals are boundedly rational has been examined and confirmed in many studies. The number of experiments reporting biases as well as the number of books just reviewing the evidence is too enormous. The article by Conlisk (1996) gives a good overview of the literature. Here it will just be stated that even in situations where one is faced with decisions that have objectively correct answers, individuals do not make the normatively correct decisions. One obvious sign for bounded rationality immanent in the data is the lack of stock-ownership for investing individuals. This fact challenges the assessment method in general. In both surveys (SCF 1995 and 1998) more than half of those people who had financial assets did not own stocks (61% and 56% in the SCF95 and SCF98 respectively). Such asset allocation habits cannot be explained by a high need for liquidity or caution due to close retirement.45 Judged from a normative point of view such allocation is suboptimal, especially in terms of diversification or the elimination of unsystematic risk. Thus, if too few investors hold stocks, a calibration will probably lead to biased re45 It has been noted before that the low propensity to hold stocks, possibly a consequence of high loss aversion or insufficient knowledge, leads to an above-average risk-premium - a phenomenon described as the ‘equity-premium puzzle’.

Chapter 2. The CAPM and two moment risk preference

49

sults, to a questionnaire with a tendency to recommend a stock ratio that is too low. The above considerations question whether the sample-method might be too susceptible to individual or cumulative errors in decision-making. They also query whether there are different degrees of bounded rationality and whether it is possible to test single observations for bounded rationality or whole samples for bias. These questions need to be discussed in the following chapter.

50

2.6. Bounded rationality and goal of methodology

Chapter 3

Empirical Analysis This chapter introduces two data sets, presents the empirical econometric analysis of a data set of individual investors and discusses what econometric models are best suited for a specific data set and problem.

3.1

The Data set

Every two years the Federal Reserve Board has a comprehensive survey carried out that collects data about financial habits and attitudes.1 Under the name “Survey of Consumer Finances” (SCF) it is sponsored and published by the Federal Reserve and the Department of Treasury. It is designed to provide detailed information on U.S. families’ balance sheets and their use of financial services, as well as on their pension rights, labor force participation, total family income, and demographic characteristics at the time of the interview. “The need to measure financial characteristics imposes special requirements on the survey design. The survey must provide reliable information both on items that are broadly distributed in the population and on items that are highly concentrated in a relatively small part of the population. To address this problem, the SCF employs a dual-frame sample design that includes a standard geographically based random sample and a special oversample of relatively wealthy families. Weights are used to combine information 1 http://www.federalreserve.gov/pubs/OSS/oss2/scfindex.html

51

52

3.1. The Data set

from the two samples for estimates of statistics for the full population.”2 The survey thus not only draws a detailed picture of the financial situation and financial attitudes of US households, its design also ensures that it is representative of the whole population. Different types of financial assets - such as cash in transaction accounts, certificates of deposit, savings bonds, bonds, stocks, mutual funds, retirement accounts, cash value of life insurance and other assets - are recorded separately. The approximate value of real estate, life insurance and expected inheritance and wage increases are asked as well. The data set further contains information about household incomes, respondent’s attitudes toward financial risk, liquidity, use of credit, and reasons for saving. This allows an easy calculation of each household’s stock ratio which will serve as the dependent variable. In fact, the SCF represents the ideal data source for a study of individual choice behavior that intends to be descriptive rather than normative, abstract rather than situational and operational rather than purely theoretical. In general the models require that individuals act rationally which essentially means that they follow a consistent and calculated decision process in which they pursue their own preferences and objectives. It excludes impulsiveness and implies consistent and transitive preferences. Under identical circumstances individuals are assumed to make the same choices. In the data sample the dependent variable’s character is continuous. In order to apply Ordered, BNL, MNL, Conditional Logit or even Nested Logit Models it had to be discretized. The advantage of models such as the MNL is that they can account for nonlinearities in the relation of the dependent variable and the independent factors. The reason for employing discrete data at all roots in practical considerations: It is generally easier for investors to determine their optimal stock ratio as an approximate interval instead of a precise, single number. In the study these intervals, later called ‘risk classes’ are represented by the discrete dependent variable Y . For a thorough analysis it was examined whether the assignment precision could be increased by creating different nests, different constellations of these classes. The different settings are represented in Appendix D. Even without discretizing the data, two distinct groups arise due to the 2 Verbatim from Kennickell, Starr-McCluer, and Sunden (1997). More details about the survey procedures and statistical measures can be found in Appendix J

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53

fact that for some people a stock ratio cannot be calculated, as they do not own any financial assets. These individuals constitute the group for which aq cat takes on the value 0. The ones who do have financial assets are assigned to 6 different groups. The first group comprises all individuals without stocks while those with stocks are subdivided into 5 groups of equal intervals with a span of 20%. Figure 3.1: The data sets: SCF 1995 and 1998 Obs.* = Number of observations. Dependent variable: Y = AQ CAT; Significance of its values: 0 1 2 3

= = = =

do not own assets 0% stocks 1% - 19% stocks 20% - 39% stocks

Panel A: SCF1995

Y 0 1 2 3

Obs.∗ 360 2399 472 334

4 = 40% - 59% stocks 5 = 60% - 79% stocks 6 = 80% - 100% stocks Panel B: SCF1998

Y

Obs.∗

4 5 6

299 216 229

Y 0 1 2 3

Obs.∗ 302 2245 349 340

Y

Obs.∗

4 5 6

364 329 375

Transforming the dependent, continuous variable into a discrete one does not come without problems and, of course, not without a loss of information. The intervals chosen to divide the IOS are admittedly arbitrary as mentioned already. However, with the transformation an error term is inevitably added

54

3.1. The Data set

to the dependent variable. This increases its noise and impedes a reliable analysis of its influencing factors. For preservation of data accuracy an analysis of the continuous form of the variable is thus clearly preferable whenever possible. Feasible models for the continuous stock ratio would be OLS or the Tobit model. For using OLS in setting 13 , the estimation of all classes in one step, the individuals without financial assets had to be coded as (−1) so that they could be included in the analysis. Of course, such arbitrary coding distorts the results and it was carried out solely to compare the OLS with the Tobit model. For such naturally censored data the Tobit model is obviously a better fit and can be expected to yield superior results to OLS. Data accuracy is not the only goal that needs to be considered. Continuous data might not always be available, as in most samples the stock ratio will be given as an approximate interval. It is conceivable that at a later time, the results of the regression analysis will be evaluated to assign clients to 5-6 different investment funds rather than to a specific stock ratio. Also, by using discretized data a vast number of additional models present themselves for analysis, such as the Ordered Logit, MNL, Discrete Choice Model and the Nested Logit. These models allow the consideration of individuals without financial assets as well as all kinds of other groups and distinctions. The models for analyzing discrete data mentioned above differ greatly in their assumptions and specifications. A comprehensive comparison will be difficult and we will therefore focus on the qualities such as the model’s proximity to the underlying financial theory, the ability of predicting the single choices correctly (out-of-sample) as well as several statistical measures like the Akaike Information Criterion (AIC).4

3 See

Appendix D. evaluating the different models the out-of-sample results are preferable, as insample estimates usually tend to overfit the data. Out-of-sample estimation was easy thanks to the high standardization of the Survey of Consumer Finances which keeps the variables virtually invariable over time. 4 When

Chapter 3. Empirical Analysis

3.2

55

Selection of factors and hypotheses

Below, a selection of those factors is given that according to theory and empirical studies have influence on the investment risk level. The selection is arbitrary and non-scientific. The factors are tested by surveys and grouped by factor analysis. Insignificant factors are omitted in later analyses. Though anonymous, the surveys include demographic data in order to control for biased sampling. Demographic factors proved insignificant and played no further role within the analysis. This was convenient, as it is not intelligable why for example a woman should invest differently from a man. The following factors are conjectured to have an impact on the choice of risk level both normatively and positively: • The number of people in the household, x101, and marital status, x7372 1, are two factors that relate to financial obligations and unpredictable events, as, on average, people who are married and have a family are financially less flexible. • Age, x8022. According to the classic life cycle hypothesis by Brumberg and Modigliani5 and according to the behavioral life cycle hypothesis by Shefrin and Thaler (1988) age impact the asset allocation over time.6 Goodfellow and Schieber (1997) confirm that empirically the percentage in fixed income increases with age while the percentage in stocks declines. They further find that higher income individuals are more inclined to invest in stocks. For the SCF 95 and 98 the influence of income was however not unambiguous. The idea of the life-cycle hypothesis is to adjust an investor’s intertemporal asset allocation to changes in his human capital. Thus, when ignoring other factors, the younger an investor the riskier he should invest. Two arguments support this hypothesis: the younger the greater the possibility to use wages to cover losses caused by higher investment 5 Due

to Brumberg’s early death not published until 1980 in Modigliani (1980) age is viewed to be negatively related with investment risk. Thus, the older the investor the lower his stock ratio should be. 6 Where

56

3.2. Selection of factors and hypotheses

risk. Human capital can be regarded as a hedge against the risk of losses from a high allocation to stocks. On the other hand, the longer an investor’s investment horizon, the more likely it is that stocks outperform any other type of asset class. Above all, the asset allocation demands most careful consideration, as the largest portion of the portfolio’s total return is “determined by its asset categories rather than by selection or timing. Whatever one’s aversion to risk, one’s age, income from employment, and specific responsibilities in life go a long way to helping one to determine the mix of assets in one’s portfolio.”7 • The level of education. Highest degree earned: x5905. A common concern about this factor is the high correlation between education, income and wealth. Evidence on the effects of education on risk taking is mixed. Riley and Chow (1992) find that risk aversion decreases with education. Surprisingly, Jianakoplos and Bernasek (1998) as well as Hersch (1996) come to the opposite result. A better and related factor seems to be financial knowledge: Bayer, Bernheim, and Scholz (1996) find that measures of savings are higher when employers offer retirement seminars. To the extent that education implies information collection, skill acquisition and general development of cognitive capacity, one’s investment in human capital lowers bounded rationality and improves investment decision. It can thus be expected that the better the education the lower the decision error. • Saving horizon and consumption patterns, x3008 1 and x3008 45. Long investment horizons indicate the ability to ‘sit out’ market downturns. Under the term “time diversification” the relevance of the investment horizon received great attention from a normative perspective, see Section 2.4. Samuelson (1963b), (1989), (1990), (1994) contributed those articles most relevant to resolve this finance puzzle. 7 Malkiel

(1999), 351ff.

Chapter 3. Empirical Analysis

57

• Tolerance towards fluctuations, patience in sitting out market downturns and willingness to take risk, x3014 1 and x3014 4. Frequent rebalancing is considered suboptimal, as transaction costs arise from speculation tax, brokerage fees and time consumption. • Low predictability of income and high dependency on savings and financial assets as a source of income: future pension: x5608, future inheritance: x5821, foreseeable expenses: x7186 and profession: x7401. These terms refer to the likelihood of being forced to liquidate investments due to insufficient funds, loss of job and probability of wage cuts. Projects and other specific needs must be funded with specific assets dedicated to that need in order to ensure their realization. If funds for specific assets are invested as risky as the free, untied part of wealth, the investor runs the risk of having to cancel the project. The old saying “never gamble with the money that you need to pay your rent” is representative for this line of argument. The best stress test of a strategy is probably: Even if the worst case occurs, the corresponding decrease in portfolio value should not affect an individual’s standard of living. • Credit behavior, x432 1 and x7131. The impact of these two factor is theoretically ambiguous. People who always pay off their credit card balance are rare in the US as these accounts are viewed as a permanent consumption credit. On the opposite, it seems as if a constant level of credit over long periods is viewed as an indicator for financial control. • Expectations on economic development, x301 1 and x301 3. Optimistic or positive expectations on the economy are an important prerequisite for investing in stocks. • Investment goal, x3006 1, x3006 5 and x7187. Aggressive goals can only be achieved by accepting a corresponding amount of risk. • Wish to leave one’s estate to others, x5832 5. The explicit desire for leaving an estate can be a reason for maintaining a high stock ratio at an old age.

58

3.3. Testing for bounded rationality

Some empirical studies on risk aversion examine independent factors such as wealth and income. Bajtelsmit (1999) used a Tobit model to analyze evidence of risk aversion in the Health and Retirement Study (Wave 2). The results expectedly indicated that individuals with greater wealth allocate a larger portion of their wealth to risky assets than those with less wealth. The percentage allocation to risky assets was also shown to be lower for those with lower education levels and for those with greater pension balances. Younger investor groups allocate a greater proportion to risky assets with the most significant difference for those in the 50-55 age group. In this study where two Surveys of Consumer Finances (SCF) were employed, categorized wealth (wealth classes) proved insignificant in explaining investment choice among the different groups of stock investors (those considered in setting 3c). Interestingly, when looking at the estimation result where wealth is the single independent factor, the coefficient for the highest risk class (80-100%) has a negative sign. This indicates that the higher the wealth, the less likely the individual is to invest very risky. Even though the factor is not significant, it shows that the assumption of decreasing relative risk aversion does not hold for the SCF95 sample of stockowners. Distribution of stockholders in the SCF95: Wealth in US$ 0 110’00050’000100’000500’0001’000’000>5’000’000

3.3

Number of people 360 1642 734 311 558 201 297 196 4299

Average age 41 43 49 54 58 58 61 64 49

Average risk class – 1.2 1.9 2.2 2.7 3.3 3.4 3.3 1.9

Average stock ratio % – 3.88% 15.04% 19.05% 26.74% 36.60% 39.23% 36.29% 25.26%

Testing for bounded rationality

It was stated that one realistic threat to the applicability of the econometric analysis of influential factors is a calibration-sample that is heavily biased. The source of the bias can be ‘irrational exuberance in the financial markets’ implying entirely overvalued stocks, it can be the opposite - a deeply depressed

Chapter 3. Empirical Analysis

59

economy of overcautious subjects8 , it can be a non-representative sample of the population or it can be the lack of investment knowledge in the majority of the population. Two basic prerequisites of the applicability and usefulness of the method discussed are a) the sample of the population must be representative and b) that financial markets must be in a non-exuberant and non-depressed state a condition which is difficult to measure. One indication for an exuberant tendency on the markets are factor coefficients that fully contradict the hypotheses in sign and magnitude. However, even if the sample is not completely biased, single hypotheses can be violated by the respondents if they are boundedly rational. The latter will be defined as the joint violation of those hypotheses specified in the tables on pp. 63ff. Such joint violations will be tested further down.9 Over the years 1992-1998 the median family income as well as median family net worth rose in constant dollars. Reasons were the excellent economic situation and the fact that both the ownership and the amount of holdings in publicly traded stocks by families expanded greatly over that period. Even though, the share of households who owned stocks increased from 34% in 92 to over 41% in 98, the vast majority of people who had assets did not invest in stocks.10 45% of households with holdings of liquid riskless assets between $60’000 and $100’000 did not directly hold stocks and 28% of households with over $100’000 in liquid riskless assets did not hold stocks directly.11 In the theory of the Consumption Capital Asset Pricing Model (CCAPM) this phenomenon is explained with monitoring costs which are assumed to be higher for stocks than for any other asset class. One indication that costly infor8 In

times of market frenzy a significant proportion of investors succumb to the temptation of ‘herding’, thereby giving up their balanced asset allocation strategies. Even a representative sample of the whole population will thus be biased towards extreme allocations. 9 The tests were carried out with boolean queries in Excel. 10 Politicians and economists alike have repeatedly argued that the average stock ratio of individual households is in general too low to secure the accustomed standard of living after retirement. 11 See Bertaut (1998) who states that “in addition to variables that capture access to investment opportunities and information about the market, lower risk aversion, higher expected future income, absence of income risk, and presence of a bequest motive will also contribute positively to the probability of holding stocks.”

60

3.3. Testing for bounded rationality

mation about stock investments deter households from participation is that household portfolios display persistent behavior: Individuals should remain stock- or non-stockholders over some period of time. Such a persistence was in fact confirmed by Bertaut (1998) in a panel study for the SCF of the years 1983 and 1992. Bertaut and Haliassos (1997) further show that the higher the education level and the stronger the bequest motive, the lower is this information or monitoring cost.12 Such costs partly justify the low ratio of stock-holding households indicating that either high education or specific motives have to be extant for coping with the higher risk of stock investment. However, the argument given does not suffice to refute the hypothesis of a biased sample and a generally too low stock-ownership. The hypothesis could be invalidated, if households without stocks were predominantly poor or lacked education. This however is not the case: Wealth and the level of education vary unsystematically with the investment risk both in the SCF 1995 and 1998. The story that the data tells us is that, in addition to the strong restraint against risky assets, there is obviously a lack of information among private households about the general characteristics of asset classes, their return and risk profile. This lack of information is not fully eliminated by a high level of education. A BA or even a PhD or MD is no guarantor for possessing financial knowledge. Thus, if the low ratio of stock-ownership is not a consequence of a conscious decision process, the level of risk chosen can be expected to be generally suboptimal.13

3.3.1

Putting things in perspective

In the preceding paragraphs, the wrongly assigned investors were interpreted as deviations from the optimal choice, as single, non-fitting observations of the model. By taking the opposite perspective, it is possible to view these differently assigned observations as a correction of the model for boundedly 12 Bertaut

(1998) thus concludes that “education and advertising campaigns can be instrumental in helping households overcome reluctance to hold stocks caused by insufficient information about the benefits, risk, and costs of market participation. 13 Another sign for this suboptimality and the poor average financial knowledge of households is the degree of diversification of the share of the risky assets: 52% of all stockholders in 1999 held less than 8 different titles.

Chapter 3. Empirical Analysis

61

rational respondents. The following paragraph analyzes what kind of profile the high-risk investors have according to the Multinomial Logit model in Setting 3c of SCF1995, a setting consisting of five risk classes: Of 1550 stockowners, 229 held more than 80% in risky assets. Only 23 of these 229 (roughly 10%), were correctly assigned to the high-risk class. The typical high-risk investor as predicted by the MNL had the following profile: • All respondents negated the question of whether their most important reason for saving was liquidity and consumption: x3006 1 = 0. • Roughly 80% opposed to having a saving and spending horizon of a few months: x3008 1 = 0. • 96% of the predicted high risk investors did not expect the US economy to perform worse in the coming 5 years: x301 3 = 0. • 85% confirmed they were willing to take financial risks: x3014 4 = 0. • 90% had to admit that they did not always pay off the total balance owed on their credit card account each month: x432 1 = 0. • 70% did not expect to inherit any amount of money: x5821 = 0. • Only 30% had earned a degree (Nursing, College, BA, MA, MBA, PhD, Law): x5905 > 0. • 95% had applied for a credit or a loan within the preceding 5 years: x7131 = 0. • 65% had wealth of less than US$50’000. • The evidence for x101, x3006 5, x3008 45, x301 1, 5825 3, x7187, x7372 3 was mixed, as they showed no unambiguous pattern. Contraintuitive results were produced for the expected amount of future pension and the goal of saving. Almost no one estimated to receive even a single dollar: x5608 = 0 and everyone saved for foreseeable expenses.

62

3.3. Testing for bounded rationality

Table 3.1: 50% of those who were characterized as high-risk investors by the MNL were between 20 and 40 years old, the average age comes to 45 years. Age

interval 0-20 21-30 31-40 41-50

Number of individuals 0 11 23 15

Age interval 51-60 61-70 71-80 81-100

Number of individuals 11 8 4 2

As a brief conclusion it can be stated that the majority of factors confirmed the hypotheses and predictions of the theory. Thus, the findings reinforce the methodology and suggest advice should best be based on the principle of exclusion of asset allocation strategies. The analysis of each group of respondents classified as a category by the model informs about common underlying characteristics. If these characteristics are not met by a new respondent, he shouldn’t be assigned to that class. For instance, a person whose saving horizon spans only a few months should never invest 80-100% in stocks. Nevertheless 12% (27 of 229) of the high-risk investors admitted that they had a saving horizon of only a few months - clearly a sign that there is a great potential for investment optimization.

3.3.2

‘Correctly’ assigned observations

In setting 3c of the SCF1998 the MNL assigned about 25% of all investors ‘correctly’ to the classes that corresponded to their actual revealed choice. ‘Correctly’ means that the regression’s assignment of these people confirmed their actual asset allocation. In order to put the results to the test, each investor’s response set is analyzed for violating the hypotheses depicted in Section 3.2.

Chapter 3. Empirical Analysis

63

Highest risk level: Class 4 In setting 3c for the SCF1998 the MNL assigned 123 people ’correctly’ to class 4. The following table lists those combinations of values for the independent variables that are inconsistent with the highest risk class. Violations considered to be grave and irreconcilable with the assignment are labeled with ‘**’ two stars. The last column indicates the number of violations for the corresponding variable.

Variable X101

X3006 1

X3008 1 X301 3

X3014 4

X432 1 X5825 3

Type of Violation > 2. a high number of dependents in connection with low wealth (< 100 000) and income leaves the household too exposed to volatility = 1. Liquidity and consumption are rather short term goals. Stock investment is less suited for such objectives. = 1. People with a very short investment horizon should not invest heavily in stock. = 1. A pessimistic view about the future economic development is possibly not the right prerequisite for excessive stock investment. = 1. Clear conflict of goals. Someone who is not willing to take any financial risks should not invest in stock. = 0. Paying off credit card debt is an indicator for having good control about financial issues. = 1. In connection with the proximity of retirement and relatively modest wealth a high stock ratio is not recommendable.

Relevance

Violations 3

**

2

**

5 22

**

3

29 **

0

Of 123 ‘correctly’ assigned observations, only 10 cases seemed to conflict with the profile of the highest risk class after accounting for overlapping violations of the above hypotheses. This signifies that the recorded violations came from only 10 different observations. In fact, the normative findings were confirmed with high validity, as the group showed great homogeneity in terms of stated risk preference (x3014), high wealth, largely managerial jobs and long investment horizons.

64

3.3. Testing for bounded rationality

Lowest risk level: Class 1 Normatively the following values can be identified as violations for the people in the lowest risk class: (inconsistency of goals and binding violations are labeled with ‘**’ two stars):

Variable X101

X3006 5 X3008 45 X301 1

X3014 1

X432 1 X5608

X5821

X7187

X7372 1

X8022

Type of Violation < 3. A low number of dependents in connection with high wealth and income leaves freedom to invest riskily =1. Retirement,Education and Family are long-term goals that allow risky investment = 1. People with a long investment horizon should invest more in stock = 1. An optimistic view about the future economic development suggests a higher stock ratio = 1. Clear conflict of goals. Someone who is not willing to take any financial risks should not invest in stock. = 1. Paying off credit card debt is an indicator for having good control about financial issues. > 3. High pension pay leaves the investor free of worries about the retirement and increases his risk taking capacity > 3. High expected inheritance pay leaves the investor free of worries about future funding and increases his risk taking capacity > 3. In connection with little wealth the necessity of a sizeable nest egg demands a higher stock ratio = 5. Singles who were never married have less financial obligations and can afford a higher stock ratio. A young person can afford a high stock ratio, especially when he does not have a family (x7372 1 = 0 and x101 = 1)

Relevance

Violations 44

**

5

**

35 12

**

0

40 0

8

**

3

10

10

The MNL model, setting 3c, SCF 1995, assigned 97 observations ‘correctly’ to the highest risk class. Each response was tested for violating the above hypotheses. Again, the normative findings were confirmed with high validity

Chapter 3. Empirical Analysis

65

with one exception: about 30% of the respondents had a long investment horizon (x3006), but held less than 20% stocks. Apart from this, the group showed great homogeneity, especially in terms of stated risk preference (x3014). No respondent showed willingness to take substantial financial risk when investing. Surprisingly, most investors in this low risk class had managerial jobs. In terms of wealth - a factor not included in the regression - the results were contraintuitive, as 91% of the investors in the low risk class had total assets of more than US$100’000.

3.3.3

‘Wrongly’ assigned observations

Highest risk level: Class 4 In setting 3c of the SCF1998 the MNL assigned 92 observations to the highest risk class (80-100% stocks). According to revealed preference (actual stock ratio), however, these should have been in the lowest risk class (0-20% stocks). The question arises as to whether these cases were ‘wrongly’ assigned. When analyzing the characteristics of those 92 respondents in detail it becomes obvious that they do not fit into the risk class they have chosen themselves. Most of them have a long investment horizon and long-term saving goals, some are even willing to take substantial risks, half are optimistic about the economic outlook and others expect a substantial windfall.

Variable X101

X3006 56 X3008 45 X301 1

X3014 1

Type of Violation < 3. A low number of dependents in connection with high wealth and income leaves freedom to invest riskily Retirement, education and family are longterm goals that allow risky investment = 1. People with a long investment horizon should invest more in stocks = 1. An optimistic view about the future economic development suggests a higher stock ratio = 1. Clear conflict of goals. Someone who is not willing to take any financial risks should not invest in stocks.

Relevance

# of cases 44

**

51

**

57 18

**

19

66

X432 1 X5608

X5821

X7187

X7372 1

X8022

3.3. Testing for bounded rationality

=1. Paying off credit card debt is an indicator for having good control over financial issues. > 3. High pension pay leaves the investor free of worries about the retirement and increases his risk taking capacity > 3. High sums of expected inheritance leaves the investor free of worries about future funding and increases his risk taking capacity > 3. In connection with little wealth the necessity of a sizeable nest egg demands a higher stock ratio = 5. Singles who were never married have less financial obligations and can afford a higher stock ratio. < 40. A young person can afford a high stock ratio, especially when he does not have a family (x7372=0 and x101=1)

15 1

9

**

2

6

2

Naturally, for some observations there are overlapping violations. However, even when accounting for these overlappings and even when only the highly relevant violations (marked with ‘**’) are considered, 88 responses seem incompatible with the lowest risk class. They thus seem to be ‘correctly’ assigned by the model when judging by the underlying hypotheses.

Lowest risk level: Class 0 In setting 3c for the SCF1998 the MNL assigned 59 observations to the lowest risk class (0-20% stocks). According to revealed preference (actual stock ratio), these should have been in the highest risk class (80-100% stocks). Again one has to examine whether these cases were really ‘wrongly’ assigned or whether the model corrected these individual choices. Even though the picture is less unambiguous than for the cases that were ’wrongly’ assigned to the highest risk class, there are many violations among these cases, too. Most of the respondents have a short investment horizon, short-term saving goals, half of them are not willing to take any financial risk and expect an economic deterioration, while others are too close to retirement to invest heavily in stock.

Chapter 3. Empirical Analysis

Variable X101

X3006 1

X3008 1 X301 3

X3014 4

X432 1 X5825 3

Type of Violation > 2. A high number of dependents in connection with low wealth and income leaves the household too exposed to volatility risk of stock investment = 1. Liquidity and consumption are rather short term goals. Stock investment is less well suited for such objectives. = 1. People with a very short investment horizon should not invest heavily in stocks. = 1. A pessimistic view about the future economic development is possibly not the right prerequisite for excessive stock investment. = 1. Clear conflict of goals. Someone who is not willing to take any financial risks should not invest in stock. = 0. Paying off credit card debt is an indicator for having good control about financial issues. = 1. In connection with the proximity of retirement and relatively modest wealth a high stock ratio is not recommendable.

67

Relevance

# of cases 20

**

0

**

8 22

**

26

11 **

10

After accounting for overlapping violations, there remain 33 observations that do not seem to belong into the highest risk class. The analysis of contradictions can be concluded by stating that the model and the empirical data strongly support the discussed hypotheses. The deviation from the hypotheses in the form of a cumulation of grave decision errors amounted to less than 3% of each risk class. This indicates that although investors commit single mistakes that concern some hypotheses and factors, they do not err cumulatively.

3.3.4

Stated and observed preferences

Various empirical studies about risk preferences of individual investors as the one by Bajtelsmit (1999) found that hypothetical questions are not a reliable source of information. Respondents of the SCF 1989, for example, were asked how much risk they would be willing to take for a certain return on a hypothetical investment. What people chose in response was not consistent

68

3.4. Structure of Analysis and Nests

with their asset allocation. Thus, surveys about stated preferences have to be judged with caution. It seems as if what people say is not always compatible with what they do. If this holds true, then approaches involving gambles won’t yield reliable results even if subjects were to act according to expected utility theory. Another reason for the inconsistency between stated and observed preference might base on a mis-understanding of the concept of risk. Possibly not all people primarily think of return volatility when asked to define risk. Communicating risk accurately and completely is thus of foremost relevance if investors’ decision making is to be improved. The incompatibility between observed and stated preferences concerning risk was less pronounced in the two surveys examined. In the SCF 95 survey, 221 of 4300 people stated that they were not prepared to take any risk at all. Half of these 221 respondents also actually held no stocks at all. Only 10% of the people showed inconsistency between stated and revealed preferences. Actual Stock Ratio 0% 1% - 20% 21% - 40% 41% - 60% 61% - 80% 81% - 100%

3.4

Number of Respondents 109 30 20 20 22 20 221

Structure of Analysis and Nests

For the analysis each sample was divided into different subsets. Each subset was then estimated separately. For an overview of the structure and nests please refer to Appendix D.1 In the first setting - labeled ‘1’ - all individuals were included in the model. Those with no assets at all and those with no stocks were examined together with stock-owners. While the first two groups were assigned quite well in this setting, the prediction of the different risk classes was quite poor. This happens often when we have groups of unequal sizes as is the case in the 1995 SCF. Cases are more likely classified to the larger groups, regardless of how

Chapter 3. Empirical Analysis

69

well the model fits. A further reason for the imbalance of predicted outcomes  xik we see that the are the weights. If they are plotted as a function of βjk  xik = 0, implying maximum weight will be given to observations for which βjk that Pt = 0.5, while relatively little weight will be given to observations for which Pt is close to 0 or 1. In the second setting, a two-step procedure was adopted: in a first step labeled ‘2a’ in Appendix D.1 - we only distinguished whether respondents had financial assets or not. In a second step - labeled ‘2b’ - we excluded respondents without financial assets from the samples. However, as the assignment to classes with high risk was still not satisfactory, a third structure was set up. In the third setting the assignment problem was divided into three steps: first we adopted the binary model from structure ‘2a’ distinguishing only whether people owned financial assets or not. In a second binary model labeled ‘3b’ in Appendix D.1, we distinguished whether people owned stocks or not. Finally in the last step - ‘3c’ -, we distinguished between the different risk levels among the stockowners. It is conceivable to combine the earlier two- and three step regressions in a single nested logit model estimating the coefficients for all nests simultaneously. This was labeled as setting ‘4’ in Appendix D.1. However, as explained in Subsection 3.5.7 the NLM is primarily used for the analysis of choice attributes rather than investors’ characteristics. It was not considered here, as the SCF did not contain any data referring to a subjective judgement of different risk classes and investment choices. A benefit of simultaneous estimation with the NLM is that it enables the consideration of different utility functions for each ‘knot’ in the structure and may obtain the effects on probabilities of all choices in the model. It is possible to calculate marginal effects of a change in any factor in the utility function for any alternative. With separate models for each step this is not feasible. A different sample, however, containing preferences on risk classes would possibly shed some light on the question why many investors refrain from stocks. When evaluating the performance of the different models characterized hereafter, it is crucial to keep in mind that not all of them can be compared with each other. For a continuous dependent variable the OLS and Tobit

70

3.5. Characterization of econometric models

model were considered. For a discrete binomial dependent variable, the WLS and the Multinomial logit model were put side by side and for a discrete multinomial dependent variable, the OLS, the Ordered Logit and the Multinomial Logit model were compared with each other.

3.5

Characterization of econometric models

This section will briefly characterize the objective function and the different models of estimation used in the empirical analysis. All models were estimated using the full version of LIMDEP7.014 . Although this program proved extremely helpful and flexible for all models considered in this study, the classification tables, AIC, out-of-sample estimation and several other calculations had to be programmed with Excel macros. In all regressions a constant term was used to account for systematic deviations not explained by the independent factors.15

3.5.1

The Objective Function

Investors are assumed to maximize expected utility. The objective function thus plays a crucial role for the estimation procedure and for the applicability of different regression models. For portfolio choice the traditional approach has been to define the objective function in terms of the moments of the return distribution. Sharpe (1964), Lintner (1965), Mossin (1966) portrayed individuals as single-period maximizers of expected utility of their future wealth. They limited their valuation model to the first two moments. This well-known mean-variance valuation model was extended by Rubinstein (1973) to a general parameterpreference model which will be applied for the second part of this study. In this first part, the objective function of the mean-variance model will suffice. The utility of investor i can thus be defined in terms of expected return µ and volatility σ of investment opportunity j. αi (3.1) Ui = µj − σj2 2 14 Developed

by Econometric Software, Inc., whose founder is Prof. W.H. Greene. Greene (1998), p. 514 suggests normalizing β0 = 0 for the MNL model in order to identify the parameters of the model. 15 Although

Chapter 3. Empirical Analysis

71

αi depicts the investor’s Pratt-Arrow measure of relative risk aversion. At the same time it is the dependent variable of the regression models to be explained by k exogenous factors xik that were presented in Section 3.2. The regression model16 therefore is Ui = µj −

 xik 2 βjk σj 2

(3.2)

In the case where discrete choice models are considered17 , a finite number of investment opportunities must be given. In setting 1 there are 7 discrete investment choices. One consisted entirely of the risk-free asset and one consisted entirely of stocks. The remaining four choices are linear combinations of these two extremes, as shown in Figure 2.1 on page 22. In combination with time series data that allows the calculation of the investment opportunities’ means and variances, threshold values of the relative risk aversion αi can be calculated. These threshold values mark those degrees of relative risk aversion for which the utilities of two adjoining investment opportunities are the same. By calculating these threshold values it is possible to regress directly onto relative risk aversion instead of making the detour via the regression (αi = β  xi ) embedded in the utility function. The threshold values for αi can be calculated by αk 2 αk 2 µk − σ = µk+1 − σ (3.3) 2 k 2 k+1 for k=1,2,...6; αk marks the relative risk aversion for which the utilities of the investment opportunities’ (k and k + 1) are equal. As an example the following yearly Swiss Market returns from 1925-1997 (source: Bank Pictet & Cie.) are depicted for six different risk classes in the model.18 k 1 2 3 16 Subscripts

µk 4.43% 5.16% 5.90%

σk 3.52% 5.47% 8.54%

k 4 5 6

µk 6.64% 7.38% 8.11%

σk 11.89% 15.34% 18.83%

j and k will be dropped hereafter. B shows how the Multinomial Logit Model can be derived from Utility Maximization and an Extreme Value Distributed Error term. 18 The time series data on market portfolio returns is analyzed in greater depth in Appendix I. 17 Appendix

72

3.5. Characterization of econometric models

For these moments of returns the following threshold values can be calculated with Equation 3.3:

k+1=k 6=5 5=4 4=3

αk 1.22 1.51 2.16

regr.value 1.00 1.37 1.84

k+1=k 3=2 2=1 1=0

αk 3.11 8.33 71.51

regr.value 2.64 5.72 39.92

The first column of the above table indicates for which two risk classes the threshold αk in the second column was calculated. Thus, for a relative risk aversion of αk = 1.22 the two investment opportunities k = 6 and k = 5 with their mean and volatility given above yield the same utility according to Equation 3.1. The goal of this proceeding is to rephrase the objective function 3.1 and to solve for relative risk aversion in order to regress directly onto αk -classes. As it is impossible in discrete choice models to regress onto non-integer values, the regression itself will be carried out using the k’s which represent different threshold values of risk aversion. The k’s are thus used as the values of a multinomial dependent variable Y in a discrete choice model and stand for classes of increasing risk. The third column labeled ‘regr.value’ gives the ‘real’ values for the relative risk aversion that are represented by k. The higher k, the higher the investment opportunity’s risk and the lower the investor’s risk aversion who chooses the corresponding asset allocation. The ’regr.value’ refers to the mean of two adjoining thresholds and can be interpreted as the risk aversion representative for the risk class k: For example, the representative relative risk aversion for the risk class 5 is [1.22 + (1.22 + 1.51)/2] = 1.37. Generally, it does not matter whether the integer values of the dependent variable in the discrete choice models are converted into a measure for relative risk aversion or a stock ratio, as these two quantities are connected anyway through Formula A.14 in Section A.1 of the Appendix. The purpose of this subsection was to show how the threshold values of relative risk aversion. These values correspond to the finite number of risk classes in the settings depicted in Figure D.1 of Appendix D.

Chapter 3. Empirical Analysis

3.5.2

73

Ordinary and Weighted Least Squares Model

Ordinary Least Squares (OLS) estimation was carried out for settings where the dependent variable Y is continuous or polytomous discrete, while Weighted Least Squares (WLS) was applied for settings with a dichotomous dependent variable19 , as the OLS does not yield efficient estimates in that case.20 Goldberger (1964) proposed the following two-step, weighted estimator to obtain unbiased and efficient estimates of the LPM: The standard linear regression model for OLS21 yi = β0 + β  xi + εi

(3.4)

where i = 1, 2, ...n and xi is a K-vector of known factors, is used to yield unbiased estimates βˆj . From these estimates, weights wi are calculated22 for each observation i wi =



 − 12 βˆ xi βˆ xi 1 −

(3.5)

All terms on both sides of the linear regression model in Eq. 3.4 are then multiplied by wi    ˆ (wi Yi ) = βwi xi + wi εi (3.6) and a second regression for these terms yields unbiased and efficient estimates. However, coefficients can no longer be interpreted. In setting 1, observations are considered for which no stock ratio could be calculated, as the corresponding individuals did not own any financial assets. For these observations the dependent variable was arbitrarily set to -1. 19 Linear regression models with a dependent variable that is either zero or one are often called linear probability models. 20 To obtain BLU estimators, serial independence and homoscedasticity of the error term as well as non-collinearity of the independent variables must be given. Clearly for a dichotomous Y the error term cannot have constant variance, as the variance of εi varies systematically with the regressors xi . 21 Subscript j on β and k on x are suppressed. 22 The weights are just the reciprocals of the estimated standard error of ε . Problems i with the formula 3.5 may arise if the predicted value β  xi is greater than 1. In that case it can be truncated to 0.999, see Aldrich and Nelson (1984), 84.

74

3.5. Characterization of econometric models

3.5.3

The Tobit Model

When the dependent variable, the percentage of wealth allocated to risky assets, is given in a continuous form in setting 1, it represents a naturally censored variable. The reason is that one cannot calculate a stock ratio for individuals who do not own any financial assets. This particularity destroys the linearity assumption so that the OLS seems clearly inappropriate. The model is censored as one can at least observe the exogenous variables. Thus, a standard Tobit model (or Type I Tobit) can be used for estimating regression coefficients: yi∗

= β0 + β  xi + εi ,

yi

=

yi∗

yi

=

0

if if

yi∗ yi∗

εi ∼ N [0, σ 2 ]

>0

(3.7) (3.8)

≤0

εi are assumed to be iid drawings from N (0, σ 2 ) and can be interpreted as the collection of all the unobservable variables that affect the utility function. We can write the likelihood function L for n independent observations of the model as  x β  −1 (yi − xi β) ] (3.9) [1 − Φ( i )] σ φ[ L= σ σ 0 1 where Φ and φ are the distribution and density function, respectively, of the standard normal variable.   ∗ 0 means the product over those i for which yi ≤ 0, and 1 means the ∗ product over those i for which yi > 0. The method of estimation is maximum likelihood. The prediction for the Tobit model in LIMDEP - which is also used for setting up the classification tables - is calculated as the conditional mean of yi given xi : E[yi |xi ] = Li ΦL + Ui (1 − ΦU ) + (ΦU − ΦL )βN xi + σi (φL − φU ) where Li = lower bound (0 in our case), Ui = upper bound (+∞ in our case)  φL = φ[ 0−βσ x) ] φU = 0

(3.10)

Chapter 3. Empirical Analysis

3.5.4

75

The Ordered Logit Model

The Ordered Logit model is based on the following specification: yi∗

= β  xi + εi ,

Yi Yi Yi

yi∗

εi ∼ N [0,

π2 ] 3

= 0 iff ≤ µ0 , = j iff µj−1 < yi∗ ≤ µj , = J iff yi∗ > µJ−1 ,

(3.11) (3.12)

where j = 0, 1, 2, ..., J and J = 6 for structure one23 The probabilities for each observation are calculated as follows24 : P [Yi = j] = f [µj − β  xi ] − f [µj−1 − β  xi ] where f with µ−1 µ0 µJ

(3.13)

= density function of logistic distribution = −∞ =0 = +∞

The particular algorithm for estimating the threshold parameters of the Ordered Logit in LIMDEP did not produce predicted choices that are evenly spread over risk classes, as in the sample. To correct for this unsatisfactory result, the threshold parameters were manually fixed to equal the appropriate coding of the risk class. For example µ1 =2, ... , µJ−1 = J. Imposing fixed values yielded significantly better classification tables for the Ordered in all settings. The Ordered Logit model can be applied since the respondents express a preference in the form of an ordinal ranking with regard to risk and return. 23 J varies among the structures depicted in Figure D.1. J = 6 for structure 1, J=5 for structure 2b, J=4 for structure 3c. 24 The algorithm used to obtain the maximum likelihood estimates is DFP (by Davidson, Fletcher and Powell, see Fletcher (1980)). Starting values are obtained by OLS. This initial regression is based on the dichotomy formed by using the binary indicator 1, as if a univariate probit model applied. For grouped data, p+ and p0 = 1 − p+ provide the dichotomy, and minimum chi-squared estimates are obtained. The constant term and the values of the thresholds are estimated by using the cell frequencies under the assumption that all of the slopes are zero. The real line is segmented in such a way that the logistic probabilities corresponding to this partition match the sample cell frequencies, cf. Greene (1998).

76

3.5. Characterization of econometric models

Therefore, “[...] the values that y takes correspond to a partition of the real line, whereas in the unordered model they correspond either to a nonsuccessive partition of the real line or to a partition of a higher-dimensional Euclidean space.”25 Amemiya (1985) urges caution in using an ordered model “because if the true model is unordered, an ordered model can lead to serious biases in the estimation of the probabilities. The cost of using an unordered model when the true model is ordered is a loss of efficiency rather than consistency.” According to the underlying financial theory, however, the employment of an Ordered model seems more obvious than the use of any discrete nonordered model (MNL, CLM), as it preserves the linearity of the risk-returnrelationship. We will discuss this point in more detail when evaluating the various models and their performance. The use of a discrete regression model, of course, has its disadvantages. The continuous dependent variable - the stock ratio - is transformed into a discrete variable representing a fund that covers a certain interval of the CML: continuous: yi

= β0 + β1 x1i + ... + ε

discrete: Yi

= β0 + β1 x1i + ... + ν

Yi

=

Yi

= k iff k + 1 >

(3.14)

0 if yi = 0 yi ≥ k for k = 1,2,3,4,5 0.2

By discretizing the dependent variable in this way, an additional error (ν − ε) is added to the error term of the continuous model: ν = ε + [yi − (0.1 + (Yi − 1) · 0.2)]

(3.15)

Thus, for the precision of estimation it is obviously more advantageous and more efficient to use a regression model that conserves the information contained in the continuous variable if possible. The Tobit model that was presented earlier certainly has that feature. It conserves both the linearity of risk as well as the significance of the unit distance. Even though the Ordered Logit model does not retain the information of the unit distance, it has one important advantage over the Multinomial, the 25 Amemiya

(1985), p.292

Chapter 3. Empirical Analysis

77

Conditional and the Nested Logit model: It conserves the ordinal ranking of the dependent variable. The linearity of risk is not lost in the Ordered Logit, while the other mentioned models assume diverse alternatives that cannot easily be ordered in one dimension.

3.5.5

The Binomial- and Multinomial Logit Model

The multinomial logit (MNL) model’s application26 to economic consumer theory is based on the random utility approach formalized by Manski (1977). It states that the individual always selects the alternative with the highest utility. These utilities, however, are not known with certainty and thus treated as random variables. Assuming iid random utilities yields a simple scalable model of choice probabilities. Appendix B illustrates how the MNL model can be derived from utility maximization with an error term that exhibits an extreme value distribution. In the MNL model the independent variables are individual specific characteristics, such as credit behavior, education, savings horizon etc. that explain the variability of tastes across the portion of the population to which our model of investment behavior applies. The vector of socio-economic characteristics is the same for all choices. In contrast to the Conditional Logit or Discrete Choice Model27 where attributes of the choices j also enter the analysis, only characteristics xi of the individual i are considered in the Multinomial Logit model. The MNL is typically employed for individual data in which the x variables are characteristics of the observed individuals, not the choices. While the independent factors are the same for all alternatives, the characteristics xi of the individual are assumed to be the same for all choices j both in the MNL and the CLM. The set of alternatives and choice characteristics can differ for each observation only in the CLM. That is why the choice subscript on x is dropped in the MNL formula below. Both the MNL and the CLM were originally derived as special cases of a general model of utility maximization where individuals i are assumed to 26 Its

original formulation is due to Luce (1959). originally referred to this model as Conditional Logit (CLM), see McFadden (1973). However, both terms - Conditional Logit (CLM) and Discrete Choice Model (DCM) - are used synonymously here. 27 McFadden

78

3.5. Characterization of econometric models

have preferences defined over a set of alternatives j: U (ij) Observed Y

 xik + εj = β0 + βjk

=

choice j if U (j) > U (k) ∀ j...k

(3.16) (3.17)

where i = 1, 2, ...n ; j = 1, 2, ..., J and k = 1, 2, ..., K. The disturbances or individual heterogeneity terms εj are assumed to be iid with extreme value distribution exp[−exp(−ε)].28 The choice probabilities of the MNL can thus be defined as: P (Yi = j)

=

exp(β0 + βj xi ) J  exp(β0 + βj xi )

(3.18)

j=0

where i = 1, 2, ...n and j = 1, 2, ..., J Both the MNL and the CLM, characterized hereafter, share one restrictive property that has come to be known as the ‘independence from irrelevant alternatives’ (iia). Amemiya (1985) notes that the MNL “implies that the alternatives are dissimilar”, as it calculates the relative probabilities between a pair of alternatives (the category j and the base category 0) ignoring the other alternatives. If two alternatives are very similar, their error terms can no longer be assumed to be independent.29 Consequently, the relative probabilities are not independent of each either and can no longer be calculated pairwise without yielding biased results.30

3.5.6

The Conditional Logit Model (CLM)

In contrast to the MNL model the CLM analyzes primarily the attributes of the choices rather than the characteristics of the individual. These choice attributes typically have a factual individual specific nature, but they can also 28 McFadden proved that the MNL is derived from Utility Maximization iff (ε ) are inj dependent and follow an extreme value or log Weibull distribution (EVD). For a proof, see Appendix B or Amemiya (1985), p.297, Equation (9.3.42) 29 See also Greene (1993). 30 The Hausman test is similar to a Likelihood Ratio test. The unrestricted model’s loglikelihood is compared with that of a restricted model that comprises a smaller set of choices. If restricting the choice set leads to a singularity, Hausman and McFadden (1984) suggest to reduce the number of regressors as well.

Chapter 3. Empirical Analysis

79

have a subjective ‘coloring’. For example, in the models analyzing transport choice variables such as the individual travel time and travel costs of alternative travel modes are factual figures, objectively assessable. The comfort, prestige and safety of a transport mode, on the other hand, are variables that possess a much more subjective element. Nevertheless it is possible to include socio-economic characteristics in the model. Apart from this, the two models (MNL, CLM) are very much alike31 Unfortunately, the SCF does not include attributes of the investment choices. Also, the number of choice attributes is limited from the perspective of financial theory. There are basically only the return moments that characterize different investment choices. These return moments could be estimated empirically and were then added into the data set. The main consideration in this context concerned the selection of the correct set of factors. It had to be a set that equally applied to all investors. The most obvious choice would be the incorporation of the moments of the return distribution of each investment class, as they are the constituent parts of all asset pricing models. Estimates for the first three moments - mean return, variance and skewness - can be obtained empirically from the S&P500 total return index for the last 25 years (1975-2000).32 These moment estimates were then included as attributes and independent variables interacting with alternative specific constants as well as with socio-economic characteristics. Thus every individual in the sample is assumed to choose from the same choice set and all alternatives are assumed to be available to all investors. Like the MNL model, the CLM can be expressed by utility U of choice j for individual i: U (ij)

= β0 + βj xi + εj

(3.19)

where i = 1, 2, ...n and j = 1, 2, ..., J. The probability that the investor chooses alternative j is Observed Y 31 The

=

choice j if U (j) > U (k) ∀ j = k

(3.20)

random, individual specific terms are assumed to be iid, each with an extreme value distribution (Gumbel). Under this condition the CLM can be derived from utility maximization just as the MNL model before, cf. Domencich and McFadden (1975), ch. 4 and 5. 32 Estimates for the Swiss Market were obtained from the MSCI Switzerland Total Return Index in Appendix I.

80

3.5. Characterization of econometric models

P (Yi = j) =

exp(βj xi ) J  exp(βj xi )

j=0

Apart from the independent variables that provide the choices attributes (return moments), socio-economic variables can be included in the model. They cannot simply be incorporated in the same way as the choices’ attributes, as the discrete choice probabilities are homogeneous of degree zero in the parameters. Attributes which are the same for all outcomes for every single individual thus drop out of the probability model. The socio-economic characteristics would clearly qualify as such attributes as they stay the same for all risk categories. They can only be integrated by using the equivalent of dummy variable interaction terms and are thus expanded by interacting with choice specific binary variables. The result of this kind of model is a large set of fixed parameters with few variables that account for variations in taste. Thus, the model was not considered in the empirical analysis.

3.5.7

The Nested Logit Model

Even though the Nested Logit model (NLM) is not used for estimation in the empirical analysis of this study, it will be briefly characterized here for the sake of completeness. Its application is well conceivable if one wishes to examine the characteristics of the different investment choices rather than those of the investor. The availability of data on the investments’ characteristics is a necessary prerequisite for the applicability of the Nested Logit model. While it is possible to include factors that portray investor characteristics, these take the role of alternative specific constants, as for a single observation they do not vary across choices. The study refrained from applying the NLM, as according to portfolio theory, investment choices are completely characterized by the moments of their return distribution - mean, variance, skewness. However, these attributes are objective in character and lack the subjective judgement of the investor. This subjective judgement, however, is exactly what the Conditional Logit and the Nested Logit mean to measure. Utilizing return moments as independent variables for explaining risky choice in these models would come down to a

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tautology. Nevertheless, there are other independent factors imaginable representing choice attributes that could be employed: the ethical purpose of a fund for example, its brand name or the name of the fundmanager or the interest in a specific industry. The choice among different investment alternatives modeled above in the MNL model and CLM may be viewed as taking place at more than one level. This so-called ‘hierarchical’ choice is comfortably handled by a nested logit model which is essentially an extension of the CLM. In contrast to the latter, the NLM, however, is not subject to the ‘independence from irrelevant alternatives’ (iia). The NLM’s Likelihood-function is given in Appendix C. Provided the data is appropriately formatted, the NLM can be estimated using NLOGIT2.033

3.6

Goodness of fit and hypotheses testing

3.6.1

Classification Tables and Error Distance

One measure for the quality and fit of the model is the number of correct predictions it makes. These can be read from classification tables that list the predicted choices of the model34 together with the actual choices of the investors (observations). An example is the following classification table from the TOBIT model in setting 1 for the SCF 1998:

33 By

Econometric Software, Inc. predicted choices had to be calculated separately from the estimation of the factor coefficients in LIMDEP. The estimates were imported into MSExcel where the predicted choices were calculated. A Macro finally assigned each observation into the appropriate cell of the classification table. 34 These

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3.6. Goodness of fit and hypotheses testing

Setting 1 - Tobit, SCF1998 Classification table Predicted Choice 0 1 2 3 4 183 71 48 0 0 455 439 1051 234 60 4 11 134 136 56 4 5 116 110 90 2 5 115 143 93 2 12 90 122 94 5 11 112 124 101 655 554 1666 869 494 Percent correct: 22.49% Error Distance: 8’967’647, ln(ED): 16.01 Actual 0 1 2 3 4 5 6

5 0 7 8 15 6 9 22 67

6 0 0 0 0 0 0 0 0

302 2246 349 340 364 329 375 4305

The last column lists the number of actual observations per category, while the last row depicts the number of predicted observations. In the diagonal there are the number of observations for which actual and predicted values of Y coincide. The sum of these values divided by the total number of observations gives the ratio of correct assignments (labeled ‘Percent correct’). That ratio in itself, though, is not a sufficient measure for comparison of different regressions, as it lacks to account for the degree of misassignment. To compare regression results, both the distance and the number of wrong assignments need to be considered. Both deviations are weighted accordingly with the Error Distance: J J

[(1 + (r − c)3 )(Prc − Trc )2 ]

(3.21)

c=0 r=0

where J = number of categories - 1, r = row in classification table, r = 0, 1 ..., J. c = column in classification table, c = 0, 1 ..., J. Trc = total number of actual observations for cell in row r in column c, Prc = total number of predicted observations for cell in row r in column c. As the resulting number for the Error Distance is rather large, its log was taken in order to ease the comparison of the models (see the previous table for example).

Chapter 3. Empirical Analysis

3.6.2

83

Akaike Information Criterion (AIC)

When choosing one regression model out of many competing models, we can use the Likelihood ratio test LRT > d where d is determined so that P [LRT > d|L(α)] = c with c a certain prescribed constant such as 5%. Such proceeding is unfortunately time consuming and often even inconclusive35 . Akaike36 formulated a loss function on the basis of the sum of squares corrected for by the number of regressors. As the loss function is to be minimized among competing models, the model with the lowest AIC is considered to yield the best results37 . AIC = −

2 2K log L(ˆ α) + T T

(3.22)

where T = number of observations, α ˆ = maximum likelihood estimators K = number of independent variables

3.6.3

Likelihood ratio tests

The likelihood ratio (LR) test in discrete choice models is used in the same way that the F-test is used in linear regression models for joint tests of several parameters.38 LIMDEP’s standard output for each model comprises a LR test of the hypothesis that all coefficients are zero. The results for these tests were not included in this study, as the null hypothesis could always be rejected at a 35 Cf.

Amemiya (1985), p.146 (1973) 37 For full details cf. Amemiya (1980) 38 Of the three classical test statistics (LR, Lagrange multiplier and Wald test) that are asymptotically equivalent, the LR test is conceptually the simplest. In general it denotes twice the difference between the restricted θˆ and unrestricted θ˜ values of the loglikelihood function 36 Akaike

ˆ − logL(θ)) ˜ 2(logL(θ) θˆ refers to the unrestricted ML estimate and θ˜ denotes the ML estimate subject to r restrictions. See Davidson and Mackinnon (1993), 275.

84

3.6. Goodness of fit and hypotheses testing

very low level of significance39 (5% and 1% respectively). It is more informative to test the null hypothesis that only some coefficients are zero or that all the coefficients except for the alternative-specific constants are zero.40 The test statistic for the latter is −2(logL(c) − logL(β))

(3.23)

with K − J + 1 degrees of freedom, where J is the number of alternatives in the choice set (7 in setting 1) and logL(c) is the log-Likelihood of a model with only constants. LogL(c) can be obtained by estimating a model with J − 1 alternative-specific constants41 or from logL(c) =

J i=1

 Ni ln

Ni N

 (3.24)

where Ni is the number of observations selecting alternative i and N is the total sample size. For the SCF samples LR tests were carried out both for single and for multiple factor coefficients. LR tests simply compare the Loglikelihoods of the restricted and the unrestricted model: The test statistic (Eq. 3.23) is distributed as χ2 (df ) under the null. For a single restriction the critical value CV (df = 1) at the 5% level is 3.841 because the probability of obtaining a random drawing from a χ2 -distribution that is greater than 3.841 is 5%. Thus, if the test statistic LR > CV , the null (that the single factor coefficient equals zero) can be rejected at the 5% level. The tables in Appendix E.3 summarize several LR tests for the MNL and Ordered Logit in different settings of the SCF 1998. For the MNL the LR tests gave the following results: • In setting 1, SCF 1998, the null that the four least significant factor coefficients jointly equal zero can not be refuted at the 5% level in the MNL. • In setting 2a and 2b, SCF 1998, the null that the nine least significant factor coefficients jointly equal zero can not be refuted at the 5% level. 39 Also

called the ‘size of test’. test was refuted for the MNL in all settings at the 5% level. 41 Cf. Ben-Akiva and Lerman (1985). 40 This

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• In setting 3b, SCF 1998, there are 12 factor coefficients for which the null cannot be refuted at the 5% level. • Finally, in setting 3c, SCF 1998, there are altogether 18 factor coefficients for which the null can not be refuted at the 5% level. The model can be estimated with only 6 variables and one alternative specific constant.

3.7

3.7.1

Results of regressions for Setting 1 - “All Categories” Coefficients

SCF1998 In-Sample-Estimation For the OLS (see Table E.1 on page 180) 17 out of 25 factors proved significant at the 5% level. 8 coefficients did not have the expected sign.42 In the Tobit model (see Table E.2 on page 182) 16 out of 25 factors were significant at the 5% level. Only 5 coefficients did not have the expected sign. Even though the AIC of the OLS is lower than the one of the Tobit, the predictive power of the Tobit as given by the classification table proves to be better than that of the OLS. The percentage of correctly classified choices is higher in the Tobit and the Error Distance is slightly lower than in the OLS. While the OLS fails to assign any choices to the two highest risk classes, the Tobit at least classifies a few observations to category 5. In the Ordered Logit model (see Table E.3 on page 184) 23 out of 25 factors are significant at the 5% level, while 6 coefficients do not have the expected sign. Even though 42% of all observations are correctly classified by the Ordered, especially the majority of cases in the low risk classes, it does not spread the predicted choices as evenly over the higher risk classes as the MNL. In the MNL model (see Table E.4 on page 186) the logits of the different risk classes had on average at least 14 significant factors. The number 42 The expected signs are discussed in Appendix F, Table F.3 on Page 258. In the tables wrong coefficient signs are encircled.

86

3.7. Results of regressions for Setting 1 - “All Categories”

of coefficients with a wrong sign varied between 7 and 9. More important than the sign, however, is the change in magnitude of each coefficient among the different risk classes. Non-monotone increasing coefficients reveal irregularities among different risk levels and can thus unveil non-linear relations between a factor and the risk level. The course of the coefficients over classes of increasing risk can be comprehended best through the graph in Figure 3.2. For setting 1 of the in-sample estimation, the MNL clearly produces the best result in terms of correctly assigned choices and lowest Error Distance.

Chapter 3. Empirical Analysis

Figure 3.2: The course of the coefficients over classes of increasing risk for setting 1, in-sample estimation of the SCF 1998 with a MNL model. The dependent variable Y represents the level of risk measured as return volatility; people for whom Y =0 do not have any financial assets; Y =1 depicts pure fixed income investors; Y =2 stands for a stock ratio of 1-20%, Y =3 ⇒ 21-40%, ... Y =6 ⇒ 81-100%.

87

88

3.7. Results of regressions for Setting 1 - “All Categories”

SCF1995 in 1998 data Out-of-Sample-Estimation

For OLS see Table E.21 on page 216, for the Tobit model see Table E.22 on page 218, for Ordered Logit see Table E.23 on page 220, for MNL see Table E.24 on page 222. Not all the coefficient estimates are significantly different from zero at the 5% or 1% levels. When LR tests were carried out for the non-significant factors in the discrete choice models, the null could not always be refuted (see Subsection 3.6.3).

3.7.2

Results of Classification Tables Empirical Analysis of Structure1: Comparison of Models Dataset: SCF 1998, In-sample Estimation Observations = 4305, Parameters = 25, Deg.Fr.= 4274 Model OLS Tobit Ordered MNL

Log L -1679.83 -2363.74 -7833.00 -5350.63

AIC 0.792 1.110 3.650 2.500

Percent 20.74% 22.49% 41.79% 54.94%

Error Dist 9’562’127 8’967’647 4’265’498 11’219’224

Ln ED 16.07 16.01 15.27 16.23

Empirical Analysis of Structure1: Comparison of Models Out-of-sample-Estimation SCF1995 in 1998 data Observations = 4305, Parameters = 25, Deg.Fr.= 4274 Model OLS Tobit Ordered MNL

Log L -1558.57 -2187.82 -7024.75 -5024.36

AIC 0.737 1.029 3.280 2.350

Percent 17.19% 16.68% 34.05% 43.46%

Error Dist 11’864’863 16’002’731 7’455’722 19’807’854

Ln ED 16.29 16.59 15.82 16.80

Chapter 3. Empirical Analysis

3.8 3.8.1

89

Results of regressions for Setting 2 - Twostep estimation Setting 2a - “Assetholder or Non-asset holder”

Coefficients SCF1998, in-sample estimation: For the WLS (see Table E.5 on page 189) 21 out of 25 factors proved significant at the 5% level. 11 coefficients did not have the expected sign.43 In the Tobit model (see Table E.6 on page 191) 16 out of 25 factors were significant at the 5% level. Only 8 coefficients did not have the expected sign. As in setting 1, the AIC of the OLS is lower than the one of the Tobit, however, the predictive power of the Tobit as given by the classification table proves to be better than that of the OLS. The percentage of correctly classified choices is higher in the Tobit and the Error Distance is slightly lower than in the OLS. In the Binomial Logit model (see Table E.7 on page 193) 18 out of 25 factors are significant at the 5% level and 9 coefficients do not have the expected sign. Even though the percentage for the number of correctly predicted choices are higher than in the Tobit, the Binomial Logit assigns too few observations to the category 0. For setting 2a of the in-sample estimation, the Tobit thus produces the best results in terms of correctly assigned choices and lowest Error Distance. SCF1995 in 1998 data Out-of-Sample-Estimation: for OLS see Table E.25 on page 225, for MNL see Table E.26 on page 227.

43 The expected signs are discussed in Appendix F, Table F.3 on Page 258. In the tables wrong coefficient signs are encircled.

90

3.8. Results of regressions for Setting 2 - Two-step estimation

Figure 3.3: The course of the coefficients over classes of increasing risk for setting 2b, in-sample estimation of the SCF 1998 with a MNL model. The dependent variable Y represents the level of risk measured as return volatility; people for whom Y =0 are pure fixed income investors; Y =1 stands for a stock ratio of 1-20%, Y =2 ⇒ 21-40%, ... Y =5 ⇒ 81-100%.

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Figure 3.4: The course of the coefficients over classes of increasing risk for setting 2b, in-sample estimation of the SCF 1998 with a MNL model. The dependent variable Y represents the level of risk measured as return volatility; people for whom Y =0 are pure fixed income investors; Y =1 stands for a stock ratio of 1-20%, Y =2 ⇒ 21-40%, ... Y =5 ⇒ 81-100%.

Results of Classification Tables Empirical Analysis of Structure2a: Comparison of Models Dataset: SCF 1998, In-sample Estimation Observations = 4305, Parameters = 25, Deg.Fr.= 4274 Model WLS Tobit BNL

Log L 183.61 -618.63 -678.87

AIC -0.074 0.299 0.327

Percent 90.94% 92.89% 93.59%

Error Dist 264’654 165’030 168’414

Ln ED 12.49 12.01 12.03

92

3.8. Results of regressions for Setting 2 - Two-step estimation Empirical Analysis of Structure2a: Comparison of Models Out-of-sample-Estimation SCF1995 in 1998 data Observations = 4305, Parameters = 25, Deg.Fr.= 4274 Model WLS BNL

3.8.2

Log L -93.26 -737.67

AIC 0.055 0.355

Percent 92.31% 92.54%

Error Dist 231’123 231’855

Ln ED 12.35 12.35

Setting 2b - “Assetholders only”

Coefficients SCF1998, In-Sample Estimation: For the OLS (see Table E.9 on page 195) only 10 out of 25 factors proved significant at the 5% level. 5 coefficients did not have the expected sign.44 In the Ordered Logit model (see Table E.11 on page 197) 22 out of 25 factors are significant at the 5% level and only 6 coefficients do not have the expected sign. Even though the bulk of cases for category 0 are correctly classified, the Ordered Logit fails to assign a satisfactory number of observations to the highest risk classes. The MNL model (see Table E.12 on page 199) however, not only classifies the bulk of cases in category 0 correctly, it also evenly spreads predicted choices over the high risk classes 4 and 5. For setting 2b of the in-sample estimation, the MNL yields the best result in terms of correctly assigned choices and lowest Error Distance. The MNL’s advantage becomes evident from the Figures in Table 3.3: No independent factor in the MNL is strictly monotone increasing or decreasing in the risk classes depicted on the x-axis. In contrast to all other models, the MNL can imitate these non-linearities between independent and dependent variables to give superior results. SCF1995 in 1998 data Out-of-Sample-Estimation: for OLS see Table E.28 on page 229, for Ordered Logit see Table E.30 on page 231, for MNL see Table E.31 on page 233. 44 The expected signs are discussed in Appendix F, Table F.3 on Page 258. In the tables wrong coefficient signs are encircled.

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Results of Classification Tables Empirical Analysis of Structure2b: Comparison of Models Dataset: SCF 1998, In-sample Estimation Observations = 4003, Parameters = 25, Deg.Fr.= 3978 Model OLS Ordered MNL

Log L -547.28 -8846.00 -4672.59

AIC 0.286 4.43 2.347

Percent 23.06% 38.67% 58.41%

Error Dist 10’113’838 6’395’874 10’996’615

Ln ED 16.13 15.67 16.21

Empirical Analysis of Structure2b: Comparison of Models Out-of-sample-Estimation SCF1995 in 1998 data Observations = 4003, Parameters = 25, Deg.Fr.= 3978 Model OLS Ordered MNL

3.9

Log L -139.87 -7672.27 -4287

AIC 0.084 3.908 2.189

Percent 22.16% 40.47% 51.04%

Error Dist 15’259’122 10’630’777 26’479’635

Ln ED 16.54 16.18 17.09

Results of regressions for setting 3 - Threestep estimation

The following analysis follows a three-step estimation of risk-taking. It was depicted in Figure D.1 as the ‘third setting’. The assignment of respondents for our data sample SCF 1995 was divided into three steps. In the first step - labeled 3a in Figure D.1 - we distinguished only between people with and without assets (coded 0 and 1 respectively). This step was already evaluated in setting 2a. In the second step - called 3b in Figure D.1 - we excluded the respondents without assets and distinguished only between non-stockholders (coded as 0) and stockholders (coded as 1). In the third step - marked 3c in Figure D.1 - we excluded both people without assets and people without stocks. The remaining respondents were analyzed for their different stock ratios. Following are the tables that summarize the regression results.

94

3.9.1

3.9. Results of regressions for setting 3 - Three-step estimation

Setting 3b - “Stock- or Non-stock holder”

Coefficients SCF1998, in-sample estimation: For the WLS (see Table E.13 on page 202) 16 out of 25 factors proved significant at the 5% level. 6 coefficients did not have the expected sign.45 In the Tobit model (see Table E.14 on page 204) 16 out of 25 factors were significant at the 5% level. Only 5 coefficients did not have the expected sign. In contrast to setting 1, the AIC of the OLS is lower than the one of the Tobit and its predictive power is better. The percentage of correctly classified choices is higher in the OLS and the Error Distance is significantly lower than in the Tobit. The Binomial Logit model (see Table E.7 on page 193) proves to be slightly better than the OLS. Its Error Distance is lower and the percentage of correctly assigned choices is higher, although only 9 out of 25 factors are significant at the 5% level. 7 coefficients do not have the expected sign. For setting 3b of the in-sample estimation, the MNL thus produces the best result in terms of correctly assigned choices and lowest Error Distance. SCF1995 in 1998 data Out-of-Sample-Estimation: for OLS see Table E.32 on page 236, for MNL see Table E.33 on page 238.

Results of Classification Tables Empirical Analysis of Structure3b: Comparison of Models Dataset: SCF 1998, In-sample Estimation Observations = 4003, Parameters = 25, Deg.Fr.= 3978 Model WLS Tobit BNL

Log L -1987.91 -3173.2 -1906.34

AIC 1.006 1.598 0.965

Percent 74.62% 71.15% 77.44%

Error Dist 1’575’318 3’245’919 1’231’107

Ln ED 14.27 14.99 14.02

45 The expected signs are discussed in Appendix F, Table F.3 on Page 258. In the tables wrong coefficient signs are encircled.

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95

Empirical Analysis of Structure3b: Comparison of Models Out-of-sample-Estimation SCF1995 in 1998 data Observations = 4003, Parameters = 25, Deg.Fr.= 3978 Model WLS BNL

3.9.2

Log L -2029.91 -1941.21

AIC 1.043 0.998

Percent 64.20% 65.88%

Error Dist 4’687’071 2’876’910

Ln ED 15.36 14.87

Setting 3c - “Stockholders only”

Coefficients For the OLS (see Table E.17 on page 208) as few as 3 factors out of 25 proved significant at the 5% level. 6 coefficients did not have the expected sign.46 Even though the AIC of the OLS was low, its predictive power as given by the classification table is so poor that it shouldn’t be considered at all for distinguishing between risky choices in this setting. In the Ordered Logit model (see Table E.19 on page 210) 7 out of 25 factors are significant at the 5% level, while 7 coefficients do not have the expected sign. Similar to the result of the OLS, the Ordered Logit produces a very poor classification table in this setting. Both, the OLS and the Ordered Logit spread the observations only over two classes. The MNL model (see Table E.20 on page 212) does a much better job in this respect. Though there are very few significant factors, the MNL spreads all observations evenly over all risk categories. It also produces the highest percentage of correctly classified cases and the lowest Error Distance. For setting 3c of the in-sample estimation, the MNL clearly produces the best result in terms of correctly assigned choices and lowest Error Distance. SCF1995 in 1998 data Out-of-Sample-Estimation: for OLS see Table E.35 on page 240, for Ordered Logit see Table E.37 on page 242, for MNL see Table E.38 on page 244. 46 The expected signs are discussed in Appendix F, Table F.3 on Page 258. In the tables wrong coefficient signs are encircled.

96

3.10. Conclusion of the two-moment setting

Results of Classification Tables Empirical Analysis of Structure3c: Comparison of Models Dataset: SCF 1998, In-sample Estimation Observations = 1758, Parameters = 25, Deg.Fr.= 1733 Model OLS Ordered MNL

Log L -330.96 -3899.90 -2766.67

AIC 0.405 3.219 3.178

Percent 21.23% 21.46% 27.95%

Error Dist 2’629’736 2’147’852 1’803’590

Ln ED 14.78 14.58 14.41

Empirical Analysis of Structure3c: Comparison of Models Out-of-sample-Estimation SCF1995 in 1998 data Observations = 1758, Parameters = 25, Deg.Fr.= 1733 Model OLS Ordered MNL

Log L -284.49 -3516.39 -2345.46

AIC 0.399 4.570 3.059

Percent 20.20% 20.94% 21.74%

Error Dist 2’231’527 4’675’546 5’743’122

Ln ED 14.62 15.36 15.56

3.10

Conclusion of the two-moment setting

3.10.1

The factors and their explanative power

The predictive power of the OLS and the Ordered Logit significantly decreased over the settings 1 - 3, as non-linearities in the independent variables became more important to predict risky choices. The MNL performed consistently well over all settings and should be the model of choice for determining risk preferences in the presented framework. In general when more than two risk classes are considered the OLS does not perform as well as the MNL. This can be attributed to the fact that the MNL considers the risk classes in a non-linear fashion. The logit of every risk class has its own coefficient for each factor.47 In this way, the model can account for non-monotone decreasing or increasing relations between Y and xik . In fact, the estimation shows that the factor combinations do not directly 47 The MNL employs the factor set for every risk class except for the base category. It also includes a constant term for every category’s logit. In this way, it uses j × k factors as compared to only k factors used in the OLS.

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97

and linearly translate into Yj , the risk classes. There are kinks in the ordinal structure of explanative variables that are best visible in the MNL regression tables.

3.10.2

Performance of the econometric models

Aim of this empirical analysis was the explanation of the choice for investment risk by investor characteristics and a comparison of models in terms of predictive power. The limited number of independent factors could expectedly not explain all of the variation in investment choice. However, given the large sample size and the diversity of tastes, intents and preferences, the MNL was able to weight the factors in such a way that the choices were evenly spread over all categories. The predictive power was also quite good. Most coefficients had the expected signs. The predictive power varied significantly between the SCF 1995 and 1998. The 1998 data produced a much better in-sample fit of the actual choice. Out-of-sample prediction with the 1995 estimates in the 1998 data gave classifications very similar to those by the 1998 in-sample estimation. The predictive power also varied greatly among the different models. For the in-sample estimation, the MNL tended to perform best, followed by the Ordered Logit, the Tobit and the OLS. This order was maintained in the outof-sample prediction, with the sole exception of setting 2b where assignment was very poor in all models. For the empirical analysis the number of risk classes chosen was arbitrary. There were seven risk classes in setting 1. Six classes were derived from the division of the CML and one class consisted of people without financial assets. Important information is lost when the continuous stock ratio is transformed into an ordinal variable. This is a plain disadvantage of working with discrete data.48 A higher number of classes would of course be also conceivable, though that would inevitably change the predictive power and the assignment precision of the models. By increasing the number of classes, more logit equations 48 The

boundaries of the classes are determined arbitrarily. An observation with a stock ratio of 21% is assigned to class 1 in setting 3c while a ratio of 19% is assigned to class 0. The true latent risk segments may thus become blurred and similarities between observations of different classes at each boundary cannot be considered in the estimation.

98

3.10. Conclusion of the two-moment setting

are created by the MNL. This leads to a decrease in the maximum probability of the predicted class. A higher number of factors per logit can be expected to be less significant implying a lower predictive power. The goodness of fit of the model will decrease at the same time.

3.10.3

Transferability of results

The aging of the population is a development all industrialized countries share - the U.S. as well as Switzerland and Germany. The proportion of families headed by individuals between 45 and 54 years has risen by around 2 percentage points between 1995 to 1998.49 The financial decisions of families with heads in this group are most likely dominated by the cost of college education for their children and the need to save for their own retirement. The predominance of these two factors are specific to the U.S., as in Europe higher education is much less costly and the public pension system still provides more security for retired people. On average, private pensions of US investors are to a higher degree invested in stocks than private investors in Germany or Switzerland. Also, it is well known that investors in the U.S. hold on average more stocks50 than European investors. Furthermore, it can only be speculated whether the investment mentality far West is more trading-orientated than in Europe. After all, speculation and short-term buying and selling involves serious tax-disadvantages in the U.S. However, there are studies indicating that the average holding period of risky assets is higher in Europe.51 49 See

Bertaut (1998) and Kennickell, Starr-McCluer, and Sunden (1997). average proportion of stocks in US private investor’s portfolio (market value) is higher than that of European investors. Also, when counting direct and indirect stockholdings, more than 45% of the US workforce holds stocks as compared to an average 30% in Europe. Surprisingly, the spread of direct stock-ownership is greater in Switzerland than in the U.S. or in Germany, see Coccart and Volkart (2001). 51 Recent behavior does not confirm the common view that investors are long term holders. According to Sanford Bernstein, a money management firm, in the U.S. the average holding period 10 years ago for a NASDAQ stock was 751 days. At the time of this study it was 181 days. ‘Equity buyers’ held a stock for an average of 8 months in 1999 as compared with an average holding period of 2 years in 1989. Those who purchased the 50 most heavily traded NASDAQ stocks kept their shares for an average of only three weeks. A decade ago, the average mutual fund was held for 11 years. Today, the lack of patience and perspective has reduced that to a 4 year average holding period, cf. Lee (2001). 50 The

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99

One aspect that might lightly hamper the transferability of results is the difference in credit-behavior and private debt, as credit-taking is generally more socially and economically accepted in the U.S. Overall, the results presented for U.S. data can be expected to have good predictive power also for Switzerland and Germany, especially as financial behavior of private investors worldwide becomes increasingly similar. Reforms of the private pension systems in Europe52 , the shift of the responsibility for retirement insurance from the public to the private sector, are indicators for a growing commitment of private investors in the stock market. The method shown can be easily transferred to fit investment behavior across different countries. All that is necessary is a representative sample53 to calibrate the weights of each question in the questionnaire.

52 From 2002 onwards statutory payments in Germany are gradually replaced by private voluntary retirement provision. The latter is financially supported by the state at 35-55%. Keeping in mind that in 2000 alone, private investors allocated DM200bn. in equity assets, this reform is bound to impact the German stockmarket. 53 ‘Small’ relates to a number of at least 500 observations in the sample.

100

3.10. Conclusion of the two-moment setting

Chapter 4

Joint estimation by gambles and observed stock ratio This chapter will extend the previously presented econometric model by incorporating gambles in order to arrive at more robust estimation results. The disadvantages of gambles have already been briefly discussed in Subsection 1.2.2. The value added of incorporating them lies in the fact that they are complementing the factor model by bringing a more direct preference test to the estimation procedure. The factor model focuses on data and characteristics that are a little more remote to the asset allocation decision, whereas the gamble directly confronts the investor to choose among risk levels. The choice, in combination with the current wealth level, yields an estimate for the Pratt-Arrow measure of an individual investor’s risk-aversion. These estimates can be used together with the presented factor model to form a joint estimation model for risk aversion. As there is no SCF1995- and SCF1998-survey data available that relates to the investors’ choice of gambles, the joint estimation method will be discussed and portrayed theoretically. 101

102

4.1. Determining Two-Moment Risk Aversion by Gambles

4.1

Determining Two-Moment Risk Aversion by Gambles

The derivation and the logic of the Pratt-Arrow measure of risk aversion has been explained in Subsection 2.2.2. Building on Formula A.14 in Appendix A.1 the investor is asked to quantify a sure outcome, called certainty equivalent CE, for a given symmetric gamble [x,p,y].1 As the main concern of the study is with return data rather than with absolute amounts, gambles for ‘payoffs in return form’ are considered in the following formulas. In the picture below, the relative risk aversion for a two moment setting is derived from a symmetric, tree-structured gamble: Two-moment risk aversion The investor I is given with his wealth W=200’000 Expected Annual Return ER=9% Volatility σ2 =22% +31% 262’000 ✒ ✒ 0.5 0.5 200’000 200’000 ❅ 0.5 ❅ 0.5 ❅ ❅ ❘ -13% ❅ ❘ 174’000 ❅ The investor declares his Certainty Equivalent CE = 5.00%

Apart from the probabilities for the outcomes, the expected return ER and volatility σ2 of the 2-moment gamble are presented to the investor. He then needs to name his Certainty Equivalent for that gamble as well as his 1 This amount, called certainty equivalent, makes the individual indifferent between playing the gamble or receiving the sure amount CE. In a more intuitive way, the certainty equivalent CE can be understood as the value the investor assigns to the gamble presented to him. If he does not approve of the gamble and its potential payoffs (x,y), he will name a low certainty equivalent for it. The higher CE, the less risk averse the investor.

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103

current wealth level. The two-moment preference is determined through the Certainty Equivalent the investor i names. For reasons of transparency and comprehensibility the gamble is portrayed by two illustrations which are complemented by the moments of the underlying distribution. The first illustration gives the payoffs in return style, the second one in absolute amounts assuming the investor considers allocating all of his wealth W .2 

The Pratt/Arrow risk premium π is − 12 σ22 W UU  . The strong assumption that must be made is: risk premium π2 = ER−CE, where ER is the Expected Return and CE stands for Certainty Equivalent. Solving for relative risk aversion αr yields αr = −W

U  ER − CE =2  U σ22

(4.1)

Dividing the above result by wealth W gives the A/P measure of absolute risk aversion αa . Based on the gamble, Equation 4.1 can be rewritten to yield a regression based relationship: 2

ER − CEi σ22

 = βjk xik + εi

(4.2)

where β  xi stands for the risk aversion measure for investor i. Different regression models can be used here to estimate the factor weights β. The model format depends on the distribution assumption of ε. For a MNL model the error term ε is assumed to be logistic distributed with mean 0 and a scale parameter λ.3 The log-likelihood function for the above model can be expressed by

logL(β|xi ) =

  ER − CEi  ln 2 − β x i σ22 i=1

I

(4.3)

2 Note: In order to avoid typical behavioral biases such as loss aversion, it would be beneficial to portray a gamble with purely positive or negative payoffs in a survey. 3 For a normal logistic distribution λ = π 2 /3. For the standardized logistic distribution √ λ = π/ 3.

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4.2

4.2. Joint estimation of econometric choice and gamble

Joint estimation of econometric choice and gamble

In Subsection 3.5.5 and Appendix B, the application of the discrete choice model to the problem of portfolio selection was justified by the basic framework of utility maximization. The investor is assumed to choose that investment that maximizes his expected utility. A conceivable utility function for his decision-making is: Ui = µj −

αri 2 σ 2 j

(4.4)

µj and σj represent the expected return and volatility of investor i’s portfolio choice j, αri stands for the investor i’s relative risk aversion. Replacing αri by β  xi yields the regression model: Ui = µj −

β  xi 2 σ 2 j

(4.5)

The weights β of the independent factors xi are to be estimated with the models described in Section 3.5.4 When applying a MNL model to the problem of utility maximization5 the choice probability for Equation 4.5 is exp Ui Pi (j) = I i=1 exp Ui

(4.7)

where ji is the choice of investor i for one of the J portfolios. The corresponding Log-Likelihood function is logL(β|xi ) =

I

lnPi (ji )

(4.8)

i=1 4 An

alternative relationship solves for the relative risk aversion and can thus be regressed directly: µj − r f β  xi = αri = (4.6) yi · σj2 µj and σj are again expected return and volatility of investment choice j, rf is the riskfree rate and yi is the investor’s current stock ratio. 5 For a derivation of the MNL from utility maximization see Appendix B.

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105

The equations 4.8 and 4.3 can be estimated jointly. It is likely that the estimates for risk aversion resulting from the two equations differ. A joint estimation of the two models will increase the precision and efficiency in the estimation of β. The joint Log-Likelihood function will be logL(β, γ|xi ) =

I  i=1

   ER − CEi  − γβ x (j ) ln 2 + lnP i i i σg2

(4.9)

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4.2. Joint estimation of econometric choice and gamble

Part II

Three-moment risk preference

107

109

In this second part the main focus of the analysis lies on the investor’s preference structure when the third moment skewness is considered in addition to the first two - expected return and variance. In the subsequent sections skewness S will be defined as the non-normalized third central moment of the gamble or the return distribution. This definition is to be distinguished from the Fisher skewness which is normalized by the cubed volatility. In the discrete case: 3 (xi − E(X)) · pi (4.10) S= i

where xi stands for return i and its probability pi . In the continuous case where simple lognormal returns are considered:

   (4.11) S = exp (σ 2 − 1) 2 + exp σ 2 where σ depicts the dispersion parameter of the normal density function corresponding with the lognormal. Two lines of thought will be intertwined in this second part: 1. How to assess an investor’s three moment risk preference for the return distribution’s moments and their trade-offs in particular. The certainty equivalence method will be employed to determine these trade-offs which indicate how much skewness is demanded by the investor per unit of expected return or variance. 2. How to formally describe the portfolio’s distribution and its modification by different option strategies. Adding options to the portfolio decreases expected return, decreases shortfall risk and increases skewness. The degree of these changes depends on the well-known parameters of the Black-Scholes pricing formula. The strike price, however, is the main driver for the change in the portfolio’s moment trade-offs. The higher the strike price of puts, the higher the degree of protection and the more costly the loss in expected return and the lower σ, µ and the higher is the skewness. These two lines of thought meet at the specific shape of the portfolio’s return distribution demanded by the investor. The distribution can be sufficiently represented by its first three moments. Different moment combinations

110

that are of equal utility for the investor can be identified by so-called moment trade-offs. The gambles define what trade-offs the investor demands and the strike price of the option strategy determines how to implement these preferences. The second part concludes with a brief theoretical instruction on how to estimate three moment preference jointly with discrete choice models and gambles.

Chapter 5

Shortcomings of two-moment asset pricing 5.1

Critique of the mean-variance approach

Traditional mean-variance analysis has a number of shortcomings that will be briefly pointed out in order to motivate a mean-variance-skewness approach. While the approximation of a utility function by a quadratic in a certain range is central to the Markowitz (1959) rational for mean and variance, normal distributions or other two-parameter families of probability distributions were not part of his justification for mean and variance. It was Tobin (1958) who offered two alternative ways of deriving mean and variance as the investor’s decision making criteria1 : (1) “the investor evaluates the future of consols only in terms of some two-parameter family of probability distributions, such as the normal, or (2) the assumption that the utility function 1 The

third moment or skewness was not considered by Markowitz or Sharpe, Lintner and Mossin for reasons of cost and inconvenience. Partly as a consequence, most work on “nonstandard” portfolio optimization has remained purely theoretical in nature, confer Mao (1970), as well as Hogan and Warren (1974). Models of portfolio optimization based on the semivariance were developed by Chen and Park (1991) and Ouederni and Sullivan (1991) to name only a few. Then as today is true that the use of variance produces the same set of efficient portfolios as the use of semi-variance iff all distributions of returns are symmetric or have the same degree of asymmetry.

111

112

5.1. Critique of the mean-variance approach

is quadratic2 , and returns do not exceed the point at which the quadratic reaches its maximum”. Thus the traditional mean-variance approach can be derived in two alternative ways: Either it must be assumed that • the investor has a quadratic utility function or that • returns are normally distributed. Unfortunately both assumptions are not only unrealistic, but prove insufficient and unsatisfactory in describing investment preferences of individuals.

5.1.1

Quadratic utility

The assumption of a quadratic utility function eliminates all moments of the return distribution that are higher than the variance from the objective function of optimal asset allocation. It rules out the possibility that investors could have any preference for skewness, e.g. an asymmetric return distribution. Any investment strategy aiming at modifying the skewness of the return distribution - such as option strategies for example - cannot be examined with the traditional mean-variance model. The quadratic utility also implies increasing absolute risk aversion, a feature that is empirically most unrealistic. The quadratic utility has been called fatuous by Pratt (1964) and Arrow (1971) for this reason. All polynomous utility functions share this grave disadvantage, as U  > 0 and U  < 0 never occur together. Quadratic utility also displays satiation. This property implies that an increase in wealth beyond the satiation point decreases utility.3 - A grave disadvantage as individuals always prefer more wealth to less and treat risky investments as normal goods. In order to circumvent this flaw the quadratic 2 Markowitz’

approach of assuming a quadratic utility did not mean to persuade the investor that some prepackaged utility function has desirable features. It refers rather to soliciting the investor’s preferences among various gambles and summarizing these in a utility function. 3 Dybvig and Ingersoll (1982) among others observe that quadratic utility implies that very high return states will have negative marginal utility and thus negative state prices which contradicts the no-arbitrage condition of equilibrium prices.

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113

utility function is usually limited to a narrow range of wealth. As a consequence it is appropriate only for relatively low returns which precludes its use from investments that like lotteries have some very high values of the potential payoff, albeit with low probabilities of occurrence.

5.1.2

Normal distribution

In answering how many moments are needed to describe the investor’s assessment of the probability distribution adequately, most asset pricing models assume that the importance of all moments beyond the variance is much smaller than that of the expected value and variance. The normality of returns relies heavily on the extent of portfolio diversification4 and the length of the holding period. For the mean-variance-results to hold, the portfolios must be very broadly diversified and the analysis must be limited to a short time interval. The assumption of perfectly diversified portfolios, however, is unsatisfactory - especially for a study on individual investors’ risk aversion. Empirical studies have clearly shown that the majority of shareholders own less than 10 different stocks. Alternatively, normality is given when it is assumed that portfolios are revised continuously. Samuelson (1972) tries to prove that disregarding moments higher than the variance will not affect portfolio choice. The flaws inherent in this paper were indicated by Loistl (1976). Brockett and Garven (1998) add essential comments to this discussion. Samuelson’s major assumption to arrive at his conclusion concerns the “compactness” of the distribution of stock returns. The distribution of the rate of return on a portfolio is said to be compact if the risk can be controlled by the investor. In general, compactness can be interpreted as the continuity of stock prices. If stock prices do not take sudden jumps, then the uncertainty of stock returns over smaller and smaller time periods decreases. Under these circumstances investors who can rebalance their portfolios frequently will do so in order to render higher moments of the stock return distribution irrelevant. 4 The Central Limit Theorem (CLT) states that the sum (or average) of a large number of - possibly dependent, possibly time-varying - random variables will converge to the Normal distribution, no matter what distribution the variables have.

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5.1. Critique of the mean-variance approach

It is not that skewness does not matter in principle. Rather, the actions of investors who are frequently revising their portfolio limits higher moments to negligible levels. Continuity or compactness, however, is not an innocuous assumption. Portfolio revisions entail transaction costs, meaning that rebalancing must necessarily be limited and that skewness and other higher moments cannot entirely be ignored. Continuous rebalancing implicitly leads to infinite large continuous trading volumes - again an assumption that is not unproblematic. Compactness rules out certain phenomena such as the major stock price jumps that occur in response to takeover attempts. It also rules out such dramatic events as the 25% one-day decline of the stock market in October 1987. The normality assumption of equity returns was also questioned by Mandelbrot (1963) and Fama (1965) upon observing the presence of leptokurtosis (fat tails) in the empirical distribution of price changes.5 Mean-variance analysis is adequate if the portfolio may be revised frequently and if there are no sudden price jumps. Unfortunately, these conditions are not satisfied. The study at hand aims at advising individual investors who consider allocating their free wealth for an undetermined period of time. These individuals cannot and will not rebalance their portfolio continuously - be it for transaction costs, time or other reasons. They are thus exposed to sudden extreme price jumps that they might want to hedge against. Simple returns in discrete time are lognormally distributed and thus positively skewed.6 The quantities decisive for the individual investor are the change in and the current level of the portfolio value. As the portfolio value can be written as the product of simple returns, the portfolio value is lognormally distributed. The product of a lognormal distribution is again a lognormal distribution. Its density function and its first four moments are characterized in Section 8.1. Given that a stock price cannot be negative, the lognormal as a representation of the return distribution is more realistic than the normal as it can be 5 The Pareto-Levy distribution originally proposed by Mandelbrot (1960) has been shown to yield empirically much better results than the Gaussian (normal) distribution. 6 The lognormal model is not fully consistent with all the properties of historical stock returns. At short horizons, historical returns show weak evidence of skewness and strong evidence of excess kurtosis, cf. Campbell, Lo, and MacKinlay (1997).

Chapter 5. Shortcomings of two-moment asset pricing

115

modified to exclude outcomes lower than -100%. The effective annual rate is re (t) = exp(rt) − 1. For short holding periods where t is small, the approximation of re (t) by rt is quite accurate and the normal distribution provides a good approximation to the lognormal. For short holding periods, therefore, the mean and standard deviation of the effective holding period returns are proportional to the mean and standard deviation of the annual, continuously compounded rate of return on the stock and to the time interval. For longer holding periods, however, the normal and the lognormal distribution differ distinctly.

5.1.3

Performance measurement of optioned portfolios

Mean-variance based pricing models (like the CAPM) will mismeasure the performance of portfolios containing fairly priced option positions.7 When skewness is positively valued, mean-variance performance measures will overrate the rebalancing or ‘enhancement’-strategies8 which reduce skewness. Equally, it will underrate the momentum strategies9 which buy skewness.10 The reason is that beta as the sole risk measure is incorrect and that the traditional CAPM formula does not hold when the market is lognormally distributed. The use of only two moments gives a distorted view of the relative performance of alternative investment strategies that result in return distributions that predominantly change the return-skewness ratio. Skewness is not considered in the world of the Capital Market Line and the mean-variance-model. As a risk-averse investor will prefer the positively skewed return distributions, it is worthwhile to quantify this characteristic. The asymmetry is relevant although it is not as important as the magnitude of the standard deviation. In reality, however, the covered-call-strategy’s immanent severe downside risk is not shown, as the third dimension, skewness is missing. The Put Option Strategy on the other hand seems to perform very poorly. It reduces the ex-ante expected return for reducing the downside-risk and thus increasing 7 Cf.

Leland (1999), 29. hold the market portfolio and write one-year covered calls on it. 9 They hold the market portfolio and buy one-year protective puts on it. 10 The standard CAPM also empirically fails to explain the asset returns of the smallest market-capitalized deciles, as they are the ones with the most skewed returns. 8 They

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5.2. Summary of critique

Figure 5.1: Distorted performance of option strategies in the traditional CAPM, source: Bookstaber and Clarke (1984). Traditional performance and risk measures do not apply to portfolios that are supplemented with options. In a mean-variance world as shown to the left, writing call options seems to have improved performance relative to the market portfolio.

skewness. It is obvious, that the complex return structure of the total optionstock portfolio, cannot be priced within the CAPM framework.

5.2

Summary of critique

The probability distribution of the rate of return can be characterized by its moments. The reward for taking risks is measured by the first moment, which is the mean of the return distribution. Higher moments characterize the volatility or risk and the asymmetry in payoffs. Investors’ risk preferences can be characterized by their preferences for the various moments of the distribution. The fundamental approximation theorem by Samuelson (1972) shows that when portfolios are revised often enough, and prices are continuous, the desirability of a portfolio can be measured by its mean and variance alone. The rates of return on well-diversified portfolios for holding periods that are not too long can be approximated by a normal distribution. For short holding periods (up to one month) the normal distribution is a good approximation for the lognormal. Based on the above, the mean-variance model is not a general model of asset choice. Its central role in financial theory can be attributed to its analytical tractability and mathematical simplicity. In situations in which variance alone is not adequate to measure risk the mean-variance assumption is potentially restrictive.

Chapter 6

Skewness The literature that deals with the role of skewness is mostly concerned with the equilibrium structure of returns. Less attention has been given to the impact of skewness on risk taking. Considering skewness in portfolio choice improves accuracy and captures investors’ preferences more comprehensively.1 Even though small losses occur more likely than with a normal distribution private investors prefer positively skewed distributions of their portfolio value. They do so because big losses are less probable than in a normally or a negatively skewed distribution. As most investors are risk averse, they view the disutility of a loss as unevenly greater than the utility of a gain in the same proportion. Kraus and Litzenberger (1976)2 show that systematic skewness3 earns a risk premium and thus matters in the composition of portfolios. As investors desire positive skewness in portfolio returns, the risky asset will be held in higher proportion than predicted by a mean-variance framework due to its 1 Even

Tobin (1958) admits that ‘mean-variance analysis is not satisfying as a framework for representing portfolio choice’. 2 They extend the CAPM by the third moment to incorporate the effect of skewness on valuation. Empirically their three-moment CAPM provides a better fit of the data than the traditional two-moment model. For their model three rates of return must be known to determine the structure of capital asset prices: market portfolio return, riskless rate of return and the return of the zero beta and non-zero gamma portfolio. 3 Ingersoll revises this view and develops a model that prices only conditional skewness, see Ingersoll (1975, 1987).

117

118

6.1. Higher moment preferences

positive skewness. Aversion to standard deviation and preference for positive skewness are general characteristics of all investors having utility functions displaying the desirable behavioral attributes of decreasing marginal utility of wealth and non-increasing absolute risk aversion.

6.1

Higher moment preferences

Under the premise that individual investors cannot achieve compactness in their portfolio distribution, their payoff structure will be asymmetric. In that case volatility4 or variance become inadequate as risk measures, as the average risk-averse investor desires low downside risk and high upside potential. Determining preferences in terms of three rather than two moments will therefore result in a more precise picture of the portfolio distribution desired by the investor.5 All the even moments represent the likelihood of extreme values. The odd moments represent measures of asymmetry. We can characterize the risk aversion of any investor by the preference scheme that the investor assigns to the various moments of the distribution.6 It must be emphasized again that both the lower and the upper end of the distribution matter for the preference structure. Risk measures such as Valueat-Risk or Lower Partial Moments focus on the lower end of the distribution and do not capture the degree of upside potential the investor desires to achieve. Skewness, by contrast, in combination with the first two moments is able to mirror the attitude towards both the upper and the lower part of the distribution.7 4 The

characterization of good and bad volatility by Sortino and van der Meer (1991) recognized the fundamental importance of downside risk. 5 Experimental tests have demonstrated with clear evidence that individuals attach significant importance to moments in final asset value higher than two (refer to Gordon, Paradis, and Rorke (1972) and Coombs and Lehner (1981)). In particular, the third moment - skewness - has been shown to have a great deal of importance in measuring what is conceived of as “risk”. 6 A number of articles has shown that moment preference does not match up with a sequence of utility derivatives: Menezes, Geiss, and Tressler (1980), Meyer (1987), Rothschild and Stiglitz (1970) and Whitmore (1970). 7 Skewness preference becomes increasingly important as a decision making criterion when the investor has to choose among different risk levels that correspond to distinctly

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119

To know if a particular moment is desirable or not, one has to know the coefficient of this moment - in other words the appropriate derivative of the utility function at the point E(W ). An increase in the expected return (ceteris paribus) of a rational individual who prefers more money to less will increase his expected utility.8 For the higher moments of the return distribution theoretical analysis reveals a preference for positive skewness as investors are loss averse. Empirically, it is possible to test for the direction of preferences with a simple regression analysis: Levy and Sarnat (1972) evaluated the impact of the distribution moments on expected utility with a sample of US mutual funds and derived the relationship of the distribution moments to investors’ utility from the regression analysis itself.9 For the period 1946-67 only the variance and the third moment are significant. For the period 43-67, the fourth moment is also significant, but the remaining higher moments are not significant in any of the time periods examined. Rsquared is very high and ranges from .65 (’56-’67) to .94 (’43-’67). In all regressions the coefficient of the variance is positive. Consequently it was concluded that investors are typically averse to variance.10 The regression coefficient of the third moment (skewness) is negative and highly significant, suggesting that the average investor likes positive and avoids negative asymmetry. Thus, increasing the skewness of the distribution reduces differently skewed return distributions. Skewness increases both with the risk level as well as with the time horizon of the investment due to the compounding effect over long time periods. 8 Brockett and Kahane (1992) and Brockett and Garven (1998) claim that expected utility preferences never universally translate into moment preferences thereby refuting the results presented by Scott and Horvath (1980). 9 Other studies examining the direction of moment preference are Arditti (1967), Arditti and Levy (1972), Beedles (1979), Booth and Smith (1987), Jean (1971), Kraus and Litzenberger (1976), Scott and Horvath (1980) and Tsiang (1972). In the regression analysis a negative coefficient for the second derivative is usually interpreted as implying an aversion to variance. A positive third derivative is seen as indicative of skewness preference. 10 A logical flaw might be inherent in this conclusion according to Brockett and Garven (1998): If the return distributions were normally distributed, actual statistical independence of the sample mean and variance should be a consequence and a characterizing property of the normal distribution. “Thus positive coefficients can arise only as a sampling artifact. However, if the distribution were not normally distributed then the sample mean and variance should be necessarily dependent. Thus, correlation may be not so much a statement indicative of a relationship between risk and return as much as a statement concerning the lack of normality of the distributions.”

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6.2. Prospect theory and adaptive aspiration

the required average return.11 The fourth moment (kurtosis) is significant in only one of the four regressions and even there its coefficient is very small. Thus, although expected utility depends on all the moments of the distribution function, it is often assumed that the first three distribution moments reflect most of the required information. This does not imply that all investors have cubic utility functions.

6.2

Prospect theory and adaptive aspiration

It was shown that the utility function and the asset pricing formulas are the core of the investor’s asset allocation decision. Utility can be expressed in terms of the moments of return or by any other function of purely arbitrary parameters. Whatever its mathematical form, its relation with wealth W matters greatly to account for behavioral phenomena. One alternative to the traditional objective function of Maximizing Expected Utility is the value function in the prospect theory of Kahneman and Tversky (1979). The function is defined with respect to a natural reference point which expresses the transition from taking chances to playing the ‘safe side’.12 As this reference point is the ‘point of inflection’ it marks the change from convexity to concavity. For deterministic choice problems final asset positions are perceived as less relevant than changes in wealth. Individuals try to minimize their losses. They take chances to decrease them, while ‘collecting’ gains instead of ‘gambling’ for higher profits. To avoid a sure loss of a given quantity, people risk incurring an even greater loss if there is a chance for a better outcome. The value function incorporates the common view that the difference between $0 and $100 is seen as greater as the difference between $1000 and $1100 - irrespective of the sign of the magnitude. In addition, the function is steeper for losses than for gains expressing the empirical finding of loss aversion. The disutility of a loss weighs 2.5 times heavier than the utility of a gain of the same magnitude. This finding is consistent with investors’ preference 11 Which partly explains why people participate in lotteries that typically have negative expected values but are characterized by high positive skewness. 12 Here as in traditional portfolio selection models it is crucial to define how the reference point (benchmark) defining gains and losses evolves over time.

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121

for positive skewness. Risk-averse individuals seek protection from loss by insurance. Stated as above the concept is static and refers to a fixed risk preference. Most studies however indicate that risk preference is not fixed, but depends on the context of a choice. The investor’s asset level and the degree of risk aversion are interdependent and should be modeled accordingly. One possibility is to assume that the reference point is changed every time the individual reaches a new level of wealth, also called aspiration level by March (1987). Observations of human decision making suggest that risk preferences do depend on the values of possible outcomes relative to levels of aspiration. “A model of variable risk preference suggests that some risk averse behavior may result from a human tendency to focus on targets and from the adaptation of those targets to experience rather than from a fixed trait of risk aversion.”13 The view that risk preferences change with respect to the level of wealth or status is in line with Friedman and Savage (1948) as Robson (1992) proved. Friedman and Savage (1948) examined the phenomenon that individuals simultaneously purchase insurance and participate in lotteries which again refers to investors’ preference of positive skewness of returns and of final wealth. Robson (1992) provides a natural explanation of the concave-convexconcave utility function by explicitly incorporating the relative standing that wealth induces. The behavioral implications of variable risk preferences help explain why investors show a strong demand for low risk alternatives. Preferred risk varies inversely with accumulated resources of the investor. Demand for riskier alternatives stems mainly from larger amounts of initial investments. The larger these amounts, the slower the elimination of decision makers with losses. Another reason for riskier alternatives are situations in which the investor faces only negative cumulative effects on expected returns. This view is consistent with studies of gamblers involved in track betting. As the likelihood of a horse winning will be monotone decreasing with the length of the odds, track betting is, on average, a losing bet. Therefore, when betting over the course of a day, the bettor will, on average, become increasingly risk prone as the day goes on. 13 March

(1987).

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6.2. Prospect theory and adaptive aspiration

“All of these findings show that the propositions about fixed risk preferences arising from exogenous forces need to be supplemented with contextdependent considerations.” Factors such as the success level at which choices occur, overconfidence and overreaction have a strong influence on investment behavior. “Aspirations and financial goals may change on the basis of experience or imitation of others. Prior histories of greater wealth produce higher aspirations and therefore a preference for greater risk.”14 In the long run, the adaptation of aspiration levels to a decision maker’s own experience tends to make risk taking independent of that person’s wealth. The theory of adaptive aspiration level picks up this finding arguing that the risk preference of a single individual depends on his current wealth level and that the observed risk seeking when facing potential losses contributes to the chances of ‘financial survival’ for the individual. Thus people who change their risk preferences according to their current level of wealth will most likely have the best investment performance. The assumption of constant relative risk aversion might therefore be suboptimal from a normative viewpoint.15 The above properties are in conformance with empirical findings by Swalm (1966). He finds that for any obtained level of wealth, a new utility schedule comes into play - a conclusion with intuitive appeal as corresponding to one’s own experience. It is a result that strengthens the plausibility of a cubic utility: Actual wealth never reaches a point where increasing marginal utility is expressed. The convex tail represents feelings about a possible richness which has extra aura. As one becomes wealthier such a level is shifted away. At the same time, this convex tail is instrumental in explaining observed behavior patterns that are shared by many individuals. Furthermore, it is a result that implies status as a critical influential factor in decision making under uncertainty. Status as a function of wealth might account for behavior that has so far been characterized as inconsistent with rationality. To summarize the results it can be stated that investors will gamble: • to avoid a sure medium-sized loss even if that implies that they run the danger of suffering a larger loss. 14 March

(1987). finance seems to suggest that risk preference is not fixed. It depends on the context of a choice. This is what Pratt/Arrow modeled as the interdependence of asset level and the degree of risk aversion. 15 Behavioral

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123

• if the outcomes are below a certain target or an aspiration level. To account for this behavioral peculiarity both the utility function of Friedman and Savage (1948) and the value function of Kahneman and Tversky (1979) have a region that is convex and then concave again. Goals are context-dependent and aspiration levels are responsive to the degree of investment success. Thus, investors’ preferences for risk depend on a target which is a function of their experience.

6.3 6.3.1

Gambling and insurance habits Cubic utility and skewness

It needs to be emphasized that truncating the Taylor approximation of the (expected) utility function after n terms does not mean that the utility function is replaced by an n-th degree polynomial. Rather U (W ) is locally approximated at every wealth level by such a polynomial. Thus the utility function is an envelope of such polynomials and not a polynomial itself. The Taylor approximation is used here to account for the impact of skewness, it is not the goal of this study to describe investors’ behavior by a cubic utility function.

6.3.2

Implications of the cubic utility

A cubic utility function typically has the following form (W again equals W0 + x): U (W ) = aW + bW 2 + cW 3 + d

(6.1)

The only separable cubic utility concave over positive wealth levels near zero that can be derived from the above equation is according to Rubinstein (1973): 2 ˜2+γ W ˜3 ˜ − γi W U (W ) = W 3

(6.2)

where γ −1 > 0 The cubic utility function exhibits several controversial properties that need to be discussed briefly.

124

6.3. Gambling and insurance habits • A cubic utility function is inconsistent with the log normality of the first differences of market price changes.16 • It does not exhibit decreasing marginal utility for all wealth levels.17 When it has decreasing marginal utility for positive wealth levels near zero, it has increasing absolute risk aversion within that same range. • The cubic function is not bounded so that in order to avoid the revival of the St.Petersburg Paradox, it must be assumed that beyond a certain point (W0 ) increments to wealth do not increase utility18 , that is, U (W ) = U (W0 ) for all W ≥ W0 . Restricting the range of permissible W in the function is also a necessary condition for its separability and a closed equilibrium solution, as it assures increasing marginal utility U  > 0. Unfortunately this implies a negative slope coefficient to the risk tolerance function which results in a negative wealth elasticity of demand for risky assets. Another price for the theoretical convenience lies in the fact that risk tolerance is non-linear which makes equilibrium valuation theoretically intractable. When properly restricted the cubic utility yields non-satiation and preference for positive skewness, it does however not exclude risk-loving for all levels of wealth. • The function has first a concave and then a convex segment. For low ˜ the investor is a risk averter and dislikes variance, but values of W ˜ the investor becomes a risk lover and for relatively high values of W consequently desires variance. Arditti (1975) criticized this characteristic, as it seems unreasonable that at high wealth levels individuals should desire investments characterized by both high variance and skewness. Wealthy individuals are per se not risk lovers. As the cubic utility does not exclude risk-loving for all levels of wealth it might seem unsuitable for purposes of portfolio analysis.

16 Arditti

(1967). (1969). 18 As there is no distribution for which the expected value of the cubic U and its cash equivalent will be infinite. 17 Levy

Chapter 6. Skewness

6.3.3

125

Insurance and Gambling

Friedman and Savage successfully argued that, while an individual in a low income group might have the type of utility function described in the section above, this same individual’s utility function becomes concave as he moves into a high income group. This finding would be consistent with an individual who buys insurance against losses, takes small bets and seeks to diversify his portfolio. It is similar in many important respects to the one presented by Friedman and Savage (1948).19 With the above utility function the purchase of insurance contracts can be reconciled with gambling, without recourse to subjective utility20 since in both cases investors prefer positive asymmetrical distributions (like a lottery’s) but dislike negative asymmetry and therefore buy insurance policies.21

19 They

attempted to reconcile the observed phenomenon that many individuals simultaneously purchase lottery tickets (or gamble) and take out insurance. The first activity hints at risk-loving behavior while the other depicts risk-averse behavior. To remove the contradiction, the utility function can be neither strictly concave nor strictly convex. 20 An approach chosen by Kahneman and Tversky (1979) whose value function accounts for the investor’s loss aversion, as it is convex below the endowment point and at the same time explains the preference for high unlikely payoffs observable in gambling. 21 For strictly concave utility functions an alternative approach to Friedman and Savage’s proposition suggests that simultaneous gambling and acquisition of insurance can be explained by assuming that subjective and objective probabilities diverge. Subjective probability are viewed as exceeding their objective counterparts when the latter are low. This assumption is sufficient to show that risk-averse individuals will also be willing to gamble or to purchase lottery tickets.

126

6.3. Gambling and insurance habits

Chapter 7

Determining skewness preference through gambles This chapter will present a novel approach of assessing an individual’s risk aversion for two moments and his preference for the third moment of the return distribution or of a gamble. The determination of the investor’s three-moment risk preference will uncover the moments’ trade-offs (units of expected return per additional unit variance, units of skewness per lost unit expected return and units of skewness per additional unit variance) and thus help decide what option strategy best suits that particular investor. In the first section of this chapter the risk premia for a three moment approximation will be derived. These are essential for the determination of the investor’s skewness preference in the second section. 127

128

7.1. Risk premia for three moment approximation

7.1

Risk premia for three moment approximation

The insufficiency of the Pratt/Arrow premium to account for higher moment preferences has already been treated in Section 2.2. The following sections will derive new risk premia that can capture trade-offs in the moment preferences of investors when payoffs are skewed. We have • W – current wealth • µ ˜ – random rate of return • π – relative risk premium 

(W ) • α = −W UU  (W ) – Pratt/Arrow measure of relative risk aversion

We are calculating the risk premium π that leaves the decision maker indifferent between risking the investment and the actuarial value of the investment: U [W (1 + µ ¯ − π)] = E{U [W (1 + µ ˜)]} (7.1) where U (•) is a von Neumann-Morgenstern utility function and µ ¯ ≡ E(˜ µ). The right-hand side represents the expected utility of the current wealth given the investment, the left-hand side is the current wealth plus the utility of the actuarial value of the investment. We can use a Taylor’s series approximation to expand the utility function of wealth around both sides: ¯) + U [W (1 + µ ¯)] − πW U  (W + µ (−1)n

π 2 W 2  U (W + µ ¯)+ (7.2) 2

∞ π n W n (n) U [W (1 + µ ¯)] = n! n=3

1 µ]2 W 2 U  (W + µ ¯) + E[˜ ¯)+ U [W (1 + µ ¯)] + µ ¯W U  (W + µ 2 ∞ 1 E[˜ µ]n W n U (n) (W + µ ¯) n! n=3 From here we can choose two roads to three-moment approximation:

Chapter 7. Determining skewness preference through gambles

129

1. We neglect terms of order π 2 and higher on the left side. In addition we neglect terms of order higher than the third moment on the right side and solve directly for the riskpremium or 2. We neglect terms of order higher than the third moment on the right side, but include π 2 on the left side. This leaves us with a quadratic equation for which there are two solutions. Following the first approach we get 1 µ]2 W 2 U  (W + µ µ ¯W U  (W + µ ¯) + E[˜ ¯) (7.3) 2 1 µ]3 W 3 U  (W + µ ¯) + E[˜ 6 ¯) 1 U  (W + µ ¯) 1 U  (W + µ − E[˜ µ]3 W 2  = − σ2 W  2 U (W + µ ¯) 6 U (W + µ ¯)

¯) = −πW U  (W + µ

π

where E[˜ µ]3 represents the non-normalized third moment of the return distribution. Following the second approach we find ¯) −πW U  (W + µ

π 2 W 2  U (W + µ ¯) 2 1 µ]2 W 2 U  (W + µ = µ ¯W U  (W + µ ¯) + E[˜ ¯) 2 1 µ]3 W 3 U  (W + µ + E[˜ ¯) 6 +

¯) and solving for π Writing U n for U n (W + µ    2   1 ± 1 + σ 2 W UU  + 13 m3 UU  UU  π =  W UU   3  1 ± 1 + αr2 σ 2 − m3 αr UU  = −αr  1 1 1 m3 U  2+ = − ± + σ αr αr2 αr 3 U 

(7.4)

(7.5)

This second approach would supposedly yield more precise results for higher moment preferences than the first approach, however it is less tractable and

130

7.2. Creating gambles for skewness preference

will for a skewness of 0 not equal the Pratt-Arrow measure of risk aversion. It is for this reason that it will not be dealt with hereafter. Instead the first approach will be pursued and employed in the subsequent analysis.

7.2

Creating gambles for skewness preference

Building on the insights of Section 2.2 that illustrated the insufficiency of the traditional measures of risk aversion, the investor’s porfolio selection model is expanded by the third moment, based on the new three-moment risk premium that derived in Equation 7.3 in Section 7.1 and the investor’s moment preferences will be derived in this section. The assessment of any individual’s risk premium is closely connected to that person’s utility function. For determining utility functions there are four principal categories that can be distinguished1 : 1. the preference comparison methods, 2. probability equivalence methods, 3. value equivalence methods, and 4. certainty equivalence methods. According to the literature2 , the approach best suited for risk premium assessment seems to be the certainty equivalence method where an individual is asked to specify a sure outcome CE for a gamble [x,p,y]. This amount, called certainty equivalent, makes the individual indifferent between playing the gamble or receiving the sure amount CE. In a more intuitive way, the certainty equivalent CE can be understood as the value the investor assigns to the gamble presented to him. If he does not approve of the gamble and its potential payoffs, he will name a low certainty equivalent for it. The higher CE, the less risk averse the investor. As the main concern of the study is with return data rather than with absolute amounts, gambles for ‘payoffs in return form’ are considered in the following formulas. 1 Farquhar (1984) gives a good, non-technical overview of all utility assessment methods and further subdivides each of the four categories. 2 See Becker, DeGroot, and Marschak (1964), Birnbaum (1992), Farquhar (1984), Hershey, Kunreuther, and Schoemaker (1982), Fishburn (1967, 1988) and Schoemaker and Hershey (1992).

Chapter 7. Determining skewness preference through gambles

131

The derivation of skewness preference The proceeding can be divided into two steps:3 In a first step, relative risk aversion for a two moment setting, a gamble without skewness, has to be derived. The investor is thus presented with a tree-structured gamble as in the Figure on page 137. Apart from the probabilities of the outcomes the investor is given the expected return ER and volatility σ2 of the 2-moment gamble. He then needs to name his Certainty Equivalent for that gamble as well as his current wealth level that will be relevant for determining his skewness preference. 2 refers Pratt/Arrow’s risk premium π2 - the subscript  to its applicability  

to 2-moment gambles or distributions - is − 21 σ22 W UU  . The strong assumption that must be made is: risk premium π2 = ER − CE, where ER stands for Expected Return and CE stands for Certainty Equivalent. Solving for relative risk aversion αr yields αr = −W

U  ER − CE =2 U σ22

(7.6)

Dividing the above result by wealth W gives the Pratt/Arrow-measure of absolute risk aversion αa . In the second step, the investor is presented with a second tree-structured gamble for which he has to name his certainty equivalent CE. This time however, the gamble is skewed and CE will contain additional information about the third moment of the gamble. To distinguish the second gamble’s parameters from those of the first one that considered only two moments, the subscript 3 was introduced (π3 , σ32 ). As the payoff structure is skewed, the Pratt/Arrow-risk premium is not applicable4 . Instead Formula 7.3 of  (W +¯ µ) section 7.1 is employed and solved for UU  (W +¯ µ) which will be abbreviated as U  U

hereafter:

3 In

  6 π3 − 12 αr σ32 U  =− U E[˜ µ]3 W 2

(7.7)

this section ‘skewness’ will denote E[˜ µ]3 , the non-normalized third moment. The traditional skewness measure, the Fisher skewness, is the third moment normalized by σ33 . 4 An exact preference ordering for risky portfolios using the first three moments of portfolio return can in general be determined only for an investor having a cubic utility function for wealth (see Subsection 6.3.1 for further discussion). Unfortunately a third degree polynomial would be an unsuitable utility function for a risk averse investor.

132

7.2. Creating gambles for skewness preference

where π3 stands for the three-moment risk premium and again π3 = ER−CE; αr depicts the relative risk aversion estimated in the first step above, σ32 represents the skewed gamble’s variance and E[˜ µ]3 its non-normalized third µ]3 will be abbreviated as m3 hereafter. moment.5 For ease of presentation E[˜   Having established U /U , it is simple to derive −U  /U  by dividing the above result by αr : 

U U   −  = UU  = U − U

6(π3 − 12 αr σ 2 ) E[˜ µ]3 W 2 −2 Wπσ2 2 2

3 σ22 =− W E[˜ µ]3



π3 σ2 − 32 π2 σ2

 (7.8)

µ]3 . where m3 = E[˜ With these formulas and after only two gambles a relatively comprehensive picture of the investor’s preferences can be drawn. Of course, this necessitates that the individual names two certainty equivalents that reliably reflect his tastes. The two newly derived ratios need to be explained: While U  /U  stands for the preference for the third moment relative to the first moment of the distribution, −U  /U  can be interpreted as the skewness ratio that reflects the investor’s preference for the third moment relative to aversion to variance.6 The lower U  /U  the more skewness is needed to compensate for a loss in expected return. Similarly, the lower −U  /U  the more skewness is needed to compensate for a rise in the gamble’s variance.7 A positive influence of skewness might offset the negative influence of the variance of the risk premium. The local three-term approximation has convex regions even though the utility function is globally concave. This may lead 5 Pratt

(1964) shows that decreasing absolute risk aversion in two moments corresponds to a decreasing risk premium. Arditti (1967) proves that a decreasing risk premium corresponds to U  (W ) > 0. This was confirmed by Bawa (1975) who applies the property to develop what came to be known as ‘stochastic dominance’. 6 −U  /U  is also known as (the degree of absolute) Prudence, a term established by Kimball (1990) relating to the desire to avoid disappointment and linked to the precautionary savings motive. Prudence implies non-increasing absolute risk aversion. Based on this concept Menezes, Geiss, and Tressler (1980) define ‘downside risk aversion’ as having a utility function with a uniformly positive third derivative. 7 Variations of financial holdings often involve trade-offs between moments. For a cubic utility the value of the trade-off between 3rd moment and the variance (−U  /U  ) decreases monotonically with W. At high levels of wealth skewness preference (U  /U  ) decreases, while third moment preference is constant.

Chapter 7. Determining skewness preference through gambles

133

to local risk seeking behavior in spite of global risk aversion. However, the question whether a utility function exhibits increasing, constant or decreasing risk aversion can only be answered with respect to a specific gamble and investor. The changes in the ratios and in the moments’ trade-offs are proportional: An investor A with a ratio U  /U  twice as high as that of investor B will demand half the skewness per unit loss of expected return. Similarly, an investor C with a ratio −U  /U  twice as high as that of investor D will demand half the skewness per unit rise in variance. In the next paragraphs these trade-offs will be derived analytically. First consider the trade-off between expected return and skewness: A decrease in expected return of one percentage point lowers the risk premium by that amount.8 To keep the risk premium constant, the compensating skewness can be calculated by expressing the new risk premium (sub-subscript 2) in terms of the original one (sub-subscript 1). Solving for the skewness of the modified gamble with lower expected return yields: π2  U 1 1 αr σ32 − W 2  m32 2 6 U m3 2

= π1 − 0.01 U  1 1 αr σ32 − W 2  m31 − 0.01 = 2 6 U 0.06U  = − m 31 W 2 U 

(7.9)

where the second subscript of m32 stands for the skewness of the gamble with a single percentage-point decrease in expected return compared with the original gamble’s skewness m31 . The change in skewness necessary to compensate for a 1% loss in expected return is thus: 0.06U  ∆m3 = 2m31 − 2  ∆ER W U

(7.10)

Next consider the trade-off between skewness and variance: An increase in variance of one percentage point raises the risk premium. To keep the risk premium constant, the compensating skewness can be calculated by equating the new risk premium (sub-subscript 3) and the original one (sub-subscript 1). 8 Empirical

studies confirm a positive tradeoff between mean return and skewness.

134

7.2. Creating gambles for skewness preference

Solving for the skewness of the modified gamble with higher variance yields: 1 U  1 αr σ32 − W 2  m31 2 6 U m33

U  1 1 αr σ32 − 0.01 − W 2  m33 2 6 U αr U  = m31 − 0.03 2  W U U  = m31 + 0.03 W U  =

(7.11)

where the second subscript of m33 stands for the skewness of the gamble with a one percentage-point increase in variance. The change in skewness necessary to compensate for a 1% increase in variance is thus: ∆m3 ∆σ 2

= =

0.03

αr 

W 2 UU  U  0.03 W U 

(7.12)

As the volatility is the more intuitive and traditional measure of risk, it might be of greater interest to know the trade-off between skewness and volatility. An increase of one percentage point in volatility necessitates the following skewness to keep the risk premium constant: U  1 1 αr σ32 − W 2  m31 2 6 U m34

U  1 1 αr (σ3 − 0.01)2 − W 2  m34 2 6 U 1 1 σ − 3 10 000 = m31 − 3αr 50 (7.13)  W 2 UU    1 3U  1 σ − = m31 + 3 W U  50 10 000 =

where the second subscript of m34 stands for the skewness of the gamble with a one percentage-point decrease in volatility. The change in skewness necessary to compensate for a 1% increase in volatility is thus: ∆m3 = 3αr ∆σ

− 101000  W 2 UU 

1 50 σ3

(7.14)

Unfortunately, even volatility is a poor risk measure for hedged (addition of puts) or enhanced (addition of calls) portfolio distributions. Lower partial

Chapter 7. Determining skewness preference through gambles

135

moments capture the modifications of put- and call strategies more appropriately and thus reflect their benefit with more accuracy. However, they cannot capture the form of the distribution’s upper end that is relevant for the propensity to gamble.

7.2.1

Interpretation of three-moment preferences

Looking at Equations 7.3, 7.12 and 7.10 it can be learned that an increase in variance requires a compensation in the form of positive skewness when assuming a concave utility function and when holding the end-of-period (eop) wealth constant. For the negative exponential utility the relative risk premium increases with the variance of returns when W increases and the distribution is symmetric (skewness=0)9 . The premium remains constant for logarithmic utility when W increases. For the negative exponential utility the variance’s compensation with skewness in Equation 7.10 decreases proportionally with W , as the implicit skewness premium for a higher variance decreases with increasing wealth. For the logarithmic utility it is constant and equals 1/2. When increasing the variance and keeping the mean constant in a gamble10 , the expected utility of the changed prospect will be lower than for the original gamble for all concave utility functions. The EU deteriorates simply by concavity as fair games are rejected by risk averters. This implies that positive skewness-variance trade-offs are rejected locally. It explains why call options with identical expected returns are priced lower the further out of the money they are. The effect of the change in variance splits investors into two groups: The first group of investors will require an increase in the risk premium to accept the second game while the second group of investors are willing to take a cut in EU because they prefer the higher positive skewness. The groups can be distinguished by −U  /U  . 9 As

the relative risk aversion increases for the negative exponential utility. other words the probability of the good outcome decreases. Tilting a prospect in such a way introduces “in the large” aspects to risk taking. For prospects of very small risks, tilting is undesirable. Remaining “in the small”, as Samuelson (1972) shows for concave utilities, leaves the first two moments as sole potential factors. 10 In

136

7.2. Creating gambles for skewness preference

The investors with more conservative utility are increasingly attracted to positive skewness while the bolder investors are invariant to it. This seems paradoxical considering Keynes’ famous quotation “The poor should not gamble while the millionaires should do nothing else”. The dollar premium for the negative exponential utility (NEU) is constant in W , increasing with the variance and decreasing with mean/variance. For the logarithmic utility it is decreasing with W , increasing with the variance and decreases with mean/variance at a decreasing rate with W . The reason is that the relative conservatism of the NEU-behavior decreases as downside risk is reduced and the ratio mean/variance increases. Aggressive behavior represented by the logarithmic utility is decreasingly concerned with downside risk as wealth increases. The relevance of this observation is exposed particularly with financial instruments such as options.

7.2.2

Example for the assessment of risk aversion

To illustrate the benefit and practical use of the above trade-off formulas, the assessment of an imaginary investor i’s preferences is simulated hereafter. In the first step, the two-moment preference is determined and the investor i is presented with a gamble. For reasons of transparency and comprehensibility the gamble is portrayed by two illustrations which are complemented by the moments of the underlying distribution. The first illustration gives the payoffs in return style, the second one in absolute amounts assuming the investor considers allocating all of his wealth W.11

11 Note: In order to avoid typical behavioral biases such as loss aversion, it would be beneficial to portray a gamble with purely positive or negative payoffs in a survey.

Chapter 7. Determining skewness preference through gambles

137

Step 1: Two-moment risk aversion The investor i is given with his wealth W =200’000 Expected Return ER2 =9% Volatility σ2 =22% 262’000 +31% ✒ ✒ 0.5 0.5 200’000 200’000 ❅ 0.5 ❅ 0.5 ❅ ❅ ❘ 174’000 ❅ ❘ -13% ❅ The investor declares his Certainty Equivalent CE2 = 5.00%

With these parameters it is possible to solve for investor i’s relative risk aversion αri and his absolute risk aversion αai : αri

αai

=

U  ER − CE 9% − 5% =2 =2 U σ22 20%2 1.653

=

8.264 · 10−6

= −W

Step 2: Three-moment risk preference Expected Return ER3 =7% Volatility σ3 =21.31% Non-normalized Skewness, 3rd moment m3 =0.0175 +65% 330’000 0.1 ✒ 0.1 ✒ 200’000

0.5✲

+9%

❅ 0.4❅ ❘ -10% ❅

200’000



0.5✲

218’000

0.4❅

❘ 180’000 ❅

The investor declares his Certainty Equivalent CE3 = 6.00%

138

7.2. Creating gambles for skewness preference

With the information of this second gamble, U  /U  can be derived:   6 π3 − 12 αr σ32 U  = − U E[˜ µ]3 W 2   6 (ER3 − CE3 ) − 12 αr σ32 = − m3 W 2   6 (7% − 6%) − 12 · 1.653 · 21.31%2 = − 0.0175 · 200 0002 = 2.360 · 10−10 Dividing U  /U  from above by αr yields −U  /U  : U  −  U

=

6(π3 − 12 αr σ32 ) E[˜ µ]3 W 2 −2 Wπσ2 2 2

3 σ22 W E[˜ µ]3 3 · = − 200 000



= −

=

 π3 σ2 − 32 π2 σ  2 2 22% (7% − 6%) 21.31%2 − 0.0175 (9% − 5%) 22%2

2.855 · 10−5

The investor i is now completely described by his relative two-moment risk aversion, his skewness-return preference and skewness-variance preference.12 The trade-offs can be calculated using the formulas 7.10, 7.12 and 7.14: The trade-off (change in skewness necessary to compensate for a 1% loss in expected return) between skewness m3 and expected return ER is: ∆m3 ∆ER

12 Note

0.06  W 2 UU 

=

2m31 −

=

2 · 0.0175 −

=

0.0286

200 0002

0.06 · 2.360 · 10−10

that for practical purposes e.g. when carrying out a survey about investment preferences, one will be especially interested in depicting the moments of empirical portfolio distributions. Portraying portfolio skewness with gambles is particularly difficult, as in tree structured gambles high skewness values cannot be achieved unless one assumes unrealistically high or low payoffs. As an alternative it might be easier to directly present graphs of different return distributions to the investor and ask his certainty equivalent for these.

Chapter 7. Determining skewness preference through gambles

139

The change in skewness necessary to compensate for a 1% increase in variance is: ∆m3 ∆σ 2

=

0.03 ·

=

0.03 ·

=

αr  W 2 UU 

1.653 200 0002 · 2.360 · 10−10 0.00525

Similarly, the change in skewness necessary to compensate for a 1% increase in volatility is: ∆m ∆σ

− 101000  W 2 UU 

1 50 σ3

=

3αr

=

3 · 1.653 ·

=

0.00229

1 1 50 21.31% − 10 000 200 0002 · 2.360 · 10−10

140

7.2. Creating gambles for skewness preference

Chapter 8

Creating portfolio skewness with options In the preceding chapter the moment preferences of investors were examined. It was shown how the trade-offs among pairs of moments desired by the investor can be determined analytically. This chapter will examine how these preferred trade-offs, or moment preferences can be implemented through option strategies. The first section deals with the question how options change the distribution of portfolio returns. Different option strategies will achieve different trade-offs among the moments of the return distribution. Investors who appreciate skewness are willing to accept lower expected portfolio return in exchange for protection from losses or in exchange for higher potential payoffs. While portfolio protection - a so-called ‘hedge’ - can be achieved via buying put options, higher potential payoffs - called ‘enhancement’ of the portfolio hereafter - are established via writing call options. In general positive skewness can be achieved by less than perfect diversification, through call and put options as well as through lotteries. Many common trading strategies also result in skewed returns. The most common is “cut your losses and let profits run”. It results in lots of small losses and a few big gains. Two other strategies that yield skewed returns are those that use leverage dynamically and those that follow a trend. The former adds to winning trades and reduces losers, the latter corresponds to posting many 141

142

8.1. Portfolio distribution

small losses in volatile markets and a few large gains in trending markets.

8.1

Portfolio distribution

The first step in modelling distributional effects of options is to choose the appropriate assumption about the form of the portfolio’s return distribution. As the study is foremost concerned with long investment horizons, single period returns and the portfolio’s terminal value at each period, the lognormal model will be employed. It assumes that continuously compounded singleperiod returns are iid normal implying that single-period gross simple returns are distributed as iid lognormal variates.1 Furthermore, it is assumed that call and put options can be bought on the particular portfolio held by the investor. This assumption is not too restricting, as today options on all major market indices with varying strike prices and maturities are available. Also, the sole investment in the market-portfolio has been an explicit assumption in the earlier empirical analysis of two-moment preference. Considering only portfolios that are tracking major indices is thus an assumption consistent with the preceding chapters. The mean, variance and skewness of simple returns are thus given by   σi2 −1 (8.1) E[Rit ] = exp µi + 2       (8.2) V ar[Rit ] = exp 2µi + σi2 exp σi2 − 1

  2  (8.3) Skew[Rit ] = exp (σ) − 1 2 + exp σ The lognormal density is 1 √

xσi 2π

 · exp

−(ln x − µi )2 2σi2

 (8.4)

On financial markets skewness is produced by sudden jumps in asset price processes. These jumps are partly perceived as momentum. In index-trackingstock-portfolios, different measures of skewness can be achieved by adding or selling options on the underlying index. The skewness preferences were 1 This model has the further advantage of not violating limited liability, since limited liability yields a lower bound of zero on gross return, see Campbell, Lo, and MacKinlay (1997), 15.

Chapter 8. Creating portfolio skewness with options

143

assessed by the gambles in Section 7. If the investor is willing to sacrifice some (Fisher) skewness to obtain a higher expected return (if both Formula 7.10 and Formula 7.14 are positive for investor i), he should write call options. If the investor primarily wants to avoid high losses and is prepared to trade in some expected return for higher positive (Fisher) skewness and less volatility (if Formula 7.14 is negative and Formula 7.10 positive for investor i), then he should buy put options.

8.2

Truncating the distribution’s lower end with Puts

Adding put-options to the portfolio truncates the lower end of the portfolio distribution. If the density’s x-axis depicts the portfolio’s value at the end of period 1, the cumulated probabilities of the portfolio realizations to the left of the strike price are shifted to the point: StrikePricePuts − TotalCostPuts . In addition to the truncation and the translation, the rest of the distribution to the right of the Strike price X is shifted left by the total cost of the put options. Figure 8.1 illustrates these changes in the return distribution of the portfolio. The total cost of the put-strategy can be calculated applying the BlackScholes’ pricing formula for a European put on a stock index (accounting for a continuous dividend yield) : p = X exp(−r(T − t))N (−d2 ) − S exp(−q(T − t))N (−d1 )

(8.5)

where X=Strike price of put option, S=index value, r=risk-free rate, (T − t)=time to maturity, N (x) is the cumulative pdf for a standardized normal variable, q=average annualized dividend yield during the life of the option, σ= volatility of the index and  S  r−q+σ2 ln X + (T − t) √ 2 (8.6) d1 = σ T −t √ (8.7) d2 = d1 − σ T − t Unfortunately the moments of the shifted portfolio’s distribution cannot be derived in closed-end form. However, with any spreadsheet program the effect

144

8.2. Truncating the distribution’s lower end with Puts

Figure 8.1: Lognormal portfolio returns hedged with puts The protective put strategy cuts off the lower end of the distribution and shifts its upper part to the left due to the incurred option premia. The put options’ strike price used for hedging the density below was at 85% of the portfolio’s value. The other parameters were determined according to empirical findings: rf = 3.5%, σ = 15.78%, (T − t) = 1 year, q = 3%. The unhedged and Mean: Volatility: FS Skewness:

hedged moments of the distribution are: unhedged=8.98% hedged=7.73% unhedged=15.39% hedged=14.82% unhedged=0.424 hedged=0.766

of the option’s pricing variables on the portfolio’s moments can be simulated. In general, the trade-off between the first and the third moment is more proportional than the trade-off between the second and the third moment. Obviously, relatively more (Fisher) skewness is needed to achieve a decrease in volatility than an increase in expected return.2 One reason is rooted in the fact that volatility alone is not an appropriate risk measure for hedging- or 2 Figures

5.1 and 8.3 clearly illustrate this fact.

Chapter 8. Creating portfolio skewness with options

145

Figure 8.2: Lognormal portfolio returns hedged with puts exhibiting different strike prices The protective put strategy cuts off the lower end of the distribution and shifts its upper part to the left by the incurred option premia. The higher the put options’ strike price used for hedging the greater is the protection against losses and the larger the cost of the options. The portfolio’s expected return µ is 8.31%, its volatility σ = 21.84%, the non-normalized third moment m3 = 0.0062 and its Fisher skewness F S = 0.5958. Calculations were carried out for the parameters rf = 3.5%, (T − t) = 1 year, and dividend yield q = 3%. The first column gives the moments of the unhedged portfolio (MuPf), the following columns show the moments of that portfolio hedged with puts whose strike prices amount to 80%, 95% and 115%, respectively, of the portfolio’s market value: MuPf 8.31% 21.84% 0.0062 0.5958

µ σ m3 FS

X=80 7.93% 20.47% 0.0075 0.8745

X=95 6.98% 17.32% 0.0074 1.4312

X=115 4.32% 11.4% 0.0044 2.9513

enhancement strategies. The tables assist in choosing the option strategy appropriate for investor i

146

8.2. Truncating the distribution’s lower end with Puts

Table 8.1: Impact of different strike prices on the moments of a lognormally distributed portfolio with high volatility. All values are calculated for the following portfolio volatility σ, riskfree rate rf , maturity T and dividend yield q for the index put options: σ = 22%, rf = 3.5%, T = 1y, q = 3% Each block of four rows depicts a different portfolio that is hedged with put options whose strike prices amount to X% of the portfolio’s current market value. The four rows of each block show the portfolio’s mean return µ, its volatility σ, its non-normalized third moment m3 and its Fisher skewness F S = m3 /σ 3 . MuPf 7.23% 21.62% 0.006 0.5958 8.31% 21.84% 0.0062 0.5958 9.4% 22.06% 0.0064 0.5958 10.5% 22.28% 0.0066 0.5958 11.61% 22.5% 0.0068 0.5958 12.73% 22.73% 0.007 0.5958

µ σ m3 FS µ σ m3 FS µ σ m3 FS µ σ m3 FS µ σ m3 FS µ σ m3 FS

X=80 6.95% 20.17% 0.0073 0.8907 7.93% 20.47% 0.0075 0.8745 8.9% 20.81% 0.0077 0.8502 9.91% 21.1% 0.0079 0.8354 10.94% 21.4% 0.008 0.8211 11.96% 21.73% 0.0082 0.7999

X=85 6.76% 19.27% 0.0075 1.044 7.64% 19.65% 0.0077 1.0133 8.59% 19.96% 0.0079 0.9951 9.51% 20.33% 0.0081 0.9665 10.49% 20.65% 0.0084 0.9495 11.45% 21.01% 0.0086 0.923

X=90 6.53% 18.16% 0.0074 1.2366 7.34% 18.56% 0.0077 1.2009 8.17% 18.96% 0.008 1.1664 9.07% 19.29% 0.0082 1.1469 9.94% 19.69% 0.0085 1.1144 10.83% 20.08% 0.0088 1.083

X=95 6.25% 16.9% 0.0071 1.4719 6.98% 17.32% 0.0074 1.4312 7.72% 17.73% 0.0078 1.3918 8.49% 18.15% 0.0081 1.3537 9.29% 18.56% 0.0084 1.3169 9.95% 18.77% 0.0087 1.3182

X=100 5.74% 15.43% 0.0065 1.7767 6.37% 15.85% 0.0069 1.7305 7.03% 16.27% 0.0073 1.6859 7.71% 16.69% 0.0076 1.6427 8.41% 17.11% 0.008 1.6009 9.13% 17.53% 0.0084 1.5602

X=105 5.07% 14.02% 0.0058 2.1058 5.72% 14.35% 0.0061 2.0828 6.29% 14.76% 0.0065 2.0316 6.88% 15.18% 0.0069 1.9823 7.48% 15.61% 0.0074 1.9346 8.11% 16.03% 0.0078 1.8885

X=110 4.46% 12.53% 0.005 2.5149 4.92% 12.94% 0.0053 2.4538 5.51% 13.26% 0.0057 2.4291 6% 13.68% 0.0061 2.3714 6.52% 14.09% 0.0065 2.316 7.05% 14.51% 0.0069 2.2626

X=115 3.85% 10.91% 0.004 3.0608 4.32% 11.4% 0.0044 2.9513 4.71% 11.8% 0.0047 2.8787 5.11% 12.2% 0.0051 2.8093 5.53% 12.61% 0.0055 2.743 6.11% 13.13% 0.006 2.6497

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Table 8.2: Impact of different strike prices on the moments of a lognormally distributed portfolio with low volatility. All values are calculated for the following portfolio volatility σ, riskfree rate rf , maturity T and dividend yield q for the index put options: σ = 16%, rf = 3.5%, T = 1y, q = 3% Each block of four rows depicts a different portfolio that is hedged with put options whose strike prices amount to X% of the portfolio’s current market value. The four rows of each block show the portfolio’s mean return µ, its volatility σ, its non-normalized third moment m3 and its Fisher skewness F S = m3 /σ 3 . MuPf 6.32% 16.03% 0.0019 0.4568 7.39% 16.2% 0.0019 0.4568 8.47% 16.36% 0.002 0.4568 9.56% 16.52% 0.0021 0.4568 10.66% 16.69% 0.0021 0.4568 11.77% 16.86% 0.0022 0.4568

µ σ m3 FS µ σ m3 FS µ σ m3 FS µ σ m3 FS µ σ m3 FS µ σ m3 FS

X=80 6.29% 15.56% 0.0023 0.6093 7.31% 15.79% 0.0023 0.5896 8.36% 15.99% 0.0024 0.5776 9.41% 16.21% 0.0024 0.561 10.48% 16.42% 0.0024 0.546 11.57% 16.62% 0.0025 0.5371

X=85 6.1% 15.01% 0.0025 0.7442 7.07% 15.27% 0.0026 0.716 8.06% 15.54% 0.0026 0.6897 9.06% 15.79% 0.0026 0.6652 10.11% 16.01% 0.0027 0.6504 11.15% 16.26% 0.0027 0.6288

X=90 5.9% 14.17% 0.0027 0.9354 6.59% 14.38% 0.0027 0.9193 7.5% 14.68% 0.0028 0.885 8.43% 14.98% 0.0029 0.8523 9.38% 15.27% 0.0029 0.8211 10.36% 15.56% 0.003 0.7914

X=95 5.37% 12.88% 0.0026 1.2374 6.16% 13.21% 0.0028 1.1944 6.98% 13.55% 0.0029 1.1529 7.76% 13.94% 0.003 1.0951 8.63% 14.27% 0.0031 1.057 9.52% 14.59% 0.0032 1.0202

X=100 4.83% 11.5% 0.0024 1.6057 5.52% 11.85% 0.0026 1.5561 6.23% 12.2% 0.0027 1.508 6.96% 12.54% 0.0029 1.4614 7.66% 12.96% 0.003 1.3926 8.45% 13.31% 0.0032 1.3492

X=105 4.18% 9.97% 0.0021 2.1012 4.75% 10.32% 0.0022 2.0417 5.26% 10.75% 0.0024 1.9502 5.88% 11.1% 0.0026 1.8964 6.69% 11.56% 0.0028 1.8131 7.37% 11.92% 0.003 1.7624

X=110 3.39% 8.45% 0.0016 2.7056 3.95% 8.89% 0.0018 2.5923 4.43% 9.24% 0.002 2.5207 4.84% 9.67% 0.0022 2.4109 5.37% 10.02% 0.0024 2.3475 5.92% 10.37% 0.0026 2.2865

X=115 2.7% 7.03% 0.0012 3.4409 3.13% 7.45% 0.0014 3.2901 3.39% 7.85% 0.0015 3.1377 3.78% 8.19% 0.0017 3.0481 4.3% 8.63% 0.0019 2.9238 4.73% 8.98% 0.0021 2.8446

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Table 8.3: Impact of different strike prices on the moments of a lognormally distributed portfolio with high volatility and higher riskfree rate. All values are calculated for the following portfolio volatility σ, riskfree rate rf , maturity T and dividend yield q for the index put options: σ = 16%, rf = 4.5%, T = 1y, q = 3% Each block of four rows depicts a different portfolio that is hedged with put options whose strike prices amount to X% of the portfolio’s current market value. The four rows of each block show the portfolio’s mean return µ, its volatility σ, its non-normalized third moment m3 and its Fisher skewness F S = m3 /σ 3 . MuPf 7.23% 21.62% 0.006 0.5958 8.31% 21.84% 0.0062 0.5958 9.4% 22.06% 0.0064 0.5958 10.5% 22.28% 0.0066 0.5958 11.61% 22.5% 0.0068 0.5958 12.73% 22.73% 0.007 0.5958

µ σ m3 FS µ σ m3 FS µ σ m3 FS µ σ m3 FS µ σ m3 FS µ σ m3 FS

X=80 7.18% 20.28% 0.0073 0.8764 8.17% 20.57% 0.0075 0.8599 9.18% 20.87% 0.0077 0.8441 10.17% 21.21% 0.0078 0.8209 10.94% 21.4% 0.008 0.8211 11.99% 21.69% 0.0082 0.8074

X=85 6.97% 19.39% 0.0075 1.0278 7.91% 19.71% 0.0077 1.0085 8.82% 20.08% 0.0079 0.9785 9.8% 20.4% 0.0082 0.9606 10.74% 20.76% 0.0084 0.9329 11.75% 21.08% 0.0086 0.9162

X=90 6.72% 18.29% 0.0075 1.2183 7.6% 18.62% 0.0077 1.1971 8.44% 19.02% 0.008 1.1619 9.29% 19.42% 0.0083 1.1281 10.23% 19.75% 0.0085 1.1087 11.12% 20.14% 0.0088 1.0769

X=95 6.51% 16.95% 0.0072 1.4705 7.24% 17.37% 0.0075 1.4289 7.99% 17.79% 0.0078 1.3887 8.77% 18.2% 0.0081 1.3499 9.56% 18.62% 0.0085 1.3124 10.45% 18.96% 0.0088 1.2921

X=100 5.99% 15.47% 0.0066 1.7777 6.63% 15.89% 0.0069 1.7306 7.29% 16.31% 0.0073 1.6851 7.97% 16.74% 0.0077 1.6412 8.68% 17.16% 0.0081 1.5986 9.48% 17.5% 0.0085 1.5775

X=105 5.56% 14.08% 0.0059 2.1137 6.12% 14.5% 0.0063 2.0598 6.69% 14.93% 0.0067 2.0078 7.29% 15.35% 0.0071 1.9577 7.91% 15.77% 0.0075 1.9094 8.64% 16.12% 0.0079 1.8873

X=110 5.05% 12.5% 0.005 2.5664 5.4% 13% 0.0054 2.4665 5.99% 13.33% 0.0058 2.4395 6.49% 13.75% 0.0062 2.3799 7.01% 14.17% 0.0066 2.3226 7.55% 14.59% 0.007 2.2675

X=115 4.4% 11.06% 0.0041 3.0484 4.77% 11.46% 0.0045 2.9702 5.16% 11.86% 0.0048 2.8955 5.69% 12.38% 0.0053 2.7923 6.13% 12.79% 0.0057 2.7246 6.58% 13.2% 0.0061 2.6598

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Figure 8.3: Assuming a portfolio of stocks with lognormally distributed returns, the following graphs show the mean, volatility and skewness of portfolios hedged with put options of strike prices that amount to 80%, 85%, ..., 115% of the portfolio’s value. The figures visualize the previous tables where the riskfree rate rf = 3.5%, the maturity of the options T = 1y and the normally distributed return had a volatility σ = 20% and a mean varying from µ = 5%, 6%, ..., 10%.

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8.2. Truncating the distribution’s lower end with Puts

according to his skewness preference that was determined with the certainty equivalence gambles. For example, an investor who stated his CE to equal 7% in gamble 1 of Section 7.2.2, Page 137, and CE = 5.25% in gamble 2 of Section 7.2.2, Page 137, exhibits the following trade-off ratios: m3 /µ = 0.00138 and m3 /σ = 0.00733. Thus, for a 1% decrease in expected return he demands an increase in skewness of at least 0.000138. For a 1% increase in volatility he expects an increase in skewness of at least 0.00733. Assuming the riskfree rate is rf = 3.5% and the time horizon of the option strategy should be T = 1y, it becomes obvious that the above ratios restrict the possible strategies depicted in Table 8.1 on Page 146. A m3 /µ-trade-off of 0.00138 excludes all put options with a strike price higher than 90% of the current portfolio price. The trade-off ratio of skewness and volatility is no restriction in this case, as the volatility of the protective put strategy decreases more than proportionally with an increasing exercise price. However, the investor is only willing to purchase out-of-the-money puts. All other options cause the mean to decrease by too much compared to the compensation with skewness. Blake (1996) finds that the cost of portfolio insurance expressed by put premiums is less than the maximum that investors would be willing to pay to avoid risk. He concludes that the demand for portfolio insurance is high in all wealth ranges. This holds even though the proportionate cost of the put premium increases with wealth, as the richer investors hold more risky assets. The expected returns increase at a faster rate than the put costs, so that the portfolio insured returns also increase monotonically with wealth. Thus, if investors were offered portfolio insurance whose costs as a proportion of their wealth (putcosts/wealthlevel) is less than π (Pratt-Arrow risk premium), we would expect them to take it.3 Judging from the above exposition and the analysis of preferences, the market’s demand for portfolio insurance should be strong. Unfortunately, it is not widely available to private investors. In considering what portfolio insurance strategy to adopt, put-options prove to be the only workable possibility. Dynamic portfolio insurance strate3 In

this context Beighley (1994) states that even though a continuous, dynamic strategy with out-of-the-money options would not be expected to pay off in every quarter, it would be expected to pay off in those quarters when protection is really needed; that is, when significant downside price movements occur.

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gies replicate put options. In contrast to the latter, however, these strategies prove to fail when they are needed most. That is, in times of market panics and crashes when market liquidity falls heavily and no one is prepared to take the other side of the sales deal. The October 1987 crash with its automated program sales of dynamic insurance strategies is a warning reminder illustrating how dynamic strategies fail to fulfill their very purpose. The question whether one wishes to simply follow a protective put strategy or whether financing the puts by writing calls additionally can be answered by the examining the investor’s skewness preference and the current market conditions for options, e.g. their prices. When considering a put option strategy over longer-term horizons (over one year) one is forced to apply a roll-over strategy. Variables that need to be decided on include: maturity T of the options used, strike price X (floor of the hedge strategy), and general strategy of the insurance program. The last variable refers to the variety of programs imaginable. Naturally, the program’s cost - the endogenous variable of the decision problem - also has to be considered. Figlewski, Chidambaran, and Kaplan (1993) consider three basic programs: the ‘fixed strike’ program which is similar to the ‘time invariant portfolio protection (TIPP)’, the ‘fixed percentage’ program which is similar to the ‘constant proportion portfolio insurance (CPPI)’ and the ‘ratchet strategy’ which employs one of the first two depending on the direction of the market movement. With the fixed strike strategy an investor selects a desired strike level which is fixed for the whole period. If the stock price rises sharply in the beginning of the strategy, rolling over the put becomes very cheap, though the strategy then proves not very protective. If the market drops heavily, the puts at roll-over will become very expensive, rendering the strategy essentially into a riskless asset. With the fixed percentage strategy the strike price is set at a given percentage of the stock price at the time of rollover. This strategy has the advantage of continued capital protection in rising markets: When the market increases, the absolute floor of the hedge program does as well. However, in times of continued downward movement, this strategy provides no adequate protection from losses, as the floor continues to drop with the market price at every rollover. The ratchet strategy sets the strike

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price of the initial put to a given percentage of the stock price, as in the fixed percentage strategy. If the stock price rises and the put matures, the fixed percentage strategy is employed. If the stock price has dropped at the time of maturity, the fixed strike strategy is applied. Thus, in case of a market rise, the strike price is raised to lock in some of the gain and the rollover is carried into a new put with a higher strike, equal to the chosen percentage of current stock price. If the stock has dropped over the option’s life, the strike is not lowered. With this program, the investor locks in early gains without accepting the downside risk of the fixed percentage strategy. For this study, put options with a maturity of three months will be used, the strike price will be arbitrarily set to 95% of the stock price at rollover and moreover, the fixed percentage strategy will be employed. The decision for three-month-options is a compromise: longer term options are more expensive and, assuming that they will be held to maturity, they will not allow continuing adaptation of the floor to the current market price. Shorter-term options will yield more losses over continued market declines. As the most severe price drops in the form of crashes usually happen within few days time and harsh market declines can prevail for longer time than one or two months, the three month-options seem to offer the best features in terms of cost, protection and potential. Furthermore, Leland (1999) found that using in-the-money options with this strategy provides the only degree of protection acceptable for a portfolio insurance strategy with options. Rolling over out-of-the-money options (5% or 10% out of the money) does not yield sufficient protection from disastrous movements at low volatility rates (sigma=10%). While a 105-strike fixed percentage strategy offers low risk, it also offers a mean return well above the riskless rate (> 2%) and good possibilities on the upside. The fixed percentage strategy in combination with three-month-options offers reasonable protection in the case of a sudden crash that happens in the course of a few days while it allows the floor to increase with the market over time. On the other hand, it does not become increasingly expensive in the case of long-term market declines like the fixed strike strategy.4 It seems 4 The diffusion process and the positive drift contribute to the price moving further and further away from its starting value, the longer the time period. This affects the performance of the fixed strike strategy significantly, as the optional character of the position disappears.

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more reasonable anyway to protect oneself against sudden market crashes than against slow steady downward movements that happen over the course of more than three months. In such a case an individual reaction like selling parts of one’s risky assets probably proves more sensible than protection via puts.

8.3

Enhancing the distribution’s upper end with Calls

Leland (1999), (1980) and Booth, Tehranian, and Trennepohl (1985) showed that an investor whose risk tolerance grows with wealth more quickly than the average investor’s risk tolerance will want portfolio insurance (exhibiting convexity); if his risk tolerance grows less quickly than the market’s, a covered call strategy (exhibiting concavity) is optimal. With growing risk tolerance the investor’s preference for greater skewness increases. Optimal strategies, therefore, are preference dependent, and no preference or performance measure that depends on the first two moments of the return distribution alone will correctly rank all alternatives for all investors. Kraus and Litzenberger (1976) - assuming investors have a cubic utility function - demonstrate that when the market portfolio has iid returns, the average investor must have a preference for skewness. This implies a positive third derivative of the investor’s utility function which in turn means that his risky investments increase as his wealth increases.5 Empirical evidence for the preference of positive skewness is omnipresent. Almost every house owner has a fire- and water- insurance; life insurances are no rarity. While the buying of insurance and simultaneous purchase of lottery tickets is incompatible with traditional utility functions, it makes perfect sense when considering positively skewed return distributions. Investors desire to The strategy is similar either to simply holding the stock (in the case of a market rise) or to holding the riskless asset (if the price falls far below the starting value), with the former dominating over the long run because of the drift. Since fixing the strike tends to lead to rolling into puts that are increasingly deep in or out-of-the-money, either the protection or the market exposure of the strategy disappears as time passes. It is therefore not a procedure that many investors would follow over the long run. 5 Pratt (1964), Arrow (1965).

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8.4. Implementation and Limits of Options strategies

sell potential losses for a certain low cost (they pay the put option premium) while at the same time they are willing to purchase high upside potential for a small price (the call option premium). The success of insurances and state lotteries are an obvious proof that most people exhibit this kind of behavior. They do not esteem low, certain incomes and are averse to losses. As a consequence they like to gamble for large amounts that can be gained only with very low probability. They appreciate high potential gains as an additional investment, even if receiving them is highly improbable.

8.4

Implementation and Limits of Options strategies

The previous sections primarily discussed the impact of option strategies on the moments of the underlying portfolio. Nothing was said about how to best design and implement an option strategy. Static protective put strategies where the level of protection is not adjusted as the underlying’s price increases, generally tend to underperform. Dynamic protective put strategies where the options are replaced with puts of higher strikes as the price of the underlying increases, effectively protect from losses, but involve significant transaction costs. A cheaper way of protection offer barrier options. However, their disadvantage comes in the form of lower liquidity, non-standardized trading and less choice and supply. In a two-moment setting a covered call strategy seems to improve the risk-adjusted performance of a portfolio, as the expected return is increased by the option premium and the volatility decreased by truncating the upper end of the distribution. This is a major flaw of the CAPM and two-moment asset pricing frameworks, as the asymmetry of risk is not considered in these models. It can easily be shown that the systematic writing of covered call options or the systematic purchase of protective puts cannot stochastically dominate the simple long stock (buy-and-hold) strategy. These ‘additions’ should only be considered as ad-hoc measures to increase performance in specific personal or particular market situations. At low exercise prices the isolated covered call strategy approximates the

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risk-free asset, while it almost resembles a stock for high strikes. The picture is reversed for the isolated protective put strategy. For low exercise prices it is similar to a pure stock, while imitating a risk-free bond for high strike prices. Increasing the strike price of a protective put position signifies moving from a pure stock to a pure bond investment.6 Finally, a general remark of caution about the efficiency of option hedges needs to be made at this point. While, compared with dynamic trading strategies such as TIPP or CPPI, option strategies are very reliable in crash situations and market crises, their protection is limited by the issuing counterparty. Put options cannot cushion the impact of an overall economic crash or worldwide depression. If a freefall similar to the one in 1929 was to happen again, the consequences on the banking system and guarantors of financial instruments in general is uncertain.

6 and

vice versa for a covered call strategy.

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Chapter 9

Joint estimation of three moment preference This last section will show theoretically how three-moment risk aversion can be estimated jointly from the checklist questions and the three moment gamble. As mentioned in subsection 3.5.1 the utility function of the general parameter preference model by Rubinstein (1973) will be employed. Limited to the first three moments, the utility of investor i is defined in terms of expected return µ, volatility σ and skewness m3 (unnormalized third moment) of investment opportunity j. Ui

1 U  1 U  2 3 W σj2 + W mj  2U 6 U 1 1 = µj + αi σj2 + ψi m3j 2 6

= µj −

(9.1)

where −W · U  /U  equals αi , the investor’s Pratt-Arrow measure of relative risk aversion. This aversion to variance, αi , is determined by the first gamble on page 137 and equation 7.6. The second determinant of the investor’s utility function is skewness preference, depicted by ψi or W 2 · U  /U  . It is assessed by the second gamble on page 7.2.2 and equation 7.7. Both quantities, αi and ψi can also be determined through two separate regressions αi = β  xi and ψi = γ  zi . Factors explaining the aversion to vari157

158

ance αi that can be used as independent factors have already been discussed. Those accounting for skewness preference will need to be determined in a separate examination and are not a subject matter of this study. The regression model is thus 1 1 = µj − β  xi σj2 + γ  zi m3j + νi j 2 6

Ui

(9.2)

where νi j is an iid Gumbel random variable capturing measurement errors not accounted for by the model. The joint estimation model for the regression and the gambles has the log-Likelihood function logL =

I

{ln [αi − ϑβ  xi ] + ln [ψi − θγ  zi ] + lnPi (ji )}

(9.3)

i=1

with αi ψi

= 2 =

ER − CEi σg2

6((ER3 − CE3i ) − 0.5αi σ32 ) E[˜ µ]W 2

σg2 stands for the gamble’s variance and is not to be confounded with the variance of the investment σj2 ; ϑ and θ are coefficients of the joint estimation  and Pi (ji ) = exp(Ui )/( exp(Ui )).

Chapter 10

Summary and Conclusion This study has presented three novel modules that were combined in an integrated approach to structuring the allocation of an investor’s free wealth in a two- and three-moment asset pricing framework. Firstly, econometric models of discrete choice were applied to portfolio selection in a two-moment, traditional mean-variance, setting. It was shown that the choice of the best model depends on the format of the data - or more specific on the dependent variable, the stock ratio. The best performance was defined a) as predictive power of the model to classify observations correctly and b) the ability to cover and seize the whole spectrum of risk classes by assigning observations evenly and consistently well over all risk segments. Both criteria a) and b) could be improved considerably by dividing the estimation in two and three steps, respectively. Especially the prediction of choices between different levels of stock ratio was significantly better when analyzed separately (setting 3c). When considering samples that contain people without any financial assets and when the dependent variable is in continuous format, then the Tobit Model performs better than the OLS. When the dependent variable is in discrete format and multinomial, the MNL Model performs better than the Ordered Logit Model. In the case where the sample contains only people that dispose of financial assets, the MNL Model performs best, regardless whether the dependent 159

160

variable is in continuous or discrete format. In the case where there are very few risk classes as in the binomial settings1 , the WLS and MNL performed almost equally well with the MNL slightly superior. Apart from a higher number of independent factors, its obvious advantage is that it can account for nonlinearities in the relation of the dependent variable and the independent factors. This advantage became evident from the Figures in Table 3.3. No independent factor in the data sample is strictly monotone increasing or decreasing in the risk classes. In addition to applying econometric models in a two-moment setting, gambles were employed to measuring risk aversion. Also, a simultaneous equation model for the two estimates of risk aversion was derived and explained theoretically. This joint estimation by stated and observed preferences given in the form of gambles and observed stock ratios leads to more robust results, as gambles can correct for biases and mistakes in the choice of the investor’s actual, current stock ratio. In the second part of the study the portfolio selection model was extended to three moments. It was shown that two moments do not suffice for portraying the investor’s preferences appropriately, as portfolios are usually not rebalanced continuously. The consideration of the third moment, skewness, matters as all investors prefer positively skewed return distributions. Some individuals will want to buy additional skewness, while others might want to trade in some skewness for obtaining a riskfree premium. These preference patterns are determined by moment preference trade-offs which can be determined and calculated for individual investors through gambles. It was further demonstrated how these preference trade-offs can be implemented in an investment strategy through option strategies that are based on different strike prices. Thirdly, the simultaneous equation model for three moments of econometric choice and the gamble was derived theoretically. The study’s goal was to survey different methods for determining individual risk aversion as well as examine and develop new ways of assessing it with 1 If the dependent variable is in continuous format, a binomial setting implies discretizing it and thus essentially increasing the error term, as pointed out in Subsection 3.5.4 on Page 76

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econometric models. Its goal did not consist in determining the factors most relevant in choosing the portfolio’s risk level. The limited number of only 25 factors in the empirical analysis of the large SCF samples was bound to result in a correspondingly modest predictive power. The likelihood ratio tests proved that in all models those variables not significant at the 5% level can be jointly left out without changing the classification results. Nevertheless, the assignment performance of the three step estimation by the Multinomial Logit model can be rated as quite good. Overall, the thesis can be characterized as a practical study on how to determine risk aversion. This topic is relevant for a number of reasons, but particularly because private clients often lack financial knowledge while investment advisors lack the standardized tools to assess their clients’ preferences. As a consequence, these preferences are never determined nor documented and investors keep their misconceptions about what their investment strategies can achieve. The evolving field of behavioral finance plays a prominent role in detecting these misconceptions and finding ways to eliminate them. The study at hand has tried to contribute its part to this mission. The possible extensions of this line of research are manifold. A more profound insight into the individual improvement of investment decisions and the change of risk taking over time is given by longitudinal studies (panel data). It is evident that there is great potential for optimizing individual portfolios given the financial objectives of investors. The internet provides an ideal platform for offering risk assessment and allocation tools to the public at no cost. These tools can also be a signal of quality for the corresponding institute and offer a good possibility to make a first contact with a new client. After all, both the individual investor and the advising financial institution have the same goal - to profoundly comprehend the individual’s preferences. “Know thyself” were the two words inscribed in the Apollo temple in Delphi. - An almost ironical welcome for visitors of an oracle, but possibly still valid, even when considering issues of asset allocation ...

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Appendix A

Asset Allocation and Higher Moment Models A.1 A.1.1

Optimal Asset Allocation in the 2-Moment Model Objective Function

We analyze the utility Uij for investor i, i = 1, ..., N who can choose k investment alternatives j = 1, ..., k. Essentially these are all funds lying on the Capital Market Line. They are correctly priced and clearly distinguishable by their risk - return profile. As all three moments of the return (expected value, standard deviation (risk) and skewness) are known, the individual’s choice depends only on his risk aversion.

We have • W0 – Wealth at the outset of the period 

(W ) • α = − UU  (W ) – the Pratt-Arrow Measure of absolute risk aversion

• µPf = rf + βPf (µM − rf ), describes the portfolio return µPf where M stands for Marketportfolio, 163

164

A.1. Optimal Asset Allocation in the 2-Moment Model

rf for riskfree rate and βPf for the beta of the portfolio (according to the CAPM) • WT = (1 + µPf )W0 is a random variable defining terminal value of wealth at time T (end of period) Assumptions 1. U (w) = − exp(−αW ), where W > 0. 2. The investor maximizes his expected utility: Max E[U (W )] 3. Stock returns are normally distributed: µi ∼ N [µ, σ 2 ] 4. Utilities of different individuals are independently distributed Assuming that a rational investor maximizes his expected utility, we have M ax E[U (WT )]

(A.1)

For clarifying the connection between expected utility and mean-variance decision criterion we approximate the investor’s utility function at E(WT ] with a Taylor Polynom U (WT ) = U (E[WT ]) + U  (E[WT ])(WT − E[WT ]) 1 + U  (E[WT ])(WT − E[WT ])2 2 ∞ 1 (i) U (E[WT ])(WT − E[WT ])i + i! i=3

(A.2)

where U (i) stands for the i-th derivative of the utility function. Assuming  that the Taylor polynom converges and that E and are interchangeable, we can write E[U (WT )]

1 2 = U (E[WT ]) + U  (E[WT ])σw 2 ∞ 1 (i) + U (E[WT ])m(i) i! i=3

(A.3)

Chapter A. Asset Allocation and Higher Moment Models

165

2 where σw describes the variance of terminal wealth and m(i) refers to the ith central moment of the probability distribution of WT : E[(WT − E(WT ))i ]. Expected utility thus depends not only on mean and variance, but in addition on higher central moments such as skewness and curtosis. For now, in order to have a mean-variance portfolio selection lead to utility maximization, we can choose between two possible roads of simplification:

A

we assume that discrete returns of risky assets are distributed multivariate normally N (µ, σ 2 ). As the normal distribution is symmetric in its mean, all higher odd central moments cancel out and all portfolios can be fully characterized by their mean-variance profile. Regardless of their specific utility function all risk averse investors will then for a given mean minimize their portfolio’s variance.

B

we can assume the investor has a quadratic utility function. In that case all moments higher n = 2 have no impact on the expected utility, as those derivatives become zero. This alternative however is not very realistic as has been pointed out. Proceeding with option A, our investor n has a utility function of Un = − exp(−αn WT )

(A.4)

where WT = W0 [ω  (µi − rf ) + rf ], with ω  being the vector of weights of the single assets i within the portfolio. The investor maximizes his expected utility, risky terminal wealth being normally distributed: Max E[− exp(−αn WT )]

(A.5)

Since we assumed WT = W0 (1+ω  (µi −rf )+rf ] to be normally distributed with mean µ and variance σ 2 , we can write the expectation as -φ(ia) where φ is the characteristic function of a normal distribution with mean and variance equal to that of the portfolio. The problem known as the “hybrid model”1 can thus be restated as Max [− exp(−αn (ω  (µi − rf ) + rf ) + 1 Spremann

(2000).

αn2 2 σ )] 2 Pf

(A.6)

166

A.1. Optimal Asset Allocation in the 2-Moment Model

The utility function to be maximized is an ordinal function of mean and variance. As it is ordinal, we can operate on it with the increasing function   1 θ(x) = − log(−x) (A.7) α As maximizing expected utility equals minimizing the argument of the exponential function, we arrive at Max [ω  (µi − rf ) + rf ] −

αn 2 σ 2 Pf

(A.8)

The above expression equals the certainty equivalent return corresponding to a risky investment that is made up of the riskfree and one efficient risky asset - namely the market portfolio. The term behind the square bracket is the Pratt-Arrow measure of the local risk premium. By substituting terminal wealth with the mean-variance features of the portfolio that the investor chooses, the objective function of the maximization problem can be rewritten as: Max E[Un ] = µPf −

αn 2 σ + εn 2 Pf

(A.9)

where εn is a random disturbance capturing measurement errors and factors not accounted for by our model.

A.1.2

Solution of the Objective Function

The following closed-end solution for 2-moment risk aversion (Pratt-Arrow measure), given the stock ratio, the portfolio’s mean expected return, its variance and the risk-free rate, is thought to be a comparison for the estimates arrived at in the regression models. It lays the foundation for using the stock ratio as a proxy for risk aversion in the two-moment model framework. As can be seen, the Pratt/Arrow 2moment risk aversion is indeed closely tied to the stock ratio. The formula has been used in various studies that estimated relative risk aversion. Among them are: Blume and Friend (1975), Morin and Suarez (1983), Siegel and Hoban (1982) as well as Riley and Chow (1992). An investor values his asset allocation strategy according to all possible, random events ZW likely

Chapter A. Asset Allocation and Higher Moment Models

167

to happen in the course of the year Max [E[WT ] −

αi 2 σ ] 2 Pf

(A.10)

The investor is given by: • his wealth w, • his absolute risk aversion as measured by Pratt/Arrow’s coefficient αA • and his asset allocation strategy x - that is, the ratio of the market portfolio (stocks) in his portfolio. According to the CAPM two types of assets are given: the risk-free described by rf , and the risky Marketportfolio given by its expected return E(rM ) and its volatility σM . With a stock ratio of x the investor yields a portfolio return of µw = rf + x(µM − rf ), the risk being σw = xσM . At the end of one year his portfolio has thus an expected value of w(1 + µw ) with a volatility of w ∗ σw . If the investor values his decision according to the classic criterion his asset allocation strategy is given by:

Max E[Un (x)]

αA 2 2 [w σw (x)] (A.11) 2 αA 2 2 2 [w x σM ] = w(1 + rf + x(µM − rf )) − 2 = w(1 + µw (x)) −

The objective function is quadratic in x. Its first-order condition 2 =0 w(µM − rf ) − αA xw2 σM

(A.12)

yields a maximum, as it is concave in x. The optimal 2-moment asset allocation is thus: µM − rf (A.13) x2m = 2 αA wσM Solving for Pratt/Arrow’s 2-moment measure of absolute risk aversion yields µM − rf αA = (A.14) 2 x2m wσM

168

A.2. Optimal Asset Allocation in the 3-Moment Model

A.2

Optimal Asset Allocation in the 3-Moment Model

We have • W0 – Wealth at the outset of the period • x – Amount of wealth to be invested in the well-diversified risky asset (Marketportfolio) • r˜Pf – Random excess portfolio return over the risk-free rate rf • W – End-of-period wealth W ≡ W0 + W0 [x(˜ rPf + rf ) + (1 − x)rf )]

Assumptions 1. The investor maximizes his expected utility: Max E[U (W )] 2. Utilities of different individuals are independently distributed

We expand expected utility to include the third moment: E{U [W (1 + x˜ rPf )]}

= U (W ) + xW U  (W )E(˜ rPf ) 1 2 2  2 rPf ) + x W U (W )E(˜ 2 1 3 + x3 W 3 U  (W )E(˜ rPf ) + ... 6

(A.15)

The first-order condition (FOC) for a maximum of the above objective function using three moment approximation is rPf ) W U  (W )E(˜

2 + xW 2 U  (W )E(˜ rPf ) 1 2 3  3 x W U (W )E(˜ rPf )=0 + 2

(A.16)

We are faced with an objective function quadratic in x. Applying the standard solution yields two results. Only one of these two has a positive

Chapter A. Asset Allocation and Higher Moment Models

169

second derivative and thus satisfies the sufficient condition for a maximum. The optimal 3-moment asset allocation is

−1 − 1 − 2x2m ξA(W ) (A.17) x3m = ξA(W ) with x2m =

r˜Pf 2 αAP wσPf

,ξ=

3 E(˜ rPf ) 2 ) E(˜ rPf

=

m3P f 2 σPf



(W ) and A(W ) = −W UU  (W )

Comparing the optimal solutions for 2- and 3-moments (equations A.17 and A.12) we find that the two are equal when the skewness measure, the third moment of the return distribution, approaches zero: limm→0 x2m = x3m . As one would expect, the allocation to the risky asset with the three moment approximation exceeds the one with the mean-variance solution when the skewness of returns is positive. The opposite holds when the skewness is negative. The existence of a three-moment solution requires that x2m ξA(W ) ≤ 0.5. This condition has to be examined more closely: When analyzing each variable separately, we see that x2m > 0 except for the unlikely case where µPf < rf . A(W ) > 0 ∀ W > 0, as U  < 0 and U  > 0.2 The sign of ξ depends solely on m3 . Thus, the 3-moment approximation always has a solution if m < 0 and µPf > rf . If m > 0, the existence of an optimal solution heavily depends on the ratio m3 σ2 . m3
0, non-satiation, positive marginal utility for wealth which means nonsatiety with respect to wealth. 2. U  (x) < 0, decreasing marginal utility for wealth, i.e. risk aversion and 3. non-increasing or decreasing absolute risk aversion i.e. risky assets are not inferior goods.

170

A.2. Optimal Asset Allocation in the 3-Moment Model

invest as much as possible in the risky asset, even if that meant taking infinite liabilities. This kind of behavior was coined ‘plunging’ by Tobin (1958). It depicts the trade-off between mean-variance and higher-moment models, as the advantages won by incorporating skewness are partly lost by the limited range of its applicability: For portfolios which are sufficiently skewed, the three-moment approximation will be inferior to the mean-variance solution.

Appendix B

Deriving the MNL model from Utility maximization The following lines are taken from Amemiya (1985), 297: Consider an individual i whose utilities associated with three alternatives j are given by Uij = xi + εij j=0,1 and 2

(B.1)

where xi is a nonstochastic function of explanatory variables and unknown parameters and εij is an unobservable random variable (we will write εj from here on for simplification). It is assumed that the individual chooses the alternative for which the associated utility is the highest. McFadden (1973) proved that the multinomial logit model is derived from utility maximization iff {εj } are independent1 and the distribution function of εj is given by exp[exp(−εj )], the Type I extreme-value distribution (Gumbel), or log Weibull distribution2 . Its density is given by f (.) = exp(εj ) exp[− exp(εj )] Proof of the iff part: The probability that the ith person chooses alterna1 IID

disturbances constrain all the disturbances to have the same scale parameter implying that the variances of the random components of the utilities are equal (homoscedastic). The independence of the random components of the utilities give rise to the IIA problem that both the MNL and the CLM share. 2 See Johnson and Kotz (1970), 272. It states that the maximum of independent Gumbel variates with a common scale parameter is itself Gumbel distributed.

171

172

tive j can be expressed as (suppressing the subscripts i from µ): P (yi = 2)

=

P (Ui2 > Ui1 , Ui2 > Ui0 )

=

P (ε2 + µ2 − µ1 > ε1 , ε2 + µ2 + µ0 > ε0 )  ε +µ −µ  ε2 +µ 2 2 1 ∞  2 −µ0 f (ε2 )  f (ε1 )dε1 · f (ε0 )dε0  dε2

=

−∞

−∞

(B.2)

−∞

∞ exp(ε2 ) exp[− exp(−ε2 )] × exp[− exp(−ε2 − µ2 − µ2 + µ1 )]

= −∞

=

× exp[− exp(−ε2 − µ2 − µ2 + µ0 )]dε2 exp(µi2 ) exp(µi0 ) + exp(µi1 ) + exp(µi2 )

This result is equal to Pij in the definition 3.18 of a multinomial logit model, if we put µi2 − µi0 = xi2 β and µi1 − µi0 = xi1 β. The expressions for Pi0 and Pi1 can be similarly derived. It has been shown that Maximization of a utility function (UM) with iid error terms that exhibit an extreme value distribution (EVD) give rise to a multinomial logit model (MLM). Or expressed with symbols: U M, IID, EV D ❀ M LM A proof utilizing the approach above shows that the same constellation conditions the indepence from irrelevant alternatives (IIA): U M, IID, EV D ❀ IIA Proof: P (Ui1 > Ui2

= =

P (µ1 + ε1 > µ2 + > ε2 ) ∞ ε1 +µ  1 −µ2 f (ε2 )dε2 f (ε1 )dε1 −∞

(B.3)

−∞

∞ exp[− exp(−ε1 + µ1 − µ2 )] exp(−ε1 ) exp[− exp(−ε1 )]dε1

= −∞



=

+∞ 1 exp[−(1 + exp(µ1 + µ2 ))] 1 + exp(−µ1 + µ2 ) −∞

Chapter B. Deriving the MNL model from Utility maximization

=

exp(µ1 ) exp(µ1 ) + exp(µ2 )

=

P (U1 > U2 |U1 > U3 or U2 > U3 )

173

Thus: P (y1 |y1 or y2 )

When carrying out the same calculations for a utility with normally distributed error terms (normal), it can be shown that: U M, IID, normal ❀ IIA Thus, the Probit model does not share the sometimes disadvantageous IIA feature with the MNL model. It poses a useful alternative for problems where the IIA is clearly not met.

174

Appendix C

Likelihood function of the Nested Logit Model Following the likelihood function for simplest nested logit model - one with three responses depicted below - is derived. The distribution of the error terms is as follows: F (ε3 )

=

F (ε1 , ε2 )

=

exp [− exp (−ε3 )]      ρ  ε1 ε2 exp − exp − + exp − ρ ρ

Figure C.1: Simplest Nested Logit Model with three responses. Utilities are assumed to be positively correlated.

175

176

The latter represents a ‘Type b’ Gumbel bivariate distribution with the correlation between ε1 and ε2 ∼ = 1 − ρ2 and 0 < ρ ≤ 1. As F (ε1 , ∞) = exp[− exp(−ε1 )] is a natural generalization of the MNL model δ2F δε1 δε2

f (ε1 , ε2 ) = f can be negative if ρ > 1. Define yji

=

1 if yj = j with j = 1, 2, 3

=

0 otherwise

Then (subscript will be omitted after the second line) L = =

Πi P (yi = 1)y1i P (yi = 2)y2i P (yi = 3)y3i y1

Π [P (y = 1|y = 1 or y = 2)P (y = 1 or 2)]

y2

· [P (y = 2|y = 1 or y = 2)P (y = 1 or 2)] =

· P (yi = 3)y3

ΠP (y = 1|y = 1 or y = 2)y1 P (y = 2|y = 1 or 2)y2 ·ΠP (y = 1 or y = 2)y1 +y2 · P (y = 3)y3

≡ L1 · L2 It follows that

 P (y = 1|y = 1 or 2)

=

= As µji = xi βj : P (y = 1|y = 1 or y = 2)

exp  

=



exp µρ1 + exp   (µ1 − µ2 ) Λ ρ 

=

µ1 ρ



µ1 ρ



ρ exp µρ1 + exp µρ2   exp( µρ1 ) + exp( µρ2 ) + exp(µ3 )  ρ   2 +1 exp(µ2 − µ3 ) exp µ1 −µ ρ  ρ   2 + 1 exp(µ2 − µ3 ) exp µ1 −µ +1 ρ

Appendix D

Structure of Estimations

177

178

Figure D.1: Structure and nests of empirical analyses

Appendix E

Empirical Results of Regressions E.1

In-sample estimation SCF1998

179

180

E.1. In-sample estimation SCF1998

Table E.1: Setting 1 - Results for OLS in-sample-regression: The dependent variable is continuous: the stock ratio is given as a percentage. Dataset: SCF 1998. Observations = 4305, Parameters = 25, Deg.Fr.= 4280 Adjusted R-squared = 0.33066, Log-L = -1679.8332 Akaike Info. Crt. = 0.792 Variable

Coeff.

Constant X101 X1706 X3006 1 X3006 5 X3008 1 X3008 45 X301 1 X301 3 X3014 1 X3014 4 X432 1 X5608 X5821 X5825 3 X5905 X7131 X7186 X7187 X7372 1 X7401 1 X7401 2 X7401 5 X8021 1 X8022

-0.1212 -0.0083 0.0144 -0.1158  0.0117 -0.0575 0.0506  0.0139 -0.0224  0.0073 -0.1661 0.1211 0.0150 0.0112 -0.0864 0.0260  0.1015  0.0307 0.0185 ⊕ 0.0517 0.0664 0.0781 ⊕ 0.0066 0.0108 ⊕ 0.0042

Standard Error 0.0325 0.0047 0.0035 0.0387 0.0157 0.0164 0.0125 0.0138 0.0131 0.0233 0.0138 0.0133 0.0102 0.0041 0.0124 0.0049 0.0127 0.0067 0.0055 0.0165 0.0153 0.0176 0.021 0.0176 0.0004

b/St.Er. -3.735 -1.751 4.169 -2.989 -0.747 -3.495 4.035 -1.012 -1.705 -0.312 -12.043 9.099 1.467 2.754 -6.957 5.293 -8.003 -4.603 3.361 3.141 4.333 4.449 0.315 0.614 9.731

P of |Z| > z 0.0002 0.0799 0 0.0028 0.4551 0.0005 0.0001 0.3117 0.0882 0.755 0 0 0.1424 0.0059 0 0 0 0 0.0008 0.0017 0 0 0.7528 0.5392 0

Mean of X 2.6492 1.2393 0.0207 0.1554 0.1619 0.4476 0.2434 0.2767 0.0609 0.3034 0.5094 0.1131 0.5422 0.3933 1.121 0.3654 1.1779 1.9331 0.5988 0.3443 0.1466 0.0904 0.7807 49.8404

Chapter E. Empirical Results of Regressions

181

Setting 1 - OLS, SCF 1998 (cont.) Classification table Predicted 2 3 10 0 667 292 100 152 96 130 90 158 73 137 81 146 1117 1015

0 1 271 21 919 297 14 15 10 13 9 12 13 12 17 18 1253 388 Percent correct: 20.74% Error Distance: 9’562’127

Actual 0 1 2 3 4 5 6

4 0 70 65 85 93 92 106 511

5 0 1 3 6 2 2 7 21

6 0 0 0 0 0 0 0 0

302 2246 349 340 364 329 375 4305

182

E.1. In-sample estimation SCF1998

Table E.2: Setting 1 - Results for Tobit in-sample-regression: The dependent variable is continuous: the stock ratio is given as a percentage. Dataset: SCF 1998. Observations = 4305, Parameters = 25, Deg.Fr.= 4280 Log-L = -2363.740, Akaike Info. Crt. = 1.110 Variable

Coeff.

Standard Error Primary Index Equation for Model Constant -0.5437 0.0612 X101 -0.0084 0.0087 X1706 0.0248 0.0053 X3006 1 -0.0101 0.0641 0.017 0.0274 X3006 5 -0.0832 0.0329 X3008 1 0.1285 0.0212 X3008 45 0.0112 0.0241 X301 1 -0.0145 0.0223 X301 3 0.0353 0.0365 X3014 1 -0.3793 0.0278 X3014 4 X432 1 0.2207 0.023 X5608 0.0175 0.0157 X5821 0.0163 0.0063 -0.1561 0.0223 X5825 3 X5905 0.046 0.0078 X7131 0.0777 0.0223 X7186 0.0306 0.0111 X7187 0.0332 0.0085 ⊕0.073 0.0295 X7372 1 0.0716 0.0252 X7401 1 X7401 2 0.069 0.0305 -0.0663 0.0428 X7401 5 0.0158 0.0335 X8021 1 X8022 ⊕0.0051 0.0008 Disturbance Standard Deviation Sigma 0.5006 0.0094

b/St.Er.

P of |Z| > z

Mean of X

-8.883 -0.963 4.698 -0.158 -0.62 -2.524 6.073 0.465 -0.65 0.966 -13.638 9.613 1.113 2.614 -6.988 5.93 -3.487 -2.743 3.908 2.478 2.838 2.266 -1.549 0.471 6.214

0 0.3357 0 0.8746 0.5356 0.0116 0 0.642 0.5156 0.3339 0 0 0.2658 0.009 0 0 0.0005 0.0061 0.0001 0.0132 0.0045 0.0234 0.1214 0.6376 0

2.6492 1.2393 0.0207 0.1554 0.1619 0.4476 0.2434 0.2767 0.0609 0.3034 0.5094 0.1131 0.5422 0.3933 1.121 0.3654 1.1779 1.9331 0.5988 0.3443 0.1466 0.0904 0.7807 49.8404

53.456

0

Chapter E. Empirical Results of Regressions

183

Setting 1 - Tobit, SCF1998 (cont.) Classification table Predicted 2 3 48 0 1051 234 134 136 116 110 115 143 90 122 112 124 1666 869

0 1 183 71 455 439 4 11 4 5 2 5 2 12 5 11 655 554 Percent correct: 22.49% Error Distance: 8’967’647 Actual 0 1 2 3 4 5 6

4 0 60 56 90 93 94 101 494

5 0 7 8 15 6 9 22 67

6 0 0 0 0 0 0 0 0

302 2246 349 340 364 329 375 4305

184

E.1. In-sample estimation SCF1998

Table E.3: Setting 1 - Results for Ordered model in-sample-regression: The dependent variable (stock ratio) is discrete, 7 risk classes are considered. Dataset: SCF 1998. Observations = 4305, Parameters = 25, Deg.Fr.= 4280 Log-L = -7833.028, Akaike Info. Crt. = 3.651 Variable

Coeff.

Standard Error Index function for probability Constant 1.112 0.0667 X101 -0.0301 0.0101 X1706 0.0896 0.0061 -0.2245 0.0773 X3006 1 0.078 0.0317 X3006 5 -0.2242 0.0371 X3008 1 0.3742 0.0236 X3008 45 X301 1 0.0039 0.0267 -0.0748 0.0255 X301 3 0.0694 0.0413 X3014 1 -0.9338 0.0314 X3014 4 X432 1 0.6213 0.0251 X5608 0.0853 0.02 X5821 0.0594 0.0069 -0.4518 0.0247 X5825 3 X5905 0.1491 0.0088 X7131 0.2805 0.0254 X7186 0.1075 0.0128 X7187 0.1188 0.0098 ⊕0.2599 0.0328 X7372 1 0.2513 0.0283 X7401 1 0.2609 0.0336 X7401 2 -0.1379 0.0495 X7401 5 0.0124 0.0365 X8021 1 X8022 ⊕0.0162 0.0009

b/St.Er.

P of |Z| > z

Mean of X

16.677 -2.972 14.738 -2.904 -2.46 -6.041 15.875 -0.145 -2.934 1.68 -29.704 24.783 4.273 8.635 -18.322 16.877 -11.021 -8.389 12.156 7.914 8.891 7.766 -2.788 0.34 17.886

0 0.003 0 0.0037 0.0139 0 0 0.8847 0.0033 0.093 0 0 0 0 0 0 0 0 0 0 0 0 0.0053 0.7337 0

2.6492 1.2393 0.0207 0.1554 0.1619 0.4476 0.2434 0.2767 0.0609 0.3034 0.5094 0.1131 0.5422 0.3933 1.121 0.3654 1.1779 1.9331 0.5988 0.3443 0.1466 0.0904 0.7807 49.8404

Chapter E. Empirical Results of Regressions

185

Setting 1 - Ordered model, SCF1998 (cont.) Variable

Coeff.

Standard b/St.Er. P of Mean Error |Z| > z of X Threshold parameters for index Mu(1) 2.0000 .... (Fixed Parameter) .... Mu(2) 3.0000 .... (Fixed Parameter) .... Mu(3) 4.0000 .... (Fixed Parameter) .... Mu(4) 5.0000 .... (Fixed Parameter) .... Mu(5) 6.0000 .... (Fixed Parameter) ....

Classification table Actual 0 1 2 3 4 5 6

0 1 2 55 241 6 56 1443 417 0 59 76 0 49 69 0 52 64 0 39 60 1 57 62 112 1940 754 Percent correct: 41.79% Error Distance: 4’265’498

Predicted 3 0 260 146 121 146 128 131 932

4 0 68 65 95 101 99 112 540

5 0 2 3 6 1 3 12 27

6 0 0 0 0 0 0 0 0

302 2246 349 340 364 329 375 4305

186

E.1. In-sample estimation SCF1998

Table E.4: Setting 1 - Results for Multinomial Logit model in-sampleregression. The standard errors of the regression coefficients are given in brackets. The significance levels are abbreviated with asterisks: ‘*’ and ‘**’ are significant at the 5, and 1 percent level. Observations = 4305, Parameters = 25, Deg.Fr.= 4280 Log-L = -5350.629, Akaike Info. Crt. = 2.497

Constant X101 X1706 X3006 1 X3006 5 X3008 1 X3008 45 X301 1 X301 3 X3014 1 X3014 4 X432 1 X5608

Prob[Y=1] 1.0538 ** (0.3994) -0.0339 (0.0523) 0.4146 ** (0.159) -1.7349 ** (0.4149) -0.1063 (0.1939) -0.4319 ** (0.1658) -0.1119 (0.1802) -0.3736 * (0.1686) -0.1474 (0.1777) -0.9411 ** (0.3438) -0.6546 ** (0.1651) 1.1629 ** (0.2551) 0.1729 (0.2175)

Prob[Y=2] -3.5672 ** (0.5752) -0.0562 (0.077) 0.5051 ** (0.1619) -1.6767 ** (0.5651) -0.284 (0.2658) -0.5417 * (0.2688) 0.2242 (0.2224) -0.4932 * (0.2331) 0.0247 (0.2239) -0.9286 * (0.416) -1.4066 ** (0.2347) 2.1681 ** (0.2936) 0.2228 (0.2423)

Prob[Y=3] -2.9594 ** (0.575) -0.1245 (0.0791) 0.5589 ** (0.162) -1.3543 * (0.5598) -0.3208 (0.2685) -0.4074 (0.2764) 0.4887 * (0.2266) -0.4579 (0.2364) 0.0681 (0.226) -0.798 (0.4084) -2.2076 ** (0.2748) 1.894 ** (0.2942) 0.3629 (0.2365)

Prob[Y=4] -2.4082 ** (0.5588) -0.1306 (0.0786) 0.4944 ** (0.1619) -1.4221 * (0.5595) 0.1105 (0.2538) -0.7436 * (0.2901) 0.4772 * (0.2235) -0.3995 (0.2287) -0.1483 (0.2272) -0.9396 * (0.4111) -2.2787 ** (0.2719) 1.9302 ** (0.2934) 0.1589 (0.2452)

Prob[Y=5] -3.1688 ** (0.5823) -0.008 (0.0775) 0.5416 ** (0.162) -1.7018 ** (0.5995) -0.229 (0.267) -0.925 ** (0.3138) 0.4886 * (0.2273) -0.2277 (0.2306) -0.1243 (0.2321) -0.7429 (0.4064) -2.2043 ** (0.2841) 1.9268 ** (0.2968) 0.3093 (0.2395)

Prob[Y=6] -2.6335 ** (0.5614) -0.0568 (0.0759) 0.5904 ** (0.1617) -1.6559 ** (0.5822) -0.2186 (0.2596) -0.7137 * (0.2873) 0.4872 * (0.2231) -0.2361 (0.2233) -0.3184 (0.2293) -0.6175 (0.3971) -2.0304 ** (0.2636) 1.9268 ** (0.2919) 0.3389 (0.2369)

Chapter E. Empirical Results of Regressions

187

Setting 1 - Results for Multinomial Logit model, SCF1998, in-sample-regression. (cont.)

X5821 X5825 3 X5905 X7131 X7186 X7187 X7372 1 X7401 1 X7401 2 X7401 5 X8021 1 X8022

Prob[Y=1] 0.2132 * (0.1026) -0.4328 ** (0.1563) 0.3968 ** (0.1418) -1.0497 ** (0.1569) -0.4994 ** (0.1042) 0.3971 ** (0.143) 0.4332 * (0.2007) 1.1768 ** (0.3245) 1.1236 ** (0.2563) 0.2441 (0.2078) 0.1976 (0.1845) 0.0387 ** (0.005)

Prob[Y=2] 0.272 * (0.11) -0.7954 ** (0.2098) 0.5928 ** (0.1489) -1.1817 ** (0.2107) -0.5603 ** (0.1256) 0.6279 ** (0.1518) 0.6744 * (0.2731) 1.6201 ** (0.3608) 1.5056 ** (0.3238) 0.2593 (0.3602) 0.5389 (0.2953) 0.0593 ** (0.0073)

Prob[Y=3] 0.292 ** (0.1096) -0.7071 ** (0.2131) 0.6075 ** (0.1493) -1.2205 ** (0.2146) -0.7976 ** (0.1263) 0.6133 ** (0.1526) 0.9834 ** (0.2836) 1.4964 ** (0.3626) 1.5388 ** (0.3243) 0.5048 (0.3545) 0.101 (0.2993) 0.0585 ** (0.0075)

Prob[Y=4] 0.2927 ** (0.1092) -1.0674 ** (0.2157) 0.6534 ** (0.1488) -1.0723 ** (0.2101) -0.7377 ** (0.1253) 0.6996 ** (0.1519) 0.811 ** (0.2785) 1.2386 ** (0.3588) 1.214 ** (0.3231) -0.1002 (0.3944) 0.0536 (0.2875) 0.0539 ** (0.0073)

Prob[Y=5] 0.2656 * (0.1099) -1.2747 ** (0.2263) 0.6775 ** (0.1495) -1.3139 ** (0.2165) -0.5239 ** (0.1277) 0.5893 ** (0.1527) 0.5703 * (0.2806) 1.3041 ** (0.3629) 1.5492 ** (0.3223) -0.1422 (0.4226) 0.234 (0.2987) 0.0594 ** (0.0076)

Prob[Y=6] 0.3092 ** (0.1087) -0.9593 ** (0.2145) 0.5593 ** (0.1487) -1.3443 ** (0.2116) -0.6094 ** (0.1251) 0.6307 ** (0.1518) 0.7814 ** (0.2759) 1.4739 ** (0.3577) 1.1924 ** (0.3242) -0.0783 (0.3834) 0.1102 (0.2902) 0.0529 ** (0.0073)

188

E.1. In-sample estimation SCF1998

Setting 1 - Multinomial model, SCF1998 (cont.) Classification table Predicted 2 3 0 0 10 10 9 17 4 30 6 17 7 13 8 25 44 112

0 1 73 229 45 2097 0 250 1 209 0 217 0 190 0 218 119 3410 Percent correct: 54.94% Error Distance: 11’219’224 Actual 0 1 2 3 4 5 6

4 0 30 24 36 65 43 38 236

5 0 22 17 24 17 30 25 135

6 0 32 32 36 42 46 61 249

302 2246 349 340 364 329 375 4305

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Table E.5: Setting 2a - Results for WLS in-sample-regression: The dependent variable is binomial, investors without assets are to be separated from investors. Dataset: SCF 1998. Observations = 4305, Parameters = 25, Deg.Fr.= 4280 Adjusted R-squared = 0.17118 , Log-L = 183.6142 Akaike Info. Crt. = -0.074 Variable WConstant WX101 WX1706 WX3006 1 WX3006 5 WX3008 1 WX3008 45 WX301 1 WX301 3 WX3014 1 WX3014 4 WX432 1 WX5608 WX5821 WX5825 3 WX5905 WX7131 WX7186 WX7187 WX7372 1 WX7401 1 WX7401 2 WX7401 5 WX8021 1 WX8022

Coeff. 1.0375 -0.0062 0.0023 -0.0764 0.0085 -0.0353 0.0032 0.0188 -0.009 0.0241 -0.0516 0.0407 0.0025 0.0028 -0.0244 0.0028 0.0656 0.0199 0.0007 ⊕0.0246 0.0398 0.0515 ⊕0.0271 0.0101 ⊕0.0024

Standard Error 0.0112 0.0012 0.0007 0.0161 0.0044 0.0051 0.0028 0.0035 0.0029 0.0052 0.004 0.0031 0.0024 0.001 0.0034 0.0009 0.0038 0.0016 0.0012 0.0043 0.0038 0.0045 0.0042 0.0051 0.0001

b/St.Er. 92.616 -4.994 -3.029 -4.739 -1.933 -6.963 -1.162 -5.454 -3.121 -4.659 -12.818 13.229 1.062 2.78 -7.267 2.973 -17.247 -12.717 -0.624 5.689 10.408 11.542 6.511 1.997 18.356

P of |Z| > z 0 0 0.0025 0 0.0532 0 0.2451 0 0.0018 0 0 0 0.2884 0.0054 0 0.0029 0 0 0.5324 0 0 0 0 0.0458 0

Mean of X 4.4055 13.9801 7.9597 0.075 0.7525 0.6012 2.7309 1.1722 1.5339 0.3655 1.0782 3.2446 0.662 2.9799 1.6916 6.9568 1.7373 6.0276 10.9689 3.6183 2.0572 0.8631 0.4756 4.4393 274.983

190

E.1. In-sample estimation SCF1998

Setting 2a - WLS,SCF1998 (cont.) Classification table Predicted 0 1 185 117 273 3730 458 3847 Percent correct: 90.94% Error Distance: 264’654 Actual 0 1

302 4003 4305

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Table E.6: Setting 2a - Results for Tobit in-sample-regression: The dependent variable is binomial, investors without assets are to be separated from investors. Dataset: SCF 1998. Observations = 4305, Parameters = 25, Deg.Fr.= 4280 Log-L = -618.6330, Akaike Info. Crt. = 0.299 Variable

Coeff.

Standard Error Primary Index Equation for Model Constant 0.8193 0.0226 X101 -0.0089 0.0033 X1706 0.0033 0.0024 X3006 1 -0.1106 0.027 0.0113 0.0109 X3006 5 -0.0485 0.0115 X3008 1 0.006 0.0087 X3008 45 0.0256 0.0096 X301 1 -0.014 0.0091 X301 3 0.0327 0.0162 X3014 1 -0.0701 0.0096 X3014 4 X432 1 0.0549 0.0092 X5608 0.0037 0.0071 X5821 0.004 0.0028 -0.035 0.0086 X5825 3 X5905 0.0037 0.0034 X7131 0.0906 0.0088 X7186 0.0281 0.0046 X7187 0.0011 0.0038 ⊕0.034 0.0115 X7372 1 0.0521 0.0107 X7401 1 X7401 2 0.0725 0.0122 0.0349 0.0146 X7401 5 0.0173 0.0123 X8021 1 X8022 ⊕0.0033 0.0003 Disturbance Standard Deviation Sigma 0.2487 0.0029

b/St.Er.

P of |Z| > z

Mean of X

36.244 -2.697 -1.363 -4.09 -1.037 -4.229 -0.685 -2.67 -1.535 -2.014 -7.299 5.941 0.521 1.402 -4.047 1.074 -10.261 -6.057 -0.277 2.971 4.891 5.944 2.387 1.408 10.743

0 0.007 0.1728 0 0.2997 0 0.4933 0.0076 0.1247 0.044 0 0 0.6027 0.1608 0.0001 0.2828 0 0 0.7818 0.003 0 0 0.017 0.1592 0

2.6492 1.2393 0.0207 0.1554 0.1619 0.4476 0.2434 0.2767 0.0609 0.3034 0.5094 0.1131 0.5422 0.3933 1.121 0.3654 1.1779 1.9331 0.5988 0.3443 0.1466 0.0904 0.7807 49.8404

86.836

0

192

E.1. In-sample estimation SCF1998

Setting 2a - Tobit, SCF1998 (cont.) Classification table Predicted 0 1 85 217 89 3914 174 4131 Percent correct: 92.89% Error Distance: 165’030

Actual 0 1

302 4003 4305

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Table E.7: Setting 2a - Results for Binomial Logit model in-sampleregression: The dependent variable (stock ratio) is discrete, 6 risk classes are considered, Dataset: SCF 1998. Observations = 4305, Parameters = 25, Deg.Fr.= 4280 Log-L = -678.8651, Akaike Info. Crt. = 0.327 Variable

Coeff.

Standard b/St.Er. Error Characteristics in numerator of Prob[Y = 1] Constant 1.148 0.3993 2.875 X101 -0.0361 0.0524 -0.688 X1706 0.4498 0.1573 2.859 X3006 1 -1.7036 0.4087 -4.169 0.1143 0.1942 -0.589 X3006 5 -0.4435 0.1662 -2.668 X3008 1 0.1799 -0.224 X3008 45 0.0403 0.3784 0.1687 -2.243 X301 1 -0.1418 0.1777 -0.798 X301 3 0.9157 0.3399 -2.694 X3014 1 -0.7956 0.1649 -4.826 X3014 4 X432 1 1.3092 0.2555 5.125 X5608 0.1844 0.2168 0.851 X5821 0.2236 0.1021 2.19 -0.4924 0.1562 -3.151 X5825 3 X5905 0.4385 0.1413 3.103 X7131 1.0615 0.1569 -6.765 X7186 0.5137 0.1042 -4.93 X7187 0.4369 0.1422 3.073 ⊕0.4644 0.2008 2.312 X7372 1 1.2243 0.3244 3.774 X7401 1 X7401 2 1.1555 0.2562 4.51 ⊕0.2235 0.2079 1.075 X7401 5 0.204 0.1846 1.105 X8021 1 X8022 ⊕0.0405 0.005 8.188

P of |Z| > z

Mean of X

0.004 0.4915 0.0042 0 0.5562 0.0076 0.8228 0.0249 0.4251 0.0071 0 0 0.3949 0.0285 0.0016 0.0019 0 0 0.0021 0.0208 0.0002 0 0.2823 0.2691 0

2.6492 1.2393 0.0207 0.1554 0.1619 0.4476 0.2434 0.2767 0.0609 0.3034 0.5094 0.1131 0.5422 0.3933 1.121 0.3654 1.1779 1.9331 0.5988 0.3443 0.1466 0.0904 0.7807 49.8404

194

E.1. In-sample estimation SCF1998

Table E.8: Setting 2a Binomial Logit model, SCF1998 (cont.) Classification table Predicted 0 1 69 233 302 43 3960 4003 112 4193 4305 Percent correct: 93.59% Error Distance: 168’414

Actual 0 1

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Table E.9: Setting 2b - Results for OLS in-sample-regression: The dependent variable is continuous: the stock ratio is given as a percentage. Dataset: SCF 1998. Observations = 4003, Parameters = 25, Deg.Fr.= 3978 Adjusted R-squared = 0.23837 , Log-L = -547.2792 Akaike Info. Crt. = 0.286 Variable

Coeff.

Constant X101 X1706 X3006 1 X3006 5 X3008 1 X3008 45 X301 1 X301 3 X3014 1 X3014 4 X432 1 X5608 X5821 X5825 3 X5905 X7131 X7186 X7187 X7372 1 X7401 1 X7401 2 X7401 5 X8021 1 X8022

0.056 0 0.0168 -0.0064 0.0011 -0.0162 0.0584 0.0125 -0.009 0.0302 -0.1085 0.069 0.0117 0.0073 -0.0569 0.022 0.0185 0.0046 0.0195 0.0188 0.0148 0.0082 -0.0283 -0.0063 ⊕0.0012

Standard Error 0.0266 0.0039 0.0027 0.0328 0.0127 0.0137 0.01 0.0112 0.0105 0.0187 0.0112 0.0105 0.008 0.0032 0.01 0.0039 0.0104 0.0053 0.0043 0.0134 0.0121 0.014 0.0174 0.0146 0.0004

b/St.Er. 2.106 -0.012 6.197 -0.194 -0.083 -1.182 5.824 1.122 -0.852 1.617 -9.669 6.555 1.454 2.291 -5.676 5.694 -1.787 -0.87 4.518 1.398 1.221 0.587 -1.629 -0.427 3.317

P of |Z| > z 0.0352 0.9907 0 0.8458 0.9337 0.2371 0 0.2618 0.394 0.1059 0 0 0.1461 0.022 0 0 0.0739 0.3841 0 0.1622 0.2221 0.557 0.1034 0.6691 0.0009

Mean of X 2.6453 1.3283 0.0187 0.1531 0.1461 0.4634 0.2343 0.277 0.0615 0.273 0.5428 0.1182 0.5723 0.3717 1.1974 0.3417 1.1511 1.9928 0.6233 0.367 0.1516 0.0859 0.7974 50.4217

196

E.1. In-sample estimation SCF1998

Table E.10: Setting 2b OLS, SCF1998 (cont.) Classification table Predicted 0 1 2 3 581 1233 396 36 10 95 200 44 7 94 183 56 6 90 204 64 6 77 179 67 8 92 195 80 618 1681 1357 347 Percent correct: 23.06% Error Distance: 10’113’838 Actual 0 1 2 3 4 5

4 0 0 0 0 0 0 0

5 0 0 0 0 0 0 0

2246 349 340 364 329 375 4003

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Table E.11: Setting 2b - Results for the Ordered model in-sampleregression: The dependent variable (stock ratio) is discrete, 6 risk classes are considered. Dataset: SCF 1998. Observations = 4003, Parameters = 25, Deg.Fr.= 3978 Log-L = -8845.996, Akaike Info. Crt. = 4.432 Variable

Coeff.

Standard Error Index function for probability Constant -0.6315 0.0616 X101 -0.0352 0.0093 X1706 0.1158 0.0053 ⊕0.1426 0.0714 X3006 1 0.0946 0.0284 X3006 5 -0.3239 0.0364 X3008 1 0.5339 0.0211 X3008 45 X301 1 0.1289 0.0241 -0.0284 0.0229 X301 3 0.2289 0.0367 X3014 1 -1.6068 0.0342 X3014 4 X432 1 0.7888 0.0222 X5608 0.105 0.0179 X5821 0.0628 0.0061 -0.5703 0.0227 X5825 3 X5905 0.2036 0.0076 X7131 0.1596 0.0234 X7186 0.1187 0.0114 X7187 0.1644 0.0085 ⊕0.2639 0.0301 X7372 1 0.1844 0.0252 X7401 1 0.1923 0.0305 X7401 2 -0.4341 0.0492 X7401 5 -0.0272 0.034 X8021 1 X8022 ⊕0.0149 0.0009

b/St.Er.

P of |Z| > z

Mean of X

-10.25 -3.781 22.032 1.998 -3.331 -8.89 25.33 5.342 -1.236 6.229 -46.95 35.577 5.878 10.276 -25.173 26.836 -6.831 -10.394 19.236 8.777 7.321 6.303 -8.816 -0.801 17.282

0 0.0002 0 0.0457 0.0009 0 0 0 0.2163 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0.423 0

2.6453 1.3283 0.0187 0.1531 0.1461 0.4634 0.2343 0.277 0.0615 0.273 0.5428 0.1182 0.5723 0.3717 1.1974 0.3417 1.1511 1.9928 0.6233 0.367 0.1516 0.0859 0.7974 50.4217

198

E.1. In-sample estimation SCF1998

Setting 2b - Ordered model, SCF1998 (cont.) Variable

Coeff.

Standard b/St.Er. P of Mean Error |Z| > z of X Threshold parameters for index Mu(1) 2.0000 .... (Fixed Parameter) .... Mu(2) 3.0000 .... (Fixed Parameter) .... Mu(3) 4.0000 .... (Fixed Parameter) .... Mu(4) 5.0000 .... (Fixed Parameter) ....

Classification table Predicted 0 1 2 3 1203 907 94 41 35 192 70 47 31 159 70 73 21 175 90 76 25 141 80 77 31 158 92 79 1346 1732 496 393 Percent correct: 38.67% Error Distance: 6’395’874 Actual 0 1 2 3 4 5

4 1 5 7 2 6 14 35

5 0 0 0 0 0 1 1

2246 349 340 364 329 375 4003

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Table E.12: Setting 2b - Results for Multinomial Logit model, SCF1998, in-sample-regression. The standard errors of the regression coefficients are given in brackets. The significance levels are abbreviated with asterisks: ‘*’ and ‘**’ are significant at the 5, and 1 percent level Observations = 4003, Parameters = 25, Deg.Fr.= 3978 Log-L = -4672.588, Akaike Info. Crt. = 2.347

Constant X101 X1706 X3006 1 X3006 5 X3008 1 X3008 45 X301 1 X301 3 X3014 1 X3014 4 X432 1 X5608

Prob[Y=1] -4.6093 ** (0.4286) -0.0214 (0.0586) 0.091 ** (0.0342) 0.0582 (0.4215) -0.1806 (0.1886) -0.1103 (0.218) 0.3361 * (0.1374) -0.1164 (0.1673) 0.1787 (0.1428) 0.0221 (0.2557) -0.7457 ** (0.1727) 1.0095 ** (0.1547) 0.0498 (0.114)

Prob[Y=2] -3.9979 ** (0.4276) -0.09 (0.0612) 0.1446 ** (0.035) 0.3789 (0.4147) -0.218 (0.192) 0.0279 (0.227) 0.6017 ** (0.1438) -0.0795 (0.1717) 0.2211 (0.1457) 0.1506 (0.2435) -1.5474 ** (0.2244) 0.7332 ** (0.1547) 0.191 (0.1009)

Prob[Y=3] -3.4461 ** (0.4057) -0.0951 (0.0605) 0.0802 * (0.0342) 0.3096 (0.415) 0.2124 (0.1712) -0.3059 (0.2436) 0.5907 ** (0.1392) -0.0235 (0.161) 0.0062 (0.1475) 0.0127 (0.249) -1.6193 ** (0.221) 0.7698 ** (0.1533) -0.0134 (0.1201)

Prob[Y=4] -4.199 ** (0.4374) 0.0266 (0.059) 0.1276 ** (0.0349) 0.0295 (0.4671) -0.1258 (0.19) -0.4935 (0.2713) 0.599 ** (0.145) 0.1481 (0.1634) 0.0304 (0.1549) 0.2103 (0.2403) -1.5441 ** (0.2357) 0.7671 ** (0.1595) 0.137 (0.1078)

Prob[Y=5] -3.6678 ** (0.4099) -0.0225 (0.057) 0.1762 ** (0.0337) 0.0746 (0.4449) -0.116 (0.1796) -0.2814 (0.2404) 0.5992 ** (0.1384) 0.1408 (0.1532) -0.1639 (0.1508) 0.3342 (0.2248) -1.3724 ** (0.2106) 0.7669 ** (0.1504) 0.1672 (0.102)

200

E.1. In-sample estimation SCF1998

Setting 2b - Results for Multinomial Logit model, SCF1998, in-sample-regression. (cont.)

X5821 X5825 3 X5905 X7131 X7186 X7187 X7372 1 X7401 1 X7401 2 X7401 5 X8021 1 X8022

Prob[Y=1] 0.0589 (0.0427) -0.3604 * (0.1458) 0.1965 ** (0.0497) -0.132 (0.147) -0.0613 (0.0736) 0.2334 ** (0.0555) 0.2264 (0.1935) 0.4369 ** (0.1678) 0.379 (0.2061) 0.0155 (0.3034) 0.3537 (0.2382) 0.0201 ** (0.0055)

Prob[Y=2] 0.0789 (0.0418) -0.2729 (0.1501) 0.2108 ** (0.0509) -0.1712 (0.1525) -0.2988 ** (0.0747) 0.2195 ** (0.0575) 0.5349 * (0.2076) 0.3134 (0.1717) 0.4122 * (0.2067) 0.2523 (0.2959) -0.0851 (0.2428) 0.0193 ** (0.0058)

Prob[Y=3] 0.0791 (0.0408) -0.6335 ** (0.1541) 0.2571 ** (0.0495) -0.0235 (0.146) -0.2385 ** (0.0732) 0.3058 ** (0.0556) 0.3618 (0.2007) 0.0548 (0.1638) 0.0868 (0.205) -0.3524 (0.3428) -0.1355 (0.2282) 0.0147 ** (0.0055)

Prob[Y=4] 0.0522 (0.0424) -0.8423 ** (0.1686) 0.2817 ** (0.0515) -0.2657 (0.1549) -0.0244 (0.0771) 0.1953 ** (0.0578) 0.1191 (0.2033) 0.1195 (0.1725) 0.423 * (0.2036) -0.3928 (0.3749) 0.047 (0.242) 0.02 ** (0.0059)

Prob[Y=5] 0.096 * (0.0395) -0.5252 ** (0.1525) 0.1626 ** (0.0492) -0.2947 * (0.1483) -0.1103 (0.0728) 0.2369 ** (0.0553) 0.3319 (0.1972) 0.2916 (0.1616) 0.068 (0.2068) -0.3254 (0.3302) -0.078 (0.2318) 0.0136 * (0.0056)

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Setting 2b - Multinomial model, SCF1998 (cont.) Classification table Predicted 0 1 2 3 2142 10 10 30 250 10 17 24 210 4 30 36 217 6 17 65 188 7 13 43 218 8 25 38 3225 45 112 236 Percent correct: 58.41% Error Distance: 10’996’615 Actual 0 1 2 3 4 5

4 22 17 24 17 31 26 137

5 32 31 36 42 47 60 248

2246 349 340 364 329 375 4003

202

E.1. In-sample estimation SCF1998

Table E.13: Setting 3b - Results for WLS in-sample-regression: The dependent variable is binomial, investors without stocks are to be separated from stock-owning investors. Dataset: SCF 1998. Observations = 4003, Parameters = 25, Deg.Fr.= 3978 Adjusted R-squared = 0.35424 , Log-L = -1987.9069 Akaike Info. Crt. = 1.006 Variable WConstant WX101 WX1706 WX3006 1 WX3006 5 WX3008 1 WX3008 45 WX301 1 WX301 3 WX3014 1 WX3014 4 WX432 1 WX5608 WX5821 WX5825 3 WX5905 WX7131 WX7186 WX7187 WX7372 1 WX7401 1 WX7401 2 WX7401 5 WX8021 1 WX8022

Coeff. 3.1433 -0.0021 0.0086 ⊕0.1686 0.0073 -0.0499 0.048 0.0063 -0.0041 0.1861 -0.1774 0.1908 0.0275 0.02 -0.059 0.0745 0.0056 0.0155 0.0032 ⊕0.1386 0.0636 0.0153 -0.034 -0.0781 ⊕0.0006

Standard Error 0.4611 0.0043 0.0042 0.0373 0.0155 0.0136 0.014 0.013 0.0125 0.0246 0.0154 0.018 0.0082 0.0034 0.0124 0.0057 0.0122 0.0069 0.0056 0.0152 0.0155 0.0166 0.0187 0.0162 0.0004

b/St.Er. 6.817 -0.477 2.05 4.519 0.473 -3.662 3.431 0.489 -0.325 7.551 -11.543 10.613 3.356 5.816 -4.756 13.077 -0.463 -2.256 0.576 9.134 -4.099 0.921 -1.819 -4.821 1.378

P of |Z| > z 0 0.6331 0.0404 0 0.6364 0.0002 0.0006 0.6247 0.7449 0 0 0 0.0008 0 0 0 0.6433 0.0241 0.5647 0 0 0.357 0.0688 0 0.1683

Mean of X 0.2056 7.929 4.2415 0.0577 0.4499 0.5235 1.3145 0.7149 0.8616 0.1654 1.105 1.4616 0.379 1.74 1.189 3.4631 1.0873 3.5573 6.1949 1.7951 1.0423 0.4486 0.2903 2.3232 151.6105

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203

Setting 3b - WLS, SCF1998 (cont.) Classification table Predicted 0 1 1805 441 575 1182 2380 1623 Percent correct: 74.62% Error Distance: 1,575,318 Actual 0 1

2246 1757 4003

204

E.1. In-sample estimation SCF1998

Table E.14: Setting 3b - Results for Tobit in-sample-regression: The dependent variable is binomial, investors without stocks are to be separated from stock-owning investors. Dataset: SCF 1998. Observations = 4003, Parameters = 25, Deg.Fr.= 3978 Log-L = -3173.196, Akaike Info. Crt. = 1.598 Variable

Coeff.

Standard Error Primary Index Equation for Model Constant -0.745 0.0925 X101 -0.0166 0.0131 X1706 0.0337 0.0079 X3006 1 ⊕0.0595 0.0985 0.0315 0.0413 X3006 5 -0.1004 0.0495 X3008 1 0.1978 0.0318 X3008 45 0.0006 0.0363 X301 1 ⊕0.022 0.0334 X301 3 0.0426 0.0555 X3014 1 -0.5708 0.0415 X3014 4 X432 1 0.365 0.0343 X5608 0.0236 0.0236 X5821 0.0241 0.0094 -0.2178 0.0335 X5825 3 X5905 0.075 0.0116 X7131 0.0822 0.0337 X7186 0.0548 0.0167 X7187 0.0606 0.0128 0.1194 0.0444 X7372 1 0.1193 0.0379 X7401 1 X7401 2 0.1229 0.0457 -0.0802 0.0641 X7401 5 0.0442 0.0507 X8021 1 X8022 ⊕0.0076 0.0012 Disturbance Standard Deviation Sigma 0.7558 0.0146

b/St.Er.

P of |Z| > z

Mean of X

-8.057 -1.266 4.268 0.604 -0.763 -2.029 6.224 0.017 0.658 0.768 -13.75 10.645 1 2.562 -6.502 6.452 -2.443 -3.278 4.743 2.691 3.145 2.688 -1.251 0.873 6.154

0 0.2055 0 0.5458 0.4456 0.0425 0 0.9861 0.5105 0.4427 0 0 0.3174 0.0104 0 0 0.0146 0.001 0 0.0071 0.0017 0.0072 0.211 0.3828 0

2.6453 1.3283 0.0187 0.1531 0.1461 0.4634 0.2343 0.277 0.0615 0.273 0.5428 0.1182 0.5723 0.3717 1.1974 0.3417 1.1511 1.9928 0.6233 0.367 0.1516 0.0859 0.7974 50.4217

51.614

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Setting 3b - TOBIT, SCF1998 (cont.) Classification table Predicted 0 1 2124 122 1033 724 3157 846 Percent correct: 71.15% Error Distance: 3’245’919 Actual 0 1

2246 1757 4003

206

E.1. In-sample estimation SCF1998

Table E.15: Setting 3b - Results for Multinomial Logit model in-sampleregression: The dependent variable is binomial, investors without stocks are to be separated from stock-owning investors. Dataset: SCF 1998. Observations = 4003, Parameters = 25, Deg.Fr.= 3978 Log-L = -1906.335, Akaike Info. Crt. = 0.965 Variable

Coeff.

Standard b/St.Er. Error Characteristics in numerator of Prob[Y = 1] Constant -2.3607 0.2484 -9.505 X101 -0.0399 0.0367 -1.088 X1706 0.1237 0.0237 5.213 X3006 1 ⊕0.1732 0.2963 0.585 0.08 0.1173 -0.682 X3006 5 -0.2148 0.1352 -1.588 X3008 1 0.5408 0.0877 6.165 X3008 45 0.02 0.102 0.196 X301 1 ⊕0.0619 0.0963 0.643 X301 3 0.1546 0.1637 0.944 X3014 1 -1.3125 0.1115 -11.766 X3014 4 X432 1 0.8141 0.0893 9.12 X5608 0.1137 0.0803 1.416 X5821 0.0742 0.029 2.556 -0.5144 0.0896 -5.744 X5825 3 X5905 0.219 0.0334 6.566 X7131 0.1738 0.0977 -1.778 X7186 0.147 0.0478 -3.077 X7187 0.2395 0.0399 5.999 ⊕0.313 0.1221 2.564 X7372 1 0.2468 0.1073 2.301 X7401 1 X7401 2 0.2726 0.1269 2.148 -0.1386 0.1714 -0.808 X7401 5 0.0233 0.1372 0.17 X8021 1 X8022 ⊕0.0175 0.0034 5.118

P of |Z| > z

Mean of X

0 0.2768 0 0.5588 0.4953 0.1122 0 0.8443 0.52 0.3451 0 0 0.1568 0.0106 0 0 0.0754 0.0021 0 0.0104 0.0214 0.0317 0.4188 0.8653 0

2.6453 1.3283 0.0187 0.1531 0.1461 0.4634 0.2343 0.277 0.0615 0.273 0.5428 0.1182 0.5723 0.3717 1.1974 0.3417 1.1511 1.9928 0.6233 0.367 0.1516 0.0859 0.7974 50.4217

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Table E.16: Setting 3b Multinomial Logit model, SCF1998 (cont.) Classification table Predicted 0 1 1831 415 2246 488 1269 1757 2319 1684 4003 Percent correct: 77.44% Error Distance: 1’231’107 Actual 0 1

208

E.1. In-sample estimation SCF1998

Table E.17: Setting 3c - Results for OLS in-sample-regression: The dependent variable is continuous: the stock ratio is given as a percentage. Dataset: SCF 1995. Observations = 1757, Parameters = 25, Deg.Fr.= 1732 Adjusted R-squared = 0.01430 , Log-L = -330.9550 Akaike Info. Crt. = 0.405 Variable

Coeff.

Constant X101 X1706 X3006 1 X3006 5 X3008 1 X3008 45 X301 1 X301 3 X3014 1 X3014 4 X432 1 X5608 X5821 X5825 3 X5905 X7131 X7186 X7187 X7372 1 X7401 1 X7401 2 X7401 5 X8021 1 X8022

0.5703 ⊕0.0059 0.0069 -0.0211 0.008 -0.0306 0.0178 0.0293 -0.0367 0.0284 -0.0657 0.0186 0.0052 0.0012 -0.0398 0.0001 0.0194 0.0102 0.0012 -0.0132 0.0231 0.0356 -0.0602 -0.0273 -0.0004

Standard Error 0.0508 0.007 0.0036 0.0457 0.0209 0.0286 0.0161 0.0185 0.0165 0.026 0.0257 0.0186 0.0104 0.0043 0.0182 0.0056 0.0168 0.0082 0.0058 0.0234 0.0189 0.0238 0.0405 0.0282 0.0007

b/St.Er. 11.234 0.841 1.913 -0.461 0.383 -1.072 1.103 1.578 -2.228 1.092 -2.561 -0.999 0.5 0.27 -2.191 0.01 -1.153 1.248 -0.211 -0.564 -1.224 -1.495 -1.488 -0.969 -0.633

P of |Z| > z 0 0.4003 0.0557 0.6447 0.7015 0.2839 0.2701 0.1145 0.0259 0.2747 0.0104 0.318 0.6171 0.7874 0.0284 0.9924 0.2491 0.212 0.8328 0.5729 0.2208 0.135 0.1367 0.3327 0.527

Mean of X 2.6693 2.2106 0.025 0.1485 0.0768 0.6181 0.2089 0.2977 0.0825 0.0916 0.7718 0.1628 0.7803 0.218 1.8036 0.3204 1.1013 2.5726 0.7513 0.5179 0.1491 0.0364 0.8839 53.2692

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Table E.18: Setting 3c OLS, SCF1998 (cont.) Classification table Predicted 0 1 2 0 67 282 0 53 287 0 44 320 0 24 305 0 30 344 0 218 1538 Percent correct: 21.23% Error Distance: 2’629’739

Actual 0 1 2 3 4

3 0 0 0 0 1 1

4 0 0 0 0 0 0

349 340 364 329 375 1757

210

E.1. In-sample estimation SCF1998

Table E.19: Setting 3c - Results for Ordered model in-sample-regression: The dependent variable (stock ratio) is discrete, 5 risk classes are considered. Dataset: SCF 1998. Observations = 1757, Parameters = 25, Deg.Fr.= 1732 Log-L = -3899.910, Akaike Info. Crt. = 4.468 Variable

Coeff.

Standard Error Index function for probability Constant 2.7241 0.1057 X101 ⊕0.0237 0.0146 X1706 0.0403 0.0077 -0.0477 0.1028 X3006 1 0.0655 0.0452 X3006 5 -0.2076 0.0574 X3008 1 0.1497 0.0334 X3008 45 X301 1 0.1761 0.039 -0.2276 0.0342 X301 3 0.1651 0.0541 X3014 1 -0.4895 0.0493 X3014 4 X432 1 0.1091 0.0388 X5608 0.042 0.022 X5821 0.015 0.009 -0.2383 0.0372 X5825 3 X5905 ⊕0.002 0.0119 X7131 0.1105 0.0356 X7186 0.0266 0.0173 X7187 0.0029 0.0125 -0.0406 0.0483 X7372 1 0.1355 0.039 X7401 1 0.1559 0.0497 X7401 2 -0.2943 0.0824 X7401 5 -0.2107 0.0599 X8021 1 X8022 -0.003 0.0014

b/St.Er.

P of |Z| > z

Mean of X

25.778 1.628 5.261 -0.464 1.449 -3.616 4.477 4.517 -6.648 3.052 -9.928 -2.81 1.904 1.663 -6.398 -0.164 -3.104 1.537 -0.229 -0.841 -3.475 -3.14 -3.573 -3.515 -2.129

0 0.1035 0 0.6424 0.1474 0.0003 0 0 0 0.0023 0 0.0049 0.057 0.0963 0 0.8693 0.0019 0.1242 0.819 0.4005 0.0005 0.0017 0.0004 0.0004 0.0333

2.6693 2.2106 0.025 0.1485 0.0768 0.6181 0.2089 0.2977 0.0825 0.0916 0.7718 0.1628 0.7803 0.218 1.8036 0.3204 1.1013 2.5726 0.7513 0.5179 0.1491 0.0364 0.8839 53.2692

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Setting 3c - Ordered model, SCF1998 (cont.) Variable

Coeff.

Standard b/St.Er. P of Mean Error |Z| > z of X Threshold parameters for index Mu(1) 2.0000 .... (Fixed Parameter) .... Mu(2) 3.0000 .... (Fixed Parameter) .... Mu(3) 4.0000 .... (Fixed Parameter) ....

Classification table Predicted 0 1 2 0 200 149 0 164 174 0 152 212 0 142 186 0 152 221 0 810 942 Percent correct: 21.46% Error Distance: 2’147’852

Actual 0 1 2 3 4

3 0 2 0 1 2 5

4 0 0 0 0 0 0

349 340 364 329 375 1757

212

E.1. In-sample estimation SCF1998

Table E.20: Setting 3c - Results for Multinomial Logit model in-sampleregression: The dependent variable (stock ratio) is discrete, 5 risk classes are considered. Dataset: SCF 1998. The standard errors of the regression coefficients are given in brackets. The significance levels are abbreviated with asterisks: ‘*’ and ‘**’ are significant at the 5, and 1 percent level Observations = 1757, Parameters = 25, Deg.Fr.= 1732 Log-L = -2766.666, Akaike Info. Crt. = 3.178

Constant X101 X1706 X3006 1 X3006 5 X3008 1 X3008 45 X301 1 X301 3 X3014 1 X3014 4 X432 1 X5608

Prob[Y=1] 0.5443 (0.5644) -0.0661 (0.0778) 0.0533 (0.0397) 0.2976 (0.4793) -0.0398 (0.2399) 0.1434 (0.2881) 0.27 (0.1759) 0.0252 (0.2144) 0.0433 (0.1746) 0.1362 (0.2957) -0.7874 ** (0.2693) -0.2777 (0.2047) 0.1354 (0.1153)

Prob[Y=2] 1.2078 * (0.5505) -0.0757 (0.0773) -0.0095 (0.0388) 0.2311 (0.4807) 0.4154 (0.2236) -0.1947 (0.3019) 0.2487 (0.1724) 0.0705 (0.2057) -0.1725 (0.1756) -0.0216 (0.3) -0.8552 ** (0.2657) -0.2365 (0.204) -0.074 (0.1334)

Prob[Y=3] 0.3806 (0.5734) 0.0513 (0.0761) 0.0366 (0.0395) -0.0454 (0.5252) 0.0404 (0.2378) -0.381 (0.3249) 0.2437 (0.1769) 0.2506 (0.207) -0.1572 (0.1814) 0.1539 (0.2921) -0.8216 ** (0.2779) -0.2182 (0.2087) 0.0864 (0.1212)

Prob[Y=4] 0.9173 (0.5536) 0.0004 (0.0742) 0.0853 * (0.0386) 0.014 (0.5065) 0.0716 (0.2296) -0.1634 (0.3001) 0.2603 (0.1719) 0.2371 (0.1991) -0.3403 (0.178) 0.3037 (0.2806) -0.6423 * (0.2575) -0.2084 (0.2017) 0.1138 (0.1163)

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Setting 3c - Results for Multinomial Logit model, SCF1998, in-sample-regression. (cont.)

X5821 X5825 3 X5905 X7131 X7186 X7187 X7372 1 X7401 1 X7401 2 X7401 5 X8021 1 X8022

Prob[Y=1] 0.0243 (0.0485) 0.0729 (0.1895) 0.0114 (0.062) -0.0444 (0.1852) -0.2312 * (0.09) -0.014 (0.0642) 0.2906 (0.2589) -0.1378 (0.2115) 0.0234 (0.2621) 0.2715 (0.4052) -0.4068 (0.3165) 0.0002 (0.0074)

Prob[Y=2] 0.0246 (0.0477) -0.2624 (0.1929) 0.057 (0.0606) 0.0982 (0.1796) -0.1709 (0.0887) 0.0702 (0.0626) 0.1277 (0.2525) -0.4232 * (0.205) -0.339 (0.2614) -0.3555 (0.4421) -0.4663 (0.3048) -0.0059 (0.0072)

Prob[Y=3] -0.0045 (0.0491) -0.4859 * (0.2049) 0.0836 (0.0623) -0.1593 (0.1866) 0.042 (0.092) -0.0421 (0.0644) -0.1301 (0.2542) -0.3301 (0.2116) 0.0466 (0.2595) -0.3637 (0.4681) -0.2715 (0.3154) 0.0005 (0.0075)

Prob[Y=4] 0.0408 (0.0465) -0.1802 (0.1921) -0.0428 (0.0603) -0.1743 (0.1813) -0.0468 (0.0885) 0.0011 (0.0621) 0.0892 (0.2498) -0.1573 (0.2035) -0.3199 (0.2628) -0.2819 (0.4336) -0.4213 (0.3088) -0.0058 (0.0073)

214

E.1. In-sample estimation SCF1998

Setting 3c - Multinomial model, SCF1998 (cont.) Classification table Predicted 0 1 2 97 48 74 56 82 76 61 46 128 47 38 88 59 59 88 320 273 454 Percent correct: 27.95% Error Distance: 1’803’590

Actual 0 1 2 3 4

3 38 45 40 61 46 230

4 92 81 89 95 123 480

349 340 364 329 375 1757

Chapter E. Empirical Results of Regressions

E.2

215

Out-of-sample estimation SCF 1995 in SCF1998

216

E.2. Out-of-sample estimation SCF 1995 in SCF1998

Table E.21: Setting 1 - Results for OLS in-sample-regression: The dependent variable is continuous: the stock ratio is given as a percentage. Dataset: SCF 1995. Observations = 4299, Parameters = 25, Deg.Fr.= 4274 Adjusted R-squared = 0.29850, Log-L = -1558.5730 Akaike Info. Crt. = 0.737 Variable

Coeff.

Constant X101 X1706 X3006 1 X3006 5 X3008 1 X3008 45 X301 1 X301 3 X3014 1 X3014 4 X432 1 X5608 X5821 X5825 3 X5905 X7131 X7186 X7187 X7372 1 X7401 1 X7401 2 X7401 5 X8021 1 X8022

-0.1925 -0.0222 0.0093 -0.0341 0.0293 -0.0412 0.012 0.0436 -0.0616 0.0014 -0.1332 0.1218 0.0007 0.0088 ⊕0.0335 0.021 0.111 0.0799 0.015 ⊕0.0964 0.0485 0.093 ⊕0.0408 0.0191 ⊕0.0044

Standard Error 0.0313 0.0048 0.0034 0.0351 0.0161 0.0166 0.0125 0.0125 0.0139 0.0247 0.0129 0.0129 0.0054 0.0041 0.0124 0.0037 0.0125 0.012 0.0053 0.0164 0.0153 0.0164 0.0196 0.0178 0.0004

b/St.Er. -6.157 -4.661 2.747 -0.972 1.823 -2.489 0.959 -3.489 -4.417 -0.055 -10.291 9.428 -0.136 2.159 2.711 5.733 -8.908 -6.642 2.81 5.884 3.177 5.685 2.079 1.073 10.204

P of |Z| > z 0 0 0.006 0.3312 0.0683 0.0128 0.3375 0.0005 0 0.9563 0 0 0.8921 0.0309 0.0067 0 0 0 0.005 0 0.0015 0 0.0376 0.2833 0

Mean of X 2.6218 1.268 0.024 0.141 0.164 0.5301 0.2901 0.2175 0.0514 0.3438 0.5194 0.2645 0.5459 0.3957 1.4836 0.3559 0.6855 1.9035 0.6215 0.3112 0.1661 0.0989 0.789 49.7704

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217

Setting 1 - OLS Out-of-Sample results SCF1995 estimates in 1998 data Classification table 0 1 2 144 42 98 568 261 784 21 21 152 12 26 124 21 16 136 24 19 122 23 26 137 813 411 1553 Percent correct: 17.19% Error Distance: 11’864’863

Actual 0 1 2 3 4 5 6

Predicted 3 15 476 115 127 134 120 131 1118

4 3 140 36 46 47 34 51 357

5 0 16 4 5 10 9 7 51

6 0 1 0 0 0 1 0 2

302 2246 349 340 364 329 375 4305

218

E.2. Out-of-sample estimation SCF 1995 in SCF1998

Table E.22: Setting 1 - Results for Tobit in-sample-regression: The dependent variable is continuous: the stock ratio is given as a percentage. Dataset: SCF 1995. Observations = 4299, Parameters = 25, Deg.Fr.= 4274 Log-L = -2187.820, Akaike Info. Crt. = 1.029 Variable

Coeff.

Standard Error Primary Index Equation for Model Constant -0.5995 0.0611 X101 -0.0375 0.0094 X1706 0.0224 0.0053 X3006 1 -0.0576 0.0609 0.0836 0.0284 X3006 5 -0.0245 0.0333 X3008 1 0.051 0.0223 X3008 45 0.0533 0.0217 X301 1 -0.0659 0.026 X301 3 0.0434 0.0397 X3014 1 -0.3094 0.0258 X3014 4 X432 1 0.1829 0.0233 X5608 0.0011 0.0089 X5821 0.0128 0.0065 ⊕0.0779 0.0213 X5825 3 X5905 0.0393 0.006 X7131 0.0416 0.0224 X7186 0.0882 0.0211 X7187 0.0407 0.0084 ⊕0.1515 0.0309 X7372 1 0.0252 0.0257 X7401 1 X7401 2 0.0888 0.0292 -0.0165 0.0402 X7401 5 -0.021 0.0352 X8021 1 X8022 ⊕0.0048 0.0008 Disturbance Standard Deviation Sigma 0.4879 0.0098

b/St.Er.

P of |Z| > z

Mean of X

-9.805 -3.989 4.201 -0.946 2.945 -0.736 2.285 -2.453 -2.532 1.094 -11.982 7.856 0.124 1.975 3.664 6.594 -1.855 -4.188 4.82 4.894 0.978 3.044 -0.41 -0.595 5.899

0 0.0001 0 0.3442 0.0032 0.4616 0.0223 0.0142 0.0113 0.274 0 0 0.9011 0.0483 0.0002 0 0.0636 0 0 0 0.3279 0.0023 0.6821 0.5515 0

2.6218 1.268 0.024 0.141 0.164 0.5301 0.2901 0.2175 0.0514 0.3438 0.5194 0.2645 0.5459 0.3957 1.4836 0.3559 0.6855 1.9035 0.6215 0.3112 0.1661 0.0989 0.789 49.7704

49.866

0

Chapter E. Empirical Results of Regressions

219

Setting 1 - Tobit Out-of-Sample results SCF1995 estimates in 1998 data Classification table 0 1 2 245 28 25 1353 306 369 136 59 96 107 57 107 96 68 126 100 51 116 113 71 108 2150 640 947 Percent correct: 16.68% Error Distance: 16’002’731

Actual 0 1 2 3 4 5 6

Predicted 3 4 150 37 43 43 34 50 361

4 0 56 14 18 19 17 19 143

5 0 10 7 7 10 9 14 57

6 0 2 0 1 2 2 0 7

302 2246 349 340 364 329 375 4305

220

E.2. Out-of-sample estimation SCF 1995 in SCF1998

Table E.23: Setting 1 - Results for Ordered model in-sample-regression: The dependent variable (stock ratio) is discrete, 7 risk classes are considered. Dataset: SCF 1995. Observations = 4299, Parameters = 25, Deg.Fr.= 4274 Log-L = -7024.7, Akaike Info. Crt. = 3.280 Variable

Coeff.

Standard Error Index function for probability Constant 0.8007 0.0733 X101 -0.0899 0.0112 X1706 0.0804 0.0066 -0.2374 0.0826 X3006 1 0.2206 0.033 X3006 5 -0.1053 0.0398 X3008 1 0.1326 0.027 X3008 45 X301 1 0.1794 0.0263 -0.2609 0.0323 X301 3 0.11 0.0453 X3014 1 -0.748 0.0314 X3014 4 X432 1 0.5372 0.0279 X5608 0.0125 0.0121 X5821 0.0382 0.0082 ⊕0.2252 0.0261 X5825 3 X5905 0.1231 0.0074 X7131 0.2727 0.0278 X7186 0.3144 0.0261 X7187 0.1228 0.0107 ⊕0.4212 0.038 X7372 1 0.0963 0.0327 X7401 1 0.3082 0.0351 X7401 2 ⊕0.0728 0.045 X7401 5 0.0047 0.0425 X8021 1 X8022 ⊕0.0163 0.001

b/St.Er.

P of |Z| > z

Mean of X

10.93 -8.057 12.114 -2.873 6.688 -2.644 4.915 -6.822 -8.069 2.429 -23.796 19.242 -1.036 4.666 8.641 16.603 -9.819 -12.028 11.506 11.089 2.95 8.78 1.617 0.111 16.18

0 0 0 0.0041 0 0.0082 0 0 0 0.0152 0 0 0.3001 0 0 0 0 0 0 0 0.0032 0 0.1059 0.9113 0

2.6218 1.268 0.024 0.141 0.164 0.5301 0.2901 0.2175 0.0514 0.3438 0.5194 0.2645 0.5459 0.3957 1.4836 0.3559 0.6855 1.9035 0.6215 0.3112 0.1661 0.0989 0.789 49.7704

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221

Setting 1 - Ordered model, SCF1995 (cont.) Variable

Coeff.

Standard b/St.Er. P of Mean Error |Z| > z of X Threshold parameters for index Mu(1) 2.0000 .... (Fixed Parameter) .... Mu(2) 3.0000 .... (Fixed Parameter) .... Mu(3) 4.0000 .... (Fixed Parameter) .... Mu(4) 5.0000 .... (Fixed Parameter) .... Mu(5) 6.0000 .... (Fixed Parameter) ....

Setting 1 - Ordered Logit Out-of-Sample results SCF1995 estimates in 1998 data Classification table 0 1 2 11 218 56 12 1158 578 0 82 123 0 74 95 0 62 109 0 66 93 0 83 110 23 1743 1164 Percent correct: 34.05% Error Distance: 7’455’722

Actual 0 1 2 3 4 5 6

Predicted 3 16 412 115 136 154 132 131 1096

4 1 83 28 33 34 33 49 261

5 0 2 1 2 5 4 2 16

6 0 1 0 0 0 1 0 2

302 2246 349 340 364 329 375 4305

222

E.2. Out-of-sample estimation SCF 1995 in SCF1998

Table E.24: Setting 1 - Results for Multinomial Logit model, SCF1995, in-sample-regression. The dependent variable (stock ratio) is discrete, 7 risk classes are considered. The standard errors of the regression coefficients are given in brackets. The significance levels are abbreviated with asterisks: ‘*’ and ‘**’ are significant at the 5, and 1 percent level Observations = 4299, Parameters = 25, Deg.Fr.= 4274 Log-L = -5024.357, Akaike Info. Crt. = 2.349

Constant X101 X1706 X3006 1 X3006 5 X3008 1 X3008 45 X301 1 X301 3 X3014 1 X3014 4 X432 1 X5608

Prob[Y=1] 1.0963 ** (0.3734) -0.0937 (0.0505) 0.5384 ** (0.1553) -0.3447 (0.4925) -0.0668 (0.1951) -0.2209 (0.1647) 0.065 (0.1656) -0.3757 * (0.1701) -0.4897 ** (0.1571) -0.5208 (0.3491) -0.5975 ** (0.1574) 1.7983 ** (0.2667) 0.0153 (0.1118)

Prob[Y=2] -3.0716 ** (0.5162) -0.2034 ** (0.0743) 0.6883 ** (0.1577) -0.416 (0.5863) -0.0914 (0.2598) -0.1019 (0.2602) 0.5442 ** (0.2094) -0.4185 * (0.2088) -0.2767 (0.2115) -0.5474 (0.413) -1.4758 ** (0.2138) 2.5944 ** (0.2967) 0.0968 (0.1198)

Prob[Y=3] -3.0243 ** (0.5484) -0.1861 * (0.0808) 0.6164 ** (0.1585) -0.4335 (0.623) 0.1636 (0.2682) -0.1342 (0.2808) 0.4095 (0.2199) -0.8029 ** (0.222) -0.6405 ** (0.2296) -0.5143 (0.4342) -1.885 ** (0.2373) 2.6838 ** (0.3084) -0.0116 (0.1272)

Prob[Y=4] -3.0706 ** (0.5744) -0.1856 * (0.0834) 0.6269 ** (0.1589) -0.2968 (0.6331) 0.2864 (0.2708) -0.6221 (0.3294) 0.1509 (0.2235) -0.7054 ** (0.2241) -0.9807 ** (0.2563) -0.5496 (0.4383) -2.1931 ** (0.2786) 2.5184 ** (0.3158) 0.1464 (0.1219)

Prob[Y=5] -3.4945 ** (0.6414) -0.2683 ** (0.0917) 0.6295 ** (0.16) -0.3621 (0.676) 0.2315 (0.2892) -0.285 (0.3514) 0.3362 (0.2432) -0.7643 ** (0.2388) -0.918 ** (0.2766) -0.0659 (0.4354) -1.9919 ** (0.3019) 2.6108 ** (0.33) -0.1374 (0.1435)

Prob[Y=6] -2.6013 ** (0.588) -0.3347 ** (0.0911) 0.7268 ** (0.1597) -1.2552 (0.7823) 0.4626 (0.2792) -0.0229 (0.2947) 0.2963 (0.2352) -0.4791 * (0.2312) -0.6895 ** (0.2558) -0.2015 (0.433) -1.6345 ** (0.2556) 2.0423 ** (0.3141) -0.0165 (0.1329)

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Setting 1 - Results for Multinomial Logit model, SCF1995, in-sample-regression. (cont.)

X5821 X5825 3 X5905 X7131 X7186 X7187 X7372 1 X7401 1 X7401 2 X7401 5 X8021 1 X8022

Prob[Y=1] 0.2739 ** (0.0936) 0.1201 (0.1815) 0.4504 ** (0.1058) -1.1994 ** (0.1464) -0.9935 ** (0.1924) 0.0047 (0.1113) 0.78 ** (0.1907) 1.1321 ** (0.3119) 0.8898 ** (0.2264) 0.4136 * (0.1989) 0.1192 (0.1774) 0.0363 ** (0.0049)

Prob[Y=2] 0.3874 ** (0.0996) 0.4924 * (0.2147) 0.6346 ** (0.1103) -1.0034 ** (0.1916) -1.2032 ** (0.2252) 0.1674 (0.1199) 1.1657 ** (0.2562) 1.3688 ** (0.342) 1.1607 ** (0.283) 0.596 (0.3102) 0.0752 (0.2689) 0.0525 ** (0.0068)

Prob[Y=3] 0.3654 ** (0.102) 0.3695 (0.2248) 0.6055 ** (0.1117) -0.9756 ** (0.2032) -1.2742 ** (0.2345) 0.236 (0.1222) 1.5339 ** (0.2908) 1.149 ** (0.3511) 1.2596 ** (0.2917) 0.1466 (0.3702) -0.4285 (0.301) 0.0574 ** (0.0073)

Prob[Y=4] 0.3479 ** (0.1024) 0.5886 * (0.2318) 0.6948 ** (0.1128) -1.133 ** (0.2127) -1.4634 ** (0.239) 0.3203 ** (0.1233) 1.285 ** (0.2913) 1.0063 ** (0.3561) 1.1629 ** (0.3018) -0.296 (0.4566) 0.0545 (0.3253) 0.0522 ** (0.0077)

Prob[Y=5] 0.3168 ** (0.1051) 0.5704 * (0.2453) 0.6084 ** (0.1146) -1.5036 ** (0.2356) -1.559 ** (0.2508) 0.2381 (0.1274) 1.8232 ** (0.336) 1.5237 ** (0.3738) 1.4387 ** (0.3284) 0.4773 (0.4193) 0.0692 (0.3976) 0.0522 ** (0.0085)

Prob[Y=6] 0.3375 ** (0.1052) 0.3715 (0.2382) 0.5897 ** (0.1138) -1.4245 ** (0.2254) -1.1777 ** (0.2483) 0.2351 (0.126) 1.2313 ** (0.2938) 0.9469 ** (0.3638) 0.9906 ** (0.311) 0.4812 (0.353) 0.1508 (0.3187) 0.0517 ** (0.0079)

224

E.2. Out-of-sample estimation SCF 1995 in SCF1998

Setting 1 - Multinomial Logit Out-of-Sample results SCF1995 estimates in 1998 data Classification table 0 1 2 32 244 0 52 1722 24 1 243 32 0 221 40 0 224 41 0 214 36 1 247 39 86 3115 212 Percent correct: 43.46% Error Distance: 19’807’854

Actual 0 1 2 3 4 5 6

Predicted 3 9 151 30 29 39 26 30 314

4 0 1 1 3 9 4 6 24

5 17 295 40 46 49 46 51 544

6 0 1 2 1 2 3 1 10

302 2246 349 340 364 329 375 4305

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225

Table E.25: Setting 2a - Results for Weighted Least Squares (WLS) in-sample-regression: The dependent variable is binomial, investors without assets are to be separated investors. The portrayed mean has no interpretation, as each variable was weighted by W , the reciprocal of the observations’ standard errors. Dataset: SCF 1995. Observations = 4299, Parameters = 25, Deg.Fr.= 4274 Adjusted R-squared = 0.19860, Log-L = -93.2645 Akaike Info. Crt. = 0.055 Variable WConstant WX101 WX1706 WX3006 1 WX3006 5 WX3008 1 WX3008 45 WX301 1 WX301 3 WX3014 1 WX3014 4 WX432 1 WX5608 WX5821 WX5825 3 WX5905 WX7131 WX7186 WX7187 WX7372 1 WX7401 1 WX7401 2 WX7401 5 WX8021 1 WX8022

Coeff. 1.0627 -0.0108 0.0021 -0.0064 0.0031 -0.0298 0.0012 0.0198 -0.0271 0.0224 -0.0504 0.0632 0.0018 0.0053 ⊕0.0078 0.0062 0.0717 0.0485 0.0066 ⊕0.048 0.0416 0.0599 ⊕0.0364 0.017 ⊕0.0026

Standard Error 0.0144 0.0016 0.0008 0.01 0.005 0.0064 0.0035 0.0036 0.0041 0.0075 0.0042 0.0038 0.0011 0.001 0.0033 0.001 0.0042 0.0036 0.0013 0.0052 0.0042 0.0052 0.0055 0.0067 0.0002

b/St.Er. 73.848 -6.856 -2.659 -0.641 -0.63 -4.69 0.331 -5.534 -6.655 -3.008 -11.86 16.677 1.731 5.239 2.378 6.387 -16.954 -13.577 -5.177 9.213 9.89 11.561 6.663 2.556 16.302

P of |Z| > z 0 0 0.0078 0.5217 0.5288 0 0.7406 0 0 0.0026 0 0 0.0835 0 0.0174 0 0 0 0 0 0 0 0 0.0106 0

Mean of X 3.7764 12.6871 7.6259 0.1169 0.6437 0.5509 2.8431 1.5168 0.9162 0.246 1.2473 3.0855 1.6937 2.9259 2.2083 8.779 1.6508 3.254 10.3022 3.514 1.8178 0.8948 0.4661 4.1698 253.8758

226

E.2. Out-of-sample estimation SCF 1995 in SCF1998

Setting 2a - WLS Out-of-Sample results SCF1995 estimates in 1998 data Classification table Predicted 0 1 Actual 0 31 271 1 60 3943 91 4214 Percent correct: 92.31% Error Distance: 231’123

302 4003 4305

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227

Table E.26: Setting 2a - Results for Multinomial Logit model in-sampleregression: The dependent variable is binomial, Assetholders are separated from people who do not own any financial assets. Dataset: SCF 1995. Observations = 4299, Parameters = 25, Deg.Fr.= 4274 Log-L = -737.6719, Akaike Info. Crt. = 0.355 Variable

Coeff.

Standard b/St.Er. Error Characteristics in numerator of Prob[Y = 1] Constant 1.1679 0.3728 3.133 X101 -0.1039 0.0506 -2.053 X1706 0.565 0.1541 3.666 X3006 1 -0.3656 0.4937 -0.741 0.0447 0.1949 -0.229 X3006 5 -0.2147 0.1648 -1.303 X3008 1 0.1024 0.1656 0.618 X3008 45 0.3944 0.1702 -2.318 X301 1 -0.5039 0.157 -3.21 X301 3 0.5118 0.3464 -1.477 X3014 1 X3014 4 -0.7152 0.1568 -4.562 1.9187 0.2668 7.191 X432 1 X5608 0.0187 0.1115 0.167 X5821 0.2814 0.0934 3.013 ⊕0.1646 0.1811 0.909 X5825 3 X5905 0.4807 0.1054 4.562 X7131 1.1953 0.1464 -8.166 X7186 1.0232 0.1923 -5.32 X7187 0.0428 0.1102 0.388 ⊕0.8307 0.1906 4.358 X7372 1 1.1408 0.3116 3.661 X7401 1 X7401 2 0.9173 0.226 4.059 ⊕0.409 0.1989 2.056 X7401 5 0.1045 0.1773 0.589 X8021 1 X8022 ⊕0.038 0.0049 7.806

P of |Z| > z

Mean of X

0.0017 0.0401 0.0002 0.4589 0.8185 0.1926 0.5363 0.0205 0.0013 0.1396 0 0 0.8672 0.0026 0.3634 0 0 0 0.698 0 0.0003 0 0.0398 0.5558 0

2.6218 1.268 0.024 0.141 0.164 0.5301 0.2901 0.2175 0.0514 0.3438 0.5194 0.2645 0.5459 0.3957 1.4836 0.3559 0.6855 1.9035 0.6215 0.3112 0.1661 0.0989 0.789 49.7704

228

E.2. Out-of-sample estimation SCF 1995 in SCF1998

Table E.27: Setting 2a - Multinomial Logit model Out-of-sample estimation SCF1995 estimates in 1998 data Classification table Predicted 0 1 Actual 0 28 274 1 47 3956 75 4230 Percent correct: 92.54% Error Distance: 231’855

302 4003 4305

Chapter E. Empirical Results of Regressions

229

Table E.28: Setting 2b - Results for OLS in-sample-regression: The dependent variable is continuous: the stock ratio is given as a percentage. Dataset: SCF 1995. Observations = 3939, Parameters = 25, Deg.Fr.= 3914 Adjusted R-squared = 0.16465 , Log-L = -139.8736 Akaike Info. Crt. = 0.084 Variable

Coeff.

Constant X101 X1706 X3006 1 X3006 5 X3008 1 X3008 45 X301 1 X301 3 X3014 1 X3014 4 X432 1 X5608 X5821 X5825 3 X5905 X7131 X7186 X7187 X7372 1 X7401 1 X7401 2 X7401 5 X8021 1 X8022

0.0376 -0.0099 0.0111 -0.0331 0.0364 -0.0022 0.0104 0.0183 -0.0263 0.0318 -0.0828 0.0412 0.0024 0.0022 ⊕0.0262 0.013 0.015 0.0218 0.0233 ⊕0.0408 0.0048 0.0184 -0.0058 -0.0053 ⊕0.0011

Standard Error 0.0241 0.0037 0.0025 0.0263 0.0122 0.0129 0.0094 0.0094 0.0107 0.0185 0.0098 0.0095 0.004 0.003 0.0092 0.0027 0.0097 0.0089 0.0039 0.0125 0.0113 0.0123 0.0151 0.014 0.0003

b/St.Er. 1.563 -2.682 4.497 -1.26 2.98 -0.166 1.1 -1.949 -2.466 1.719 -8.413 4.327 -0.611 0.746 2.853 4.858 -1.555 -2.432 5.928 3.257 -0.43 1.499 -0.383 -0.378 3.363

P of |Z| > z 0.118 0.0073 0 0.2075 0.0029 0.868 0.2712 0.0513 0.0137 0.0856 0 0 0.5411 0.4559 0.0043 0 0.1199 0.015 0 0.0011 0.6675 0.1338 0.7019 0.7053 0.0008

Mean of X 2.6111 1.379 0.0244 0.1391 0.146 0.5519 0.2917 0.2031 0.0518 0.3112 0.5626 0.2777 0.5826 0.4174 1.6078 0.3343 0.6677 1.9667 0.654 0.3359 0.1731 0.0955 0.8116 50.5463

230

E.2. Out-of-sample estimation SCF 1995 in SCF1998

Table E.29: Setting 2b OLS, Out-of-sample estimation SCF1995 estimates in 1998 data Classification table Predicted 2 3 198 1 66 0 81 0 96 0 80 1 94 0 615 2

0 1 545 1502 22 261 17 242 18 250 17 231 19 262 638 2748 Percent correct: 22.16% Error Distance: 15’259’122 Actual 0 1 2 3 4 5

4 0 0 0 0 0 0 0

5 0 0 0 0 0 0 0

2246 349 340 364 329 375 4003

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231

Table E.30: Setting 2b - Results for Ordered model in-sample-regression: The dependent variable (stock ratio) is discrete, 6 risk classes are considered. Dataset: SCF 1995. Observations = 3939, Parameters = 25, Deg.Fr.= 3914 Log-L = -7672.327, Akaike Info. Crt. = 3.908 Variable

Coeff.

Standard Error Index function for probability Constant -1.1537 0.0705 X101 -0.1321 0.0112 X1706 0.1025 0.0056 -0.1893 0.0699 X3006 1 0.3785 0.0306 X3006 5 -0.1808 0.0422 X3008 1 0.2403 0.0244 X3008 45 X301 1 0.2711 0.0239 -0.337 0.0312 X301 3 0.2123 0.0413 X3014 1 -1.2624 0.0323 X3014 4 X432 1 0.716 0.0257 X5608 0.0076 0.0104 X5821 0.0453 0.0071 0.3611 0.0233 X5825 3 X5905 0.1459 0.0065 X7131 0.1071 0.0264 X7186 0.362 0.0238 X7187 0.1843 0.0094 ⊕0.5389 0.0367 X7372 1 0.0491 0.0284 X7401 1 0.2868 0.0325 X7401 2 -0.2179 0.0493 X7401 5 -0.0525 0.0418 X8021 1 X8022 ⊕0.0171 0.001

b/St.Er.

P of |Z| > z

Mean of X

-16.361 -11.84 18.153 -2.708 12.383 -4.286 9.853 -11.354 -10.795 5.14 -39.06 27.895 -0.731 6.407 15.53 22.602 -4.055 -15.238 19.578 14.674 1.726 8.835 -4.42 -1.255 17.583

0 0 0 0.0068 0 0 0 0 0 0 0 0 0.465 0 0 0 0.0001 0 0 0 0.0843 0 0 0.2095 0

2.6111 1.379 0.0244 0.1391 0.146 0.5519 0.2917 0.2031 0.0518 0.3112 0.5626 0.2777 0.5826 0.4174 1.6078 0.3343 0.6677 1.9667 0.654 0.3359 0.1731 0.0955 0.8116 50.5463

232

E.2. Out-of-sample estimation SCF 1995 in SCF1998

Setting 2b - Ordered model, SCF1995 (cont.) Variable

Coeff.

Standard b/St.Er. P of Mean Error |Z| > z of X Threshold parameters for index Mu(1) 2.0000 .... (Fixed Parameter) .... Mu(2) 3.0000 .... (Fixed Parameter) .... Mu(3) 4.0000 .... (Fixed Parameter) .... Mu(4) 5.0000 .... (Fixed Parameter) ....

Out-of-sample estimation SCF1995 estimates in 1998 data Classification table Predicted 0 1 2 3 1389 803 44 9 125 206 14 4 93 226 16 5 90 250 16 8 91 216 16 5 113 236 17 9 1901 1937 123 40 Percent correct: 40.47% Error Distance: 10’630’777 Actual 0 1 2 3 4 5

4 1 0 0 0 1 0 2

5 0 0 0 0 0 0 0

2246 349 340 364 329 375 4003

Chapter E. Empirical Results of Regressions

233

Table E.31: Setting 2b - Results for Multinomial Logit model, SCF1995, in-sample-regression. The dependent variable (stock ratio) is discrete, 6 risk classes are considered. The standard errors of the regression coefficients are given in brackets. The significance levels are abbreviated with asterisks: ‘*’ and ‘**’ are significant at the 5, and 1 percent level Observations = 3939, Parameters = 25, Deg.Fr.= 3914 Log-L = -4287.000, Akaike Info. Crt. = 2.189

Constant X101 X1706 X3006 1 X3006 5 X3008 1 X3008 45 X301 1 X301 3 X3014 1 X3014 4 X432 1 X5608

Prob[Y=1] -4.1796 ** (0.3719) -0.1088 (0.0564) 0.1492 ** (0.0302) -0.0396 (0.3404) -0.0252 (0.1797) 0.1144 (0.2091) 0.4733 ** (0.1349) -0.0465 (0.1275) 0.2182 (0.1492) -0.0379 (0.2419) -0.8767 ** (0.1514) 0.7962 ** (0.1396) 0.0806 (0.0476)

Prob[Y=2] -4.1282 ** (0.416) -0.0914 (0.0648) 0.0775 * (0.0341) -0.0513 (0.4009) 0.2315 (0.1916) 0.0802 (0.234) 0.3379 * (0.1507) -0.4292 ** (0.1481) -0.1448 (0.174) -0.0035 (0.2773) -1.2879 ** (0.1832) 0.8879 ** (0.1632) -0.0282 (0.064)

Prob[Y=3] -4.175 ** (0.4487) -0.0906 (0.0677) 0.0877 * (0.0355) 0.0885 (0.4152) 0.3545 (0.1944) -0.4109 (0.2899) 0.0786 (0.1552) -0.3324 * (0.1508) -0.4853 * (0.2075) -0.0407 (0.2816) -1.5963 ** (0.234) 0.7236 ** (0.1754) 0.1304 * (0.0521)

Prob[Y=4] -4.6043 ** (0.5314) -0.1731 * (0.0777) 0.0904 * (0.0404) 0.0151 (0.4773) 0.2983 (0.2191) -0.0713 (0.3145) 0.2654 (0.1822) -0.3914 * (0.1716) -0.4234 (0.2318) 0.4424 (0.2765) -1.3914 ** (0.2609) 0.814 ** (0.1995) -0.1536 (0.092)

Prob[Y=5] -3.7115 ** (0.4689) -0.2386 ** (0.0774) 0.1878 ** (0.0399) -0.885 (0.6208) 0.5274 * (0.208) 0.1924 (0.2516) 0.2249 (0.1729) -0.1072 (0.1626) -0.194 (0.2083) 0.304 (0.2773) -1.0345 ** (0.2072) 0.2446 (0.1733) -0.0331 (0.0751)

234

E.2. Out-of-sample estimation SCF 1995 in SCF1998

Setting 2b - Results for Multinomial Logit model, SCF1995, in-sample-regression. (cont.)

X5821 X5825 3 X5905 X7131 X7186 X7187 X7372 1 X7401 1 X7401 2 X7401 5 X8021 1 X8022

Prob[Y=1] 0.1131 ** (0.0368) 0.3781 ** (0.1224) 0.1857 ** (0.0339) 0.1967 (0.1305) -0.207 (0.1237) 0.1635 ** (0.0496) 0.3758 * (0.1792) 0.2417 (0.1496) 0.2773 (0.1789) 0.1905 (0.2481) -0.0324 (0.2111) 0.0162 ** (0.0049)

Prob[Y=2] 0.0912 * (0.0427) 0.2551 (0.1393) 0.1565 ** (0.0383) 0.2231 (0.1469) -0.2769 * (0.1399) 0.2318 ** (0.0553) 0.7445 ** (0.2259) 0.02 (0.1688) 0.3723 (0.1922) -0.2616 (0.3195) -0.5392 * (0.2506) 0.021 ** (0.0056)

Prob[Y=3] 0.0734 (0.0435) 0.4752 ** (0.1499) 0.2459 ** (0.0411) 0.0681 (0.1594) -0.4664 ** (0.1469) 0.3166 ** (0.0574) 0.4935 * (0.2259) -0.1212 (0.1786) 0.2797 (0.2067) -0.6962 (0.4162) -0.0561 (0.2787) 0.0157 * (0.0061)

Prob[Y=4] 0.0428 (0.0495) 0.4559 ** (0.1698) 0.1594 ** (0.0458) -0.304 (0.1888) -0.5629 ** (0.1654) 0.2348 ** (0.0654) 1.0315 ** (0.2812) 0.3953 (0.2116) 0.5539 * (0.2437) 0.071 (0.3747) -0.0417 (0.3604) 0.0159 * (0.0071)

Prob[Y=5] 0.0636 (0.0502) 0.2563 (0.1611) 0.1402 ** (0.0444) -0.2269 (0.1774) -0.1855 (0.163) 0.2323 ** (0.0638) 0.4405 (0.2308) -0.1811 (0.1946) 0.1042 (0.2214) 0.0792 (0.3008) 0.0376 (0.2728) 0.0155 * (0.0064)

Chapter E. Empirical Results of Regressions

235

Setting 2b - Multinomial model Out-of-sample estimation, SCF1995 estimates in 1998 data Classification table Predicted 0 1 2 3 1999 0 2 0 312 3 0 0 292 7 1 0 305 3 0 2 286 4 0 1 321 5 0 2 3515 22 3 5 Percent correct: 51.04% Error Distance: 26’479’635 Actual 0 1 2 3 4 5

4 245 34 40 54 38 47 458

5 0 0 0 0 0 0 0

2246 349 340 364 329 375 4003

236

E.2. Out-of-sample estimation SCF 1995 in SCF1998

Table E.32: Setting 3b - Results for WLS in-sample-regression: The dependent variable is binomial, investors without stocks are to be separated from stock-owning investors. The portrayed mean has no interpretation, as each variable was weighted by W , the reciprocal of the observations’ standard errors Observations = 3939, Parameters = 25, Deg.Fr.= 3914 Adjusted R-squared = 0.30814 , Log-L = -2029.9107 Akaike Info. Crt. = 1.043 Variable WConstant WX101 WX1706 WX3006 1 WX3006 5 WX3008 1 WX3008 45 WX301 1 WX301 3 WX3014 1 WX3014 4 WX432 1 WX5608 WX5821 WX5825 3 WX5905 WX7131 WX7186 WX7187 WX7372 1 WX7401 1 WX7401 2 WX7401 5 WX8021 1 WX8022

Coeff. 5.1786 -0.0169 0.0247 -0.0186 0.0239 ⊕0.0273 0.0346 0.0417 -0.0125 0.024 -0.1559 0.1238 0.0094 0.0101 ⊕0.075 0.0415 0.0145 0.0261 0.0428 ⊕0.1137 0.0093 0.0566 -0.0243 -0.0511 ⊕0.0013

Standard Error 1.8554 0.0031 0.004 0.0264 0.0145 0.0122 0.0103 0.0109 0.0097 0.0263 0.0126 0.015 0.0051 0.0039 0.0131 0.0039 0.0095 0.0117 0.0052 0.0136 0.0153 0.0144 0.0123 0.0143 0.0004

b/St.Er. 2.791 -5.393 6.161 -0.704 1.645 2.231 3.357 -3.815 -1.283 0.914 -12.379 8.235 1.849 2.576 5.707 10.681 1.528 -2.223 8.159 8.342 0.609 3.915 -1.983 -3.568 3.586

P of |Z| > z 0.0053 0 0 0.4814 0.1001 0.0257 0.0008 0.0001 0.1993 0.3607 0 0 0.0645 0.01 0 0 0.1265 0.0262 0 0 0.5423 0.0001 0.0473 0.0004 0.0003

Mean of X 0.0478 7.9872 3.779 0.0773 0.4007 0.5181 1.5073 0.8188 0.6828 0.1365 1.3259 1.3829 0.768 1.6019 1.1176 4.1887 1.0113 2.1663 5.7473 1.8505 0.8597 0.4959 0.3654 2.3439 152.4247

Chapter E. Empirical Results of Regressions

Setting 3b - WLS Out-of-sample estimation SCF1995 estimates in 1998 data Classification table Predicted 0 1 Actual 0 2047 199 2246 1 1234 523 1757 3281 722 4003 Percent correct: 64.21% Error Distance: 4’687’071

237

238

E.2. Out-of-sample estimation SCF 1995 in SCF1998

Table E.33: Setting 3b - Results for Multinomial Logit model in-sampleregression: The dependent variable is binomial, investors without stocks are to be separated from stock-owning investors. Dataset: SCF 1995. Observations = 3939, Parameters = 25, Deg.Fr.= 3914 Log-L = -1941.207, Akaike Info. Crt. = 0.998 Variable

Coeff.

Standard b/St.Er. Error Characteristics in numerator of Prob[Y = 1] Constant -2.5324 0.2452 -10.328 X101 -0.1303 0.0391 -3.328 X1706 0.1209 0.0226 5.342 X3006 1 -0.1209 0.265 -0.456 0.2434 0.1203 2.024 X3006 5 ⊕0.0198 0.1388 0.143 X3008 1 0.303 0.092 3.293 X3008 45 0.2366 0.0917 -2.579 X301 1 -0.1238 0.109 -1.136 X301 3 0.1061 0.177 0.6 X3014 1 X3014 4 -1.1626 0.1043 -11.146 0.708 0.0904 7.834 X432 1 X5608 0.0326 0.0375 0.87 X5821 0.0843 0.0288 2.922 ⊕0.3587 0.086 4.173 X5825 3 X5905 0.18 0.0247 7.291 X7131 0.0535 0.0963 0.555 X7186 0.3121 0.0873 -3.574 X7187 0.2273 0.0382 5.954 ⊕0.5568 0.1276 4.364 X7372 1 0.0788 0.1078 0.731 X7401 1 X7401 2 0.3022 0.1209 2.498 -0.0593 0.1652 -0.359 X7401 5 -0.1413 0.1442 -0.98 X8021 1 X8022 ⊕0.0171 0.0034 5.018

P of |Z| > z

Mean of X

0 0.0009 0 0.6483 0.043 0.8865 0.001 0.0099 0.2559 0.5487 0 0 0.3845 0.0035 0 0 0.5787 0.0004 0 0 0.4646 0.0125 0.7196 0.327 0

2.6111 1.379 0.0244 0.1391 0.146 0.5519 0.2917 0.2031 0.0518 0.3112 0.5626 0.2777 0.5826 0.4174 1.6078 0.3343 0.6677 1.9667 0.654 0.3359 0.1731 0.0955 0.8116 50.5463

Chapter E. Empirical Results of Regressions

Table E.34: Setting 3b Multinomial Logit model Out-of-sample estimation SCF1995 estimates in 1998 data Classification table Predicted 0 1 Actual 0 1449 797 2246 1 569 1188 1757 2018 1985 4003 Percent correct: 65.88% Error Distance: 2’578’910

239

240

E.2. Out-of-sample estimation SCF 1995 in SCF1998

Table E.35: Setting 3c - Results for OLS in-sample-regression: The dependent variable is continuous: the stock ratio is given as a percentage. Dataset: SCF 1995. Observations = 1550, Parameters = 25, Deg.Fr.= 1525 Adjusted R-squared = 0.01510 , Log-L = -284.4867 Akaike Info. Crt. = 0.399 Variable

Coeff.

Constant X101 X1706 X3006 1 X3006 5 X3008 1 X3008 45 X301 1 X301 3 X3014 1 X3014 4 X432 1 X5608 X5821 X5825 3 X5905 X7131 X7186 X7187 X7372 1 X7401 1 X7401 2 X7401 5 X8021 1 X8022

0.5217 -0.0114 0.0004 -0.0563 0.0498 -0.0191 0.0318 0.0065 -0.0517 0.0472 -0.0349 0.0427 0.0105 0.0073 -0.0038 0.003 0.0459 0.0078 0.0111 ⊕0.0143 0.0287 0.0116 ⊕0.0015 0.0185 -0.0002

Standard Error 0.0532 0.0081 0.0039 0.0467 0.0226 0.0302 0.0184 0.017 0.0216 0.0295 0.0241 0.0204 0.0066 0.0046 0.017 0.0045 0.0177 0.0172 0.006 0.0262 0.0199 0.0239 0.0389 0.0317 0.0007

b/St.Er. 9.812 -1.4 -0.091 -1.206 2.2 -0.633 -1.727 -0.385 -2.394 1.602 -1.448 -2.095 -1.592 -1.574 -0.221 -0.678 -2.589 -0.452 1.843 0.545 -1.444 -0.486 0.038 0.583 -0.341

P of |Z| > z 0 0.1615 0.9277 0.2279 0.0278 0.527 0.0842 0.7006 0.0166 0.1091 0.1475 0.0362 0.1113 0.1155 0.8247 0.4975 0.0096 0.6514 0.0653 0.5859 0.1489 0.6271 0.9694 0.5597 0.733

Mean of X 2.5555 2.3277 0.0271 0.1477 0.0845 0.6794 0.3077 0.1574 0.0723 0.1284 0.7871 0.3368 0.8 0.5916 2.4135 0.3626 0.6381 2.5852 0.7723 0.4729 0.1761 0.0458 0.8852 54.6277

Chapter E. Empirical Results of Regressions

241

Table E.36: Setting 3c OLS Out-of-sample estimation SCF1995 estimates in 1998 data Classification table Predicted 0 1 2 1 126 221 0 109 230 0 119 244 0 109 219 0 123 250 1 586 1164 Percent correct: 20.20% Error Distance: 2’231’527

Actual 0 1 2 3 4

3 1 1 1 1 2 6

4 0 0 0 0 0 0

349 340 364 329 375 1757

242

E.2. Out-of-sample estimation SCF 1995 in SCF1998

Table E.37: Setting 3c - Results for Ordered model in-sample-regression: The dependent variable (stock ratio) is discrete, 5 risk classes are considered. Dataset: SCF 1995. Observations = 1550, Parameters = 25, Deg.Fr.= 1525 Log-L = -3516.390, Akaike Info. Crt. = 4.570 Variable

Coeff.

Standard Error Index function for probability Constant 2.4061 0.1076 X101 -0.0748 0.0164 X1706 0.0064 0.0081 -0.2652 0.1013 X3006 1 0.337 0.047 X3006 5 -0.0581 0.059 X3008 1 0.0382 X3008 45 0.2067 X301 1 0.1048 0.0345 -0.3872 0.0435 X301 3 0.2985 0.0587 X3014 1 -0.2645 0.0477 X3014 4 X432 1 0.299 0.0404 X5608 0.0743 0.0133 X5821 0.0411 0.0092 ⊕0.0064 0.035 X5825 3 X5905 0.0107 0.0093 X7131 0.3311 0.0362 X7186 0.1189 0.0355 X7187 0.0728 0.0125 ⊕0.1639 0.0509 X7372 1 0.1918 0.0416 X7401 1 0.0605 0.05 X7401 2 -0.1248 0.0738 X7401 5 0.0548 0.0624 X8021 1 X8022 -0.0005 0.0014

b/St.Er.

P of |Z| > z

Mean of X

22.354 -4.561 -0.782 -2.619 7.17 -0.984 -5.41 -3.042 -8.907 5.089 -5.55 -7.396 -5.603 -4.474 0.181 -1.157 -9.141 -3.345 5.828 3.217 -4.61 -1.21 -1.692 0.879 -0.33

0 0 0.4344 0.0088 0 0.3252 0 0.0023 0 0 0 0 0 0 0.856 0.2471 0 0.0008 0 0.0013 0 0.2261 0.0906 0.3795 0.7413

2.5555 2.3277 0.0271 0.1477 0.0845 0.6794 0.3077 0.1574 0.0723 0.1284 0.7871 0.3368 0.8 0.5916 2.4135 0.3626 0.6381 2.5852 0.7723 0.4729 0.1761 0.0458 0.8852 54.6277

Chapter E. Empirical Results of Regressions

243

Setting 3c - Ordered model, SCF1995 (cont.) Variable

Coeff.

Standard b/St.Er. P of Mean Error |Z| > z of X Threshold parameters for index Mu(1) 2.0000 .... (Fixed Parameter) .... Mu(2) 3.0000 .... (Fixed Parameter) .... Mu(3) 4.0000 .... (Fixed Parameter) ....

Out-of-sample estimation SCF1995 estimates in 1998 data Classification table Predicted 0 1 2 0 323 24 0 313 26 0 310 53 0 292 35 0 328 46 0 1566 184 Percent correct: 20.94% Error Distance: 4’675’546

Actual 0 1 2 3 4

3 2 1 1 2 1 7

4 0 0 0 0 0 0

349 340 364 329 375 1757

244

E.2. Out-of-sample estimation SCF 1995 in SCF1998

Table E.38: Setting 3c - Results for Multinomial Logit model in-sampleregression: The dependent variable (stock ratio) is discrete, 5 risk classes are considered. Dataset: SCF 1995. The standard errors of the regression coefficients are given in brackets. The significance levels are abbreviated with asterisks: ‘*’ and ‘**’ are significant at the 5, and 1 percent level Observations = 1550, Parameters = 25, Deg.Fr.= 1525 Log-L = -2345.463, Akaike Info. Crt. = 3.059

Constant X101 X1706 X3006 1 X3006 5 X3008 1 X3008 45 X301 1 X301 3 X3014 1 X3014 4 X432 1 X5608

Prob[Y=1] 0.1129 (0.5181) 0.009 (0.0792) -0.0775 * (0.0379) 0.0291 (0.4399) 0.2607 (0.2304) -0.0058 (0.2883) -0.1432 (0.1817) -0.3845 * (0.1693) -0.3547 (0.2009) 0.0849 (0.3074) -0.454 * (0.2236) 0.0796 (0.2024) -0.1078 (0.0686)

Prob[Y=2] 0.0138 (0.5435) 0.0203 (0.0818) -0.0679 (0.0388) 0.2046 (0.4456) 0.3768 (0.2311) -0.5741 (0.3335) -0.4317 * (0.1842) -0.272 (0.17) -0.7043 ** (0.2289) 0.0417 (0.3092) -0.743 ** (0.2678) -0.0686 (0.2113) 0.0522 (0.0569)

Prob[Y=3] -0.4407 (0.6183) -0.0772 (0.0907) -0.0684 (0.0438) 0.0805 (0.5081) 0.3226 (0.2528) -0.205 (0.3564) -0.2222 (0.2093) -0.3285 (0.1896) -0.6356 * (0.2525) 0.5145 (0.306) -0.5089 (0.2922) 0.0251 (0.2335) -0.2318 * (0.0958)

Prob[Y=4] 0.4576 (0.5773) -0.1354 (0.0919) 0.0334 (0.044) -0.8612 (0.6473) 0.5412 * (0.2483) 0.1135 (0.3062) -0.24 (0.2034) -0.069 (0.1853) -0.3611 (0.2352) 0.3458 (0.314) -0.1988 (0.2484) -0.5659 ** (0.2117) -0.1235 (0.0801)

Chapter E. Empirical Results of Regressions

245

Setting 3c - Results for Multinomial Logit model, SCF1995, in-sample-regression. (cont.)

X5821 X5825 3 X5905 X7131 X7186 X7187 X7372 1 X7401 1 X7401 2 X7401 5 X8021 1 X8022

Prob[Y=1] -0.0213 (0.045) -0.1321 (0.1641) -0.0353 (0.0434) 0.0588 (0.1692) -0.0835 (0.1684) 0.0787 (0.0589) 0.3586 (0.2631) -0.2267 (0.1924) 0.0897 (0.2337) -0.5399 (0.3838) -0.5145 (0.304) 0.0048 (0.0067)

Prob[Y=2] -0.0373 (0.0455) 0.1095 (0.1735) 0.0591 (0.046) -0.1225 (0.1777) -0.2242 (0.1726) 0.1547 ** (0.0598) 0.1602 (0.2605) -0.35 (0.2002) -0.0048 (0.2437) -0.9636 * (0.4676) -0.1026 (0.3272) -0.0001 (0.0071)

Prob[Y=3] -0.0671 (0.0513) 0.1147 (0.1926) -0.0178 (0.0505) -0.5052 * (0.2053) -0.3583 (0.1906) 0.0801 (0.0677) 0.6719 * (0.3126) 0.1345 (0.2321) 0.2497 (0.2779) -0.1089 (0.433) -0.0763 (0.4014) 0.0009 (0.008)

Prob[Y=4] -0.048 (0.053) -0.1157 (0.1865) -0.0482 (0.0493) -0.4059 * (0.1993) -0.0026 (0.1927) 0.0739 (0.0677) 0.0548 (0.2724) -0.4321 * (0.2168) -0.2006 (0.2605) -0.1223 (0.3743) 0.0643 (0.3285) 0.0004 (0.0076)

246

E.2. Out-of-sample estimation SCF 1995 in SCF1998

Setting 3c - Multinomial Logit model Out-of-Sample estimation SCF1995 estimates in 1998 data Classification table Predicted 0 1 2 231 16 18 216 21 25 213 17 39 209 14 21 226 17 28 1095 85 131 Percent correct: 21.74% Error Distance: 5’743’122

Actual 0 1 2 3 4

3 79 75 89 78 91 412

4 5 3 6 7 13 34

349 340 364 329 375 1757

Chapter E. Empirical Results of Regressions

E.3

247

Likelihood Ratio Tests

All LR tests are set up in such way that the null could not be refuted for the maximal number of least significant factor coefficients. LR tests, setting 1 In setting 1 the dependent variable has 7 categories. The number of observations are 4305, there are 25 parameters and thus 4280 degrees of freedom. MNL model. H0 : CoefficientVarX =0, Var X: X101, X3006 5, X301 3, X5608. 24 Restrictions: 4 variables in 6 logits. P-value=0.345

Var X see above

Likelihood Ratio test LR Test θ˜ θˆ -5350.63 -5363.71 26.16

Tabl-V. 36.42

Result n.r.

θ˜ = LogL of unrestricted model, θˆ = LogL of restricted model Tabl-V. = Critical Value of chisquare distribution ref. = refute the null, n.r. = the null cannot be refuted

Ordered Logit model. H0 : CoefficientVarX =0. 6 Restrictions: X3006 1, X3006 5, X301 1, X3014 1, X7401 5, X8021 1. P-Value=0.14

Var X see above

Likelihood Ratio test LR Test θ˜ θˆ -7833.03 -7837.90 9.74

Tabl-V. 12.59

Result n.r.

LR tests, setting 2a In setting 2a the dependent variable has 2 categories. The number of observations are 4305, there are 25 parameters and thus 4280 degrees of freedom. MNL model. H0 : CoefficientVarX =0, VarX: X101, X3006 5, X3008 45, X301 3, X5608, X7401 5, X8021 1, X5821, X301 1. Nine Restrictions. P-value=0.05

Var X see above

Likelihood Ratio test LR Test Tabl-V. θ˜ θˆ -678.87 -687.32 16.918 16.92

Result n.r.

248

E.3. Likelihood Ratio Tests

LR tests, setting 2b In setting 2b the dependent variable has 6 categories. The number of observations are 4003, there are 25 parameters and thus 3978 degrees of freedom. MNL model. H0 : CoefficientVarX =0, VarX: X101, X3006 1, X3008 1, X301 1, X3014 1, X5608, X7131, X7186, X7401 5. Nine Variables, 6 Risk Classes = 45 Restrictions. P-value=0.14

Var X see above

Likelihood Ratio test ˜ LR Test θ θˆ -4672.59 -4700.15 55.13

Tabl-V. 61.66

Result n.r.

Ordered Logit model. H0 : CoefficientVarX =0. 5 Restrictions: X101, X3006 1, X3006 5, X301 3, X8021 1. P-Value=0.13

Var X see above

Likelihood Ratio test LR Test θ˜ θˆ -8846.00 -8850.23 8.47

Tabl-V. 11.07

Result n.r.

LR tests, setting 3b In setting 3b the dependent variable has 2 categories. The number of observations are 4003, there are 25 parameters and thus 3978 degrees of freedom. MNL model. H0 : CoefficientVarX =0, VarX: X101, X3006 1, X3006 5, X3008 1, X301 1, X301 3, X3014 1, X5608, X7131, X7401 2, X7401 5, X8021 1. 12 Restrictions. P-value=0.08

Var X see above

Likelihood Ratio test LR Test θ˜ θˆ -1906.34 -1915.93 19.18

Tabl-V. 21.03

Result n.r.

LR tests, setting 3c In setting 3c the dependent variable has 2 categories. The number of observations are 1757, there are 25 parameters and thus 1732 degrees of freedom. MNL model. H0 : CoefficientVarX =0, VarX: X101, X3006 1, X3006 5, X3008 1, X301 1, X301 3, X3014 1, X432 1, X5608, X5821, X5825 3, X5905, X7131, X7187, X7401 1, X7401 2, X7401 5, X8022. 18 Variables, 5 Risk classes = 72 Restrictions. P-value=0.47

Chapter E. Empirical Results of Regressions

Var X see above

Likelihood Ratio test LR Test θ˜ θˆ -2766.66 -2802.76 72.18

249

Tabl-V. 92.81

Result n.r.

Ordered Logit model. H0 : CoefficientVarX =0, VarX: X101, X3006 1, X3006 5, X5608, X5821, X5905, X7131, X7186, X7187, X7372 1, X7401 2, X8022. 12 Restrictions. P-value=0.19

Var X see above

Likelihood Ratio test LR Test θ˜ θˆ -3899.91 -3907.92 16.02

Tabl-V. 21.03

Result n.r.

250

E.3. Likelihood Ratio Tests

Appendix F

Independent Factors From a total of over 3’000 SCF-variables, 150 were selected for the sample. 30 of these were chosen according to the study’s hypotheses and significance tests. The best 17 out of these 30 were chosen as the Independent Variable Set. For most of the variables the original coding as given by the SCF had to be modified: Codes were combined, values adjusted to yield an ordinal ranking and continuous variables had to be discretized to end up with a limited number of separate classes with an approximately equal number of observations per class. The variables are: x101, x1706, x3006, x3008, x301, x3014, x432, x5608, x5821, x5825, x5905, x7131, x7186, x7187, x7401, x8021, x8022 Some of the above are multidimensional variables, their dimensions not obeying any ordinal ranking. They had to be split into several dummy variables to identify the effect of each dimension. The number of factors defined in the following thus increased from 17 to 21.

251

252

Table F.1: Calculation of financial assets Composition of total financial assets and the stock ratio. tmv = total market value tdv = total dollar value Variable

Element

Stock ratio Amount invested in stocks Financial Assets

= Amount in stocks / Total financial assets = + + = + + + + + + + + + + + + + + + + + + + + + +

x3822 x3915 x6704 x3506 x3529 x3610 x3706 x3718 x3721 x3804 x3818 x3822 x3824 x3826 x3828 x3830 x3902 x3915 x3930 x6704 x6706 x7635 x7636 x7637 x7638 x7639

Significance

tmv of all Stock Funds tmv of publicly traded stock tmv of all of the mutual funds amount in checking account amount in all other checking accounts how much in IRA or KEOGH? amount in tax-free money market account amount in remaining money market accounts tdv of all these CDs amount in savings account amount in all your remaining savings accounts total market value of all of the Stock Funds tmv of all of the Tax-Free Bond Funds tmv of all of the Government Bond Funds tmv of all of the Other Bond Funds tmv of any other mutual funds total face value of all the savings bonds tmv of publicly traded stock tdv of all the cash or call money accounts tmv of all of the mutual funds tmv of all of your bonds tmv of all of the Mortgage-backed bonds tmv of US Gov bonds tmv of these state/municipal bonds tmv of your foreign bonds tmv of all of the other type of bonds

Chapter F. Independent Factors

253

Table F.2: Overview of variables Overview of variables used in the empirical analysis as independent factors. Variable

Significance

Coding

x101

Number of people in the household Do you have any lines of credit, not counting credit cards? How much is your real estate property worth if sold today

Code amount

x1101

x1706

x1715

How much is still owed of the mortgage on the property?

x3006

What are your most important reasons for saving?

x3006 1

Is your most important reason for saving liquidity and consumption?

0 - Yes 1 - No 0 1 2 3 4 5 6 0 1 2 3 4 5 6 0 1 2 3 4 5 6 0 1

-

$0 $1 to $10’000 $10’000 to $50’000 $50’000 to $100’000 $100’000 to $500’000 $500’000 to $1’000’000 over $1’000’000 $0 $1 to $10’000 $10’000 to $50’000 $50’000 to $100’000 $100’000 to $500’000 $500’000 to $1’000’000 over $1’000’000 do not save Liquidity Consumption Events House, Car Education, Family Retirement, Reserves No Yes continued on next page

254

Variable

Significance

Coding

x3006 5

Is your most important reason for saving education and family? In planning your saving and spending, which of the time periods is most important for you? In planning your saving and spending, are the next few months most important for you? In planning your saving and spending, do you have a longer than 5 years? Do you expect the US economy to perform better in the next 5 years than it has over the last 5? Do you expect the US economy to perform better in the next 5 years than it has over the last 5? What kind of financial obligations will you have to meet in the next five years? What amount of financial risk would you be willing to take when you save or make investments?

0 - No 1 - Yes

x3008

x3008 1

x3008 45

x301 1

x301 3

x3011 4

x3014

1 2 3 4 5 0 1

-

Next few months Next year Next few years Next 5-10 years Longer than 10 years No Yes

0 - No 1 - Yes 0 - else 1 - better

0 - else 1 - worse

0 - else 1 - education

1 - substantial risks to earn substantial returns 2 - above average risks to earn above average returns 3 - average risks to earn average returns continued on next page

Chapter F. Independent Factors

Variable

x3014 1

x3014 4 x401 0

x413

x432

x432 1

x432 3

x5318

Significance

Would you be willing to take substantial risks to earn substantial returns when you save or make investments? You are not willing to take any financial risks? How do you judge buying things on the installment plan? After the last payments were made on these accounts roughly what was the balance still owed on these accounts?

Do you pay off the total balance owed on your credit card account each month? Do you always pay off the total balance owed on your credit card account each month? Do you hardly ever pay off the total balance owed on your credit card account each month? How much social benefits are received each month or

255

Coding 4 - Not willing to take any financial risks 0 - No 1 - Yes

0 1 0 1

-

Wrong Right else good idea

0 1 2 3 4 5 6 1 2 3

-

$0 $1 to $499 $500 to $999 $1’000 to $1’999 $2’000 to $4’999 $5’000 to $9’999 over $10’000 Always or almost always Sometimes Hardly ever

0 - No 1 - Yes

0 - No 1 - Yes

0 - $0 1 - $1 to $499 continued on next page

256

Variable

Significance

Coding

year?

2 - $500 to $999 3 - $1’000 to $1’999 4 - $2’000 to $4’999 5 - $5’000 to $9’999 6 - over $10’000 0 - $0 1 - $1 to $499 2 - $500 to $999 3 - $1’000 to $1’999 4 - $2’000 to $4’999 5 - $5’000 to $9’999 6 - over $10’000 0 - $0 1 - $1 to $10’000 2 - $10’000 to $50’000 3 - $50’000 to $100’000 4 - $100’000 to $500’000 5 - $500’000 to $1’000’000 6 - over $1’000’000 0 - Wrong 1 - Right 1 - Nursing, Chiropratic, other 2 - Associate’s, junior college 3 - Bachelor’s degree 4 - MA, MS, MBA 5 - PhD, MD, Law, JD, other doct. 0 - Yes 1 - No

x5608

About how much do you expect your future pension to be?

x5821

About how much in future inheritance (or transfer of assets) do you expect?

x5825 3

You do not expect to leave a sizable estate to others? What is the highest degree you earned?

x5905

x7131

x7141

Have you applied for any type of credit or loan in the last 5 years? How much did you borrow the last time you used your largest

0 - $0 1 - $1 to $999 2 - $1’000 to $4’999 continued on next page

Chapter F. Independent Factors

Variable

x7186 x7187

257

Significance

Coding

line of credit secured by the equity in your home?

3 - $5’000 to $9’999 4 - $10’000 to $49’999 5 - $50’000 to $100’000 6 - over $100’000 0 - Yes 1 - No 0 - $0 1 - $1 to $10’000 2 - $10’000 to $50’000 3 - $50’000 to $100’000 4 - $100’000 to $500’000 5 - $500’000 to $1’000’000 6 - over $1’000’000 1 - married 2,3,4 - separated, divorced, widowed 5 - never married 0 - No 1 - Yes 1 - Managerial, Executive 2 - Technical, Sales, Administrative 5 - Operator, Fabricator, Laborer 1 - male 2 - female 0 - No 1 - Yes

Are you saving for foreseeable expenses now? About how much do you think you need to have in savings for emergencies and other unexpected things that may come up?

x7372

Marital status

x7372 1

Are you married?

x7401

Is the official title of your job: (y/n) ?

x8021

Gender

x8021 1

Are you male?

x8022

Age in years

258

Table F.3: Expected signs of factor coefficients Based on the hypotheses in Section 3.2 the estimated coefficients of the independent variables are expected to have the following signs: Variable

Sign

Justification

X101



X1706

+

X3006 1



X3006 5

+

X3008 1



X3008 45

+

X301 1

+

X301 3



X3014 1 X3014 4 X432 1

+ − +

X5608

+

X5821

+

The higher the number of people in the household the higher the financial obligations and the lower will be the ability to take risks More wealth in the form of real estate property implies higher risk taking capacity Liquidity and consumption are short-term saving goals and call for less risky investments Long-term saving goals such as education and family allow for higher risk in the portfolio A short investment horizon requires a less risky asset allocation A long investment horizon makes higher stock ratio in the portfolio possible Optimistic economic expectations are one prerequisite for taking a higher investment exposure Pessimistic economic expectations impede high stakes in risky assets The investor is willing to take substantial risks The investor is not willing to take any financial risks Investors who always pay off their credit signal financial discipline and thus the ability to take financial risks The higher the expected future pension the higher the risk to be taken Future windfalls in the form of inheritance allow an investor to take higher risks continued on next page

Chapter F. Independent Factors

259

Variable

Sign

Justification

X5825 3



X5905

+

X7131

+

X7186

+

X7187

+

X7372 1



X7401 1

+

X7401 2

+

X7401 5



X8021 1



X8022



Those who do not plan to leave an estate will consume their wealth and should take increasingly less risk as they grow older The higher the educational background the higher the probability that the investor has an understanding about the financial markets Investors who haven’t applied for credit or loan are more likely to be in control of their finances and thus can take higher risks Investors not saving for foreseeable expenses are considering to invest their free wealth and thus focus on a longer time perspective The higher the savings needed the more risky one needs to invest Married investors are less independent and have a lower potential for risky investments Managers and Executives are willing to take more risks in order to earn higher returns Just as Managers and Executives, Technical, Sales and Administrative employees due to relatively high wages also have a higher ability to compensate for losses in their investments Operators and Laborers have less leeway to compensate for losses with their income Even though there is no unambiguous interpretation of the gender coefficient, some studies found that men in their futile attempts to time the market trade more often than female investors. Male investor should thus hold less risky portfolios The older the investor the lower his capacity to offset investment losses by human capital (future income)

260

Appendix G

Various Riskrulers G.1

Risk Quota by Fidelity Investments

G.2

Allianz Anleger Analyse

G.3

Union Investment

261

262

G.3. Union Investment

Figure G.1: ‘Risk Quota by Fidelity Investments’: Example of traditional risk rulers

Chapter G. Various Riskrulers

263

Figure G.2: ‘Allianz Anleger-Analyse’: Example for a psychologically motivated risk ruler. These are typically not empirically tested with quantitative methods.

264

G.3. Union Investment

Figure G.3: ‘Union Investment - Test zur pers¨ onlichen Risikobereitschaft’: Example for a psychological risk ruler. Answering Sheet.

Chapter G. Various Riskrulers

265

Figure G.4: Example for a psychological risk ruler: ‘Union Investment - Test zur pers¨onlichen Risikobereitschaft’. Evaluation Sheet.

266

G.3. Union Investment

Appendix H

Practical implementation of a Risk Ruler H.1

Proceeding when developing a Risk Ruler

The first step when developing a questionnaire for the assessment of risk aversion or risk preferences in general is to consult the underlying theory and empirical studies on the topic in order to derive workable hypotheses about financial behavior. On the basis of these hypotheses the factors and indicators most relevant for financial decision making can be determined. In a first number of estimations these selected factors will be tested for their significance and collinearity. Insignificant factors can be dropped and similar, collinear factors can be condensed to create new ones. In order to minimize cost and effort these pretests can be carried out using existing sample data. With the final set of factors a new questionnaire is set up that can be used in a survey. The survey design must ensure that the resulting sample is representative. The collected answers are then used to estimate the weights of each factor e.g. question in the questionnaire. In this sense, the data serves the purpose of a calibration sample. 267

268

H.2

H.2. Example of an interactive Risk Ruler

Example of an interactive Risk Ruler

For experimental reasons I set up a Sample-Risk Ruler under the Web-address http://riskaversion.hypermart.net. Any investor can visit this site and determine his optimal stock ratio and skewness preference by answering the questions on that page.1 The questions differ slightly from the ones in the SCF used for the empirical analysis in the sense that they are more investmentspecific and thus have hopefully more predictive power. Alternatively, the Risk Ruler test can be carried out by oneself using the same questions displayed below. After answering all questions, the responses must be weighted by using the scheme in Figure H.2. The questionnaire leaves the investor with three stock ratios: The econometric model yields the optimal stock ratio according to the questions answered. The second estimate for the optimal stock ratio comes from the gamble’s certainty equivalent and the third stock ratio is the one the investor currently holds. Typically this last figure is not a result of conscious decision-making. Portfolios evolve over time as an investor buys new shares due to capital increases, inherits financial assets or disinvests because of changing circumstances. The result of the gamble captures usually a rough estimate of one’s risk aversion. However, for an average investor gambles are difficult to understand. An accurate answer for the certainty equivalent will thus result seldomly. The most reliable estimate for the optimal stock ratio is therefore the result of the questionnaire. The other two numbers provide useful comparative values. The online Risk Ruler was programmed in PERL and set up on a Host that supports cgi-scripts thus enabling the processing and saving of the input data. In order to ensure that the sample is at least partly representative, demographic factors were included as questions. Also, the input file automatically stored the visitor’s IP address. In that way, experimenting visitors who are sending multiple rounds of answers can be eliminated, as they would otherwise distort the sample.

1 The corresponding risk aversion coefficients are not displayed, but can easily be calculated using the equations in Chapter 7.

Chapter H. Practical implementation of a Risk Ruler

Your Financial Risk Preference By filling in the following questionnaire you can determine your personal risk preference (risk aversion) measured as the stock ratio optimal for you. The ratio of stocks is the percentage of stocks to your total financial assets. Apart from your optimal exposure you will learn about your degree of skewness preference regarding the distribution of investment returns. In other words you will find out whether an option strategy would be suitable for you either to prevent you from downside risk or to enhance your upside potential. Your results have been calculated utilizing a multinomial logit model and factor analysis based on a data set collected on the internet. Bold text in [brackets] symbolizes the key for each question. The sample data and regression results later refer to these keys. 1. Please state the country of your main residence: [nat] ...................................................... 2. Please name your age in years: [age] (a) < 20 (b) 21-34 (c) 35-44 (d) 45-54 (e) 55-64 (f) 65-74 (g) > 75 3. Are you ... [sex] (a) female (b) male 4. What is your marital status? [mar] (a) single (b) married (c) divorced (d) widowed

269

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5. Do you have any children? [chil] (a) None (b) One or Two (c) More than two 6. What educational degree did you accomplish? [degr] (a) Highschool (b) Professional degree (c) BA / MA (d) PhD or higher 7. Please name your profession: [prof ] (a) Managerial, Executive, Professional (b) Technical, Sales, Administrative (c) Operator, Fabricator, Laborer 8. When answering the following question please consider that you are investing only that part of your wealth that is not bound to finance planned projects (such as a house, car etc.). Consider investing that part of your wealth which does not have to be liquidated within the next 5-10 years. Below you are given the distributions of the yearly returns of 6 different investment opportunities. Please select >the one< that you think fits your risk tolerance level best. [risk]

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9. Which describes best your attitude regarding fluctuations of your portfolio value: [fluc] (a) I couldn’t accept a daily loss of 1%, monthly loss of 5% or a yearly loss of 16% of my portfolio value. (b) I couldn’t accept a daily loss of 1.5%, monthly loss of 9% or a yearly loss of 32% of my portfolio value. (c) I couldn’t accept a daily loss of 2%, monthly loss of 12% or a yearly loss of 40% of my portfolio value. (d) I could accept any of the above knowing I’m able to wait out even longer periods of a downmarket. 10. Concerning Gains: Imagine you could invest part of your portfolio, say $10’000, in one of the following lotteries that after 3 years result in the given additional payoffs. Which lottery would you choose? (All have the same expected value). You >will not lose< your principal of $10’000. You will get that amount back at the end of 3 years! [lot1] (a) 50% probability for a gain of 4’000 and 50% probability for a gain of 3’000 (b) 30% probability for a gain of 9’100 and 70% probability for a gain of 1’100 (c) 20% probability for a gain of 17’500 and 80% probability for no gain 11. Concerning Losses: Imagine you could invest yet another part of your portfolio, say $10’000, in one of the following lotteries that after 3 years result in the given additional payoffs. Which lottery would you choose? You will ¿not¡ lose your principal of $10’000!You will get it back after the 3 years! [lot2] (a) 50% probability for a gain of 5’500 and 50% probability for a loss of -3’900 (b) 40% probability for a gain of 4’000 and 60% probability for a loss of -1’400 (c) 30% probability for a gain of 2’500 and 70% probability for no loss 12. What percentage of your total financial assets do you currently have invested in stocks (or riskier assets such as options, hedge funds)? Your total financial assets consist of all your liquid wealth:checking and saving accounts, bonds, stocks, funds, options etc. [curr]

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(a) 0% - 20% (b) 21% - 40% (c) 41% - 60% (d) 61% - 80% (e) 81% - 100% 13. How would you rate the current risk level of your portfolio: [levl] (a) safe (b) moderate risk (c) considerable risk (d) very high-risk 14. What percentage of stocks of your total financial assets do you consider optimal for yourself for the next 5-10 years? [opt] (a) 0% - 20% (b) 21% - 40% (c) 41% - 60% (d) 61% - 80% (e) 81% - 100% 15. According to you: what will be the path of development for the economy of your country over the next 3-5 years? [econ] (a) considerable higher growth than during the last few years (b) growth will be a bit better than during the last few years (c) there will be no additional growth (d) growth will slow down (e) growth will slow down considerably 16. Since how many years do you trace the development of the financial markets or since how many years do you carry out financial market transactions? [finex] (a) less than a year (b) 1 - 2 years (c) 3 - 4 years

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(d) 3 - 4 years (e) 5 - 10 years (f) more than 10 years 17. Imagine that after allocating all your financial assets, your portfolio - after an initial gain - lost considerably in value. What do you do? [aftls] (a) I switch to safer assets. As I check the prices of my investments at least several times a month, I can sell quickly if they begin to lose money. (b) Daily losses in the value of my investments make me uncomfortable but do not cause me to sell immediately. If my investments suffer a substantial loss over a full quarter, however, I am likely to sell. (c) I realize there may be substantial day-to-day changes in the value of my investments. Although I focus on quarterly performance trends, I usually wait an entire year before making any changes. (d) If my investments suffered significant losses over a given year (in a down market), I would continue to follow a consistent long-term investment plan and maintain my asset mix. 18. The primary goal of my investment strategy is ... [invgol] (a) ... mainly to protect conservatively the principal value of my investments. I accept lower returns of a conservative strategy in order to minimize the danger of a loss. (b) ... long-term protection of the principal value. Daily temporary fluctuations of my portfolio value are acceptable, as I achieve higher returns with a little more risk. (c) ... long-term growth through higher exposure in the stock market. Moderate fluctuations of my portfolio value are a daily routine for me and do not bother me. (d) ... aggressive growth through maximum exposure in the stock market. Considerable fluctuations of my portfolio value are a daily routine for me and I accept them. (e) ... speculation: I constantly track gains and losses of my position trying to sell winners before a downward movement gathers momentum and trying to convert losses into money before they become substantial. 19. What role does income tax play within your investment decision? [tax]

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(a) taxes play a central role in deciding over my asset allocation. (b) taxes need to be considered, however they are not as important as other factors (such as safety of my investments). (c) taxes play a neglectable or no role for my asset allocation. 20. What is your primary source of information when making investment decisions? [isrc] (a) friends, family, relatives, own research (b) media:magazines, newspapers, advertisements (c) specialists’ journals, investment seminars or clubs (d) banks, brokers, financial advisors 21. Do you think you have the temperament and time to sit through a long downturn - sometimes maybe years of losses - without selling? [tprt] (a) no, I would sell my stocks and invest in fixed-income assets. I wouldn’t have the patience to wait. (b) yes, I wouldn’t sell my stocks even if a down market prevailed for as long as five years 22. Do you tend to sell stocks if they have performed well and made you the profit you wanted and at the same time keep the stocks that have already experienced a loss in the hope that they’d come up again? [sellos] (a) yes, I tend to keep stocks that performed badly and sell stocks that made a good profit. (b) no, my decision over buying or selling the stocks of a specific company depends solely on the economic situation of the company, analyst reports and the overall economic outlook. 23. How long could you live from your financial assets, if you stopped receiving any income from working today? [dep] (a) less than 3 months (b) 3 - 6 months (c) 6 - 12 months (d) 1 - 5 years (e) more than 5 years

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24. In planning your saving and spending, which of the time periods listed below is most important to you? [savhor] (a) next few months (b) next year (c) next few years (d) next 5 - 10 years (e) longer than 10 years 25. How high do you think are the odds of beating a money-market account with a pure stock investment in any given year from now? [perf ] (a) < 50% (b) 50% (c) > 50%

H.2.1

Small sample Internet survey

An internet survey on the above questionnaire resulted in the small sample printed on the pages 276-278. The following explanations apply to all tables and figures in this section: Question 14, the optimal stock ratio class, was chosen as the dependent variable. The variable ‘nationality’, with the key ‘[nat]’, was coded as: 1 = Switzerland, 2 = Germany, 3 = USA/Canada, 4 = Japan, 5 = India, 6 = UK, 7 = Australia, 8 = China, 9 = Portugal. The codes for all other questions correspond to the letter of the subchoice, e.g. ‘a)’ for example translates into ‘1’.

H.2.2

Calculating the predicted choice

In the Evaluation Sheet on page 281 all answers need to be multiplied with the rounded coefficients and summed up vertically. These vertical sums are the logits βjk · xik of each risk class j (where the number of individuals i = 1, 2, ...79, the number of risk classes j = 0, 1, ..., 5 and the number of factors k = 0, 1, ...14; k = 0 depicts the alternative specific constant). The predicted choice is the one with the highest calculated probability P rob[Y = j]: exp(β  x) Prob[Y=j] = J  j=0 exp(β x)

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H.2. Example of an interactive Risk Ruler

Figure H.1: Sample Data on above questionnaire collected by an Internet survey.

Chapter H. Practical implementation of a Risk Ruler

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H.2. Example of an interactive Risk Ruler

Chapter H. Practical implementation of a Risk Ruler

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Table H.1: Setting 2b - Results for Multinomial Logit model, Internet Survey for Questionnaire on Page 269, in-sample-regression. The standard errors of the regression coefficients are given in brackets. The names of the factors refer to the [keys] given in the questionnaire. The significance levels are abbreviated with asterisks: ‘*’ and ‘**’ are significant at the 5, and 1 percent level. Few estimates prove significant due to the small sample size. Observations = 79, Parameters = 15, Log-L = -40.74,

Constant NAT AGE SEX ECON FINEX AFTLS INVGOL TAX ISRC TPRT SELLOS DEP SAVHOR PERF

Prob[Y=1] -2.3393 (3.2555) 0.1163 (0.486) 1.6714 * (0.8231) 1.1775 (1.1415) -0.5189 (0.4126) 0.4082 (0.411) -0.9862 (0.7167) 1.2263 (1.0289) 0.4674 (0.8801) -0.6322 (0.4618) 0.1585 (1.0306) -0.6461 (0.9436) 0.437 (0.4084) -0.7542 (0.4348) -0.0854 (0.6832)

Prob[Y=2] -63.0035 (33.8814) -0.2131 (1.0249) 3.8266 * (1.5819) 7.2082 * (3.1556) 1.8349 (1.6637) -1.9749 (1.1658) -3.4778 (1.8582) 10.4711 (6.4943) 0.5292 (1.3102) -0.3862 (0.6645) 0.7303 (1.6275) 9.5108 (5.7451) 1.2171 (0.6751) 1.3851 (1.1408) -2.9827 (1.9595)

Prob[Y=3] -8.0629 (9.4166) -0.4891 (0.752) 2.3133 (1.3083) -0.6264 (2.797) -2.0392 (1.1158) 0.7545 (1.1107) 0.564 (1.5494) 4.3067 (2.8606) -1.6319 (2.3037) -1.4987 (0.9294) -0.6025 (2.2954) -1.2642 (2.4853) 1.4342 (0.9864) -1.9625 (1.0817) 2.9755 (1.9896)

Prob[Y=4] -1266 (97150376) 8.0704 (3811863) -84.4998 (16528327) 180.1105 (27177377) 31.4971 (4708341) -8.5671 (6745442) -12.3785 (3892892) 88.3713 (8490158) 19.8785 (8144898) 77.7727 (5824206) 69.0922 (17522556) -2.6697 (11264384) 75.1689 (5896293) 31.9328 (4581164) -9.2831 (8964300)

Prob[Y=5] -196.3973 (443980640) 9.8495 (7162340) 22.8121 (13715327) -13.5277 (101743810) -5.3666 (16386961) 7.0477 (21673740) 0.4784 (25293093) 22.3252 (47691625) 10.672 (98016705) -2.5162 (34863411) 1.1108 (39984903) -18.534 (47807819) -14.0191 (23613952) 8.2517 (16895304) 8.3509 (17744862)

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The previously estimated coefficients produced the following Classification Table for the small internet sample: Predicted 0 1 2 3 23 2 1 1 5 10 2 1 1 0 12 1 1 1 1 6 0 0 0 0 0 0 0 0 30 13 16 9 Percent correct: 78.48%

Actual 0 1 2 3 4 5

4 0 0 0 0 9 0 9

5 0 0 0 0 0 2 2

27 18 14 9 9 2 79

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Figure H.2: Evaluation Sheet for Answers on Internet Questionnaire of Page 269, for explanations please see the previous pages.

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H.2. Example of an interactive Risk Ruler

Appendix I

Empirical performance of risk classes Descriptive Statistics - Monthly time series data for the time period 1975-2000 on the Swiss market produced the following moment estimates for the 6 buy-and-hold strategies motivated in Figure 2.1. As the risky asset the MSCI Switzerland Total Return Index and as the riskfree asset the 1M EUROCHF rate was used:

0% Stocks 20% Stocks 40% Stocks 60% Stocks 80% Stocks 100% Stocks

Range Statistic 0.00956 0.08345 0.16163 0.23981 0.31799 0.39617

0% Stocks 20% Stocks 40% Stocks 60% Stocks 80% Stocks 100% Stocks

Std. Dev. Statistic 0.00224 0.00952 0.01891 0.02838 0.03788 0.04738

Minimum Statistic 0.00008 -0.05105 -0.10437 -0.15769 -0.21100 -0.26432

Maximum Statistic 0.00964 0.03240 0.05726 0.08212 0.10698 0.13185

Mean Statistic 3.45E-03 4.81E-03 6.17E-03 7.52E-03 8.88E-03 1.02E-02

Skewness Statistic 0.660 -1.121 -1.127 -1.120 -1.115 -1.111

Std. Err. 0.14 0.14 0.14 0.14 0.14 0.14

Kurtosis Statistic -0.469 5.300 5.153 5.040 4.973 4.930

283

Std. Err. 1.29E-04 5.48E-04 1.09E-03 1.63E-03 2.18E-03 2.73E-03

Std. Err. 0.28 0.28 0.28 0.28 0.28 0.28

Std.Dev. Statistic 2.24E-03 9.52E-03 1.89E-02 2.84E-02 3.79E-02 4.74E-02

284

Investment performance for 6 buy-and-hold strategies exhibiting linearly increasing risk. For the risky asset the Swiss Stock Market represented by the MSCI Switzerland Total Return Index over the period 1975-2000 was chosen. The riskless asset is the 1M EUROCHF rate. All time series in monthly data for the time period 1975-2000. The second column depicts the continuously compounded monthly returns which give the multiplicators in the third column. Lastly, the terminal wealth for an initial investment of 100 in 1975 is given.

0% Stocks 20% Stocks 40% Stocks 60% Stocks 80% Stocks 100% Stocks

Sum of ccmr 1.0433 1.4528 1.8622 2.2716 2.6810 3.0904

Multiplicator 2.83868 4.27489 6.43775 9.69490 14.59999 21.98678

Total after 25 years 284 427 644 969 1460 2199

The multiplicator is simply: exp(sum of returns). The multiplicators mpf of the mixed strategies (20% - 80% stocks) can be calculated according to mpf

=

exp(smm · pctmm ) + exp(ssm · pctsm )

=

exp(1.0433 · pctmm ) + exp(3.0904 · pctsm )

smm stands for the sum of returns of the money market investment, pctmm represents the percentage of investment in the money market, while ssm stands for sum of returns of the stock market investment and pctsm represents the percentage of investment in the stock market.

Chapter I. Empirical performance of risk classes

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Figure I.1: Distribution of returns for portfolios with stock ratios of 0% to 40%: Swiss Market, yearly data, 1925-1997, Source: Pictet.

Panel A: Stock Ratio=0% Expected return=4.4% p.a., Max.values: -4% and 15% p.a., Volatility=3.5% p.a., Probability for loss=3%, Growth over last 25 years: from 100 to 283.

Panel B: Stock Ratio=20% Expected return=5.2% p.a., Max.values: -6.5% and 20% p.a., Volatility=5.5% p.a., Probability for loss=14%, Growth over last 25 years: from 100 to 420.

Panel C: Stock Ratio=40% Expected return=5.9% p.a., Max.values: -15% and 25% p.a., Volatility=8.5% p.a., Probability for loss=19%, Growth over last 25 years: from 100 to 607.

286

Figure I.2: Distribution of returns for portfolios with stock ratios of 60% to 100%: Swiss Market, yearly data, 1925-1997, Source: Pictet.

Panel D: Stock Ratio=60% Expected return=6.6% p.a., Max.values: -23% and 31% p.a., Volatility=11.9% p.a., Probability for loss=26%, Growth over last 25 years: from 100 to 850.

Panel E: Stock Ratio=80% Expected return=7.4% p.a., Max.values: -32% and 40% p.a., Volatility=15.3% p.a., Probability for loss=26%, Growth over last 25 years: from 100 to 1’160.

Panel F: Stock Ratio=100% Expected return=8.1% p.a., Max.values: -40% and 48% p.a., Volatility=18.8% p.a., Probability for loss=27%, Growth over last 25 years: from 100 to 1’530.

Appendix J

Survey of Consumer Finances: Details The following explanations on the procedures and statistical measures used for the development of the Survey of Consumer Finances were taken from Kennickell, Starr-McCluer, and Sunden (1997): Since 1989, the questionnaires for the SCF have changed only slightly. Generally, changes have been introduced to gather additional information needed to understand other data in the survey - for example, the 1995 SCF introduced a question on uses of funds for mortgages that were taken out after the time a primary residence was purchased. Also, the major aspects of the sample design have been fixed over this time. Thus, the information obtained by the survey is comparable over 1989-95. The survey is intended to provide an adequate descriptive basis for the analysis of family assets and liabilities. To address this requirement, the SCF combines two types of samples. First, a standard multi-stage area-probability design is selected to provide good coverage of characteristics, such as home ownership, that are broadly distributed in the population. Second, a special list sample is included to oversample wealthy families, who hold a disproportionately large share of such assets as noncorporate businesses and tax-exempt bonds. This list sample is drawn from a sample of tax records made available for this purpose under strict rules governing confidentiality, the rights 287

288

of potential respondents to refuse participation in the survey, and the types of information from the interview that can be made generally available. Of the 3,906 completed interviews in the 1992 SCF, 2,456 families were from the area-probability sample and 1,450 were from the list sample; the comparable figures for the 4,299 interviews completed in 1995 are 2,780 families from the area-probability sample and 1,519 from the list sample. A very important factor in the ability to conduct surveys is the generosity of the public in giving their time for an interview. In the 1995 SCF, the average interview required 90 minutes. However, for some particularly complicated cases, the amount of time needed was substantially more than two hours. Data for the 1992 and 1995 surveys were collected by the National Opinion Research Center at the University of Chicago (NORC) between the months of June and December in each of the two years. The great majority of interviews were conducted in person, although interviewers were allowed to conduct telephone interviews if that was a better arrangement for the respondent. In the 1995 survey, one important change was the introduction of laptop computers for use in administering the questionnaire. This change increased the length of the interview a little, and it may also have had some effects on the quality of information collected. Nonetheless, the effects of the change in the mode of questionnaire administration appear to be fairly small. Errors may be introduced into survey results at many stages. Sampling error, the variability expected to occur in estimates based on a sample instead of a census, is a particularly important source of error. Such error may be reduced either by increasing the size of the sample or by designing the sample to reduce important types of variability; the latter course has been chosen for the SCF. Estimation of sampling error in the SCF is described further below. Interviewers may introduce errors, though SCF interviewers are given lengthy project-specific training to minimize this problem. In addition, computer control of the 1995 survey greatly reduced technical errors made by interviewers. Respondents may introduce errors by understanding a question in a sense different from that which was intended by the survey designers. For the SCF, extensive pretesting and other review of questions tend to reduce this source of error. Nonresponse - either complete nonresponse to the survey or nonresponse to selected items within the survey - may be another important source of

Chapter J. Survey of Consumer Finances: Details

289

error. As noted in more detail below, the SCF uses weighting adjustments to compensate for complete nonresponse. To deal with missing information on individual items, the SCF uses statistical methods to impute missing data. Response rates differ markedly in the two parts of the SCF sample. In both 1992 and 1995, about 70 percent of families selected for the area probability sample actually completed interviews. The overall response rate in the list sample was about 34 percent. Detailed analysis of the data suggests that the tendency to refuse participation in an interview is highly correlated with wealth. The response rates for both samples are low by the standards of other major government surveys. However unlike other surveys, which almost certainly also have differential nonresponse by wealthy families, the SCF sample frame provides a basis for adjusting for nonresponse by such families. To provide a measure of the frequency with which families similar to the sample families could be expected to be found in the population of all families, analysis weights are computed for each case to account for both the systematic properties of the design and for nonresponse. A major part of research by SCF staff is devoted to adjustments for nonresponse through the analysis weights for the survey. One possibility to make the key findings of a study more reliable is the trimming of the weights: weights can be adjusted to decrease the possibility that the results are overly affected by a small number of observations. Such influential observations can be detected using a graphica1 technique to inspect the underlying data. Most of the cases to be found will be holders of an unusual asset or liability or members of demographic groups for which such holdings are rare.

290

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Index Active strategies, 11 Adaptive aspiration, 121 Akaike Information Criterion, 83 Allais, Maurice, 5 Allianz Anleger Analyse, 4, 263 Arbitrage Pricing Theory, 10 Asset allocation intertemporal, 12, 41 Objective function, 163 Optimal three-moment, 169 Optimal two-moment, 167 Assets free, 15 tied, 15 Attributes Choice-, 77 Investors’, 77

Distribution Compactness of, 113 Lognormal, 114 lognormal, 142 logWeibull,EVD, 78, 171 Normal, 113 Draw-down criterion, 42 Empirical Analysis Data sample, 53 Discretizing data, 76 Hypotheses, 55 Independent factors, 16, 251, 252 Results, 85, 193, 247 Structure, 68, 179 Equity-premium puzzle, 7 Error Distance, 82

Bernoulli principle, 4, 46 Bounded rationality, 6, 44 testing for, 58 Buy-and-hold, 23

Fallacy of large numbers, 38 Gambles, 4 critique of, 5 fair, 26

Capital Market Line, 22 CAPM, 21 Certainty equivalent, 27, 130 Classification table, 81 Conditional Logit model, 77, 78

Habit models, 9 Herding, 45 Hypotheses 306

INDEX

Testing, 84 Independence from irrelevant alternatives (IIA), 78 Insurance and gambling, 125 Internet survey, 268 Self-test, 269 Intuitive Risk Rulers, 3 Joint estimation, 105, 158 Least Squares Ordinary, 73 Weighted, 73 Likelihood ratio test, 83, 247 LIMDEP7.0, 70, 83 Mean reversion, 10, 13, 41 Mean-variance approach Critique, 111 Moment trade-offs, 133 Multinomial Logit model, 77 Utility maximization, 171 Myopic loss aversion, 47 Nested Logit model, 80 Likelihood function, 175 Objective function, 70 Ordered Logit model, 75 Predictive power, 16, 97, 99 Preferences observed, 68 stated, 68 Prospect theory, 120

307

Prudence, 132 Random walk, 12 Rebalancing, 23 Returns predictable, 11 skewed, 14 Risk aversion Empirical findings, 7 absolute, 30, 43 Assessment example, 136 relative, 30, 43 Two-moment, 25, 71, 103 Risk classes Empirical performance, 283 Risk premium Markowitz, 28, 32 Pratt-Arrow, 29, 32 Three-moment, 129 Risk Quota by Fidelity, 3, 262 Safety-first, 16 Settings ‘1-4’, 68 Shortfall-approach, 15 Skewness, 115 Fisher, 109 Implementation, 141 Preference, 117, 119, 131, 135 Strategies Enhancement, 115 Strategy balanced, 35 Covered-Call, 153 Protective Put, 143 switching, 35 Survey of Consumer Finances, 51

308

INDEX

Details, 287 Time diversification, 38, 41 Time horizon effect, 12, 38 Timing, 24 Tobit model, 74 Transferability of results, 99 Two-moment risk aversion, 167 Utility cubic, 123 isoelastic, 28 quadratic, 112 Utility function Von Neumann-Morgenstern, 45 Wealth free, 15 reserved, 15

Curriculum Vitae

Name: Born: Nationalities:

Fabian Wenner 1.2.1972 in Frankfurt/Main, Germany Swiss and German

Education 1999 - 2000

1992 - 1998 1996 - 1998 1996 1982 - 1991

Visiting Fellow at the Department of Economics, Stanford University, CA, USA. Supervisor: Prof. Takeshi Amemiya. Studies of Finance and Capital Markets at the University of St.Gallen, degree: lic.oec. HSG. CEMS Studies and exams, degree: CEMS Master. Exchange Program at Copenhagen Business School, Denmark Humboldt-Gymnasium, Ulm

Practical Experience 1997 - 1999 1995 1994 1993 1991 - 1992

Research assistant for Prof. Dr. K. Spremann at the Swiss Institute for Banking and Finance, St.Gallen Mercedes Benz Italia SpA, Rome, Marketing department UBS, Zurich, FX Dealing and Treasury Deutsche Shell AG, Hamburg, law and controlling department Qualified radio operator (10wpm), German Air Force.