  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.
