Longevity hedging is becoming increasingly attractive to pension schemes and life insurance companies. This is partly a result of broader moves to de-risking. However, falls in long-term interest rates have increased the impact of changing mortality improvement rates on liabilities.

Liability-specific solutions to managing longevity risk do exist, most obviously the buyout of pension liabilities by insurance companies. However, they are inflexible and involve a one-time-only decision to offload a set of liabilities. They are also capital-intensive — the treatment of longevity risk is tied to the investment decision, in particular in relation to the investments used to price the annuities.

There are also liability-specific solutions that are less capital intensive — longevity swaps. These involve the pension scheme or insurance company paying over a series of cash flows representing the expected pensions or annuities — plus a margin to reflect the risk being removed — and receiving a series of cash flows representing the actual payments made to the pensioners or annuitants. However, these solutions are available only for large portfolios of lives, so are not suitable for small pension schemes. What would be more useful for them is an index-based swap. Furthermore, such swaps would also be useful for larger institutions wanting to make short-term amendments to their levels of longevity hedging, and also for other investors wanting to take a tactical view on longevity improvements, such as hedge funds. However, such swaps require longevity indices.

**Why longevity indices are attractive**

The nature of such indices is important — any index series should be attractive enough to encourage its use by longevity market participants. But what would make an index series attractive? One criterion is that indices should be unambiguous. It is important that the reference population on which any indices are based should be defined in detail. This includes how individuals can enter and leave the index. The period over which calculations are made should also be specified, as should the method used to calculate index values.

If the raw index values are thought to contain too much random variation, then it might be desirable to smooth the results, probably across ages rather than over years. In this case, transparency is vital — the methods used to graduate mortality rates should be clear. Furthermore, any graduation methods used should be objective, with as little subjective input as possible.

The longevity experience used to construct an index should also be measurable. This means that it is possible to independently verify published index data, an important way of ensuring that any indices are credible. The experience should also be available shortly after its effective date, ensuring that indices are produced in a timely manner. This enables the prompt settlement of derivatives based on the indices. However, timeliness is not enough on its own — it is also important that indices are published regularly, in accordance with a pre-arranged timetable, since it is impossible to value derivatives based on future index values accurately without this information.

Appropriateness is also important. One measure of this is the extent to which an index follows the mortality experience of the portfolio of lives being hedged. However, a compromise is needed between the extent to which an index reflects the experience of a particular set of liabilities — with more indices providing better cover — and the range of liabilities for which a particular index provides a reasonable reflection of experience. The advantage of many indices, each of which closely reflects experience is clear: each index provides a good hedge. Having fewer indices means each set of liabilities gets a less effective hedge, but all things being equal it will increase the number of trades done on a particular index. This increased liquidity makes the few indices attractive to counterparties, thus reducing bid/offer spreads. The popularity of an index series is therefore also important.

It is also helpful to consider how appropriate a longevity index is when taking into account the volatility of a set of liabilities resulting from random variation. This measure of relevance can be assessed by comparing two measures: the variability of the liabilities being hedged relative to a tailored longevity index; and the variability of these liabilities relative to a broad population-based measure of longevity. For a longevity index to be relevant, the former measure should be lower than the latter.

Similarly, the longevity index and the liabilities being hedged should differ in similar ways from the broader population. This can be measured by comparing the correlation between two further measures: the difference between the longevity index and a broad population-based measure of longevity; and the difference between the liabilities being hedged and a broad population-based measure of longevity. The correlation should be strongly positive.

The constituents of longevity indices should be stable, changing only infrequently. Index constituents should also be specified in advance, so that derivatives based on index values in the distant future can be priced with some degree of confidence. Finally, the structure of an index should reflect the use to which the index will be put. This means that it should reflect the needs of those using such an index to hedge a set of liabilities.

**Index metrics**

This final criterion raises a number of interesting issues. The most important ones are around the number of indices to be quoted and the function of mortality rates to be used. When considering index structure, it is helpful to consider the similarity between mortality rates and interest rates. In particular, the rate of survival to a particular age is analogous to the rate of interest payable over different terms, shown in Figure 1.

When considering the rate of survival for a group of individuals of different ages, this picture must be expanded into three dimensions, as shown in Figure 2, with the third dimension being current age. This still leaves a number of potential structures for indices. The most obvious might relate to the cash flows required. However, since most longevity risk arises dues to changes in expected future mortality rates cash flow matching is not necessary. If swaps for only a few ages were purchased, say current ages 60, 70, and 80, then the proportions of these swaps could be targeted so that for a change in expected mortality rates over a given range of ages at a given point in the future, the value of the swaps and the value of the pension scheme liabilities would change by similar amounts.

A handful of terms could be chosen, say 10, 20 and 30 years into the future. This would mean that liabilities could be hedged by using single payments for each of current age 60, 70 and 80 for a future term of 10, 20 and 30 years. Even within this range not all combinations would be needed — the maximum combination of current age and future term could be limited to 90 years of age. This would leave the swaps for which indices would be required, as shown in Table 1.

The proportions of the swaps would be adjusted and combined with cash so that the sensitivity of the swap portfolio to changes in future expectations of life expectancy would be the same as that of the portfolio of lives. Having a small number of indices would also increase the liquidity in contracts based on these indices.

This leaves the function of mortality rates to be used. A swap designed for pension schemes could be designed around single payments to an individual currently aged x years at a point n years in the future. However, for a lives assured — who might provide a hedge for annuitants — it might be preferable to use the probability of survival for an individual currently aged x for n-1 years, with death occurring in the nth year. It is also worth considering a simpler metric, the probability of mortality in year n for an individual currently aged x, a ‘q-forward’.

Using a simple stochastic mortality model to test the effectiveness of these hedging structures utilising the six swaps outlined in Table 1 gives some interesting results. In particular, the pension-based swap performs consistently better than either of the other two formulations on both an annuitant portfolio and a portfolio of lives assured. The life assurance-based swap gives the next best hedge, then the q-forward. This pension-based swap performs well because the presence of a final mortality term at the end of a sequence of survival terms essentially weakens the response of the life assurance-based swap to changes in mortality improvement rates.

**Conclusion**

Longevity indices could serve a useful role in hedging longevity risk, although the characteristics of good indices are numerous. Furthermore, good hedging results can be achieved using a small number of pension-based swap contracts at key combinations of age and term. Such an approach would help to develop a liquid market in such swaps.

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The full version of this paper, together with the presentation given at ICA2010, is available at *www.ica2010.com/abstracts_details.php?abstracts=149&id=149v*

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*Paul Sweeting is a professor of Actuarial Science at the University of Kent*