Correlated age-specific mortality model: an application to annuity portfolio management

Abstract

This article models the dynamics of age-specific incremental mortality as a stochastic process in which the drift rate can be simply and effectively modeled as the average annual improvement rate of a group time trend for all ages and the distribution of residuals can be fitted by one of the Gaussian distribution and four non-Gaussian distributions (Student t, jump diffusion, variance gamma, and normal inverse Gaussian). We use the one-factor copula model with six distributions for the factors (normal–normal, normal–Student t, Student t–normal, Student t–Student t, skewed t–normal, and skewed t–Student t) to capture the inter-age mortality dependence. We then construct three annuity portfolios (Barbell, Ladder, and Bullet) with equal portfolio value (total net single premium) and portfolio mortality duration but different portfolio mortality convexities. Finally, we apply our model to managing longevity risk by an approximation to the change in the portfolio value in response to a proportional or constant change in the force of mortality, and by estimating Value at Risk for the three annuity portfolios.

Appendix

Given $\ln (m_{x,\,t})$ data in the age-year window $[x_L,\,x_U]\times [t_L,\,t_U]$, the non-parametric Bühlmann estimate $\Delta \hat{Y}_{x,\,t_U+1}{\mathop {=}\limits ^{\bigtriangleup }} \ln (\hat{m}_{x,\,t_U+1})-\ln (m_{x,\,t_U})$ for age $x=x_L,\ldots , x_U$ in year $t_U+1$ is given by Tsai and Lin [39] as

$$\begin{aligned} \Delta \hat{Y}_{x,\,t_U+1}=Z\cdot \Delta \overline{Y}_x+(1-Z)\cdot \Delta \overline{Y}, \end{aligned}$$

(A.1)

where $\Delta \overline{Y}_{x}=\frac{1}{n-1}\sum _{t=t_L+1}^{t_U}\Delta Y_{x,\,t}$ with $n-1=t_U-t_L$ and $\Delta Y_{x,\,t}{\mathop {=}\limits ^{\bigtriangleup }}\ln (m_{x,\,t})-\ln (m_{x,\,t-1})$, and $\Delta \overline{Y}=\frac{1}{m}\sum _{x=x_L}^{x_U} \Delta \overline{Y}_x$ with $m=x_U-x_L+1$. Then the non-parametric Bühlmann estimate of $\ln (m_{x,\,t_U+1})$ for age x in year $t_U+1$ is

$$\begin{aligned} \ln (\hat{m}_{x,\,t_U+1})=\ln (m_{x,\,t_U})+\Delta \hat{Y}_{x,\,t_U+1}. \end{aligned}$$

(A.2)

The estimate of the Bühlmann credibility factor Z is given as

$$\begin{aligned} \hat{Z}=\frac{(n-1)}{(n-1)+\hat{v}/\hat{a}}, \end{aligned}$$

where $\hat{v}=\frac{1}{m}\sum _{x=x_L}^{x_U}\hat{v}_x$ with $\hat{v}_{x}=\frac{1}{n-2}\sum _{t=t_L+1}^{t_U}(\Delta Y_{x,\,t}-\Delta \overline{Y}_{x})^2$, and $\hat{a}=\frac{1}{m-1}\sum _{x=x_L}^{x_U} (\Delta \overline{Y}_x-\Delta \overline{Y})^2-\frac{\hat{v}}{n-1}$. Note that it is possible that $\hat{a}$ could be negative due to a substraction. When that happens, it is customary to set $\hat{a}=0$, implying $\hat{Z}=0$.

The non-parametric Bühlmann credibility estimate $\Delta \hat{Y}_{x,\,t_U+\tau }$ for year $t_U+\tau$ ($\tau \ge 1$) is expressed by

$$\begin{aligned} \Delta \hat{Y}_{x,\,t_U+\tau }=Z(t_U+\tau )\cdot \Delta \overline{Y}_{x}(t_U+\tau ) +[1-Z(t_U+\tau )]\cdot \Delta \overline{Y}(t_U+\tau ), \end{aligned}$$

(A.3)

where $Z(t_U+\tau )$, $\Delta \overline{Y}_{x}(t_U+\tau )$ and $\Delta \overline{Y}(t_U+\tau )$ are Z, $\Delta \overline{Y}_{x}$ and $\Delta \overline{Y}$ for year $t_U+\tau$, respectively. Then the non-parametric Bühlmann credibility estimates of $\ln (m_{x,\,t_U+\tau })$ for age x in year $t_U+\tau$ is given by

$$\begin{aligned} \ln (\hat{m}_{x,\,t_U+\tau })=\ln (m_{x,\,t_U}) +\sum _{t=1}^{\tau }\Delta \hat{Y}_{x,\,t_U+t}. \end{aligned}$$

(A.4)

We adopt the expanding window strategy to determine $\Delta \hat{Y}_{x,\,t_U+\tau }$ for $\tau =2, 3,\ldots$. We first add the Bühlmann credibility estimate $\Delta \hat{Y}_{x,\,t_U+1}$ to $\{\Delta Y_{x,\,t_L+1},\ldots ,\Delta Y_{x,\,t_U}\}$ such that the initial year span is expanded by one year to $[t_L,\,t_U+1]$. Then we obtain

$$\begin{aligned} \Delta \overline{Y}_x(t_U+2)= & {} \frac{1}{n}\left[ \sum _{t=t_L+1}^{t_U}\Delta Y_{x,\,t}+\Delta \hat{Y}_{x,\,t_U+1}\right] ,\\ \Delta \overline{Y}(t_U+2)= & {} \frac{1}{m} \sum _{x=x_L}^{x_U}\Delta \overline{Y}_x(t_U+2) \end{aligned}$$

and

$$\begin{aligned} Z(t_U+2)=\frac{n}{n+\hat{v}/\hat{a}}. \end{aligned}$$

The non-parametric Bühlmann credibility estimate $\Delta \hat{Y}_{x,\,t_U+2}$ is given by (A.3) with $\tau =2$. Repeat the same procedure to get

$$\begin{aligned} \Delta \overline{Y}_x(t_U+\tau )= & {} \frac{1}{n+\tau -2}\left[ \sum _{t=t_L+1}^{t_U}\Delta Y_{x,\,t}+\sum _{t=t_U+1}^{t_U+\tau -1}\Delta \hat{Y}_{x,\,t}\right] ,\\ \Delta \overline{Y}(t_U+\tau )= & {} (1/m)\sum _{x=x_L}^{x_U}\Delta \overline{Y}_x(t_U+\tau ) \end{aligned}$$

and

$$\begin{aligned} Z(t_U+\tau )=\frac{n+\tau -2}{n+\tau -2+\hat{v}/\hat{a}} \end{aligned}$$

using the data in the year span $[t_L, t_U+\tau -1]$ for $\Delta \hat{Y}_{x,\,t_U+\tau }, \tau =3, 4,\ldots$.

Tsai and Lin [39] prove that

$$\begin{aligned} \Delta \hat{Y}_{x,\,t_U+\tau }=\Delta \hat{Y}_{x,\,t_U+1} {\mathop {=}\limits ^{\bigtriangleup }}\hat{\theta }^B_{x} \end{aligned}$$

(A.5)

for $\tau >1$ under the expanding window strategy. As a result, the non-parametric Bühlmann estimates of $\ln (m_{x,\,t_U+\tau })$ for $\tau \ge 1$ can be obtained from (A.4) as

$$\begin{aligned} \ln (\hat{m}_{x,\,t_U+\tau })=\ln (m_{x,\,t_U})+\tau \cdot \hat{\theta }^B_{x}. \end{aligned}$$

(A.6)