2. Definitions

We denote by $V_{x,n}$ the random variable representing the expected present value of a liability contingent on a forecast of future mortality rates for a person aged x in n years’ time. In this paper, $V_{x,n}$ will be either the reserve for an annuity in payment or else a deferred annuity. We assume that all basis elements for the calculation of $V_{x,n}$ are known apart from the future mortality improvement rates, i.e., that $V_{x,n}$ is random with respect to longevity trend risk only (we will discount cash flows throughout the paper at a constant net rate of 1% per annum). The best-estimate of the liability is assumed to be ${\mathbb{E}}[V_{x,n}]$ and the risk measures of interest are

(1) \begin{align} {\text{VaR}}_{\alpha,x,n} &= \left(\frac{Q_{\alpha,x,n}}{{\mathbb{E}}[V_{x,n}]}-1\right)\times 100\%\end{align}

(2) \begin{align} {\text{CTE}}_{\alpha,x,n} &= \left(\frac{{\mathbb{E}}[V_{x,n}|V_{x,n}>Q_{\alpha,x,n}]}{{\mathbb{E}}[V_{x,n}]}-1\right)\times 100\%\end{align}

where $Q_{\alpha,x,n}$ is the $\alpha$ -quantile of the distribution of $V_{x,n}$ , i.e., $\Pr(V_{x,n}\le Q_{\alpha,x,n}) = \alpha$ . ${\text{VaR}}_{\alpha,x,n}$ is the percentage extra capital on top of the best-estimate to cover a proportion $\alpha$ of losses that might occur over the following n years due to a change in the best-estimate assumption. ${\text{CTE}}_{\alpha,x,n}$ is the percentage extra capital on top of the best-estimate to cover the average expected loss if an extreme event with probability $1-\alpha$ occurs in the next n years. See Hardy (Reference Hardy2006) for discussion and contrast of the properties of VaR- and CTE-style approaches to risk measurement. $Q_{a,x,n}$ is an order statistic, and in this paper, we will estimate it using the methodology from Harrell & Davis (Reference Harrell and Davis1982).

In the EU, the Solvency II regime for insurer solvency calculations is based on ${\text{VaR}}_{99.5\%,x,1}$ , while the Swiss Solvency Test is based on ${\text{CTE}}_{99\%,x,1}$ . Both are 1-year views of risk and both typically produce values in the interval $(0\%, 10\%)$ for longevity trend risk.

Solvency II further specifies an Own Risk and Solvency Assessment (ORSA) for regular use in business planning. This is also a ${\text{VaR}}_{\alpha,x,n}$ calculation, but with n typically 3–5 years. A specific value for $\alpha$ is not laid down, but we will use $\alpha=95\%$ in this paper for ORSA-style calculations.

3. Data

We use data from the Human Mortality Database (HMD) for the United Kingdom and the Netherlands at ages 50–104 years over the period 1971–2016. The data are (i) $d_{x,y}$ , the deaths classified by age last birthday, x, and year of occurrence, y, and (ii) $E^c_{x,y}$ , the corresponding central exposed to risk. Data are available separately for males and females and lend themselves to modelling the central rate of mortality, $m_{x,y}$ , without modification. The data also lend themselves to modelling $\mu_{x+\frac{1}{2},y+\frac{1}{2}}$ , the force of mortality at age $x+\frac{1}{2}$ at time $y+\frac{1}{2}$ . In this paper, we will model the force of mortality, and we will drop the $+\frac{1}{2}$ for simplicity.

There are two aspects of the data which affect comparisons of the results in this paper with those in earlier works. The first aspect concerns the UK data only, where population estimates were updated following the 2011 census; see Cairns et al. (Reference Cairns, Blake, Dowd and Kessler2015). Figure 1 shows the ratio of population estimates for 2010 before and after the census, showing large changes (reductions) in the estimates at advanced ages. This will change the calibration of models in earlier papers that used population estimates made prior to the 2011 census.

Table 1. Models used in this paper

Figure 1 Population estimates for males in England & Wales in 2010: ratio of post-census estimate to pre-census estimate. After the 2011 census, the population estimates above 90 years of age were revised downwards, thus increasing stated mortality rates (the deaths data were unchanged). Source: own calculations using ONS data.

The second aspect is that the HMD revised its methods protocol in 2017 to improve estimates of exposures using the distribution of births for each cohort; see Wilmoth et al. (Reference Wilmoth, Andreev, Jdanov, Glei and Riffe2017). Updated population estimates for the UK and the Netherlands became available in May 2018; see Boumezoued (Reference Boumezoued2020) for further details of how exposure estimates are enhanced using fertility data, and why this is necessary. The effects of this change are twofold: first, false cohort effects are removed from the data; second, the volatility of mortality rates is reduced. Both of these effects will change model fits, projection behaviour, and thus the results of any VaR assessments. Removal of false cohort effects will obviously change the parameterisation of any model incorporating a cohort term, while the reduction in volatility will change the calibration of any projection model based on a time series. As shown in Kleinow & Richards (Reference Kleinow and Richards2016, Section 7), the volatility of the forecasting process plays a dominant role in the 1-year VaR capital requirements for longevity trend risk.

4. Models and Their Fitting

We assume that the number of deaths at each age x in year y, $d_{x,y}$ , has a Poisson distribution with mean $E^c_{x,y}\mu_{x,y}$ . Models are fitted by maximising the log-likelihood with the functional forms for $\mu_{x,y}$ shown in Table 1. Smoothing is achieved with the method of P-splines of Eilers & Marx (Reference Eilers and Marx1996). Here, parameters for individual ages are replaced by the coefficients in a regression model where the regression matrix is computed with an appropriate basis of B-splines; the coefficients are chosen by optimising a penalised likelihood. Over-dispersion of the Poisson death counts is allowed for directly in the 2DAP and 2DAC models.

The LC, DDE, LC(S) and APC(S) models project $\kappa_y$ as an ARIMA time series, with the order chosen by minimising the AIC (Akaike Reference Akaike1987) with a small-sample correction (Macdonald et al., Reference Macdonald, Richards and Currie2018, p 98). ARIMA models are fitted with R’s arima() command using a limited number of attempts with different initial values and we select the fit that has the smallest AIC (see also Appendix A). Standard linear identifiability constraints were used, although the precise choice is irrelevant to both fit and forecast; see Currie (Reference Currie2020). The M5(S) model is a P-spline-smoothed variant of Cairns et al. (2006), although $\{\kappa_{0,y}, \kappa_{1,y}\}$ is still projected using a bivariate random walk with drift. The 2DAP and 2DAC models are projected using a second-order penalty functions, i.e., a linear extrapolation on a logarithmic scale. Further technical details of the models and their fitting can be found in Richards et al. (Reference Richards, Currie and Ritchie2014, Appendices 1–4) and Richards et al. (Reference Richards, Currie, Kleinow and Ritchie2019, Appendices B, C & D).

For the VaR calculations, we extend the method of Richards et al. (Reference Richards, Currie and Ritchie2014) as follows. Data one selected for a given age range and date range, e.g., ages $x_{\text{min}}-x_{\text{max}}$ over the years $y_{\text{min}}-y_{\text{max}}$ as depicted in Figure 2. A stochastic projection model is fitted to this data and the parameters are estimated; we call this the baseline model. The baseline model is then used to generate sample path realisations of mortality rates over the following n years. The details of this for the various projection methods are described in Richards et al. (Reference Richards, Currie and Ritchie2014, Section 8). Where projections are carried out by an ARIMA(p, d, q) model for $\kappa,$ we restrict our attention to cases with $d=1$ and where the fitted ARIMA model has a valid estimate of the parameter covariance matrix, i.e., where the leading diagonal contains only positive values. These simulated mortality rates are then used to simulate the population experience from one year to the next, i.e., generate $d_{x,y}$ and $E^c_{x,y}$ for the shaded region in Figure 2. The simulated pseudodata is then appended to the real data and the model is refitted. In most of the calculations in this paper, the refitted model has the same structure and nature as the baseline model. However, in Section 8, we consider the impact of refitting a model of a different nature to the baseline model.

Figure 2 Simulation of n years of population experience data. Rolling forward the population results in an increasingly large triangle of missing pseudodata in the upper right. This region can be weighted out in the subsequent model refits, but as n increases, it becomes trickier to fit models containing cohort terms.

The repeated model-fitting that underlies the methodology here is computationally intensive. Other authors have developed approximations for financial functions under specific models, such as Cairns et al. (Reference Cairns, Blake, Dowd, Coughlan, Epstein and Khalaf-Allah2011) for the model of Cairns et al. (2006) and Liu & Li (2019) for simple drift models for the time index in the models of Lee & Carter (Reference Lee and Carter1992) and Renshaw & Haberman (Reference Renshaw and Haberman2006). However, we want to keep our approach general and to be able to use any model, and we particularly want to use ARIMA models, not drift models, for the time index of the Lee–Carter model (see Kleinow & Richards Reference Kleinow and Richards2016, for a discussion). For these reasons, we use parallel processing to complete the calculations in a manageable time frame (see Appendix A) and we do not re-estimate the smoothing parameters between simulations (see Appendix B).

5. Insurer Solvency

Solvency II regulations in the EU specify a ${\text{VaR}}_{99.5\%,x,1}$ approach to longevity trend-risk capital. Figure 3 shows the capital requirements as a percentage of best-estimate reserves for males and females in the UK and Netherlands for a selection of four different models.

Figure 3 ${\text{VaR}}_{99.5\%,x,1}$ capital requirements as percentage of best-estimate reserve for a selection of models from Table 1. The age distribution of liabilities is important, but model risk is key – solvency capital requirements have to be set by actuarial judgement. Source: 1,000 refits for each model, immediate annuity liabilities with cash flows discounted at 1% p.a.

As has been documented elsewhere (Richards et al. Reference Richards, Currie and Ritchie2014), the capital requirements in Figure 3 vary considerably by age and model choice. The variation by age means that an insurer’s longevity trend-risk capital will depend on the age distribution of liabilities. The variation by model means that the final choice of trend-risk capital must be a matter for actuarial judgement – the quite different shapes in Figure 3 show that this actuarial judgement must be informed by the results of a variety of models. Bank of England Prudential Regulatory Authority (2015, p. 10) and Woods (Reference Woods2016, p. 8) describe using members of four model families, and the four models in Figure 3 were chosen for their very different nature.

Solvency capital for longevity trend risk is different for deferred annuities. Figure 4 shows the ${\text{VaR}}_{99.5\%,x,1}$ capital requirements for annuities where payment is deferred until age 67. As a rule of thumb, whatever the choice of model, the deferred annuity capital requirements at age 50 are around double those for immediate annuities at the same age. This is one of a number of reasons why insurers and reinsurers prefer to transact business related to annuities in payment, rather than deferred annuities or pensions. Insurers often have a fixed capital budget to support new business, and a greater volume of business can be written if that capital is applied to write immediate annuities rather than deferred annuities. This difference affects insurers’ appetite for deferred annuities and thus the keenness of their pricing. As a result, pension schemes may find that it helps to let their deferred liabilities mature before approaching an insurer, i.e., allowing the proportion of deferred liabilities to shrink by waiting for more lives to reach retirement age. This is considered further in Section 7.

Figure 4 ${\text{VaR}}_{99.5\%,x,1}$ capital requirements as percentage of best-estimate reserve. Deferred annuities have up to double the capital requirements for longevity trend risk at the age of 50 years. Source: 1,000 simulations for each model, the Netherlands data, immediate and deferred annuity liabilities with cash flows discounted at 1% p.a.

One thing that immediate and deferred annuities share is that ${\text{VaR}}_{99.5\%,x,1}$ and ${\text{CTE}}_{99\%,x,1}$ produce comparable capital requirements. Figure 5 shows how closely the ${\text{CTE}}_{99\%,x,1}$ capital requirements shadow the ${\text{VaR}}_{99.5\%,x,1}$ ones. The picture is similar for the Netherlands females and for males and females in the UK (not shown). The ${\text{CTE}}_{99\%,x,1}$ capital requirements are in fact slightly higher, but the difference is negligible in the context of the actuarial judgement that needs to be exercised when considering model risk. For all practical purposes, ${\text{VaR}}_{99.5\%,x,1}$ and ${\text{CTE}}_{99\%,x,1}$ can be considered equivalent for longevity trend risk.

Figure 5 ${\text{VaR}}_{99.5\%,x,1}$ and ${\text{CTE}}_{99\%,x,1}$ capital requirements as percentage of best-estimate reserve. The two approaches produce comparable results, with model risk clearly dominant. Source: 1,000 simulations for each model for Netherlands males, immediate- and deferred-annuity liabilities with cash flows discounted at 1% p.a.

Figure 6 ${\text{VaR}}_{95\%,x,n}$ capital requirements for immediate annuities to single-life males as percentage of best-estimate reserve for $n\in\{1, 3, 5\}$ using various models. The capital requirements have similar shapes across time horizons, and the figures for a 3-year horizon are generally equidistant between the 1- and 5-year horizons (with the exception of the 2DAC model). As always, model risk is very material. Source: 1,000 refits for each model, data for UK males.

6. Insurer Business Planning

Beyond the 1-year solvency calculations, Solvency II also mandates the form of near-term business planning. Insurers are supposed to carry out VaR-style investigations over the short term using lower p-values under the ORSA. Fixed details are not laid down, but in this section, we consider VaR capital requirements for annuities at a p-value of 95% over 1-, 3- and 5-year horizons. Figures 6 and 7 show the results for UK males and females, respectively. As with the 1-year solvency capital calculations in Section 5, model risk is a major source of difference, as is variation by age. One new feature is that the capital requirements for a 3-year horizon are not always equidistant between the 1- and 5-year horizons. While the LC(S), M5(S) and APC(S) models have roughly equidistant gaps, the 2DAC model requires relatively little extra for the jump from a 3- to a 5-year horizon. As with solvency capital calculations, business planning capital is a question of actuarial judgement.

Figure 7 ${\text{VaR}}_{95\%,x,n}$ capital requirements for immediate annuities to single-life females as percentage of best-estimate reserve for $n\in\{1, 3, 5\}$ using various models. The capital requirements have similar shapes across time horizons, and the figures for a 3-year horizon are often equidistant between the 1- and 5-year horizons, albeit not for the 2DAP and 2DAC models. As always, model risk is very material. Source: 1,000 refits for each model, data for UK females.

7. Pension Scheme Funding Targets

Many defined benefit pension schemes find themselves wanting to buy out their liabilities with a life insurer, but have insufficient funds to afford this in the near term. To achieve the buyout goal, the scheme’s funding target can be defined as the buyout premium in n years’ time. The scheme’s funding rate and investment policy are set accordingly, and the scheme’s buyout deficit is monitored over time. This situation is sometimes referred to as the “glide path” to buyout, although the actual financial journey may be rather bumpier than this expression implies.

However, the buyout premium in n years, time is not a static target. Among many other things, insurer buyout prices will be influenced by yield curves, both for nominal and index-linked securities, and forecasts of mortality improvements. To calculate an estimated buyout premium in n years, the pension scheme can use futures prices for the economic assumptions. However, no such market prices exist for mortality improvements, so how can a pension scheme judge what assumptions insurers might use for pricing in n years’ time? This obviously cannot be predicted precisely, but a pension scheme can get an idea of the uncertainty and variability of insurers’ future mortality forecasts by using the same VaR procedure with the time horizon extended to match the buyout goal. For funding purposes, the pension scheme is also likely to use a lower p-value, say 90% instead of 99.5%. The two panels of Figure 8 illustrate the potential levels of uncertainty caused by longevity trend risk over a 10-year to buyout term.

Figure 8 ${\text{VaR}}_{90\%,x,10}$ excess over best-estimate for pensions in payment to males. According to these four models, there is a 10% chance of buyout prices being at least 4% more expensive in 10 years’ time due solely to changed expectations of longevity trend risk. Source: 1,000 refits of each model valuing immediate annuities with cash flows discounted continuously at 1% per annum.

On a practical note, the number of models in Figure 8 is rather limited. Specifically, the only model with any year of birth or cohort patterns is the 2D P-spline model of Richards et al. (Reference Richards, Kirkby and Currie2006). This is because the P-spline models can cope with the missing cohort triangle in Figure 2, which becomes an increasingly challenging problem for models with individual cohort terms, such as the APC(S) model. Another practical issue is that longer terms for n increase the risk of a restatement of the data of the kinds described in Section 3.

8. Valuing Index-Based Hedging Instruments

A relatively recent financial innovation for managing longevity risk is the index-based hedge. Several such hedges have been transacted in recent years by insurers in the Netherlands; see Blake et al. (Reference Blake, Cairns, Dowd and Kessler2018) for an overview. The basic idea is that the owner of longevity risk, e.g., an insurer or pension scheme, uses a hedging instrument defined with reference to population mortality. This critically assumes that the longevity trend risk in the owner’s portfolio moves in parallel with the longevity trend risk in the reference population; if it does not, the owner is exposed to basis risk. This basis risk is magnified by the tendency of portfolio liabilities to be heavily concentrated among a select subset of lives; see Richards & Currie (Reference Richards and Currie2009, Table 1) for an example of this. The owner of the longevity risk is also left with the volatility arising from individual lifetimes being longer or shorter than expected, known variously as idiosyncratic risk or binomial risk. An index-based hedging instrument is therefore an imperfect risk management tool compared with the more traditional practice of directly (re)insuring the entire longevity risk in the portfolio, which covers trend risk, basis risk and idiosyncratic risk. Such all-encompassing (re)insurance is sometimes known as an indemnity arrangement to contrast it with hedging instruments.

An index-based hedging contract needs a risk-taker on the other side. To meet the needs of such investors, index-based hedges usually have a fixed term, n years say, as opposed to the perpetual nature of an indemnity arrangement. In order to close out the hedging contract, some financial metric needs to be calculated that bears a sufficiently strong relationship to the insurer’s balance-sheet liability. An example might be that, at the end of n years, a specified mortality-forecasting model would be fitted to the data available and an annuity factor would be calculated using the resulting best-estimate projection. In such circumstances, the financial metric is the now-familiar annuity factor, $V_{x,n}$ . Since there is no market price for the hedge contract, we need to use a model to value it (so-called “mark-to-model”). However, while the model used for the closeout calculation will be written into the hedge contract, there is no guarantee that the mortality rates over the contract term will follow this model. In this section, we will explore the impact of model risk on the pricing and valuation of such index-based contracts.

For the purpose of illustration, we assume a term of $n=15$ years. Since most recent index-based hedges have been written with Dutch insurers (Blake, et al. Reference Blake, Cairns, Dowd and Kessler2018, pp. 44, 48, 57–58), we will use mortality data for the Netherlands. For our closeout forecast model, we will use the unsmoothed Lee–Carter model for its simplicity and widespread acceptance, although Currie (Reference Currie2013) notes that a smoothed version of the Lee–Carter model is unambiguously superior in terms of forecasting quality. For our closeout metric, we will use an annuity factor for a life aged x in n years, with cash flows discounted continuously at an interest rate of 1% per annum. Since this annuity factor is unknown at the point of valuation of the index-based hedge, we treat it as a random variable, $V_{x,n}$ , whose value will be observed after n years when the final forecast is made and the annuity factor can be calculated. We will follow Cairns & El Boukfaoui (Reference Cairns and El Boukfaoui2019) in not using the annuity factor “as is”, but instead define the hedge pay-off function, $h(V_{x,n})$ , as follows:

(3) \begin{equation}h(V_{x,n}) = \max\left(0,\min\left(\frac{V_{x,n}-AP}{EP-AP},1\right)\right)\end{equation}

where AP is the attachment point, i.e., the minimum value the annuity factor has to reach before any payment is made to the insurer, and EP is the exhaustion point, i.e., the annuity factor above which the payout is capped. $h(V_{x,n})$ therefore takes values in [0, 1], as depicted in Figure 9, and it can be scaled by a nominal amount to provide a suitable hedge value.

Figure 9 Example hedge pay-off function, $h(V_{x,n})$ , showing the attachment and exhaustion points for the contract analysed in Table 2 for $x=70$ .

Using the same unsmoothed Lee–Carter model for both generating the sample paths and calculating the closeout value of the hedge, we further follow Cairns & El Boukfaoui (Reference Cairns and El Boukfaoui2019) in setting $AP=Q_{60\%, n}$ and $EP=Q_{95\%,n}$ . If we use the same model to generate the sample paths over the following $n=15$ years as the model used to set the attachment point, then there is by definition a probability of 0.4 of a non-zero pay-off. Using 1,000 15-year simulations of sample paths according to the unsmoothed Lee–Carter model, we agree the payout probability of 0.4; the average pay-off of $h(V_{70,10})$ is 0.338 (conditional on there being a payout, i.e., ${\mathbb{E}}[V_{70,10}|V_{70,10}>AP]$ ). The insurer and investor can then negotiate what price should be paid to the investor to enter into this contract, allowing for further items like expenses, discounting and profit margins.

But how do these figures change if mortality rates over the next n years follow a different process than the one implied by the closeout forecasting model? To illustrate this, we use the unsmoothed Lee–Carter model for both the pay-off calculation and setting the attachment and exhaustion points, but vary the model generating the sample paths over the n years between setting up the hedge and closing it out. Tables 2 and 3 show the resulting pay-off probabilities and mean pay-offs under some alternative models for males and females.

Table 2. Impact of different sample-path models on pay-off of 15-year longevity hedge.

Source: own calculations using population data for males in the Netherlands, ages 50–104, 1971–2016. The attachment point is 14.61, being the 60^th percentile of the annuity factor under the unsmoothed Lee-Carter model, while the exhaustion point is 15.31, being the 95^th percentile. The hedge pay-off function is shown in Figure 9.

Table 3. Impact of different sample-path models on pay-off of 15-year longevity hedge.

Source: own calculations using population data for females in the Netherlands, ages 50–104, 1971–2016. The attachment point is 17.02, being the 60^th percentile of the annuity factor under the unsmoothed Lee-Carter model, while the exhaustion point is 17.90, being the 95^th percentile.

The diverse results in Tables 2 and 3 do not necessarily affect the hedge contract’s effectiveness. If the portfolio’s longevity trend truly moves in parallel with that of the reference population, then the hedge contract will be effective if the insurer’s projection basis in n years’ time is the Lee–Carter model. However, what Tables 2 and 3 do show is the difficulty in placing a value on the hedge contract, either for the purpose of pricing at outset or for putting on the balance sheet once written. Model risk is very significant in valuing the hedge pay-off – if mortality rates were to follow the 2DAC model, then the payout probability would be more than doubled for males but essentially zero for females. Of course, the converse could also apply: if the hedge contract were defined with reference to the 2DAC model and mortality rates actually followed the Lee–Carter model, the payout probability and mean pay-off would both be substantially lower for males, but considerably higher for females.

Placing a value on such a hedge contract is therefore not a purely mathematical exercise, but necessarily involves substantial actuarial judgement. Such judgement needs to take account of model risk by exploring the results under various different models, but it is clear that valuing an index-based hedge is even more sensitive to the choice of model than setting solvency capital requirements for either immediate or deferred annuities (see Section 5). A follow-on question for regulators is what solvency capital relief (if any) can be granted for holding such index-based contracts? A further conclusion from Tables 2 and 3 is that increasing the number of simulations would be irrelevant; there is little benefit in increasing the accuracy of estimated payout probabilities in the face of such overwhelming model risk.

As with the pension fund “glide path” to buyout in Section 7, the choice of models that can be used in Tables 2 and 3 is limited by the large number of missing cohorts in the shaded area of Figure 2. Of course, the data in this triangle will not be missing at the point the hedge contract is closed. However, the risk of a restatement of the past data – of the kinds described in Section 3 – is even greater with longer terms like $n=15$ . As a result, hedge contracts need carefully drafted clauses to handle restated data. A particularly challenging scenario would be where a hedge contract terminated around the same time as data were restated, as the opposing parties would favour different versions of the data.