The values of life insurance and annuity liabilities move in opposite directions in response to a change in the underlying mortality. Natural hedging utilizes this to stabilize aggregate liability cash flows. We find empirical evidence that suggests that annuity writing insurers who have more balanced business in life and annuity risks also tend to charge lower premiums than otherwise similar insurers. This indicates that insurers who have a natural hedge have a competitive advantage. In addition, we show how a mortality swap might be used to provide the benefits of natural hedging.

If future mortality improves relative to current expectations, life insurer liabilities decrease because death benefit payments will be later than expected. However, annuity writers have a loss relative to current expectations because they have to pay annuity benefits longer than expected. If the mortality deteriorates, the situation is reversed: life insurers have losses and annuity writers have gains. Natural hedging utilizes this interaction of life insurance and annuities to a change in mortality to hedge against unexpected changes in future benefit payments.

The purpose of this paper is to study natural hedging of mortality risks and to propose mortality swaps as a risk management tool. Few researchers investigate the issue of natural hedging. Most of the prior research explores the impact of mortality changes on life insurance and annuities separately, or investigates a simple combination of life and pure endowment life contracts (Frees, Carrière, and Valdez 1996; Marceau and Gaillardetz 1999; Milevsky and Promislow 2001; Cairns, Blake, and Dowd 2004). Studies on the impact of mortality changes on life insurance focus on "bad" shocks, while those on annuities focus on "good" shocks.

Wang, Yang, and Pan (2003) analyze the impact of the changes of mortality factors and propose an immunization model to hedge risks based on the mortality experience in Taiwan. However, life insurance and annuity mortality experience can be very different, so there is "basis risk" involved in using annuities to hedge life insurance mortality risk. Their model cannot pick up this basis risk.

Marceau and Gaillardetz (1999) examine the calculation of the reserves in a stochastic mortality and interest rates environment for a general portfolio of life insurance policies. In their numerical examples, they use portfolios of term life insurance contracts and pure endowment polices, similar to Milevsky and Promislow (2001). They focus on convergence of simulation results. There is a hedging effect in their results, but they do not pursue the issue.

Our paper proceeds as follows: In Section 2 we use an example to illustrate the idea of natural hedging. In Section 3, using market quotes of single-premium immediate annuities (SPIAs) from A. M. Best, we find empirical support for natural hedging. That is, insurers who have better naturally hedge mortality risks tend to have a competitive advantage over otherwise similar insurers. In Sections 4 and 5 we propose and price a mortality swap between life insurers and annuity insurers. Section 6 is the conclusion and summary.

2. Introductory Example

If mortality improves, what happens to the insurer's total liability? We know that, on average, the insurer will have a loss on the annuity business but not on the life insurance business. And if mortality declines, the effects are interchanged. This section illustrates the idea of a natural hedge.

2.1 The Portfolio

At time 0, consider an insurer's portfolio of life contingent liabilities consisting of whole life policies written on lives at ages 25, 30, 35, 40, 45, 50, 55, and 60 and single-premium immediate life annuities written on lives at ages 65, 70, 75, 80, 85, 90, 95, and 100. The interest rate is flat at 8%. Life insurance premiums and annuity benefits are paid annually. Death benefits are paid at the end of the year of death.

The net annual premium rate for one unit of whole life benefit is determined so that the present value of net premiums is equal to the present value of benefits. The annuity policy is purchased with a single payment. The expected loss of the portfolio is zero. However, this expectation is calculated under the assumption that the mortality follows the tables assumed in setting the premiums. If we replace the before-shock lifetimes with the after-shock lifetimes, what happens to the loss?

The overall mortality shock effect on an insurer is determined by its business composition, that is, the ratio of annuity business to life business. To study different outcomes under different business compositions and illustrate the idea of natural hedging, we introduce a variable r that is the ratio of annuity business to whole life business. For example, given r = 1 at time 0, if death benefits and annuity annual payments follow Table 1 for different ages, the present values of liabilities (or premiums) of whole life insurance and of annuities are equal. Suppose there is a mortality shock striking all ages evenly. We expect that an insurer with such a balanced business (where r = 1) will have the lowest volatility in cash flows. If r = 0.5 (or r = 2), it means the annuity business is half of (or twice) life business. A mortality shock in this case may impose a bigger problem in cash flow stability than when r = 1. However, in reality mortality operates within a complex framework and is influenced by socioeconomic factors, biological variables, government policies, environmental influences, health conditions, and health behaviors (Rogers 2002). Therefore, we use past mortality shocks to illustrate our idea of natural hedging.