Forward Mortality Rates in Discrete Time II: Longevity Risk and Hedging Strategies

Abstract

Longevity risk has emerged as an important risk in the early 21st century for the providers of pension benefits and annuities. Any changes in the assumptions for future mortality rates can have a major financial impact on the valuation of these liabilities and motivates many of the longevity-linked securities that have been proposed to hedge this risk. Using the framework developed in Hunt and Blake (2020c), we investigate how these assumptions can change over a one-year period and the potential for hedging longevity risk in an illustrative annuity portfolio and find that relatively simple hedging strategies can significantly mitigate longevity risk over a one-year period.

ACKNOWLEDGMENTS

Work in this paper was presented at the 49th Actuarial Research Conference in Santa Barbara, CA, in July 2014; the Tenth International Longevity Conference in Santiago, Chile, in September 2014; and the Society of Actuaries Longevity Seminar in Chicago, IL, in February 2015. We are grateful to participants at these conferences for their comments and suggestions, to Andrew Cairns and Pietro Millossovich for their helpful review on an earlier draft of this article, and to Andrés Villegas for many useful discussions on this and related topics.

DISCLAIMER

This study was performed when Dr. Hunt was a Ph.D. student at Cass Business School, City University of London, and therefore the views expressed within it are held in a personal capacity and do not represent the opinions of Pacific Life Re and should not be read to that effect.

Discussions on this article can be submitted until October 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 Longevity zeros are zero-coupon bonds whose principal is proportional to the survivorship of a specified cohort of individuals.

2 We adopt the convention that the subscript on operators ${\mathbb{E}}_{\tau }(.),\mathbb{V}a{r}_{\tau }(.)$ or $\mathbb{C}o{v}_{\tau }(.)$ denotes conditioning on the information available at time τ, i.e., ${\mathcal{F}}_{\tau }.$

3 This also assumed that ${\mathbb{E}}_{\tau }\hspace{0.17em}\text{exp}\hspace{0.17em}({\mu }_{x,t})=\hspace{0.17em}\text{exp}\hspace{0.17em}({\mathbb{E}}_{\tau }{\mu }_{x,t}),$ which is the upper bound given by Jensen’s inequality. When this assumption was tested in Hunt and Blake (2020c), it was found to be reasonable across almost all ages and years of interest.

4 These can be either non-parametric in the sense of Hunt and Blake (2020a) as being one fitted without any imposing any a priori structure across ages, or parametric, in the sense of having a specific functional form, ${\beta }_x^{(i)}=f^{(i)}(x;{\theta }^{(i)}),$ selected a priori.

5 In the sense of Hunt and Blake (2020d,e) that the trends, X_t, are selected so that the projected mortality rates do not depend on any identifiability constraints imposed to fit the model in Equation (3)(3) $$\text{ln}\hspace{0.17em}({\mu }_{x,t})={\eta }_{x,t}={\alpha }_x+{\mathit{\beta }}_x^{\top }{\mathit{\kappa }}_t+{\gamma }_{t-x}$$(3) to historical data.

6 For example, the classic random walk with drift process has a constant trend, X_t = 1, with the “drift”, μ, found be regressing $\Delta {\mathit{\kappa }}_t$ on this trend. Similarly, the random walk with linear drift introduced in Hunt and Blake (2020e) has constant and linear trends, $X_t={(\begin{array}{c}1,t\end{array})}^{\top },$ with the drifts found by regressing $\Delta {\mathit{\kappa }}_t$ against X_t in a similar fashion.

7 These depend upon our identifiability constraints. In practice, we impose a set of identifiability constraints such that β = 0 to simplify matters considerably.

8 Collectively, we denote these market prices of longevity risk as $\{{\lambda }^{(j)}j=1,\dots ,N,\gamma \}.$

9 We also implicitly assume that processes governing the evolution of mortality rates are independent of other financial risks. This is in common with the majority of studies, such as Cairns, Blake, and Dowd (2006) and Bauer et al. (2008), and with the available evidence to date, as discussed in Loeys, Panigirtzoglou, and Ribeiro (2007). Although there may be some situations where longevity risk is not independent of other financial risks in the real-world measure, as in the examples of Miltersen and Persson (2005), we believe that these situations are relatively extreme and are better considered by scenario analysis rather than through a stochastic model. Furthermore, Dhaene et al. (2013) show that independence between longevity risk and financial risks in the real-world measure does not automatically ensure independence in the market-consistent measure. However, more complicated models are required in order to allow for any dependence between longevity and investment risks, which require more market information for calibration. Therefore, we believe that the assumption of independence between longevity risk and other financial risks is both necessary and justifiable at this early stage of development of the longevity risk market.

10 A similar line of reasoning can be found in Tan et al. (2014), which used the “time-invariant” property of the period functions in some mortality models to investigate the hedging of longevity risk.

11 Such as longevity zeros (based on survivorship), q-forwards (based on probabilities of death), e-forwards (based on period life expectancy), and other securities based on bespoke indices.

12 For example, the Black-Scholes stock option price is a martingale in the risk-neutral measure by construction. When performing risk management on stock options in the real-world measure, the options prices will not be martingales (in general, we would expect to see the value of a call option increase with time, because the share price is expected to grow at a faster rate than the risk-free rate).

13 Note that because the drift of the random walk process, μ, is assumed to be known, the forward mortality framework does not allow for what was termed “recalibration” risk in Cairns et al. (2013); that is, the risk that one year’s new information will cause a reappraisal of the drift term. This may have the effect of understating the risk in long-term projections of mortality rates. We leave the inclusion of recalibration risk in the framework as future work.

14 Note that we use indicator variables to remove the large outliers due to the cohort anomalies in 1919/20 and 1946/47 when estimating the time series dynamics for the cohort process. This is because we believe them to be artifacts of the data collection process (see Richards 2008; Cairns et al. 2016), rather than genuine features of mortality for these cohorts.

15 Introduced in Cairns, Blake, and Dowd (2006) and defined as “a method of reasoning used to establish a causal association (or relationship) between two factors that is consistent with existing medical knowledge.”

16 We therefore see that an annuity is equal to a portfolio of longevity zeros, as defined in Blake, Cairns, and Dowd (2006) and used in Hunt and Blake (2020c).

17 In Equation (24)(24) $$B{(\tau ,\tau +1)}_1{p}_{x,\tau }\left(1+a_{x+1}(\tau +1)\right).$$(24) and subsequently, ${}_t{p}_{x,\tau }$ is the probability that an individual aged x at τ has survived to age x + t at $\tau +t,$ which is realized at time $\tau +t$ but is a random variable before then.

18 The standard deviation of the annuity value divided by its expectation.

19 Note that the s-forward is defined on a reference cohort aged x at the inception data, $t_0\le \tau ,$ and therefore the survivorship of this cohort is a product of the observed survivorship from t₀ to τ, given by ${}_{\tau -t_0}{p}_{x,t_0},$ and the anticipated survivorship from τ to maturity, t, given by ${}_{t-\tau }{P}_{x+\tau -t_0,\tau }^{\mathbb{Q}}.$ For the purposes of this article, we assume $t_0=\tau .$

20 Typically, $q_{x,t}$ will be in the range $[0.005,0.05]$ for most ages of interest, whilst ${}_{t-t_0}{p}_{x,t_0}$ will be in the range $[0.1,0.9],$ and $e_{x,t}$ will be in the range $[10,30].$

21 We draw a slight distinction between such a strategy and purchasing a single customized asset without the intention of rebalancing the hedge in future. Examples of these customized assets include bespoke longevity swaps and highly customized bespoke options on mortality, such as those discussed in Michaelson and Mulholland (2014). However, we feel that this alternative strategy has more in common with a reinsurance policy than truly hedging risk using capital market securities.

22 For examples of using q-forwards as cashflow hedges, see Li and Luo (2012).

23 We have tested the hedging strategies using the simpler models of the short rate of mortality discussed in Hunt and Blake (2020c) and obtain even higher reductions in risk. In particular, we observed perfect correction between the liabilities and securities, and therefore perfect hedges, when using the Lee-Carter model as the underlying mortality model, because this model only possesses one age/period term and hence only one source of risk.