In recent years there have been a number of longevity transactions by institutions wishing to protect themselves from unexpected increases in life expectancy. The vast majority of these have been bespoke transactions referencing the mortality of the specific pool of individuals for whom longevity is being hedged. In this article we focus on index-based hedges, which differ fundamentally from indemnity-based transactions in that they reference a large pool of individuals for whom mortality rates are readily available and often published.

The most commonly perceived disadvantage of index-based hedges compared to bespoke transactions is that due to basis risk. As the reference population of the index is not identical to the target population being hedged, it is expected that the hedge will not be exact when compared with a bespoke hedge. Nevertheless, this should not form a basis on which to reject the idea of hedging using indices. The reduced hedge effectiveness relative to the bespoke hedge must be assessed against the cost saving made by transacting an index-based hedge.

This is not a simple analysis but the groundwork is starting to be laid, and many similarities can be drawn with the early stages of the hedging of financial risks — the future looks promising.

**Types of basis risk**

The drivers of the mismatch between a pension fund or annuity pool’s projected liabilities and a longevity hedge that references an index can be broadly categorised into three main types of basis risk: ‘demographic risk’, ‘sampling risk’ and ‘structural risk’.

Demographic risk describes a fundamental difference in the mortality behaviour of the two populations that are being compared. It arises because the target population, such as a group of pensioners, often represents a subset (or sub-population) of the general population, on which the currently available indices are based. The sub-population can demonstrate differences in mortality patterns driven by characteristics such as socio-economic status and lifestyle.

Sampling risk, also often referred to
as Poisson risk, is simply the risk coming from the volatility in mortality rates of the target population. This risk diminishes as size of the target population increases and the law of large numbers takes over,
driving the observed mortality rates to their ‘true’ expected value.

Structural risk arises from the choice of hedge structure, for instance, in terms of maturity and the number of hedge contracts implemented. In structuring a hedge, it is not possible to perfectly match the target population’s characteristics, and choices have to be made regarding the reference age and gender, as well as the duration and payoff characteristics of the instrument to be used.

The selection of the optimal hedge also relies heavily on both the models used to simulate future mortality rates and the models used for valuation at future dates, which can introduce significant model risk. The selection and availability of data used to calibrate the models, the frequency of recalibration of the models, and parameter uncertainty can all have important impacts on the resulting calculations.

**Assessing basis risk**

Having gained an understanding of the types of basis risk arising from a hedging process, the next step is to try and quantify the risk. The possible approaches can broadly be categorised as historical, stochastic and deterministic methods of assessment.

The lack of historical data for most target hedge populations presents a major challenge to completing a basis risk analysis using an historical approach. Historical data for target populations – typically pension schemes or insurance annuity books — is often limited and contains a lot of noise.

Stochastic analysis presents another route to comparing and projecting the mortality for the two populations of interest.
However, stochastic modelling of the mortality for the target population poses a significant challenge, again due to the often limited historical data for the target population, which makes accurate model calibration difficult.

Several stochastic two-population mortality models have been developed to circumvent this issue and model the mortality for the general population and a sub-population concurrently1.

Deterministic analysis involves risk reduction quantification and scenario testing of the hedging strategy. Choice of metric is itself a fundamental component of basis risk assessment. Both longevity hedgers and their counterparties need to establish what their relevant metrics are, making sure they are aligned with the hedging objective, and assess them accordingly. If the goal of the hedge is to mitigate the variability in cashflows, a measure based on the variance of unexpected cashflows might be selected. However, if the aim is more to mitigate value and capital requirements, a Value at Risk measure might be more appropriate.

**Final thoughts**

Mounting regulation such as Solvency II is likely to provide further stimulus to the longevity hedging market in general, as well as encouraging practitioners to investigate alternative risk mitigation strategies such as index-based longevity hedging to optimise their required capital. However, in order for such a hedge to be an effective and useable form of risk management for the industry,
the basis risk has to be accounted for.
Figure 1 shows a summary of the key features of the basis risk landscape described thus far. It suggests that getting to grips with basis risk from longevity hedging with indices is no easy task.

Yet the market has historical parallels to other markets for which similar obstacles existed, and these could offer a glimpse of the future. For example, take the market for inflation risk hedging as it appeared several years ago to pension schemes and insurance companies. The available hedging tools, namely inflation swaps, appeared as unfamiliar then as do longevity derivatives to the market today. Also at that time there was often a series of possible differences between what was being hedged and the indices and hedge instruments being used.

Many European pension schemes adopted the use of European inflation indices to manage local inflation risk on the basis that general trends in those indices should be highly correlated with domestic inflation. Hedge instruments themselves (inflation swaps) didn’t precisely match the inflation-linked liabilities they were being used to hedge — often the latter had greater maturities than the swaps.

The types of basis risk summarised here were equally applicable to the problem of inflation hedging, with the exception of that due to sample size, which is arguably the easiest component of longevity basis risk to quantify, and is diversified away for larger portfolios.
These challenges were eventually confronted and financial hedging and the framework for its assessment have since become commonplace such that some degree of basis risk is the norm. There are no fundamental reasons why similar developments cannot take place for the hedging of longevity risk using index-based solutions.

Most academic work to date that has attempted to address the question of basis risk from index-based longevity hedging suggests a real benefit in terms of risk reduction. Furthermore, there is a significant body of academic research currently focusing on the subject, and solid frameworks for assessing the risk are materialising. As more practitioners take the time to translate these complex, theoretical solutions into practical applications, a broader understanding of the risk will be developed and diffused across the industry, much as it was for the market for hedging financial risks such as interest rates and inflation.

*(1) Cairns, A. et al., 2011, Bayesian Stochastic Mortality Modelling for Two Populations; Jarner, S. F. and Kryger, E. M., 2009, Modelling Adult Mortality in Small Populations:
The Saint Model; Salhi, Y.
and Loisel, S., 2010, Longevity basis risk modeling: A Co-Integration Based Approach*

Jessica Mosher (left) is a life actuary at AXA Group Risk management in Paris. Pretty Sagoo is a director in the Longevity Markets group at Deutsche Bank. Both are members of the Technical Committee of the Life and Longevity Markets Association.