Basis Risk in Index-Based Longevity Hedges: A Guide for Longevity Hedgers

Abstract

This article considers the assessment of longevity basis risk in the context of a general index-based hedge. We develop a detailed framework for measuring the impact of a hedge on regulatory or economic capital that takes population basis risk explicitly into account. The framework is set up in a way that accommodates a variety of regulatory regimes such as Solvency II as well as local actuarial practice, attempting, therefore, to bridge the gap between academia and practice. This is followed by a detailed analysis of the capital relief resulting from a hedge that uses a call spread as the hedging instrument. We find that the impact of population basis risk on capital relief (expressed in terms of a “haircut” relative to the case with no population basis risk) depends strongly on the exhaustion point of the hedge instrument. In particular, in a Solvency II setting, if the exhaustion point lies well below the 99.5% Value-at-Risk, population basis risk has a negligible impact and the haircut is zero.

ACKNOWLEDGMENTS

The authors thank Philippe Coulomb for his input and support during the process of writing this article, and Raymond Waucomont, Wilbert Ouburg, and Alan Rae for their helpful comments in the early stages of the work. All opinions expressed in the article are, however, those of the authors and not those of their employers or the contributors above.

Discussions on this article can be submitted until September 1, 2020. The authors reserve the right to reply to any discussion. Please see the Instructions for Authors found online at http://www.tandfonline.com/uaaj for submission instructions.

Notes

1 The risk transfer chain consists of a sequence of transactions, each of which results in the transfer of some measure of longevity risk from one counterparty to another; for example, first from a pension plan to an insurer and then to the capital markets.

2 Some of the process (especially those elements discussed in Sections 2 and 3) will be familar to market participants, and other elements along with the rigourous detail will be new.

3 Good documentation reflects good internal risk management practice but also helps facilitate discussions with regulators.

4 For simplicity in our later numerical examples, we do not include spouses’ pensions. Additionally, to focus on longevity risk and for consistency within the Solvency II regime, payments are not inflation linked.

5 A set of model points is a collection of hypothetical or sample insurance policies that, in aggregate, exhibit characteristics very similar to the true portfolio of insurance policies. See, for example, Richards (2010).

6 Note that we are not including the idiosyncratic risk associated with the individual random lifetimes given the simulated mortality rates in each stochastic scenario. In proportional terms, the additional idiosyncratic risk will become smaller the larger the underlying number of lives (assuming no concentration of risk by amounts).

7 Independence is assumed here for simplicity within the context of the larger framework. The broader framework can, of course, accommodate dependence between primary and secondary lives.

8 We assume here for simplicity of exposition that payments are made at the end of each year. The framework can, of course, be adapted to account for monthly or other payment patterns.

9 The L in the superscript LG is included to emphasize that the projection is for liability valuation.

10 For consistency with the standard formula approach in Solvency II, the discount curve used here is $P(0,t)/P(0,T)$ rather than P(T, t). This provides a capital requirement for longevity risk on its own, which is then separately blended in with the capital requirements for other risks, including interest rate risk.

11 Again, the LG superscript indicates that this projection is for liability valuation purposes.

12 For example, the payoff will be non-zero if X(T) exceeds AP and reaches its maximum value if X(T) exceeds EP.

13 The capital charge might, alternatively, and depending on the view of regulators, refer to the median rather than the mean of L(T).

14 Note that the indexation here to x and t contrasts with the earlier model point-oriented mortality rates such as $q_1^G(i,y),$ which are cohort specific.

15 In live situations, hedgers should take responsibility for establishing what stochastic model is appropriate for their specific population as well as their chosen general population. As a recent example, Cairns et al. (2019) develop a more general multipopulation model for 10 distinct subgroups of the Danish male population, with the underlying data serving as a testbed for longevity hedge innovators. For a review of multipopulation models for the assessment of basis risk, see Villegas et al. (2017).

16 In practice, benefits will be payable on a more regular basis and therefore more evenly spread through the year. The contractual structure of the hedge index, $\tilde{L}(T),$ reflects this simple annual payment assumption.

17 The best estimate differs from the mean of $\tilde{L}(T)$ for two reasons. First, the distribution of $\tilde{L}(T)$ might not be symmetrical. Second, and more important, the sliding window for recalibration at time T results in a distribution for the central trend after time T (ν in Eq. [15(15) $$K(t)=K(t-1)+\nu +{\sigma }_0{Z}_0(t)$$(15) ]) that is not centered around the trend estimated at time 0.

18 Note that $\mathit{VaR}(\tilde{L}(T)-H(T),0.995)$ and $\mathit{VaR}(L(T)-H(T),0.995)$ are therefore both EP – AP lower than their unhedged counterparts. From the hedger’s perspective, this represents the true release of capital. The reductions in the corresponding SCRs are, in contrast, lower because there is a matching drop of $E[H(T)]=87$ in the benchmark mean.

19 Figure 6 also shows what the outcomes would look like in the absence of population basis risk (green x’s). In this case, population basis risk is removed by ordering the simulated L(T) and $\tilde{L}(T)$ and then pairing off the ordered samples so that L(T) and $\tilde{L}(T)$ are comonotonic.

20 Note that the middle kinked section of the line of green crosses is sloping slightly upwards. This is the result of $\tilde{L}(T)$ having a higher variance than L(T).

21 See van Loon et al. (2015) for a discussion of the illiquidity premium.

22 The results of steps 8 and 9 will be used to calculate the hedge instrument payoff, and calculate liability values at T.

23 The Kannisto method carries out a linear regression on the logit $m^G(x,t)$ over ages 80 to 90, and then extrapolates the fitted line to higher ages.

24 This step is only relevant if death rates below age 40 are required in the liability valuations.

Additional information

Funding

AJGC acknowledges funding from the Actuarial Research Centre of the Institute and Faculty of Actuaries through the “Modelling Measurement and Management of Longevity and Morbidity Risk” research program. This article has been commissioned and partialy funded by Societe Generale and is co-authored by Andrew Cairns and Ghali El Boukfaoui. The views and opinions expressed in this article are those of the authors and do not necessarily reflect those of Societe Generale or any of its affiliates.