Hedging Annuity Risks with the Age-Period-Cohort Two-Population Gravity Model

Abstract

We consider the effectiveness of an illustrative annuity hedging problem in which a forward annuity predicated on one population is hedged by a position in a forward annuity predicated on another population. Our analysis makes use of the age-period-cohort two-population gravity model that takes account of the observed interdependence between the two populations’ mortality rates; it also considers the implications of parameter uncertainty, individual death or Poisson risk, and interest-rate risk for hedge effectiveness. We consider horizons of up to 20 years. For the most part, our results are robust and indicate strong hedge effectiveness, with estimates of relative risk reduction varying from about 70% in the least effective case to well over 95% in the most effective cases.

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Notes

1 Alternatively, we could have used the Markov-chain Monte Carlo (MCMC) two-population model of Cairns et al. (2011a) or the cointegration model of Li and Hardy (2011). An analysis of model risk would be a natural extension to the analysis presented here.

2 Our work also complements the earlier analyses of Coughlan et al. (2011) and Cairns et al. (2014), both of which have considered hedging problems somewhat similar to those considered here.

3 By Poisson death risk, we mean the random variation risk associated with death rates in a small sample of the overall population. If the sample is small, as will be the case in a small pension fund, the realized pattern of deaths could differ significantly from that of the national population. Since we assume that actual deaths follow a Poisson distribution, we call the random variation risk “Poisson risk.”

4 See Coughlan et al. (2004) for more details of optimizing hedge effectiveness in a general setting and Coughlan et al. (2007a, 2007b) for the first applications to longevity hedging.

5 We take these to be the 95% expected shortfalls (ES) relative to the median, but the standard deviation or the Value at Risk (VaR) relative to the median would also do and would appear to give much the same results, presumably because of the underlying assumption of bivariate Gaussianity. Similar results also hold for different prediction intervals.

6 We say “at best” because simulation exercises suggest that the optimal hedge ratio is indeed zero; other hedge ratios appear to produce RRRs that are negative. This also makes intuitive sense: Adding a substantial position to the first position that is uncorrelated with the first position merely adds noise.

7 These values are taken from Dowd et al. (2011b).

8 See llma.org

9 For example, we might hedge each of the two annuities using bonds of the same value and duration. For more on fixed-interest hedging, see, e.g., Fabozzi (2000).

10 Such possibilities include rapid inflation, which would undermine our interest rate assumptions, or mass defaults, which might cause the institution that issued the hedge to itself default.

11 For more details of all these transactions, see Blake et al. (2018).

12 See Hunt and Blake (2018) for more details.

13 Another consideration is that having extra Poisson risk in the E(t, x) as well as D(t, x) would not have a significant impact on the D(t, x) Poisson randomness, except perhaps at the high ages and as E(t, x) gets very small.

14 For example, with 1000 simulation paths in each stage for each of the mortality state variables, this would require 2 million simulation paths for the mortality state variables alone; combined with all the other calculations required, this implies a computational burden that is not practically feasible under real-time constraints.

15 Cairns (2007) actually suggests a second-order Taylor series expansion, but we use a first-order expansion here because the Δ parameters of the second population are sometimes very unstable when we allow for Poisson risk.

16 To be precise, if $R(T)$ follows a CIR process, then $(4\alpha R(T))/\{{\sigma }^2(1-\hspace{0.17em}\text{exp}[-\alpha T])\}$ has a noncentral chi-squared distribution with $4\alpha \overline{R}/{\sigma }^2$ degrees of freedom and a noncentrality parameter equal to $(4\alpha R(0))/\{{\sigma }^2(1-\hspace{0.17em}\text{exp}[-\alpha T])\}$ (Cairns 2004, Theorem 4.8 (c)).

17 Estimating (D3) for T = 0 gives plausible $${\Delta }_1^{(i)}$$ results for our dataset, but in principle any T value would do, provided the $${\Delta }_1^{(i)}$$ values are plausible and fairly stable.