In a financial market with one riskless asset and n risky assets whose prices follow geometric Brownian motions, we solve the problem of a pension fund maximizing the expected CRRA utility of its surplus. We consider a unique optimization problem for both the accumulation phase and the decumulation phase, and find a closed form solution to the allocation problem when the stochastic death time of the fund member is distributed as a Gompertz-Makeham random variable. We show that the optimal asset allocation during these two phases must be different. In particular, the optimal portfolio starts from the allocation prescribed by Merton's theory. Then during the first phase, the investment in the risky assets must decrease through time, while during the second phase, it must increase. Our findings also suggest that it is not optimal to manage the two phases separately, and that outsourcing of allocation decisions either during the accumulation or the decumulation phase should be avoided.

Keywords : pension fund, mortality risk, asset allocation.

JEL : G23, G11.
MSC 2000 : 62P05, 91B28, 91B30, 91B70, 93E20.