Market pricing of longevity-linked securities

ABSTRACT

One way of mitigating longevity risk is constructing a hedge using longevity- or mortality-linked securities. A fundamental question is how to price these securities in an incomplete life market where liabilities are not liquidly traded. Although various premium principles have been developed in the literature, no consensus has been reached on the best choice to price longevity risk. This study explores the impact of mortality model uncertainty and pricing rule uncertainty on the valuation of longevity-linked securities. Twelve premium principles based on risk-neutral and real-world measures are investigated under the Lee-Carter model and the generalised CBD model. Calibration constraints are set using the quotations of UK pension annuities to incorporate the market view of longevity risk. Different premium principles and model assumptions are tested and compared based on the estimated prices of S-forwards and longevity swaps with different maturities.

KEYWORDS:

• Premium principles
• model uncertainty
• longevity-linked securities
• longevity risk

Acknowledgments

The authors would like to thank the editor and the referees for their valuable comments and suggestions, which greatly enhance the presentation of this paper.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Notes

1 One may also classify different pricing rules by the kind of method in developing each of them (Young 2006). There are three categories – Ad-hoc method, Characterisation method, and Economic method from which premium principles are derived based on intuitive grounds, required properties, and economic theories respectively. Yet, under such classification, groups may overlap with one another. For example, the Wang transform discussed in Section 2 belongs to all three methods.

2 We also obtain risk premiums from the zero-utility principle with an exponential utility function (not presented in the paper). The results are close to zero and sometimes even negative under both mortality models, which are in line with our consideration.

3 Some earlier analyses have been performed on q-forwards (Coughlan et al. 2007, Barrieu & Veraart 2016), longevity bonds (Denuit et al. 2007, Kogure & Kurachi 2010), and longevity swaps (Dowd et al. 2006, Zhou & Li 2013, Li & Tan 2018, Li et al. 2019).

4 The identification issue exists in the estimation of the model parameters ${\beta }_x$ and ${\kappa }_t$. Specifically, there are infinitely many combinations of ${\beta }_x$ and ${\kappa }_t$ resulting in the same value of $\ln (m_{x,t})$. We follow Hunt & Villegas (2015) and set $\sum _x{\beta }_x=1,\sum _t{\kappa }_t=0,\sum _c{\gamma }_c=0$.

5 Note that we denote the median absolute deviation from the median as MAD, but not the average absolute deviation from the median or mean.

6 For presentation purpose, we only show the abbreviation of each principle in the following tables and figures. Specifically, the Wang transform, proportional hazard transform, dual-power transform, Gini principle, Denneberg's absolute deviation principle, exponential transform, logarithmic transform, canonical valuation (maximum entropy), Esscher transform, standard deviation principle, variance principle, and median absolute deviation principle are denoted as Wang, PH, DP, Gini, Denne, Exp, Log, ME, Ess, sd, var, and mad, respectively. See Wang et al. (1997), Wirch and Hardy (2001), Dickson (2005) and Young (2006) for more formal proofs or counter examples for many of these properties under different premium principles.

7 See Wang (2003) for a counter example.

8 Wang (1995) proved the sub-additivity for the proportional hazard transform, which can be generalised to any distortion measures with increasing and concave distortion functions (Wang 1996). For the standard deviation principle, $\begin{array}{rl}{\mathrm{\Pi }}_{X_1+X_2}& =E[X_1+X_2]+\lambda \ast SD[X_1+X_2]\\ & =E[X_1]+E[X_2]+\lambda \ast \sqrt{VAR[X_1+X_2]}\\ & =E[X_1]+E[X_2]+\lambda \ast \sqrt{VAR[X_1]+VAR[X_2]+2\ast SD[X_1]SD[X_2]COR[X_1,X_2]}\\ & \le E[X_1]+E[X_2]+\lambda \ast \sqrt{VAR[X_1]+VAR[X_2]+2\ast SD[X_1]SD[X_2]}\\ & =E[X_1]+E[X_2]+\lambda \ast \sqrt{{(SD[X_1]+SD[X_2])}^2}\\ & =E[X_1]+\lambda \ast SD[X_1]+E[X_2]+\lambda \ast SD[X_2]={\mathrm{\Pi }}_{X_1}+{\mathrm{\Pi }}_{X_2}.\end{array}$

9 Annuity quotations were obtained from https://www.sharingpensions.co.uk. Note that the standard rates are based on Central London survival probabilities which tend to be higher than the country-wide experience. The estimated risk loading parameters may then be biased due to the potential mismatch in populations. Specifically, the simulated present values of life annuities are underestimated because of the use of country-wide mortality data. The risk loading parameters in the premium principles are estimated by matching these underestimated simulated annuity values with the London standard annuity rates, leading to an overestimation of the risk loading parameters. However, the corresponding S-forward cashflow projections and simulations are also based on the country-wide experience. As a consequence, these two effects would be ‘counterbalanced’ with each other, and the resulting S-forward forward rates can still serve as a useful proxy. Unfortunately, there is not enough data of London mortality experience to obtain more accurate London-specific S-forward forward rates.

10 Aquilina et al. (2017) investigated that the money's worth for 65-year-old male annuitants is 94% in the UK for a £50,000 pension from 2006 to 2014. We acknowledge that this information does not match with our data exactly (e.g., the size of the pension pot), while it is the most relevant one we can find as a proxy. Gallagher (2003) reported in his conference paper that the expense assumption of UK annuities is usually 1% to 3% of the total price. An analysis of the European insurance industry by Oliver Wyman (Whitworth & Byron 2012) also suggested an equal annuity margin for operation expense and for longevity risk. We borrow this information and adopt a 3% expense loading in our analysis.

11 The cohort aged 65 in 2018 is not employed, as the HMD data is available up to only 2016. Besides, the several annuity quotations from 2017 to 2018 were of comparable levels.

12 The original form is $\frac{K(T)}{{(1+r)}^T}=\frac{S^c(T)}{{(1+r)}^T}{e}^{\delta T},$ where the discounting factors can be cancelled.

13 As at 12th June 2020, the standard annuity rate for a £100,000 pension fund was £4,192 for a single life aged 60. The discount rate is set to 1%, considering the low levels of 15-year gilt rates in 2019 and 2020.

14 The baseline risk premiums refer to those calculated using the risk-free discounts rate.

15 Note that the empirical cumulative distribution is based on scenarios sorted by simulated annuity prices. The real-world probability of each scenario is equal to $(1/10000)$ (given 10,000 simulations).