# A sensitivity analysis of the long-term expected utility of optimal portfolios

## TL;DR

This paper analyzes how small changes in market factors influence the long-term expected utility of optimal portfolios for investors with constant relative risk aversion, using eigenpair analysis in factor models.

## Contribution

It introduces a method to determine long-term sensitivity of utility via eigenpairs of an operator in incomplete markets, with explicit results for specific models.

## Key findings

- Eigenpair characterizes long-term utility behavior
- Long-term sensitivity is determined by a specific eigenpair
- Explicit results provided for Kim-Omberg and Heston models

## Abstract

This paper discusses the sensitivity of the long-term expected utility of optimal portfolios for an investor with constant relative risk aversion. Under an incomplete market given by a factor model, we consider the utility maximization problem with long-time horizon. The main purpose is to find the long-term sensitivity, that is, the extent how much the optimal expected utility is affected in the long run for small changes of the underlying factor model. The factor model induces a specific eigenpair of an operator, and this eigenpair does not only characterize the long-term behavior of the optimal expected utility but also provides an explicit representation of the expected utility on a finite time horizon. We conclude that this eigenpair therefore determines the long-term sensitivity. As examples, explicit results for several market models such as the Kim--Omberg model for stochastic excess returns and the Heston stochastic volatility model are presented.