Optimal consumption and investment in pooled annuity funds with and without fund managers

In this paper, we study two schemes of pooled annuity funds: pools with and without fund managers. In the pool without managers, there are no artificial rules or management cost and we introduce Nash equilibrium to depict the simultaneous non-cooperative game among participants. Given others' strategies, participants choose optimal consumption and investment strategies to maximize their own utility. In the pool with managers, the fund manager makes artificial rules for consumption and investment, ensuring cooperation among participants. The manager then determines optimal consumption withdrawal and investment strategies to maximize the total utility of all participants. By using variational methods and recursive algorithms, semi-analytical solutions in both pools are obtained. Due to prisoner's dilemma, larger pools without managers induce more intense competition in consumption and reduce utility. Thus, there exists an optimal pool size and only homogeneous participants form the pool. Meanwhile, larger pools with managers diversify investment and longevity risks more sufficiently and the subsidy effect among participants is mitigated. Therefore, the perfect pool with infinite participants is recommended as the optimal pool. Moreover, the preference between the two optimal pools depends on risk aversion degree, management cost and flexibility in selecting the strategies.