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Univariate and multivariate measures of risk aversion and risk premiums with joint normal distribution and applications in portfolio selection models

University of British Columbia

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 unidimensional 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 approximations 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 normal 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 preferences 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 operator 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 investments. 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.

Text

2010-06-30

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http://hdl.handle.net/2429/26110

Business Administration

University of British Columbia

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