Abstract
In the paper we consider two types of utility functions often used in portfolio allocation problems, i.e. the exponential utility and the quadratic utility. We link the resulting optimal portfolios obtained by maximizing these utility functions to the corresponding optimal portfolios based on the minimum value-at-risk (VaR) approach. This allows us to provide analytic expressions for the risk aversion coefficients as functions of the VaR level. The results are initially derived under the assumption that the vector of asset returns is multivariate normally distributed and they are generalized to the class of elliptically contoured distributions thereafter. We find that the choice of the coefficients of risk aversion depends on the stochastic model used for the data generating process. Finally, we take the parameter uncertainty into account and present confidence intervals for the risk aversion coefficients of the considered utility functions. The theoretical results are validated in an empirical study.
Highlights
The von Neumann–Morgenstern expected utility theory (see, von Neumann and Morgenstern (1944)) is often used to find the optimal portfolio weights
The results are initially derived under the assumption that the vector of asset returns is multivariate normally distributed and they are generalized to the class of elliptically contoured distributions thereafter
We argue that the risk aversion coefficient can be linked to the distribution used as a model for the data generating process and to the level of the value-at-risk (VaR) in which the investor is interested in or is required to report
Summary
The von Neumann–Morgenstern expected utility theory (see, von Neumann and Morgenstern (1944)) is often used to find the optimal portfolio weights. The authors show that using VaR as an additional constraint within a mean-variance portfolio strategy clearly reduce the impact of estimation risk in the parameters of asset returns. Alexander and Baptista (2011) derived an explicit expression for the implied risk aversion coefficient as a function of the VaR confidence level. This framework allows to incorporate the mental accounts and objectives of investors. Note that in both papers the authors assume Gaussian asset returns and the quadratic utility function.
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