In a collectivised pension fund, investors agree that any money remaining in the fund when they die can be shared among the survivors.
We compute analytically the optimal investment-consumption strategy for a fund of $n$ identical investors with homogeneous Epstein--Zin preferences, investing in the Black--Scholes market in continuous time but consuming in discrete time. Our result holds for arbitrary mortality distributions.
We also compute the optimal strategy for an infinite fund of investors, and prove the convergence of the optimal strategy as $nto infty$. The proof of convergence shows that effective strategies for inhomogeneous funds can be obtained using the optimal strategies found in this paper for homogeneous funds, using the results of [2].
We find that a constant consumption strategy is suboptimal even for infinite collectives investing in markets where assets provide no return so long as investors are "satisfaction risk-averse." This suggests that annuities and defined benefit investments will always be suboptimal investments.
We present numerical results examining the importance of the fund size, $n$, and the market parameters.