Local risk minimization, consistent interest-rate modeling, and applications to life insurance

This thesis studies local risk minimization, consistent interest-rate modeling, and their applications to life insurance.

Part I considers local risk minimization, which is one possible approach to price and hedge claims in incomplete markets. In this first part, our two main results are Propositions 3.6 and 4.3: they provide an easy way to compute locally risk-minimizing hedging strategies for common life-insurance products in discrete time and in continuous time, respectively.

Part II considers consistent interest-rate modeling; that is, interest-rate models in which a change in the yield curve can be explained by a change in the state variable, without changing the parameters of the model. In this second part, we present a single-factor interest-rate model (jointly specified under the physical and the risk-neutral probability measures), which allows for observation errors. Our main result is an algorithm to estimate the hidden values of the state variable, as well as the five parameters of our model. We also outline how our results can be extended to the multi-factor case.

Part III combines the results of Parts I and II in a numerical example. In this example, we compute a locally risk-minimizing hedging strategy for a life annuity under stochastic interest rates. We assume that the insurance company is trying to hedge this product by trading zero-coupon bonds of various maturities. Since a perfect hedge is impossible in this case, we obtain (by simulation) the distribution of the cost resulting from the ``mis-hedge''. This distribution is with respect to the physical probability measure, while most of the existing literature considers it under a risk-neutral measure.