2. Definitions

Insurers have to hold capital against various risks, including economic and demographic risks as well as many others; see Kelliher et al. (Reference Kelliher, Wilmot, Vij and Klumpes2013) for a review based on classifications used in the UK and Germany and for two large UK insurance groups. Even within a given risk, there are sub-risks; Table 1 gives an example of the components of longevity risk for a UK insurer. This list is not intended to be exhaustive, and insurers may have additional, portfolio-specific sub-components. Note that different definitions are possible; the CMI’s Mortality and Morbidity Steering Committee (2020, section 9) reviews five published taxonomies for components of longevity risk and proposes a sixth.

Table 1. Specimen Itemisation of the Components of Longevity Risk. Source: adapted from Richards (Reference Richards2016, Table 1)

We assume that we have a data set spanning the time period $[{y_0},{y_1})$ and that we are interested in the mis-estimation risk of the liabilities at time ${y_1}$ . Denote by $\hat \theta $ the maximum-likelihood estimate of a parameter vector $\theta $ that is calibrated to a model for the mortality of the lives observed over $[{y_0},{y_1})$ . Let ${\hat \theta ^{(n)}}$ be the revised maximum-likelihood estimate of $\theta $ from the addition of n years of further experience data after ${y_1}$ .

Denote by $V(\theta ,{y_1})$ the scalar value of life-contingent liabilities in-force at time ${y_1}$ using the mortality rates effective at time ${y_1}$ according to the model specified by the parameter vector $\theta $ . We assume that all basis elements for the calculation of $V(\theta ,{y_1})$ are known apart from the level of mortality rates at ${y_1}$ , i.e. $V(\theta ,{y_1})$ is a deterministic function of $\theta $ , but uncertainty is introduced through uncertainty over $\theta $ . Throughout this paper, mortality improvements will be modelled up to ${y_1}$ , but no future mortality improvements after ${y_1}$ will be assumed – the mortality rates used will be those applying at ${y_1}$ with no allowance for future changes in time apart from the ageing of the lives assured. We are concerned in this paper with a value-at-risk assessment of the mis-estimation risk in $V(\hat \theta ,{y_1})$ only; for a value-at-risk approach to longevity trend risk after time ${y_1}$ , see Börger (Reference Börger2010), Plat (Reference Plat2011) or Richards et al. (Reference Richards, Currie, Kleinow and Ritchie2020).

(1) $$V(\theta ,{y_1}) = \displaystyle\sum\limits_i a(i,{y_1},\theta )$$

(2) $$a(i,{y_1},\theta ) = w_i\displaystyle\int_0^\infty {}_tp_{{x_i},{y_1},\theta }v^tdt$$

In this paper, V will be the reserve for a portfolio of continuously paid single-life annuities, as defined in equations (1) and (2). In equation (2), ${w_i}$ is the level annuity paid to life i aged ${x_i}$ at time ${y_1}$ and $_t{p_{x,{y_1},\theta }}$ denotes the t-year survival probability at outset age x at time ${y_1}$ under parameter vector $\theta $ . ${v^t}$ is a discount function, which can be adapted to allow for escalating benefit payments if necessary. In this paper, we will mainly discount at a constant net rate of 0.75% per annum, applied continuously, although section 10 considers the impact of different discount rates.

We value liabilities with ${\hat \theta ^{(n)}}$ at time ${y_1}$ , rather than at time ${y_1} + n$ , because we are interested in the potential impact on current liabilities of n additional years of experience after time ${y_1}$ (a risk against which we can hold additional capital). Note that insurers under Solvency II must also hold capital against idiosyncratic risk, which for annuities would be the risk that more annuitants survived the following n years than expected. This idiosyncratic risk capital would often be determined separately from the mis-estimation risk capital, but the two are obviously correlated. One could use the approach outlined in this paper to carry out a combined assessment of mis-estimation risk and idiosyncratic risk by valuing at time ${y_1} + n$ instead of at time ${y_1}$ , a subject touched on in section 12.

The risk measure of interest is ${\rm VaR}_p[V({\hat \theta ^{(n)}},{y_1})]$ defined in equation (3), i.e. the proportion of the best-estimate needed to cover a proportion p of losses that might occur due to a change in the best-estimate assumption caused by an additional n years of experience data after time ${y_1}$ . ${{\rm{Q}}_p}[V({\hat \theta ^{(n)}},{y_1})]$ is the p-quantile of the distribution of liability $V({\hat \theta ^{(n)}},{y_1})$ , which we will estimate according to Harrell & Davis (Reference Harrell and Davis1982). In the UK and the European Union (EU), the Solvency II regime for insurer solvency calculations is based on $n = 1$ and $p = 99.5\%$ , i.e. ${\rm VaR}_{99.5\%}[V({\hat \theta ^{(1)}},{y_1})]$ .

(3) $${\rm VaR}_p[V({\bf{\hat \theta}} ^{(n)},{y_1})] = {{{{\rm Q}_p[V({\bf{\hat \theta}}^{(n)},{y_1})]}}\over{{{\rm E}[V({\bf{\hat \theta}}^{(n)},{y_1})]}}} - 1$$

(4) $$\Pr [V({\hat \theta ^{(n)}},{y_1}) \le {{\rm{Q}}_p}[V({\hat \theta ^{(n)}},{y_1})]] = p$$

In equation (3), the best-estimate of the liability at time ${y_1}$ is taken to be the expected value, ${\rm{E}}[V({{\hat \theta}^{(n)}},{y_1})]$ , which for annuities work we have found to be indistinguishable from $V(\hat \theta ,{y_1}),\forall n$ . However, this is not guaranteed, even assuming that $\hat \theta $ and ${\hat \theta ^{(n)}}$ have the same distribution: for a random vector X and non-linear scalar function f, ${\rm E}[f(X)] \ne f({\rm E}[X])$ in general (Perlman, Reference Perlman1974). The relationship between ${\rm{E}}[V({{\hat \theta}^{(n)}},{y_1})]$ and $V(\hat \theta ,{y_1})$ depends on the concavity or convexity of the liability function, V, and the size of the gap ${\rm{E}}[V({{\hat \theta}^{(n)}},{y_1})]-V({\hat \theta},y_1)$ depends on the covariance matrix of $\hat \theta $ . This is considered further in section 7.

3. Parameter Risk and Mis-Estimation

We broadly recapitulate the mis-estimation methodology of Richards (Reference Richards2016). We assume that we have a log-likelihood function, $\ell (\theta )$ , that is twice differentiable so that a Hessian matrix may be calculated (McCullagh & Nelder, Reference McCullagh and Nelder1989, page 6). The true underlying value of $\theta $ is unknown and is denoted ${\theta ^*}$ . The maximum-likelihood estimate of ${\theta ^*}$ is $\hat \theta $ . Under the maximum-likelihood theorem $\hat \theta \approx {\cal N}({\theta ^*},{{\cal I}^{ - 1}})$ , where ${\cal I}$ is the Fisher Information (Cox & Hinkley, Reference Cox and Hinkley1996, Chapter 9). In practice, we replace the unknown ${\theta ^*}$ with $\hat \theta $ , i.e. $\hat \theta \approx {\cal N}(\hat \theta ,{{\cal I}^{ - 1}})$ . Parameter uncertainty is summarised in ${{\cal I}^{ - 1}}$ , i.e. not just the parameter variances along the leading diagonal but also the covariances between parameter estimates (Richards et al., Reference Richards, Kaufhold and Rosenbusch2013, Table 14). To explore parameter risk, we can generate an alternative parameter vector, $\theta '$ , that is consistent with the data and model from $\theta ' = \hat \theta + Az$ , where A is the Cholesky decomposition (Kreyszig, Reference Kreyszig1999, pp. 896–898) of ${{\cal I}^{ - 1}}$ and ${\bf{z}}$ is a vector of independent ${\cal N}(0,1)$ variates of the same length as $\theta $ .

For assessing mis-estimation risk in run-off, Richards (Reference Richards2016) looked at the variation in $V(\theta ',{y_1})$ from repeated simulation of z. The alternative valuations were normalised by dividing by the mean of the values for V and various quantiles computed to express mis-estimation risk as a percentage of the expected reserve. There are several important conditions for a model to be suitable for this kind of approach, and a summary overview is given in Table 2.

Table 2. Assumption Checklist for Mis-Estimation Methodology

4. A Value-at-Risk Approach to Mis-Estimation

The methodology in section 3 was used in Richards (Reference Richards2016) for what might be called pricing mis-estimation, such as when transferring a block of liabilities. The experience data of the block up to time ${y_1}$ are used to calibrate a model for best-estimate purposes, and the pricing risk is represented by the range of values for V consistent with $\hat \theta $ and the estimated covariance matrix, ${{\cal I}^{ - 1}}$ .

However, regulatory frameworks like Solvency II view risk over a fixed horizon, a feature that is absent from the run-off approach in section 3. We can however adapt the methodology to an n-year view of mis-estimation risk in two stages. In the first stage, we use the model to simulate the future lifetimes of the survivors and censor those surviving more than n years; Richards (Reference Richards2012, Table 7) lists formulae for simulating future lifetimes under various mortality models. As per Table 3, we have two options for simulation: we can either use the best-estimate parameters, $\hat \theta $ , or else we can use the perturbed parameters, $\theta '$ . Using the latter will increase the variability in the survival times and corresponds to what one would intuitively understand to drive mis-estimation. The only point in this paper where we will not include parameter risk is in section 6, where we will switch it off to quantify the relevant contributions of parameter risk and idiosyncratic risk.

Table 3. Parameter Options for Simulating Individual Lifetimes

In the second stage, we take the real experience data up to time ${y_1}$ , add the additional n years of simulated pseudo-experience and refit the model; this will yield an alternative parameter vector, ${\hat \theta ^{(n)}}$ . ${\hat \theta ^{(n)}}$ can be viewed as the response of the parameter estimates to n years of new experience data, and we use it to value the liabilities at time ${y_1}$ . We repeat this process of simulating lifetimes, refitting the model and revaluing the liabilities to collect (say) 10,000 realisations of the liability value, ${V_j},j = 1, \ldots ,10,000$ . The approach of repeatedly refitting models and revaluing liabilities is necessarily computationally intensive, so we use parallel processing over 63 threads to reduce run-times (Butenhof, Reference Butenhof1997).

It is worth noting that viewing mis-estimation risk over a finite horizon in this manner is perhaps more accurately described as model recalibration risk (Cairns, Reference Cairns2013). Since the simulated individual lifetimes will affect the recalibration of the model, idiosyncratic risk will also be driving the portfolio valuations. Indeed, the liability valuations are only affected by parameter estimation risk if we use $\theta '$ in Table 3 for simulating individual lifetimes. In contrast, if we use $\hat \theta $ in Table 3 for simulating the individual lifetimes, then what is notionally a VaR mis-estimation risk exercise actually has no parameter risk element at all (in the sense of the covariance matrix for $\hat \theta $ being entirely unused) and is therefore purely about recalibration risk. The importance of this distinction is shown in Figure 5.

5. Model Structure

(5) $$\ell = -\displaystyle\sum\limits_i {H_{{x_i},{y_i}}}({t_i}) + \displaystyle\sum\limits_i {d_i}\log {\mu _{{x_i} + {t_i},{y_i} + {t_i}}}$$

(6) $${H_{x,y}}(t) = \displaystyle\int_0^t {\mu _{x + s,y + s}}ds$$

(7) $${\mu _{x,y}} = {e^{{\alpha _i} + {\beta _i}x + \delta (y - 2000)}}$$

Following Macdonald et al. (Reference Macdonald, Richards and Currie2018), we use survival models for individual lifetimes. We maximise the log-likelihood shown in equation (5), where ${\mu _{x,y}}$ is the mortality hazard at exact age x and calendar time y. Life i enters observation at exact age ${x_i}$ at time ${y_i}$ and is observed for ${t_i}$ years. ${d_i}$ is an indicator variable taking the value 1 if life i is dead at $x_i+t_i$ , and zero otherwise. ${H_{x,y}}(t)$ is the integrated hazard function, defined in equation (6). We have a wide choice of functional forms for ${\mu _{x,y}}$ for post-retirement mortality and Richards (Reference Richards2012) reviews seventeen such models applied to the mortality of UK annuitants. However, we start with the simple and familiar model of Gompertz (Reference Gompertz1825) in equation (7), where the offset of $ -2000$ is to keep the parameters well-scaled. $\delta $ represents the portfolio-specific time trend; here, it is common to all lives, but it could be interacted with any risk factor (such as with gender to estimate separate time trends for males and females).

(8) $${\alpha _i} = {\alpha _0} + \displaystyle\sum\limits_{j = 1}^m {\alpha ^{(j)}}z_i^{(j)}$$

(9) $${\beta _i} = {\beta _0} + \displaystyle\sum\limits_{j = 1}^m {\beta ^{(j)}}z_i^{(j)}$$

${\alpha _i}$ and ${\beta _i}$ are parameters for life i structured as in equations (8) and (9), where ${\alpha ^{(j)}}$ is the main effect of risk factor j and ${\beta ^{(j)}}$ is the interaction of the ${j^{{\rm{t}}h}}$ risk factor with age. $z_i^{(j)}$ is an indicator variable taking the value 1 if life i has risk factor j and zero otherwise. Using the data in Appendix A, we fit a model with age, gender (male or female), early-retirement status (pension commencing before or after age 55), widow(er) status, pension size and calendar time as explanatory variables. Pension size is treated as an ordinal factor with three levels: below £ 5,385 p.a., £ 5,385–12,560 p.a. or above £ 12,560 p.a. (these being the rounded boundaries of an optimised assignment of pension vigintiles to the three factor levels). For multi-level factors, we adopt a policy of making the most numerous level the reference value, i.e. the baseline case is a male first life retiring after age 55 with a small pension. We therefore have parameters for those retiring early, widow(er)s, females and those with medium or large pensions. We estimate these parameter values by maximising the log-likelihood in equation (5), with the results shown in Table 4. Of particular note are the different mortality characteristics of those with the largest pensions. Uncertainty over the mortality of this small sub-group could potentially have a disproportionately large impact on uncertainty over the liability value, V.

Table 4. Parameter Estimates under the Gompertz (Reference Gompertz1825) Model. The Estimate Column is $\hat \theta $ in the Sense of sections 3 and 4, while the Standard-Error Column Contains the Square Roots of the Entries in the Leading Diagonal of ${{\cal I}^{ - 1}}$ . Source: own calculations fitting model in equations (5)–(9) to the data in Appendix, using data for ages 60–105 over 2001–2009

We need to check the validity of our assumptions using the checklist in Table 2 before performing any mis-estimation assessments. We have already deduplicated the data, as described in Appendix A, so the independence assumption holds true.

Regarding the assumption of a multivariate normal distribution for the parameter estimates, we can see in Figure 1 that all profile log-likelihoods are suitably quadratic around the maximum-likelihood estimates. Note that the choice of horizontal scale is important, as it would be possible to find a quadratic-like shape for even poor models by selecting a suitably narrow range; in contrast, in Figure 1 the horizontal range is determined by the estimated standard error of the parameter.

Figure 1. Profile log-likelihoods around estimates in Table 4. The horizontal scale is determined as two standard errors on either side of the joint maximum-likelihood estimate of each parameter.

Since we are only interested in detecting exceptions to the quadratic shape, we can create a space-saving signature for the profile log-likelihoods in Figure 1 by plotting them one after another without labels to see if there are any that do not have a cleanly inverted U-shape; Figure 2 shows this summary graphic of the shapes in Figure 1. This approach becomes particularly useful when the number of parameters grows large. For an example where a parameter fails this quadratic test, see the signature for the Makeham-Perks model in Table 7.

Figure 2. “Signature” formed from profile log-likelihoods in Figure 1.

The model in Table 4 has a time-trend parameter, so the only remaining item to check is financial suitability. Figure 3 shows that the residuals by pension size band are plausibly drawn from a N(0,1) distribution (Macdonald et al., Reference Macdonald, Richards and Currie2018, section 6.5); the apparent outlier is not unduly extreme and is of minimal financial significance. Furthermore, the bootstrapping procedure of Richards (Reference Richards2016, section 8.3) shows that on average the model in Table 4 predicts 100.1% of lives-based mortality and 98.9% of amounts-based mortality. The model is therefore broadly suitable for financial purposes, and so all four validity criteria in Table 2 are fulfilled.

Figure 3. Deviance residuals by pension size band ( $1 \equiv 5\%$ of lives with smallest pensions, $20 \equiv 5\%$ of lives with largest pensions).

Before considering the value-at-risk approach to mis-estimation of section 4, we first consider the run-off approach of section 3. Figure 4 shows the distribution of reserves with 10,000 simulations of the parameter column in Table 4. The 99.5% point of the distribution is 2.61% above the mean, with a 95% confidence interval of 2.55–2.67%. On a run-off basis, the mis-estimation capital requirement would therefore be around 2.6% of mean liability.

Figure 4. Distribution of 10,000 simulations of $V(\theta ',2010)$ for model in Table 4 applied to survivors at 1st January 2010 for portfolio in Appendix. Mortality rates are at 1st January 2010 with no further improvements.

The run-off approach to mis-estimation is useful for setting a confidence interval on the pricing basis for a bulk annuity or longevity swap. For a best-estimate basis, the mean of the distribution in Figure 4 can be back-solved to a given percentage of a chosen table (or done separately for the reserves for males and females). For example, if we use the S2PA table (CMI Ltd, 2014) the equivalent best-estimate percentages are 109.7% for males and 100.1% for females. We can further use the 2.5% and 97.5% points of the distribution in Figure 4 to form a 95% confidence interval for this basis: back-solving leads to 95% confidence intervals of 103.6–116.3% for males and 95.3–105.0% for females. Note that the confidence interval is not symmetric around the central estimate in part because the distribution of reserves is not normal – the p-value of the test statistic from Jarque & Bera (Reference Jarque and Bera1987) is 0.2581 for males, but 0.0009158 for females. One clearly cannot simply assume normally-distributed liabilities for mis-estimation.

6. The Roles of VaR Horizon and Parameter Risk

Using the model from Table 4, we turn to the question of the n-year value-at-risk capital assessment; this is shown in Figure 5 for the same portfolio. The impact of horizon, n, is shown both with parameter risk (simulating lifetimes with $\theta '$ as per Table 3) and without (simulating lifetimes with $\hat \theta $ only). The capital requirements without parameter risk in Figure 5 are fairly flat, as the simulated experience is similar to the real data. In contrast, the capital requirements including parameter risk rise with increasing horizon; most of this difference is driven by the uncertainty over the experienced time trend, which makes the estimate of mortality levels at 1st January 2010 more uncertain.

A comparison of the two series in Figure 5 shows the relative role of parameter risk over a short horizon and reveals an oddity: the value of ${\rm VaR}_{99.5\%}[V({\hat \theta ^{(1)}},2010)]$ would be used for Solvency II mis-estimation capital, yet most of the capital requirement is clearly driven by the idiosyncratic variation in the simulated experience, not parameter risk: we have 1.27% of mean liability with parameter risk, but we still have 1.08% without it. The run-off or pricing mis-estimation assessment at the end of section 5 was driven solely by estimation error in $\theta $ , but 85% of the one-year VaR mis-estimation capital stems from idiosyncratic risk. This counter-intuitive aspect of the value-at-risk approach is not an anomaly: Kleinow & Richards (Reference Kleinow and Richards2016, Table 5) found that most of the value-at-risk capital for longevity trend risk was similarly driven by the simulated experience, not parameter risk. Thus, what is described as a value-at-risk approach to either longevity trend risk (Richards et al., Reference Richards, Currie and Ritchie2014) or mis-estimation risk (this paper) is in fact largely recalibration risk in both cases (Cairns, Reference Cairns2013).

Table 5. Options for Measuring Best-Estimate Liability in Denominator of equation (3)

Figure 6 shows the distribution of $V({\hat \theta ^{(1)}},2010)$ from which ${\rm VaR}_{99.5\%}[V({\hat \theta ^{(1)}},2010)]$ was calculated. As with Figure 4, we can use percentiles to back-solve to a percentage of a standard table: the 99.5% reserves for males and females equate to 106.1% of S2PA for males and 97.0% for females. Compared with the central estimates in section 5, a shorthand 99.5% stress for Solvency II mis-estimation risk would then be −3.6% of S2PA for males and −3.1% for females (larger portfolios would likely have smaller mis-estimation stresses).

Figure 5. ${\rm VaR}_{99.5\%}[V({\hat \theta ^{(n)}},2010)]$ capital requirements as percentage of the mean reserve, with ( $ \bullet $ ) and without ( $ \circ $ ) parameter risk in simulation of additional $n$ years of experience data. 95% confidence intervals are marked with -.

Figure 6. Distribution of 10,000 simulations of $V({\hat \theta ^{(1)}},2010)$ for model in Table 4 applied to survivors at 1st January 2010 for portfolio in Appendix A. Mortality rates are at 1st January 2010 with no further improvements.

7. Choice of Best-Estimate Liability

In equation (3), we used ${\rm E}[V({\hat \theta}^{(n)},{y_1})]$ as our best-estimate of the liability. However, there are at least three options, as described in Table 5, and there is no guarantee that they are the same.

Table 6 shows the values of each of the three measures of best-estimate liabilities for the portfolio in Appendix A, assuming level single-life annuities discounted at 0.75% per annum. As can be seen from the p-values of the t statistics, there is no meaningful difference between the three measures of best-estimate liability for this portfolio.

Table 6. Measures of Best-Estimate Liability in Denominator of equation (3). (b), (c) and (d) are Calculated from the 10,000 1-year VaR Simulations with Parameter Risk from section 6

8. The Role of Liability Concentration

One question is the extent to which the relative mis-estimation VaR capital requirements are driven by the concentration of liabilities in a small sub-group of lives with large pensions. To investigate this, Figure 7 shows the relative VaR mis-estimation capital requirements using actual pension amounts ( ${w_i}$ in equation (2)) and with ${w_i} = 1$ for all lives. The differences are surprisingly modest.

Figure 7. Mis-estimation ${\rm VaR}_{99.5\%}[V({\hat \theta ^{(n)}},2010)]$ capital requirements with actual pension amounts ( $ \circ $ ) and homogeneous pensions ( $ \times $ ).

9. The Role of Portfolio Size

Section 6 found that parameter risk played a surprisingly small role in one-year 99.5% VaR capital requirements and that most of these capital requirements were driven by the idiosyncratic risk in simulated lifetimes. This raises the question of how VaR mis-estimation capital requirements vary with portfolio size. To investigate this, we created an artificial second portfolio by repeating the records ten times. The new pseudo-portfolio is then ten times the size in terms of number of lives, but has the same profile and characteristics (such as concentration of liabilities). Figure 8 shows the VaR mis-estimation capital requirements for the real portfolio and the scaled portfolio, showing that portfolio size has a large impact and that this impact increases with horizon.

Figure 8. Mis-estimation ${\rm VaR}_{99.5\%}[V({\hat \theta ^{(n)}},2010)]$ capital requirements with actual portfolio records ( $ \bullet $ ) and with each record repeated ten times to create a larger portfolio with the same profile ( $ \circ $ ). Source: 10,000 simulations with parameter risk of model fitted to data for UK pensioner liabilities in Appendix.

10. The Role of Discount Rate

Figure 9 shows the VaR mis-estimation capital requirements using various discount rates. For a given horizon, risk capital increases as the discount rate falls. Mis-estimation assessments clearly need to be regularly updated as the shape or level of the yield curve changes.

Figure 9. Mis-estimation ${\rm VaR}_{99.5\%}[V({\hat \theta ^{(n)}},2010)]$ capital requirements using various discount rates. Source: 10,000 simulations with parameter risk of model fitted to data for UK pensioner liabilities in Appendix.

With historically low interest rates and inflation-proofed defined-benefit pensions, annuity liabilities for bulk buy-outs will at times have to be valued using a net negative discount rate, implying possibly even higher relative mis-estimation capital requirements than those in Figure 9.

11. The Role of Mortality Law

(10) $$\mu _{x,y} ={ {e^{\alpha + \beta x + \delta (y - 2000)}}\over{1 + e^{\alpha + \beta x + \delta (y - 2000)}}}$$

(11) $$\mu _{x,y} ={ {e^{\alpha + \beta x + \delta (y - 2000)}}\over{1 + e^{\alpha + \beta x + \rho + \delta (y - 2000)}}}$$

(12) $$\mu _{x,y} = {{e^\epsilon + e^{\alpha + \beta x + \delta (y - 2000)}}\over{1 + e^{\alpha + \beta x + \delta (y - 2000)}}}$$

The results in this paper have so far all been based on the Gompertz (Reference Gompertz1825) mortality law. In this section, we explore some alternative mortality laws, starting with the simplified logistic model from Perks (Reference Perks1932) in equation (10). It has the same number of parameters as the Gompertz law, but captures the tendency for log(mortality) to increase less slowly than linearly at advanced ages; see Barbi et al. (Reference Barbi, Lagona, Marsili, Vaupel and Wachter2018) and Newman (Reference Newman2018) for the ongoing debate as to the validity of this phenomenon. Individual mortality differentials are handled in the same way as the Gompertz model with equations (8) and (9). A variation on the logistic Perks law is the model from Beard (Reference Beard, Wolstenholme and O’Connor1959) in equation (11). The Beard and Gompertz laws are linked: if individual mortality follows a Gompertz law, but there is also Beta-distributed heterogeneity in $\alpha $ , then observed mortality will follow the Beard law; see Horiuchi & Coale (Reference Horiuchi and Coale1990). Another variation on equation (10) is to add a constant Makeham-like term (Makeham, Reference Makeham1860), as in equation (12).

(13) $$\log {\mu _{x,y}} = \alpha {h_{00}}(t) + {m_0}{h_{10}}(t) + \omega {h_{01}}(t)$$

A more recent option is the Hermite-spline model of Richards (Reference Richards2020) in equation (13), where the Hermite basis-spline h functions are shown in Figure 10. Hermite splines are defined for $t \in [0,1]$ and so we map age x onto $[0,1]$ with $t = (x - {x_0})/({x_1} - {x_0})$ with pre-defined values of ${x_0} = 50$ and ${x_1} = 105$ . We assume ${\mu _x} = {\mu _{{x_0}}},x \le {x_0}$ and ${\mu _x} = {\mu _{{x_1}}},x \ge {x_1}$ .

Figure 10. Hermite basis splines for $t \in [0,1]$ .

This new class of Hermite-spline mortality models is highly parsimonious when modelling different risk factors. It was specifically designed for modelling post-retirement mortality such that differentials automatically narrow with increasing age. This reduces the number of parameters compared to the other four models, and further avoids the crossing-over of fitted mortality rates at advanced ages; see Richards (Reference Richards2020, section 1). Individual mortality differentials are therefore handled with just equation (8) – narrowing age differentials are handled automatically and so there is no need for equation (9) with the Hermite family.

Richards (Reference Richards2020) modelled age-related mortality changes with a peak improvement at an age estimated from the data. Here we instead extend equation (13) for time variation as follows:

(14) $$\log {\mu _{x,y}} = (\alpha + \delta (y - 2000)){h_{00}}(t) + ({m_0} + m_0^{\rm trend}(y - 2000)){h_{10}}(t) + \omega {h_{01}}(t)$$

where $\delta $ plays a similar role to equations (10)-(12) by changing the level of mortality in time and $m_0^{\rm trend}$ changes the shape at younger ages. A common feature to both parameters is the automatic reduction in influence with age when multiplying by the Hermite spline functions ${h_{00}}$ and ${h_{10}}$ . We find that $\delta $ in equation (14) does not improve the fit for the data set in Appendix, so we use the simpler version in equation (15):

(15) $$\log {\mu _{x,y}} = \alpha {h_{00}}(t) + ({m_0} + m_0^{\rm trend}(y - 2000)){h_{10}}(t) + \omega {h_{01}}(t)$$

Equations (7), (10), (11) and (12) are therefore all one-parameter approaches to changes in mortality level: the parameter $\delta $ adds or subtracts to the level set by $\alpha $ . Figure 11 shows the implied annual mortality improvements by age for the baseline combination of risk factors in Table 4 (the implied improvements under the Beard model are not shown as they are indistinguishable from those of the Perks model). Each model has a single-parameter allowance for mortality change, but clearly some models have a more reasonable shape for improvements by age than others. At one extreme, the Gompertz model in equation (7) has a simple-but-unrealistic constant rate of improvement at all ages. In contrast, the Hermite model of equation (15) has perhaps the most realistic near-zero mortality improvement leading up to age 100. The Hermite model also has zero improvement at age 50 by design because there is too little experience data at the youngest ages to carry out reliable estimation (see Figure 13). Since we are modelling mortality from age 60 only, this design decision has minimal impact.

Figure 11. Modelled percentage mortality improvements per annum by age. Source: own calculations of $100\% \times (1 - {\mu _{x,2001}}/{\mu _{x,2000}})$ for male first lives with the smallest 75% of pensions who retired after age 55. The period covered by the data is 2001–2009.

Figure 12 shows the relative mis-estimation capital requirements for each of the five models. We see that the parsimony of the Hermite model in equation (15), and the more-realistic allowance for improvements in Figure 11, results in less mis-estimation risk and thus less capital. Table 7 shows that there are no unwelcome compromises: the Hermite model has the lowest AIC (Akaike, Reference Akaike1987) of the five, while having an ability to predict lives- and pension-weighted variation as good as any of the other models.

Figure 12. ${\rm VaR}_{99.5\%}[V({\hat \theta ^{(n)}},2010)]$ capital requirements for various mortality laws. Source: 10,000 simulations with parameter risk of model fitted to data for UK pensioners, single-life immediate-annuity cash flows discounted at 0.75% p.a.

Figure 13. Distribution of deaths (top) and time lived (bottom) for 2001–2009 after data validation and deduplication.

One possibility might have been that the relative Hermite mis-estimation capital requirements were lower because the reserves themselves were higher to start with. Table 8 shows that the opposite is the case for this portfolio – the Hermite model has both the lowest mean reserve (males and females combined) and the lowest absolute one-year mis-estimation capital requirement.

Table 7. Summary of Model Fits. Note that one of the Makeham-Perks parameters, $\varepsilon $ , does not have a Properly Quadratic Profile in the Log-Likelihood Signature, Although the Impact is Minimal. Source: own calculations fitting to data in Appendix; bootstrap percentages are the mean ratio of actual deaths v. model-predicted deaths from 10,000 samples of 10,000 lives (sampling with replacement)

Table 8. Best-Estimate Reserve at 1st January 2010, Together with One-Year 99.5% VaR Mis-Estimation Capital, Sorted by Ascending Total

12. Correlation with Idiosyncratic Risk

Table 1 lists specimen longevity-related risks in an annuity portfolio. Under Solvency II, a capital amount for each component is itemised separately and then aggregated into a single longevity capital requirement using a correlation matrix. However, in section 6 we noted that the one-year mis-estimation capital requirements were largely driven by the recalibration of the model, rather than the parameter uncertainty. This raises a question regarding a value-at-risk approach to capital setting: to what degree are the mis-estimation capital requirements correlated with those for adverse experience over the same period? To investigate this, we took the mis-estimation reserves at time ${y_1} = 2010$ with parameter risk behind Figure 5 and looked at the correlation of these reserves with three alternative measures of the pseudo-experience in $[{y_1},{y_1} + n)$ used to recalibrate the model. Table 9 shows the results. As expected, higher numbers of deaths are strongly negatively correlated with the mis-estimation VaR reserves, although this correlation reduces with the VaR horizon. The correlation over a one-year horizon between mis-estimation VaR reserves and the payments made is perhaps weaker than might have been expected.

Table 9. Correlations between the Mis-Estimation VaR Reserves at ${y_1} = 2010$ and Three Measures of the Simulated Pseudo-Experience Underlying the Recalibration