Replicating Portfolios: $\mathscr {L}^1$ Versus $\mathscr {L}^2$ Optimization

Abstract

Currently, the major challenge in the life insurance sector is to find a numerically efficient and precise method for the estimation of the fair value of future liability cash flows. Besides least square Monte Carlo algorithms, the construction of replicating portfolios is very popular. However, there has been a debate as to how diversions between future discounted cash flows of the replicating portfolio and liabilities ought to be penalized. A frequently used argument against squared error penalization is that a few scenarios with abnormally high interest rates will cause big discrepancies between future cash flows. These scenarios will therefore dominate in the minimization with the consequence that the replicating portfolio badly approximates liabilities in the average scenario. In this article we undermine this argument by showing that the described observation will not take place when discounting with the appropriate numéraire.