A new immunization inequality for random streams of assets, liabilities and interest rates

In this paper, we investigate the problem of immunization of insurers' surplus when liabilities are financed by a stream of assets. The term structure of interest rates is assumed to be random, as are the streams of assets and liabilities. A new inequality for changes in the portfolio surplus in response to changes in the term structure of interest rates is proven. A comparison with other immunization inequalities shows that it gives better lower bounds for a wide variety of scenarios. The inequality is sharp in the sense that the lower bound is attainable for some interest rate perturbations. Whenever net insurance premiums are considered, it is factorized into a product of two terms: one depending only on the change of interest rates, and the other depending only on the portfolio structure. Hence the second term may be treated as a measure of the interest rate risk. We call it L2-measure, because it is related to the second order distance between assets and liabilities. Explicit formulas for this measure for portfolios of some life products vs streams of net premiums are given. Applications to the Merton's, Vasicek's and simple log-normal models of interest rate are also provided.