Theory and Validation of Replicating Portfolios in Insurance Risk Management

Abstract

The Solvency II framework challenges insurers to evaluate and manage their embedded balance sheet risks appropriately. However, insurances hold balance sheet items, for which closed-form solutions and market prices are not available. Pure Monte Carlo valuation requires nested simulations, which are too time-intensive. Therefore, methods that project these balance sheet items into functional representations, which simplify and enhance risk analysis, have been suggested. Among these, replicating portfolios are widely applied in practice, though their validity and properties have not been fully examined yet. This paper corrects this shortcoming and proposes a mathematical framework within which the asymptotic properties of replicating portfolios are analyzed. It is shown that the replicating portfolio problem is mathematically well-defined and asymptotically converges to the true solution. Hence, this paper provides a general mathematical validation for replicating portfolios applied in insurance. A typical path-dependent insurance policy is discussed within the framework and numerical results are presented.