  We consider a discrete-time dividend payout problem with risk sensitive
shareholders. It is assumed that they are equipped with a risk aversion
coefficient and construct their discounted payoff with the help of the
exponential premium principle. This leads to a non-expected recursive utility
of the dividends. Within such a framework not only the expected value of the
dividends is taken into account but also their variability. Our approach is
motivated by a remark in Gerber and Shiu (2004). We deal with the finite and
infinite time horizon problems and prove that, even in general setting, the
optimal dividend policy is a band policy. We also show that the policy
improvement algorithm can be used to obtain the optimal policy and the
corresponding value function. Next, an explicit example is provided, in which
the optimal policy of a barrier type is shown to exist. Finally, we present
some numerical studies and discuss the influence of the risk sensitive
parameter on the optimal dividend policy.
