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The Actuary The magazine of the Institute & Faculty of Actuaries

Basis risk: the long game

Jackie Li and Leonie Tickle discuss recent research results on measuring basis risk and hedge effectiveness in longevity risk transfer for pension plans


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If improvements
in capital requirements turn out to be possible, insurers and consultants can work together to explain to clients how the hedging mechanism can help achieve the best pricing

Since the first pension buy-in was transacted in 2007, there has been much development and strong growth in the longevity de-risking market. From 2011-2015, the average annual volume of pension buy-ins, buy-outs, and longevity swaps in the UK reached over £18bn, but this is still only around 1.5% of the total private sector defined benefit pension assets. From 2006-2015, the proportion of pension plan assets in bonds and other matching assets nearly doubled; replacing these with longevity risk transfer products is a natural next step for many pension plans.

While the insurer and reinsurer capacity for taking on longevity risk has recently increased, providers will need more capital to maintain sufficient supply and affordable prices to meet the strong growth in demand. Standardised, index-based longevity products could be the key to their accessing the wider capital market. 

The key feature of index-based securities is that their cashflows are linked to a selected reference or index population, rather than being tailored to the population underlying the pension plan being hedged. A notable example is the €12bn longevity swap offered by Deutsche Bank to Dutch insurer Aegon in 2012, where the index was taken as the Dutch population, and the entire trade was targeted specifically at capital market investors. 

There is a potential mismatch between the indexed hedge-based securities and the pension plan, owing to demographic differences. 

In addition, a small plan usually has high sampling variability, which makes it more likely to deviate from the index population. 

The payoff structures, in terms of amount and timing, would also be different between the indexed securities and the pension plan. All these discrepancies mean that the index-based hedge will not be perfect, resulting in residual longevity risk for the pension plan. This is referred to as longevity basis risk and is under intense research at the moment.

The major types of standardised index-based securities proposed in the longevity literature are longevity bond, longevity swap, q-forward, S-forward, K-forward, mortality option, and survivor option (Li et al. 2017). Some of these have been issued and tested in practice, with different levels of success.

In phase one of a research project on assessing longevity basis risk (Haberman et al. 2014) a decision tree framework was developed as a practical guide on how to select a two-population mortality model. It includes the M7-M5 and CAE+Cohorts models, and the characterisation approach. 

Phase two of the project, focusing on measuring longevity basis risk in realistic scenarios under practical circumstances, has recently been published (Li et al. 2017) and the key results are briefly discussed below.

Approach adopted

Broadly speaking, there are three main sources of longevity basis risk: demographic basis risk (demographic or socioeconomic differences), sampling basis risk (random outcomes of individual lives), and structural basis risk (differences in payoff structures). Figure 1 summarises how these three risk components can be modelled and addressed:

  • Demographic basis risk: process error (variability in the time series), parameter error (uncertainty in parameter estimation), and model error (uncertainty in model choice) need to be taken into account via appropriate modelling assumptions
  • Sampling basis risk: can be allowed for by using a binomial distribution to model the future number of lives
  • Structural basis risk: can be minimised by numerically optimising the weights of the index-based securities or using mortality duration matching to determine the weights. 

Figure 1	Modelling longevity basis risk
Click to download full res image (Figure 1: Modelling longevity basis risk)

The effectiveness of an index-based longevity hedge can be described as how much longevity risk is transferred away through the hedge. The remaining portion of the risk can then be seen as caused by longevity basis risk. In phase two, the level of longevity risk reduction for a longevity hedge on a pension or annuity portfolio is defined as:


The terms risk(unhedged) and risk(hedged) are the portfolio’s longevity risk before and after taking the hedge. This metric gives the proportion of the portfolio’s initial longevity risk that is removed by the hedge. Three risk measures are considered for the metric: the standard deviation, 99.5% Value-at-Risk (VaR), and 99.5% expected shortfall (conditional VaR). The 99.5% VaR is of particular interest in practice, as this concept is embedded in the assessment of the solvency capital requirement under Solvency II. 

Hypothetical cases of pension portfolios, based on UK and Australian industry datasets, were studied, where standardised longevity swaps were used to build the longevity hedge. A range of hedging scenarios were examined for each subgroup, such as a single or multiple cohorts, an open or closed pension plan, varying portfolio sizes, and different levels of precision in the use of longevity swaps. 

Results and conclusions

The major finding, from applying the two-population mortality models from phase one, is that the levels of risk reduction (based on the present value of the hedged position) are often around 50-80% for a large portfolio and usually less than 50% for a small portfolio. The precise risk reduction depends on the particular hedging scenario under consideration.

An extensive sensitivity analysis on the hedging results was carried out by changing the initial model settings and assumptions, the most important of which were found to be:

  • The coherence property; a constant ratio of mortality rates between the two populations at each age in the long run
  • Behaviour of simulated future variability, portfolio size, data size and characteristics, type of hedging instrument, simulation method, and additional model features, such as mortality structural changes. 

Some back tests and scenario tests were performed, which suggest that the hedging strategy works reasonably well when there are sizeable unanticipated mortality improvements. On the other hand, if the major mortality trends are well captured by the modelling process and these trends endure over time, the longevity hedge would not have much impact on the pension portfolio. 

When the longevity shocks are more significant for the portfolio than for the index population, owing to longevity basis risk, the reduction in the portfolio loss from the hedge would still be considerable, given that the longevity shocks on the index and the portfolio are in the same direction. 

A sensitivity analysis on time series modelling was also performed. As the data length of the portfolio is short, the feasible choices of time series processes are limited, though several modifications were still made to those assumed in phase one. The results reveal that the time series modelling assumptions in decreasing order of importance are: (1) the behaviour of simulated future variability of the book component, (2) the pace of reaching coherence, and then (3) the other correlation assumptions.

Appropriate judgment, reference materials, experts’ opinions, and thorough testing are needed in making time series modelling assumptions in practice. Further research is also required when more pensioner and annuitant data can be collected over longer periods and for different categories, such as industries, pension amounts, residential areas and others.

Potential quick guides

A qualitative assessment table or a simple ‘rule-of-thumb’ formula can be developed as a possible quick guide for practitioners and regulators to assess the effectiveness of an index-based longevity hedge. An example is illustrated in Figure 2, under which a rank of 0-10 (from mild to significant) is given to each question.

Figure 2	Qualitative assessment of longevity risk reduction

For instance, consider a hypothetical pension plan having 30,000 male pensioners over multiple cohorts. Suppose it is estimated that the plan’s longevity risk is about £250m. After investigating the underlying factors and conditions, the pension actuary has given a rank of ‘6’ to the relationship between the pension plan and the index population, a rank of ‘5’ to how fast the two populations’ mortality trends would move back in line, and a rank of ‘4’ to the possible occurrence of mortality structural changes and their potential impact on the two populations. 

So the overall risk reduction level is computed as 65% (= 50 + 6 + 5 + 4). The plan’s residual longevity risk after hedging, owing to longevity basis risk, is then about £87.5m (= 250 × 35%). 

As another example, a simple linear regression can be applied to the risk reduction estimates using 10 explanatory variables, in which the first five variables refer to the pension plan and the hedging environment, and the others reflect the model settings and assumptions. This approach takes all major factors into account simultaneously in a single equation. 

Regardless of which approach is adopted, it is important to note that these quick guides are model-dependent and are highly specific to the datasets being modelled. If suitable resources and expertise are available, practitioners are encouraged to use the detailed technical information in phase two or other relevant references to construct their own models for more accurate and specific calculations.

Future developments

This longevity risk reduction effect has the potential to allow a capital reduction for insurers, either through greater confidence in their own pricing or possibly favourable treatment from the regulator. It is of critical importance to test the index-based solutions thoroughly on more data and scenarios to identify how exactly they can help in capital assessment and pricing, and then communicate the results properly with different stakeholders. Insurers may be interested in working with consultants to discuss with the regulator what relief may be given, how to measure the impact on capital requirements, and what further steps are needed to obtain the relief.

If improvements in capital requirements turn out to be possible, insurers and consultants can work together to explain to their clients how the hedging mechanism can help achieve the best pricing. Investment banks may also exploit the findings to assist in pricing hedging products or structured investment products and help them market the products to potential investors. 

There is huge potential in what the de-risking market can finally offer. When global economic conditions improve, hedging instruments become more widely affordable, financial institutions offer more innovative products, and pension plans sponsors and capital market investors have a better understanding of longevity risk transfer, the de-risking market will have every opportunity to flourish. 

Jackie Li is an associate professor in actuarial studies at Macquarie University

Leonie Tickle is a professor and interim associate dean of the Faculty of Business and Economics at Macquarie University


Haberman et al (2014). Longevity Basis Risk: A Methodology for Assessing Basis Risk. Cass Business School, Hymans Robertson LLP, Inistitute & Faculty of Actuaries (IFoA) and Life and Longevity Markets Association (LLMA)

Li et al (2017). Assessing Basis Risk for Longevity Transactions – Phase 2. Macquarie University, IFoA and LLMA