Regarding the barrier-fitting algorithm in Tang et al. (Reference Tang, Li and Tickle2022), it is worth noting that for a single-sex data set the penalty is not required and the model in equation (8) is then just a generalised linear model (GLM) (McCullagh & Nelder, Reference McCullagh and Nelder1989). Furthermore, the models specified in equation (8) require no identifiability constraints. This can be proved by showing that the rank of the model matrix, $\boldsymbol{X}$ , equals the length of the parameter vector, $\boldsymbol{B}$ (Currie, Reference Currie2020). This is a feature shared only with the stochastic mortality model of Cairns et al. (Reference Cairns, Blake and Dowd2006), which is itself also a GLM. The models of Tang et al. (Reference Tang, Li and Tickle2022) are therefore easily implemented for single-sex datasets.

Like the Gompertz model of Cairns et al. (Reference Cairns, Blake and Dowd2006), the Hermite-spline models of Tang et al. (Reference Tang, Li and Tickle2022) can extrapolate mortality rates beyond the upper age of the available data. This is a particularly useful feature for actuarial calculations involving annuities and pensions. For example, in Figure 1, the calibrating data stop at age 105, but extrapolation to higher ages was achieved simply by setting $x_1=120$ . In the case of females for England and Wales, Table 1 shows that using $x_1=120$ also markedly reduces the AIC (Akaike, Reference Akaike1987) compared to using $x_1=105$ . The Gompertz model of Cairns et al. (Reference Cairns, Blake and Dowd2006) extrapolates an ever-increasing mortality hazard with age, as advocated by Gavrilov and Gavrilova (Reference Gavrilov and Gavrilova2015). In contrast, the Hermite-spline models of Richards (Reference Richards2020) and Tang et al. (Reference Tang, Li and Tickle2022) extrapolate to a mortality plateau, as advocated by Gampe (Reference Gampe2010). In Figure 1 the limiting mortality hazard is around 1.089, corresponding to a limiting annual mortality rate of 66%. This compares with annual mortality rates of 61–63% at age 119 in the mortality tables used by UK actuaries for pension and annuity calculations (CMI, 2020). In contrast, Gampe (Reference Gampe2010) found a limiting annual mortality rate of 50%, which is the rate assumed from age 112 by US actuaries for similar calculations (PBGC, 2023).

Figure 1 Observed, fitted and extrapolated mortality rates in 2019 for females in England & Wales, ages 50–105. Source: own calculations for HS2 GLM of Tang et al. (Reference Tang, Li and Tickle2022) using data at ages 50–105, 1971–2019.

For mortality projections it is necessary to have a clear time signal in the parameters. However, not every parameter vector for every data set will exhibit this. An example is shown in Figure 2 for females in England & Wales – there is a clear time signal for $\{\hat \alpha _t\}$ , and a relatively clear signal for $\{\hat s_{0,t}\}$ after the mid-1990s, but not for $\{\hat \omega _t\}$ . The estimated drift term, $\hat \mu$ , for the $\{\hat \omega _t\}$ process is 0.002 with a standard error of 0.0095, suggesting that $\{\hat \omega _t\}$ is merely a random walk without drift. For a long-term forecast it therefore makes sense to adopt a simplifying assumption of $\omega _t=\omega$ as follows:

(1) \begin{equation} \log \boldsymbol{M} = \boldsymbol{XB} = [{\boldsymbol{I}_{{\boldsymbol{n}}_{\boldsymbol{y}}}}\otimes {\boldsymbol{h}_{\textbf{00}}} \;:\; {\boldsymbol{I}_{\boldsymbol{n}_{\boldsymbol{y}}}}\otimes \;:\; {\boldsymbol{h}_{\textbf{10}}} \;:\; {\boldsymbol{h}_{\textbf{01}}}]\boldsymbol{B} \end{equation}

where ${\boldsymbol{h}_{\textbf{00}}}$ , ${\boldsymbol{h}_{\textbf{10}}}$ and ${\boldsymbol{h}_{\textbf{01}}}$ denote the column vectors of Hermite splines in (Tang et al., Reference Tang, Li and Tickle2022, equation (3)). For a single-sex data set equation (1) is also a GLM that requires no identifiability constraints. Like the corresponding CBD model, the mortality rates in equation (1) would also be forecast with a bivariate random walk with drift for $(\alpha _t, s_{0,t})$ . The long-term projection of mortality rates under equation (1) will be simpler and more stable than the trivariate HS2 model, albeit at the cost of a poorer fit to the data, as shown in Table 1.

However, closeness of fit to data is not the sole criterion (or even necessarily the best one) when choosing a forecasting model. The quality of the forecast (Cairns et al., Reference Cairns, Blake, Dowd, Coughlan, Epstein, Ong and Balevich2009) is also a consideration, and the forecast values of $s_{0,t}$ in Figure 2 will eventually turn negative, thus causing projected period mortality rates at young ages to reduce with increasing age, as shown in Figure 3. This minor defect in the forecast can be corrected by replacing the $s_{0,t}$ multiplier of $h_{10}$ with $e^{s_{0,t}^*}$ . This adjusted model is not a GLM, but as long as $\hat s_{0,t}\gt 0$ in equation (1), then we can derive $\hat s_{0,t}^* = \log \hat s_{0,t}$ . The other parameters and the model fit overall are unchanged, but the bivariate random walk with drift applied to $(\alpha _t, s_{0,t}^*)$ leads to non-decreasing mortality rates at all periods in the forecast, as shown in Figure 3. Thus, forecast quality can be improved at no change to the fit as long as $\hat s_{0,t}\gt 0$ .

The choice of which HS2 parameterisation to use – the trivariate $(\alpha _t, s_{0,t}, \omega _t)$ model of Tang et al. (Reference Tang, Li and Tickle2022) or the bivariate model $(\alpha _t, s_{0,t}^*)$ based on equation (1) – will depend on the application. For a long-term forecast of period mortality, one would probably use $(\alpha _t, s_{0,t}^*)$ . However, with short-term value-at-risk calculations for the likes of Solvency II (Richards et al., Reference Richards, Currie, Kleinow and Ritchie2020) it would be important to fully express the short-term variability in $\omega _t$ , and so one would probably use the trivariate parameterisation of Tang et al. (Reference Tang, Li and Tickle2022) for sample paths.

Table 1. AICs for various GLMs fitted to data for females in England & Wales, ages 50–105, 1971–2019.

Figure 2 Parameters for HS2 GLM behind Figure 1.

Figure 3 Forecast mortality rates in 2100 (i.e. time $n_y+81$ ) for females in England & Wales using alternative multipliers for the $h_{10}$ Hermite spline. Source: own calculations using data at ages 50–105, 1971–2019.