Mean-variance versus expected utility in dynamic investment analysis

Given the existence of a Markovian state price density process, this paper extends Merton’s continuous time (instantaneous) mean-variance analysis and the mutual fund separation theory in which the risky fund can be chosen to be the growth optimal portfolio. The CAPM obtains without the assumption of log-normality for prices. The optimal investment policies for the case of a hyperbolic absolute risk aversion (HARA) utility function are derived analytically. It is proved that only the quadratic utility exhibits the global mean-variance efficiency among the family of HARA utility functions. A numerical comparison is made between the growth optimal portfolio and the mean-variance analysis for the case of log-normal prices. The optimal choice of target return which maximizes the probability that the mean-variance analysis outperforms the expected utility portfolio is discussed. Mean variance analysis is better near the mean and the expected utility maximization is better in the tails.

[1]  N. H. Hakansson..  Capital Growth and the Mean-Variance Approach to Portfolio Selection , 1971, Journal of Financial and Quantitative Analysis.

[2]  David M. Kreps,et al.  Martingales and arbitrage in multiperiod securities markets , 1979 .

[3]  Etienne Pardoux,et al.  Stochastic partial differential equations, a review , 1993 .

[4]  David Heath,et al.  Coherent multiperiod risk adjusted values and Bellman’s principle , 2007, Ann. Oper. Res..

[5]  J. Cox,et al.  Optimal consumption and portfolio policies when asset prices follow a diffusion process , 1989 .

[6]  W. Ziemba,et al.  Worldwide asset and liability modeling , 1998 .

[7]  H. Markowitz,et al.  Mean-Variance versus Direct Utility Maximization , 1984 .

[8]  W. Sharpe.  CAPITAL ASSET PRICES: A THEORY OF MARKET EQUILIBRIUM UNDER CONDITIONS OF RISK* , 1964 .

[9]  W. Ziemba,et al.  Calculation of Investment Portfolios with Risk Free Borrowing and Lending , 1974 .

[10]  William T. Ziemba,et al.  Growth versus security tradeoffs indynamic investment analysis , 1999, Ann. Oper. Res..

[11]  J. Lintner.  THE VALUATION OF RISK ASSETS AND THE SELECTION OF RISKY INVESTMENTS IN STOCK PORTFOLIOS AND CAPITAL BUDGETS , 1965 .

[12]  R. C. Merton,et al.  Continuous-Time Finance , 1990 .

[13]  W. Ziemba,et al.  Stochastic optimization models in finance , 2006 .

[14]  L. Breiman.  Optimal Gambling Systems for Favorable Games , 1962 .

[15]  R. C. Merton,et al.  AN INTERTEMPORAL CAPITAL ASSET PRICING MODEL , 1973 .

[16]  Robert C. Merton,et al.  Samuelson, Paul A. , 2010 .

[17]  Suleyman Basak,et al.  A General Equilibrium Model of Portfolio Insurance , 1995 .

[18]  R. C. Merton,et al.  Optimum consumption and portfolio rules in a continuous - time model Journal of Economic Theory 3 , 1971 .

[19]  R. C. Merton,et al.  Lifetime Portfolio Selection under Uncertainty: The Continuous-Time Case , 1969 .

[20]  Klaus Reiner Schenk-Hoppé,et al.  Evolutionary stability of portfolio rules in incomplete markets , 2005 .

[21]  Stephen A. Ross,et al.  THE CURRENT STATUS OF THE CAPITAL ASSET PRICING MODEL (CAPM) , 1978 .

[22]  Sanford J. Grossman,et al.  Equilibrium Analysis of Portfolio Insurance , 1996 .

[23]  T. Cover,et al.  Asymptotic optimality and asymptotic equipartition properties of log-optimum investment , 1988 .

[24]  N. H. Hakansson.,et al.  On the use of mean-variance and quadratic approximations in implementing dynamic investment strategies: a comparison of returns and investment policies , 1993 .

[25]  W. Ziemba,et al.  Growth versus security in dynamic investment analysis , 1992 .

[26]  P. Samuelson.  General Proof that Diversification Pays , 1967, Journal of Financial and Quantitative Analysis.

[27]  J. Tobin.  Liquidity Preference as Behavior towards Risk , 1958 .

[28]  William N. Goetzmann,et al.  Active Portfolio Management , 1999 .

[29]  Stanley R. Pliska,et al.  A Stochastic Calculus Model of Continuous Trading: Optimal Portfolios , 1986, Math. Oper. Res..

[30]  William T. Ziemba,et al.  Capital Growth Theory , 2011 .

[31]  J. Kallberg,et al.  Comparison of Alternative Utility Functions in Portfolio Selection Problems , 1983 .

[32]  A Comparison of Growth Optimal and Mean Variance Investment Policies , 1981 .

[33]  William T. Ziemba,et al.  A stochastic programming model using an endogenously determined worst case risk measure for dynamic asset allocation , 2001, Math. Program..

[34]  M. David.  HARRISON, J. Michael, and KREPS, . Martingales and Arbitrage in Multiperiod Securities Markets, Journal of Economic Theory, , . , 1979 .

[35]  J. Harrison,et al.  Martingales and stochastic integrals in the theory of continuous trading , 1981 .