Long-term portfolio management: a mathematical finance approach.

Summary We propose mathematical models applied to the management of pension funds, and more generally to the management of financial institutions with a long time horizon (30 or 40 years). In the first part we explain the optimal investment strategy for the consumption-investment problem when interest rates follow the (stochastic) dynamics proposed by Cox, Ingersoll and Ross (1985) and when the utility function is of the HARA or exponential type. The approach used is the martingale approach developed by Karatzas et al ( 1987) in the context of complete markets. We then extend the results in two directions: multifactor rate structure and more general utility functions. As an application, we consider the case of pension funds with defined contributions and a guaranteed minimum. In the second part, we focus on market incompleteness. We introduce a criterion, which we call Conditional Dominance, that allows us to obtain bounds on the value of any contingent asset. These bounds do not depend on the characteristics of the agents and in some cases they are strictly better than those obtained by over-replication. As an application, we consider the case of a defined benefit pension fund. Using the conditional dominance criterion, we can explicitly find the upper bound for the defined benefit value, as well as the hedging strategy.