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Univariate and Multivariate Measures of Risk Aversion and Risk Premiums with Joint Normal Distribution and Applications in Portf

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Univariate and Multivariate Measures of Risk Aversion and Risk Premiums with Joint Normal Distribution and Applications in Portf UNIVARIATE AND MULTIVARIATE MEASURES OF RISK AVERSION AND RISK PREMIUMS WITH JOINT NORMAL DISTRIBUTION AND APPLICATIONS IN PORTFOLIO SELECTION MODELS by YUMING LI B.Sc, Shanghai Jiao Tong University, 1983 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE IN BUSINESS ADMINISTRATION in THE FACULTY OF GRADUATE STUDIES (The Faculty of Commerce and Business Administration) We accept this thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA September 1987 ©Yuming Li, 1987 In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Department of COMMERCE The University of British Columbia 1956 Main Mall Vancouver, Canada V6T 1Y3 Date October 1, 1987 DE-6(3/81) Abstract This thesis gives the formal derivations of the so-called Rubinstein's measures of risk aversion and their multivariate generalizations. The applications of these measures in portfolio selection models are also presented. Assuming that a decision maker's preferences can be represented by a unidimen• sional von Neumann and Morgenstern utility function, we consider a model with an uninsurable initial random wealth and an insurable risk. Under the assumption that the two random variables have a bivariate normal distribution, the second-order co- variance operator is developed from Stein/Rubinstein first-order covariance operator and is used to derive Rubinstein's measures of risk aversion from the approxima• tions of risk premiums. Rubinstein's measures of risk aversion are proved to be the appropriate generalizations of the Arrow-Pratt measures of risk aversion. In a portfolio selection model with two risky investments having a bivariate nor• mal distribution, we show that Rubinstein's measures of risk aversion can yield the desirable characterizations of risk aversion and wealth effects on the optimal portfolio. These properties of Rubinstein's measures of risk aversion are analogous to those of the Arrow-Pratt measures of risk aversion in the portfolio selection model with one riskless and one risky investment. In multi-dimensional decision problems, we assume that a decision maker's pref- - iii - erences can be represented by a multivariate utility function. From the model with an uninsurable initial wealth vector and insurable risk vector having a joint normal distribution in the wealth space, we derived the matrix measures of risk aversion which are the multivariate extension of Rubinstein's measures of risk aversion. The derivations are based on the multivariate version of Stein/Rubinstein covariance op• erator developed by Gassmann and its second-order generalization to be developed in this thesis. We finally present an application of the matrix measures of risk aversion in a portfolio selection model with a multivariate utility function and two risky invest• ments. In this model, if we assume that the random returns on the two investments and other random variables have a joint normal distribution, the optimal portfolio can be characterized by the matrix measures of risk aversion. - iv - Table of Contents Abstract ii Table of Contents iv Acknowledgements vi § 1. Introduction 1 § 2. Rubinstein's Measures of Risk Aversion and Risk Premiums 2.1 Review of the Arrow-Pratt Measures of Risk Aversion 7 2.2 The Mathematical Properties of Two Covariance Operators 10 2.3 The Derivation of Rubinstein's Measure of Absolute Risk Aversion 15 2.4 Risk Aversion with Quadratic Utility Function 24 2.5 Rubinstein's Measure of Relative Risk Aversion 29 Appendix 2.1 32 Appendix 2.2 34 § 3. An Application of Rubinstein's Measures in the Portfolio Theory 38 3.1 Formulation of the Portfolio Model 40 3.2 Necessary and Sufficient Conditions for Diversification 40 3.3 Risk Aversion and the Choice between Two Risky Investments 47 3.4 Wealth Effect on the Choice between Two Risky Investments 51 - v - 3.5 Remarks 55 § 4. Multivariate Measures of Risk Aversion and Risk Premiums 56 4.1 Introduction to the Multivariate Measures of Risk Aversion 58 4.2 The Mathemetical Properties of Two Multivariate Covariance Operators 60 4.3 The Derivation of Matrix Measures of Risk Aversion 64 4.4 Directional Matrix Measures of Risk Aversion 70 Appendix 4.1 74 § 5. An Application of Matrix Measure of Risk Aversion in the Portfolio Theory 77 5.1 Formulation of the Model 78 5.2 Risk Aversion and the Optimal Portfolio 79 References 82 - vi - Acknowledgements I am grateful to my supervisor, Professor W.T. Ziemba for his skilled direction throughout the preparation of this thesis. I would also like to thank the other com• mittee members, Professor S. Brumelle and Professor G. Sick, for their suggestions for this thesis. -1 - § 1. Introduction In an uncertain world, a decision maker's preferences for the various consequences of any action can be represented by a utility function. The von Neumann and Morgen- stern utility function was derived on the basis of a set of axioms about the decision maker's preferences. Different sets of axioms that imply the existence of utilities with the property that expected utility is an appropriate guide for consistent decision making are presented by von Neumann and Morgenstern[51], Savage[47], Luce and Raiffa[2l], Pratt, Raiffa, and Schlaifer [38] and Fishburn[10]. The axioms of util• ity theory assume that for the possible consequences of any action, either objective probabilities exist, or that the subjective probabilities are known to a decision maker. Under the expected utility hypothesis, if an appropriate utility value is assigned to each consequence and the expected utility is calculated, then the best course of action is the alternative with the highest expected utility. The utility theory is the basis of many mathematical or stochastic models in economics and finance. Consider a portfolio selection problem with two investments. We assume that an investor's preference can be represented by a von Neumann and Morgenstern utility function. He has an initial wealth w0 to be allocated to the two possibly risky investments with the objective of maximizing the expected utility of final wealth. One formulation of the problem is max U(\) = E[u(w0(\(X -Y) + Y))] where u is the investor's utility function, X and Y are (one plus) the returns on the two investments, X and 1 - A are the proportions of the initial wealth to be allocated - 2 - to the two investments, and E[-] represents the mathematical expectation of a random outcome. In the portfolio selection problem, an interesting question is to determine the relationship between the investor's preference towards risk and his optimal portfolio. Pratt[37] and Arrow[4] developed two measures which attempt to characterize these attitudes: One is the absolute risk aversion measure r(i) = -u"(x)/u'(x), (1.1) and the other is the relative risk aversion measure r*(x) = -xu'\x)/u'{x). (1.2) Under the assumption that all of the returns on the different investments are jointly normally distributed or the investor's preference can be represented by a quadratic utility function, Rubinstein[41] employed the alternative measure of risk aversion: R(W) = -E[u"{W)]/E[u\W)] (1.3) where If is a random variable representing an investor's final wealth. The other version of Rubinstein's measure[23] is R*(w0, Z) = -w0E[u"{w0Z)]/E[u'{wQZ)} (1.4) where w0 is the investor's initial wealth and Z is (one plus) the return on the investor's portfolio. If one of the two investments is riskless, the Arrow-Pratt measure of relative (absolute) measure of risk aversion is shown to give the desirable characterization of - 3 - risk aversion and wealth effect on the proportional (absolute) allocation of investments in the portfolio[37],[4],[7]. But if both of the investments are risky, as shown by Cass and Stiglitz[7], Ross[40], and Kira and Ziemba[30], the Arrow-Pratt measures of risk aversion cannot yield an unambiguous relationship between risk preferences and portfolio allocations without any additional conditions. However, under the assumption of joint normality of returns on different invest• ments, it is shown by Kallberg and Ziemba that the investors with the same Rubin• stein's measure of risk aversion (as given by 1.4) have the same optimal portfolio. The different powers of alternative risk aversion measures in determining the risk preferences in portfolio selection problems with different assumptions give rise to the following question: Why are the Arrow-Pratt measures of risk aversion useful only in the portfolio selection problem with one riskless and one risky investments, whereas Rubinstein's measures of risk aversion are powerful in the portfolio selection problem with many risky investments having joint normally distributed returns ? As we know, the Arrow-Pratt measures of risk aversion were derived from the context with only one random outcome[37]. Specifically, let ir(x,Y) be the risk pre• mium a decision maker with non-random initial wealth x is willing to pay to avoid a gamble Y, which satisfies E[u{x + Y)] = u{x + E[Y] - ir(x, Y)). (1.5) Then the Arrow-Pratt measure of absolute risk aversion r(x) could be derived from the approximation: ir(x,Y) « ^aYr(x) where the approximation is of order o(aY). The corresponding Arrow-Pratt measure - 4 - of relative risk aversion was derived in the same fashion.
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