Actuarial applications of the linear hazard transform in mortality immunization

- Magnitude-free mortality durations and convexities are defined and derived.
- The weights of a portfolio are determined by duration/convexity matching strategies.
- The matching strategies can be used for hedging mortality/longevity risks.
- A portfolio of two products of life insurance and annuity is always feasible.
- Three-product portfolios are not always feasible under some conditions.

In this paper, we apply the linear hazard transform to mortality immunization. When there is a change in mortality rates, the respective surplus (negative reserve) changes for life insurance and annuity policies lead to oppositive sign changes, which provides mortality hedging strategies with a portfolio of life insurance and annuity policies. We first show that by the strategy of matching duration of the weighted surplus at time 0, the surplus changes at time 0 for both portfolios PTP (the n-year term life insurance and the n-year pure endowment) and PWA (the n-payment whole life insurance and the n-year deferred whole life annuity) in response to a proportional or parallel shift in the underlying force of mortality are always negative. Next, we prove that the term life insurance, the whole life insurance and the deferred whole life annuity cannot always form a feasible portfolio (feasibility means that all the weights of the product members of a portfolio are positive) by the strategy of matching two durations or one duration and one convexity of the weighted surplus at time 0. Finally, numerical examples including figures and tables are exhibited for illustrations.