Forward Mortality Rates in Discrete Time I: Calibration and Securities Pricing

Abstract

Many users of mortality models are interested in using them to place values on longevity-linked liabilities and securities. Modern regulatory regimes require that the values of liabilities and reserves are consistent with market prices (if available), though the gradual emergence of a traded market in longevity risk needs methods for pricing new types of longevity-linked securities quickly and efficiently. In this study, we develop a new forward mortality framework to enable the efficient pricing of longevity-linked liabilities and securities in a market-consistent fashion. This approach starts from the historical data of the observed mortality rates, i.e., the force of mortality. Building on the dynamics of age/period/cohort models of the observed force of mortality, we develop models of forward mortality rates and then use a change of measure to incorporate whatever market information is available. The resulting forward mortality rates are then used to value a number of different longevity-linked securities, such as q-forwards, s-forwards, and longevity swaps.

ACKNOWLEDGMENTS

Work in this paper was presented at the 49h 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.

APPENDIX A. IDENTIFIABILITY AND MORTALITY FORWARD RATES

In Hunt and Blake (2020d) and Hunt and Blake (2020e), we discuss the identifiability issues in AP and APC mortality models, respectively. In particular, we find that almost all APC mortality models possess “invariant” transformations, i.e., transformations of the parameters of the model which leave the fitted mortality rates unchanged. In order to find a unique set of parameters, we impose a set of identifiability constraints on them. Typically, these are chosen to give a particular demographic significance to each term in the model. However, since any interpretation of demographic significance is subjective, it is important that our choice of identifiability constraints does not have any impact on any conclusions we draw about historical or projected mortality rates. For instance, we discuss in Hunt and Blake (2020d,e) how to ensure that projected force of mortality is independent of the choice of identifiability constraint.

It is also important that the forward mortality rate framework described in this study is independent of the choice of identifiability constraints used when fitting the underlying APC model to historical data. However, due to our definitions of the forward mortality rates in Equation (11)(11) $${\nu }_{x,t}^{\mathbb{Q}}(\tau )={\mathbb{E}}^{\mathbb{Q}}{}_{\tau }{\mu }_{x,t}.$$(11) , we see that ${\nu }_{x,t}^{\mathbb{P}}(\tau )$ in the real-world measure is automatically independent of the identifiability constraints if the distribution of ${\mu }_{x,\tau }$ is also independent of the identifiability constraints. We therefore do not need to do any additional work to ensure identifiability in the forward rates once the methods used to project the force of mortality are well-identified.

We also need to ensure that the forward mortality surface in the market-consistent measure is also independent of the choice of arbitrary identifiability constraints. This is mostly straightforward, as we see that Equation (31)(31) $$\begin{array}{c}{\nu }_{x,t}^{\mathbb{Q}}(\tau )={\mathbb{E}}^{\mathbb{Q}}{}_{\tau }{\mu }_{x,t}\\ ={\mathbb{E}}^{\mathbb{Q}}{}_{\tau }\hspace{0.17em}\text{exp}\hspace{0.17em}({\eta }_{x,t})\\ =\frac{{\mathbb{E}}^{\mathbb{P}}{}_{\tau }\hspace{0.17em}\text{exp}\hspace{0.17em}(-Z_{x,t}{\eta }_{x,t})}{{\mathbb{E}}^{\mathbb{P}}{}_{\tau }\hspace{0.17em}\text{exp}\hspace{0.17em}(-Z_{x,t})}\\ =\frac{{\mathbb{E}}^{\mathbb{P}}{}_{\tau }\hspace{0.17em}\text{exp}\hspace{0.17em}({\alpha }_x+{\left({\mathit{\beta }}_x-\mathit{\lambda }\right)}^{\top }{\mathit{\kappa }}_t+(1-{\lambda }^{(\gamma )}){\gamma }_{t-x})}{{\mathbb{E}}^{\mathbb{P}}{}_{\tau }\hspace{0.17em}\text{exp}\hspace{0.17em}(-{\mathit{\lambda }}^{\top }{\mathit{\kappa }}_t-{\lambda }^{(\gamma )}{\gamma }_{t-x})}\\ =\hspace{0.17em}\text{exp}\hspace{0.17em}({\alpha }_x+{\mathit{\beta }}_x^{\top }{\mathbb{E}}^{\mathbb{P}}{}_{\tau }{\mathit{\kappa }}_t+\frac{1}{2}{\mathit{\beta }}_x^{\top }\mathbb{V}a{r}_{\tau }^{\mathbb{P}}({\mathit{\kappa }}_t){\mathit{\beta }}_x+{\mathbb{E}}^{\mathbb{P}}{}_{\tau }{\gamma }_{t-x}\\ \hspace{1em}+\frac{1}{2}\mathbb{V}a{r}_{\tau }^{\mathbb{P}}({\gamma }_{t-x})-\frac{1}{2}{\mathit{\beta }}_x^{\top }\mathbb{V}a{r}_{\tau }^{\mathbb{P}}({\mathit{\kappa }}_t)\mathit{\lambda }-\frac{1}{2}{\mathit{\lambda }}^{\top }\mathbb{V}a{r}_{\tau }^{\mathbb{P}}({\mathit{\kappa }}_t){\mathit{\beta }}_x-{\lambda }^{(\gamma )}\mathbb{V}a{r}_{\tau }^{\mathbb{P}}({\gamma }_{t-x}))\\ =\hspace{0.17em}\text{exp}\hspace{0.17em}\left(-{\mathit{\beta }}_x^{\top }\mathbb{V}a{r}_{\tau }^{\mathbb{P}}({\mathit{\kappa }}_t)\mathit{\lambda }-{\lambda }^{(\gamma )}\mathbb{V}a{r}_{\tau }^{\mathbb{P}}({\gamma }_{t-x})\right){\nu }_{x,t}^{\mathbb{P}}(\tau )\end{array}$$(31) depends upon the forward mortality rates in the real-world measure (which should be independent of the identifiability constraints for the reasons discussed above), the variances of the period and cohort functions (which are independent of the allocation of any levels and linear trends if the projection methods are well-identified, as discussed in Hunt and Blake [2020e]) and the market prices of longevity risk. However, we note that if the model is transformed using $$\{{\widehat{\mathit{\beta }}}_x,{\widehat{\mathit{\kappa }}}_t\}=\{{\left(A^{-1}\right)}^{\top }{\mathit{\beta }}_x,A{\mathit{\kappa }}_t\},$$ then the market prices of risk are also transformed in the model to $\widehat{\mathit{\lambda }}={(A^{-1})}^{\top }\mathit{\lambda }.$ Hence we see that, not only are the values of the market prices of risk dependent upon the underlying APC model used for the force of mortality, they will also depend upon the normalisation scheme and specification of the age function in the model, and so are not the same across all models which give the same fitted mortality rates.

APPENDIX B. IMPACT OF JENSEN’S INEQUALITY

In Section 2.2, it was argued that (35) $$\begin{array}{c}{}_t{P}_{x,\tau }={\mathbb{E}}_{\tau }\left[\text{exp}\hspace{0.17em}\left(-\sum _{u=1}^t{\mu }_{x+u,\tau +u}\right)\right]\\ \approx \hspace{0.17em}\text{exp}\hspace{0.17em}\left(-\sum _{u=1}^t{\mathbb{E}}_{\tau }{\mu }_{x+u,\tau +u}\right),\end{array}$$(35) due to the relatively low degree of variability in ${\mu }_{x,t},$ and hence it was shown in Section 2.2 that $${\nu }_{x,t}(\tau )\approx {\mathbb{E}}_{\tau }{\mu }_{x,t}.$$ This assumption can be tested numerically, as follows.

For simplicity, we consider $P_{x,t}={\mathbb{E}}_{\tau }\hspace{0.17em}\text{exp}\hspace{0.17em}(-{\mu }_{x,t}).$ Therefore $$P_{x,t}={\mathbb{E}}_{\tau }\hspace{0.17em}\text{exp}\hspace{0.17em}\left(-\hspace{0.17em}\text{exp}\hspace{0.17em}\left({\eta }_{x,t}\right)\right)$$ In Section 2.3, we assume that $${\eta }_{x,t}\sim N({\mathcal{M}}_{x,t},{\mathcal{V}}_{x,t}),$$ and therefore (36) $${\mathbb{E}}_{\tau }\hspace{0.17em}\text{exp}\hspace{0.17em}(-{\mu }_{x,t})\approx \hspace{0.17em}\text{exp}\hspace{0.17em}\left(-{\mathbb{E}}_{\tau }{\mu }_{x,t}\right)=\hspace{0.17em}\text{exp}\hspace{0.17em}\left(-\hspace{0.17em}\text{exp}\hspace{0.17em}\left({\mathcal{M}}_{x,t}+0.5{\mathcal{V}}_{x,t}\right)\right).$$(36) Holland and Ahsanullah (1989) discussed the log-log distribution, where X is such that $$\text{ln}\hspace{0.17em}(-\hspace{0.17em}\text{ln}\hspace{0.17em}(X))\sim N(\mathcal{M},\mathcal{V}).$$ We therefore see that $P_{x,\tau }(\tau )$ is given by the mean of the log-log distribution if ${\eta }_{x,t}$ is normally distributed. However, the moments of this distribution do not have a closed form solution. Holland and Ahsanullah (1989) showed that the r^th raw moment of the distribution is given by $$\mathbb{E}{X}^r=\frac{1}{\sqrt{2\pi }}{\int }_{-\infty }^{\infty }\hspace{0.17em}\text{exp}\hspace{0.17em}\left(-0.5x^2-r\hspace{0.17em}\text{exp}[\mathcal{M}+x\sqrt{\mathcal{V}}]\right)dx,$$ which can be computed numerically.

From Section 2.3, we see $$\begin{array}{c}{\mathcal{M}}_{x,t}={\alpha }_x+{\mathit{\beta }}_x^{\top }{\mathbb{E}}_{\tau }{\mathit{\kappa }}_t+{\mathbb{E}}_{\tau }{\gamma }_{t-x}\\ {\mathcal{V}}_{x,t}={\mathit{\beta }}_x^{\top }\mathbb{V}a{r}_{\tau }({\mathit{\kappa }}_t){\mathit{\beta }}_x+\mathbb{V}a{r}_{\tau }({\gamma }_{t-x}).\end{array}$$ Hence we can use the results of Holland and Ahsanullah (1989) to compute $P_{x,t}$ numerically, without recourse to the approximation in Equation (36)(36) $${\mathbb{E}}_{\tau }\hspace{0.17em}\text{exp}\hspace{0.17em}(-{\mu }_{x,t})\approx \hspace{0.17em}\text{exp}\hspace{0.17em}\left(-{\mathbb{E}}_{\tau }{\mu }_{x,t}\right)=\hspace{0.17em}\text{exp}\hspace{0.17em}\left(-\hspace{0.17em}\text{exp}\hspace{0.17em}\left({\mathcal{M}}_{x,t}+0.5{\mathcal{V}}_{x,t}\right)\right).$$(36) . Using this, we calculate (37) $$\begin{array}{c}{P}_{x,t}={\mathbb{E}}_{\tau }\hspace{0.17em}\text{exp}\hspace{0.17em}(-{\mu }_{x,t})\\ =\frac{1}{\sqrt{2\pi }}{\int }_{-\infty }^{\infty }\hspace{0.17em}\text{exp}\hspace{0.17em}\left(-0.5z^2-\hspace{0.17em}\text{exp}[{\mathcal{M}}_{x,t}+x\sqrt{{\mathcal{V}}_{x,t}}]\right)dz,\end{array}$$(37) numerically and compare it with the values assumed in Equation (36)(36) $${\mathbb{E}}_{\tau }\hspace{0.17em}\text{exp}\hspace{0.17em}(-{\mu }_{x,t})\approx \hspace{0.17em}\text{exp}\hspace{0.17em}\left(-{\mathbb{E}}_{\tau }{\mu }_{x,t}\right)=\hspace{0.17em}\text{exp}\hspace{0.17em}\left(-\hspace{0.17em}\text{exp}\hspace{0.17em}\left({\mathcal{M}}_{x,t}+0.5{\mathcal{V}}_{x,t}\right)\right).$$(36) . This gives us a check on the accuracy of the approximation in Equation (36)(36) $${\mathbb{E}}_{\tau }\hspace{0.17em}\text{exp}\hspace{0.17em}(-{\mu }_{x,t})\approx \hspace{0.17em}\text{exp}\hspace{0.17em}\left(-{\mathbb{E}}_{\tau }{\mu }_{x,t}\right)=\hspace{0.17em}\text{exp}\hspace{0.17em}\left(-\hspace{0.17em}\text{exp}\hspace{0.17em}\left({\mathcal{M}}_{x,t}+0.5{\mathcal{V}}_{x,t}\right)\right).$$(36) , which underpins the forward mortality framework.

Figure B.1 shows the ratio of the numerical value of $P_{x,t}$ calculated using Equation (37)(37) $$\begin{array}{c}{P}_{x,t}={\mathbb{E}}_{\tau }\hspace{0.17em}\text{exp}\hspace{0.17em}(-{\mu }_{x,t})\\ =\frac{1}{\sqrt{2\pi }}{\int }_{-\infty }^{\infty }\hspace{0.17em}\text{exp}\hspace{0.17em}\left(-0.5z^2-\hspace{0.17em}\text{exp}[{\mathcal{M}}_{x,t}+x\sqrt{{\mathcal{V}}_{x,t}}]\right)dz,\end{array}$$(37) and the approximate value calculated using Equation (36)(36) $${\mathbb{E}}_{\tau }\hspace{0.17em}\text{exp}\hspace{0.17em}(-{\mu }_{x,t})\approx \hspace{0.17em}\text{exp}\hspace{0.17em}\left(-{\mathbb{E}}_{\tau }{\mu }_{x,t}\right)=\hspace{0.17em}\text{exp}\hspace{0.17em}\left(-\hspace{0.17em}\text{exp}\hspace{0.17em}\left({\mathcal{M}}_{x,t}+0.5{\mathcal{V}}_{x,t}\right)\right).$$(36) for the five mortality models considered in this paper (in the real-world measure). We can assume that in the vast majority of cases, the difference that the assumption makes is less than 0.2% (i.e., ratios less than 1.002) and for no ages and years does the approximation make more than a 1.5% difference to the forward mortality rates. This is consistent with the results of a simulation exercise, which showed that forward mortality rates estimated using Monte Carlo simulations were within the sampling error of those calculated using the algebraic approximation.

The mortality rates which are most affected by the approximation are those at the highest ages and the years of projection furthest into the future, which makes sense as these are the mortality rates with the greatest levels of uncertainty attached to them. However, they are also the least economically important, since any cashflows that would be affected by these mortality rates would be in respect of individuals who are very old (and so there is very little survivorship to these ages) and far into the future (which means that the present value of the affected cashflows would be very small due to discounting). This gives us reassurance that the approximation in Equation (35)(35) $$\begin{array}{c}{}_t{P}_{x,\tau }={\mathbb{E}}_{\tau }\left[\text{exp}\hspace{0.17em}\left(-\sum _{u=1}^t{\mu }_{x+u,\tau +u}\right)\right]\\ \approx \hspace{0.17em}\text{exp}\hspace{0.17em}\left(-\sum _{u=1}^t{\mathbb{E}}_{\tau }{\mu }_{x+u,\tau +u}\right),\end{array}$$(35) does not systematically distort the results found using the forward mortality framework derived in this paper, compared with those which could be found using an exact but considerably more complicated framework which does not make this assumption.

Notes

1 Demographic significance is defined in Hunt and Blake (2020a) as the interpretation of the components of a model in terms of the underlying biological, medical, or socio-economic causes of changes in mortality rates that generate them.

2 These can be non-parametric in the sense of being one fitted without imposing any a priori shape for the function across ages or be parametric in the sense of having a specific functional form, ${\beta }_x^{(i)}=f^{(i)}(x;{\theta }^{(i)}),$ selected a priori. Potentially, parametric age functions can have free parameters ${\theta }^{(i)}$ that are set with reference to the data.

3 The forward mortality framework described in this study is not significantly affected if the cohort parameters are modulated by an age function, ${\beta }_x^{(0)},$ as in the model of Renshaw and Haberman (2006). However, for simplicity and the reasons discussed in Hunt and Blake (2020a), we do not consider such models in this study.

4 $p_{x,t}=1-q_{x,t},$ the one-year probability of death.

5 ${}_0{p}_{x,\tau }=1$ trivially.

6 In this article, we use the term “security” to refer to any tradable financial contract, and this also includes derivative securities such as forwards and options in this definition.

7 Longevity zeros were also used to define forward mortality rates in Barbarin (2008) for use in a Heath-Jarrow-Morton framework and in Cairns (2007) and Alai, Ignatieva, and Sherris (2013) to develop extensions of the Olivier-Smith model.

8 It is important that the security used to define the forward mortality rates depends purely on the systematic component of longevity risk, rather than on the idiosyncratic time of death of any individual lives, in order to avoid the potential for conflicting definitions of the forward rates described in Norberg (2010).

9 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; that is, ${\mathcal{F}}_{\tau }.$

10 This approximation is tested numerically in Appendix B.

11 It is also true for the valuation of annuities for reserving purposes, since idiosyncratic risk is not allowed for in this context.

12 Note that, if we were using ${\eta }_{x,t}=\text{logit}(q_{x,t})$ in conjunction with a binomial model for the death count, then $q_{x,t}$ would follow a “logit-normal” distribution (see Frederic and Lad 2008). Unfortunately, this is not analytically tractable and does not possess closed-form expressions for the expectation. Therefore, we are unable to define a forward mortality framework in the logit-link function/binomial death count model as we can in the log-link function/Poisson death count model.

13 Introduced in Cairns, Blake, and Dowd (2006a) 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.”

14 Note that we assume that the drifts μ are known at time τ and will not be re-estimated on the basis of new information arising in the future. Therefore, the forward mortality framework described in this article and in Hunt and Blake (2020c) does not allow for “recalibration” risk as defined in Cairns (2013); that is, the risk caused by the uncertainty in the drift. This risk is potentially substantial, as discussed in N. Li, Lee, and Tuljapurkar (2004) and J. Li (2014).

15 In general, these have a similar form to the deterministic functions for the period parameters, X_t, in Section 2.4.1.

16 Typically, cohort parameters for the last few years of birth are not estimated due to the lack of data; for instance, see Renshaw and Haberman (2006).

17 Note that the drifts, β, depend upon the arbitrary identifiability constraints chosen. In practice, we therefore impose a set of identifiability constraints such that β = 0 to simplify matters considerably.

18 That is, the simplification of the main model discussed in Plat (2009) without the third, high-age term or, equivalently, an extension of the CBDX model with a cohort term.

19 See Hunt and Blake (2014) for full details of the construction of the GP model. For all models, we also select age functions that are normalized so that $\sum _x\left|{\beta }_x\right|=\sum _x|f(x)|=1.$ This involves either including normalization constants or choosing age functions that are “self-normalizing” in the sense of Hunt and Blake (2020e). However, for clarity, these are not shown, although they are taken into account in the fitting algorithms.

20 In a sense, the difference between the external and internal markets for longevity risk could be compared to the difference between using mark-to-market and mark-to-model valuation methods when valuing securities in company accounts, depending upon whether deep and liquid markets exist for them.

21 The use of the CMI Projection Model in this context is purely illustrative and should not imply that we believe that this is the best model to use for pricing longevity-linked securities. Newer versions of the CMI Projection Model have been issued since the work in this study was performed.

22 This value of 1.75% can be compared with the assumption of a long-term rate of improvement of 1.5% used for the fixed leg of the index-based longevity swap above. The long-term rate of improvement is likely to be higher on an annuity reserving basis than for valuing a longevity swap, since it is common practice, in our experience for annuity providers to include an implicit margin for prudence in their mortality projection. In contrast, the assumption used in a longevity swap typically reflects a best estimate of future mortality improvements and risk is explicitly allowed for via the swap premium rather than an implicit margin in the mortality assumption.

23 There will therefore be a distinction between the price an annuity is sold to the public for and the amount it is reserved for by the life insurer, with the additional margin for idiosyncratic mortality risk charged to the individual forming part of the profit margin of the product.

24 Annuities are valued using a real discount rate of 1% p.a.

25 See Hunt and Blake (2015) for a further discussion of the Swiss Re Kortis bond and its construction.

26 Before 2020, the Kortis index is based partly on projected and partly on observed mortality rates, and hence exhibits more variability than after 2020.