Constructing Out-of-the-Money Longevity Hedges Using Parametric Mortality Indexes

Abstract

Proposed by Chan, Li, and Li, parametric mortality indexes (i.e., indexes created using the time-varying parameters in a suitable stochastic mortality model) can be used to develop tradable mortality-linked derivatives such as K-forwards. Compared to existing indexes such as the Life and Longevity Markets Association’s LifeMetrics, parametric mortality indexes are richer in information content, allowing the market to better concentrate liquidity. In this article, we further study this concept in several aspects. First, we consider options written on parametric mortality indexes. Such options enable hedgers to create out-of-the-money longevity hedges, which, compared to at-the-money-hedges created with q-/K-forwards, may better meet hedgers’ needs for protection against downside risk. Second, using the properties of the time series processes for the parametric mortality indexes, we derive analytical risk-neutral pricing formulas for K-forwards and options. In addition to convenience, the analytical pricing formulas remove the need for computationally intensive nested simulations that are entailed in, for example, the calculation of the hedging instruments’ values when a dynamic hedge is adjusted. Finally, we construct static and dynamic Greek hedging strategies using K-forwards and options, and demonstrate empirically the conditions under which an out-of-the-money hedge is more economically justifiable than an at-the-money one.

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Notes

1 For instance, the assumed model in the setup of Bauer, Börger, and Ruß (2010) may yield negative forces of mortality.

2 The logit transformation of a real number w is given by $\text{ln}\hspace{0.17em}(w/(1-w)).$

3 We fully acknowledge that the calibrated market prices of risk are subject to some limitations: (1) the longevity bond was not actually traded, so that its announced price may not truly reflect the market participants’ aversion to longevity risk and (2) the bond was announced more than 10 years ago, so the information contained in its announced price may not be up to date. To mitigate these limitations, we sensitivity test (in Section 7.3) the hedging results using a wide range of market prices of risk, including the extreme case when ${\lambda }^{(1)}={\lambda }^{(2)}=0.$

4 For simplicity, we ignore the potential lag in the availability of the index data. This assumption is made throughout the rest of the article.

5 Parameter estimation is discussed in Section 7.1.

6 We do not incorporate sampling risk (which arises from the fact that the portfolio of annuity has a finite size) into the definition of ${\mathcal{S}}_{x,t}(s)$ and subsequent analyses. This piece of uncertainty is diversifiable and is not intended to be hedged using the index-based hedging instruments.

7 The evaluation results for $L_{t_0+s},s=2,3,\dots$ are similar and therefore are not shown.

8 The 1,000 pairs of $({\kappa }_{t_0+1}^{(1)},{\kappa }_{t_0+1}^{(2)})$ are simulated under the real-world probability measure.

9 We may further incorporate recalibration (drift change) risk into the empirical distributions by (1) recalibrating the assumed model at time $t_0+1$ using the historical data plus the realized mortality experience for year $t_0+1$ and then (2) calculating the realized values of $H_{t_0+1}^{(i)}(T^{(i)},K^{(i)})$ and $L_{t_0+1}$ in the second step using the recalibrated model. We refer readers to Cairns (2013) for a discussion on recalibration risk.

10 Moneyness is defined in Equation (8)(8) $$\text{Moneyness}=\frac{K-{\mathrm{E}}_t^{(Q)}\left[{\kappa }_T^{(i)}\right]}{\sqrt{{\text{Var}}_t^{(Q)}\left[{\kappa }_T^{(i)}\right]}},$$(8) .

11 For a static cash flow hedge, the expected cost of hedging is calculated as the real-world expected value of (11) given information up to and including time t₀.

12 The 45 degree line may not pass through the origin due to a non-zero hedge cost.

13 Cairns (2011) and Zhou and Li (2017b) did not incorporate costs of hedging into their results.

14 For a dynamic cash flow hedge, the expected cost of hedging is computed as the real-world expected value of (13) given information up to and including time t₀.

15 For a value hedge, the expected cost of hedging is calculated as the negative of the real-world expected value of (16) given information up to and including time t₀.

16 VARIMA stands for vector autoregressive integrated moving-average.

17 The definitions of A, $\mathit{\mu },$ and ${\mathit{z}}_t$ are provided in Section 3.

Additional information

Funding

The authors gratefully acknowledge financial support from SCOR on the 2017 Longevity Risk Project, from which this article is extracted. This project has been implemented under the Insurance Risk and Finance Research Centre at Nanyang Business School, Singapore.