  This study investigates the influence of risk tolerance on the expected
utility in the long run. We estimate the extent to which the expected utility
of optimal portfolios is affected by small changes in the risk tolerance. For
this purpose, we adopt the Malliavin calculus method and the Hansen--Scheinkman
decomposition, through which the expected utility is expressed in terms of the
eigenvalues and eigenfunctions of an operator. We conclude that the influence
of risk aversion on the expected utility is determined by these eigenvalues and
eigenfunctions in the long run.
