We consider a portfolio selection problem for an investor who consumes at the end of a finite horizon. With important qualifications on the sufficiency part, we show that convergence of the optimal investment policy as the horizon becomes distant occurs if and only if the corresponding Arrow-Pratt coefficient of relative risk aversion converges as wealth increases. A major step in the proof shows that convergence of the Arrow-Pratt coefficient of relative risk aversion is equivalent to regular variation of the marginal utility function.