All roads lead to risk preference: a turnpike theorem for conditionally independent returns

An individual is repeatedly offered the opportunity to invest in a risky asset whose return distribution is unknown. Because the return distribution is constant over time, however, he is able to learn about that distribution by observing investment outcomes. Results are presented regarding the asymptotically optimal investment behavior of a risk-averse individual under these circumstances when his aim is to maximize the expected utility of his end-of-horizon wealth. For the class of isoelastic utility functions with constant relative risk averson less than one, the optimal investment approaches a constant proportion of wealth. The limiting proportion is the same as the proportion put up by the most optimistic individual when the investment opportunity is offered only once. These results are then extended by generalizing the class of utility functions under consideration and, at the same time, restricting the class of possible investment schemes. This paper is distinguished from the previous literature by the assumption of conditional independence of returns across periods.