Coherent extrapolation of mortality rates at old ages applied to long term care

Abstract

In an insurance context, Long-Term Care (LTC) products cover the risk of permanent loss of autonomy, which is defined by the impossibility or difficulty of performing alone all or part of the activities of daily living (ADL). From an actuarial point of view, knowledge of risk depends on knowledge of the underlying biometric laws, including the mortality of autonomous insureds and the mortality of disabled insureds. Due to the relatively short history of LTC products and the age limit imposed at underwriting, insurers lack information at advanced ages. This represents a challenge for actuaries, making it difficult to estimate those biometric laws. In this paper, we propose to complete the missing information at advanced ages on the mortality of autonomous and disabled insured populations using information on the global mortality of the portfolio. In fact, the three previous mortality laws are linked since the portfolio is composed only of autonomous and disabled policyholders. We model the two mortality laws (deaths in autonomy and deaths in LTC) in a Poisson Generalized Linear Model framework, additionally using the P-Splines smoothing method. A constraint is then included to link the mortality laws of the two groups and the global mortality of the portfolio. This new method allows for estimating and extrapolating both mortality laws simultaneously in a consistent manner.

Appendix A: Convergence of the Newton Raphson algorithm

To be the maximum penalized likelihood estimator of $\varvec{\theta }$, the Hessian matrix $H_{\widehat{\varvec{\theta }}}$ at the final step of the algorithm has to be negative semi-definite.

Let us analyse the Hessian matrix.

• The first term $-B^T W W_{\varvec{\theta }}B$ is negative semi-definite for all $\varvec{\theta }$. Indeed, recalling that $W_{\varvec{\theta }}$ is diagonal with only non-negative terms,

  $$\begin{aligned} h^TB^TW_{\varvec{\theta }}Bh = (Bh)^T W_{\varvec{\theta }}(Bh) \ge 0 \, \quad \forall h \in \mathbb {R}^{2M}. \end{aligned}$$

• The second term $-P$, which does not depend on $\varvec{\theta }$, is also negative semi-definite. Indeed, from 2.3, we know that $P_d = D_d^T D_d$. Therefore, $h^T P_d h \ge 0 \, \quad \forall h \in \mathbb {R}^{2M}$.

• The third term $- KB^T \left[ W_{\varvec{\theta }} \left( [(\tilde{W}_3^{-1})^2 W_{\varvec{\theta }}^Q ] \otimes I_2 \right) \right] B$ is not necessarily negative semi-definite for all $\varvec{\theta }$. In fact, the weight matrix $\left[ W_{\varvec{\theta }} \left( [(\tilde{W}_3^{-1})^2 W_{\varvec{\theta }}^Q ] \otimes I_2 \right) \right]$ is diagonal, but not all coefficients are greater than 0 for some $\varvec{\theta }$. The terms of the diagonal matrix are non-positive if some terms of $W_{\varvec{\theta }}^Q$ are non-negative. This is the case when

  $$\begin{aligned} \lambda ^{A}_{\varvec{\theta },x} e^A_{x} + \lambda ^D_{\varvec{\theta },x} e^D_{x} \le \lambda _{x}^{gen} [ e^A_{x} + e^D_{x} ],\hbox { for some } x_{min} \le x \le x_{max}. \end{aligned}$$

• The fourth term $-K \begin{bmatrix} \tilde{W}_3^{-1} W_{\varvec{\theta }}^{A} B_{A}&\tilde{W}_3^{-1} W_{\varvec{\theta }}^{D} B_{D} \end{bmatrix}^T \begin{bmatrix} \tilde{W}_3^{-1} W_{\varvec{\theta }}^{A} B_{A}&\tilde{W}_3^{-1} W_{\varvec{\theta }}^{D} B_{D} \end{bmatrix}$ is negative semi-definite for all $\varvec{\theta }$. In fact,

  $$\begin{aligned}&h^T \begin{bmatrix} \tilde{W}_3^{-1} W_{\varvec{\theta }}^{A} B_{A}&\tilde{W}_3^{-1} W_{\varvec{\theta }}^{D} B_{D} \end{bmatrix}^T \begin{bmatrix} \tilde{W}_3^{-1} W_{\varvec{\theta }}^{A} B_{A}&\tilde{W}_3^{-1} W_{\varvec{\theta }}^{D} B_{D} \end{bmatrix} h \\&\quad = \left\| \begin{bmatrix} \tilde{W}_3^{-1} W_{\varvec{\theta }}^{A} B_{A}&\tilde{W}_3^{-1} W_{\varvec{\theta }}^{D} B_{D} \end{bmatrix} h \right\| _2^2 \\&\quad \ge 0 . \end{aligned}$$

Then, a sufficient condition for $H_{\varvec{\theta }}(l_{pen})$ to be negative semi-definite and therefore for $\widehat{\varvec{\theta }}$ to be the optimal parameter is that the third term is negative semi-definite. The condition is given by

$$\begin{aligned} \lambda ^{A}_{\varvec{\theta },x} e^A_{x} + \lambda ^D_{\varvec{\theta },x} e^D_{x} \le \lambda _{x}^{gen} ( e^A_{x} + e^D_{x} ) , \quad \forall x_{min} \le x \le x_{max} . \end{aligned}$$

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This means that the sum of the predicted number of deaths in states A and D has to be lower than or equal to the predicted number of deaths of the overall population.