On immunization, stop-loss order and the maximum Shiu measure

Abstract

Some main immunization results are derived in a straightforward way using some well-known equivalent characterizations of the stop-loss order by equal means, also called convex order. In a special case, the conditions by Fong and Vasicek [A risk minimizing strategy for multiple liability immunization. Unpublished; Return maximization for immunized portfolios. In: Kaufmann, G.G., Bierwag, G.O., Toevs, A. (Eds.), Innovation in Bond Portfolio management: Duration Analysis and Immunization. JAI Press, London, pp. 227–238] and Shiu [Insur. Math. Econ. 7 (1988) 219], are extended to a necessary and sufficient condition for immunization under arbitrary convex shift factors of the term structure of interest rates. Based on a linear control problem with the Shiu measure as objective function, we analyze in detail the recent bounds by Uberti [Insur. Math. Econ. 21 (1997) 195] on the change in portfolio value of Shiu decomposable portfolios under α-convex and convex-β shift factors. In particular, we determine corresponding maximal and minimax bounds using notions of absolute maximum and minimax Shiu measure. Finally, we consider partial cash flow matching immunization strategies, which allow to reduce the maximum Shiu measure over a fixed period to the maximum Shiu measure over shorter periods.