Exponential utility maximization in small/large financial markets
Miklós Rásonyi, Hasanjan Sayit
TL;DR
This work studies utility-maximizing portfolios in markets where risky asset returns follow normal mean-variance mixture (NMVM) models. For exponential utility, it derives a closed-form, near-closed-form solution that reduces the optimization to a one-dimensional minimization over a parameter $\theta$, with the optimum expressed as $x^{*}= frac{1}{aW_0}[\Sigma^{-1}\gamma - q_{min}\Sigma^{-1}(\mu - r_f\mathbf{1})]$, where $q_{min}$ minimizes a function $Q(\theta)$ determined by the Laplace transform of the mixing variable $Z$. The paper extends these results to large financial markets (countably infinite assets), proving convergence of small-market optima to an overall best portfolio in the limit and establishing a two-fund separation description in the limit. It also provides a broad suite of applications and examples (GH/GIG and other NMVMs), numerical experiments, and a stochastic-dominance framework to compare frontier portfolios under mean-variance and CVaR criteria. The key contribution is a versatile, implementable approach that links mean-CVaR frontier problems under NMVM dynamics to tractable mean-variance problems with adjusted moment parameters, with rigorous results for both finite and infinite asset markets and practical guidance via CVaR-frontier computations.
Abstract
Obtaining utility maximizing optimal portfolios in closed form is a challenging issue when the return vector follows a more general distribution than the normal one. In this note, we give closed form expressions, in markets based on finitely many assets, for optimal portfolios that maximize the expected exponential utility when the return vector follows normal mean-variance mixture models. We then consider large financial markets based on normal mean-variance mixture models also and show that, under exponential utility, the optimal utilities based on small markets converge to the optimal utility in the large financial market. This result shows, in particular, that to reach optimal utility level investors need to diversify their portfolios to include infinitely many assets into their portfolio and with portfolios based on any set of only finitely many assets, they never be able to reach optimum level of utility. In this paper, we also consider portfolio optimization problems with more general class of utility functions and provide an easy-to-implement numerical procedure for locating optimal portfolios. Especially, our approach in this part of the paper reduces a high dimensional problem in locating optimal portfolio into a three dimensional problem for a general class of utility functions.