【CS2】Ch06 Survival models 生存模型

Jackie

IFoACM1CS2

发布于：2021年1月18日

186 次浏览

本文介绍生存模型，属于CS2第六章的内容，也是学习CM1的寿险精算部分的先修内容。

本文不仅适用于英国精算师的考试，Jackie 在写这篇推文时也参考了北美体系的内容，所以考SOA的同学也不妨看一下。

Notation

$(x)$ 表示当前为 $x$ 岁的个体，$T_x$表示(x)的余命的随机变量(Future lifetime rv.)，$T_0$ 为新生儿的余命的随机变量。$K_x$表示 $(x)$ 的取整余命的随机变量(Curtate future lifetime rv.)。

$T_0$的分布

cdf of $T_0$:

$$F_0(x)=P(T_0\le x)$$

sf of $T_0$:

$$S_0(x)=P(T_0>x)=1-F_0(x)$$

pdf of $T_0$:

$$f_0(x)=\frac{d}{dx}{F}_0(x)=-\frac{d}{dx}{S}_0(x)$$

Relationship between $T_0$ and $T_x$

Events that are equivalent (important relationship between $T_0$ and $T_x$): $$T_x\le t\iff T_x=T_0-x\le t|T_0>x$$

$T_x$的分布

cdf of $T_x$:

$F_x(t)=P(T_x\le t)=\frac{F_0(x+t)-F_0(x)}{1-F_0(x)}$

sf of $T_x$:

$S_x(t)=P(T_x>t)=1-F_x(t)=\frac{S_0(x+t)}{S_0(x)}$

pdf of $T_x$:

$f_x(t)=\frac{d}{dt}{F}_x(t)=\frac{f_0(x+t)}{S_0(x)}$

Chaining Survival

$$S_0(x+t)=S_0(x)S_x(t)$$ $$S_x(t+u)=S_x(t)S_{x+t}(u)$$

Conditions on $S_x(t)$

$$S_x(0)=P(T_x>0)=1$$ $$\underset{x\to \mathrm{\infty }}{lim}{S}_x(t)=\underset{x\to \mathrm{\infty }}{lim}P(T_x>t)=0$$ $S_x(t)$ is non-increasing in $t$

Assumptions on $S_x(t)$

$S_x(t)$ is differentiable $\mathrm{\forall }t>0$

$$E[T_x]<\mathrm{\infty }$$ $$Var[T_x]<\mathrm{\infty }$$

The force of mortality

${\mu }_x$，称为force of mortality, hazard rate, transition intensity, 定义为： $$\begin{array}{rl}{\mu }_x& =\underset{dx\to 0^{+}}{lim}\frac{P(T_0\le x+dx|T_0>x)}{dx}\\ \\ & =\underset{dx\to 0^{+}}{lim}\frac{P(T_x\le dx)}{dx}\end{array}$$ 需要掌握的几个公式： $${\mu }_x=\frac{f_0(x)}{S_0(x)}=-\frac{\mathrm{\partial }}{\mathrm{\partial }x}log{S}_0(x)$$

$${\mu }_{x+t}=\frac{f_x(t)}{S_x(t)}=-\frac{\mathrm{\partial }}{\mathrm{\partial }x}log{S}_x(t)$$

$$S_x(t)=exp(-{\int }_0^t{\mu }_{x+r}dr)$$

Typical ${\mu }_x$ curve for humans

${\mu }_x$ on $lo{g}_{10}$ scale (ELT15 (Males) Mortality Table) The main features of ${\mu }_x$ are:

• High infant mortality
• An ‘accident hump’ at ages around 20
• The nearly exponential increase at older ages

Some Common Mortality Models

• Exponential Model: Constant force of mortality
• De Moivre Model: $T_x\sim Uni(0,\omega -x)$
• Compertz Model
• Generalized (or Modified) De Moivre Model

Actuarial Notation

Survival:

$${\phantom{p}}_t^{}{p}_x^{}=P(T_x>t)=S_x(t)$$

Mortality:

$${\phantom{q}}_t^{}{q}_x^{}=P(T_x\le t)=F_x(t)$$

Deferred Mortality:

$${\phantom{q}}_{t|u}^{}{q}_x^{}=P(t<T_x\le t+u)$$

It's convention that $t=1$ year, we omit the 1, i.e.

${\phantom{p}}_1^{}{p}_x^{}={\phantom{p}}_{}^{}{p}_x^{}$, ${\phantom{q}}_1^{}{q}_x^{}={\phantom{q}}_{}^{}{q}_x^{}$, ${\phantom{q}}_{t|1}^{}{q}_x^{}={\phantom{q}}_{t|}^{}{q}_x^{}$.

3 ways to calculate ${\phantom{q}}_{t|u}^{}{q}_x^{}$

$${\phantom{q}}_{t|u}^{}{q}_x^{}=F_x(t+u)-F_x(t)={\phantom{q}}_{t+u}^{}{q}_x^{}-{\phantom{q}}_t^{}{q}_x^{}$$ $${\phantom{q}}_{t|u}^{}{q}_x^{}=S_x(t)-S_x(t+u)={\phantom{p}}_t^{}{p}_x^{}-{\phantom{p}}_{t+u}^{}{p}_x^{}$$ $${\phantom{q}}_{t|u}^{}{q}_x^{}={\phantom{p}}_t^{}{p}_x^{}\cdot {\phantom{q}}_{x+t}^{}{q}_u^{}$$

Relationship between F and f

$${\phantom{q}}_t^{}{q}_x^{}={\int }_0^t{\phantom{p}}_u^{}{p}_x^{}{\mu }_{x+u}du$$

Approximate $q_x$ with ${\mu }_{x+u}$

For small $q_x$:

$$\begin{array}{r}{q}_x\approx {\int }_0^{1}1\cdot {\mu }_{x+u}du\approx {\mu }_x\end{array}$$

$q_x$ can also be approximated by ${\mu }_{x+\frac{1}{2}}$.

Distributional Quantities of $T_x$

Mean of $T_x$

Complete expectation of life: $${\mathring{e}}_x=E[T_x]={\int }_0^{\mathrm{\infty }}t{f}_x(t)dt={\int }_0^{\mathrm{\infty }}t\cdot {\phantom{p}}_t^{}{p}_x^{}{\phantom{\mu }}_{}^{}{\mu }_{x+t}^{}dt={\int }_0^{\mathrm{\infty }}{\phantom{p}}_t^{}{p}_x^{}dt$$

Median of $T_x$

This is the value $m$ s.t. $P(T_x>m)=0.5$. i.e. ${\phantom{p}}_m^{}{p}_x^{}=0.5$

Variance of $T_x$

$$Var{T}_x=E[{T_x}^2]-(E{T}_x{)}^2$$

Distribution of $K_x$

pmf:

$$P(K_x=k)=P(k\le T_x\le k+1)={\phantom{q}}_{k|1}^{}{q}_x^{}={\phantom{q}}_{k|}^{}{q}_x^{}$$

cdf

$$P(K_x\le k)=1-{\phantom{p}}_{k+1}^{}{p}_x^{}={\phantom{q}}_{k+1}^{}{q}_x^{}$$

sf:

$$P(K_x>k)={\phantom{p}}_{k+1}^{}{p}_x^{}$$

Mean of $K_x$

$$e_x=E[K_x]=\sum _{k=0}^{\mathrm{\infty }}k\cdot P(K_x=k)=\sum _{k=1}^{\mathrm{\infty }}{\phantom{p}}_k^{}{p}_x^{}$$

Variance of $K_x$

$$Var[K_x]=2\sum _{k=1}^{\mathrm{\infty }}k\cdot {\phantom{p}}_k^{}{p}_x^{}-e_x-(e_x{)}^2$$

Relationship between $e_x$ and ${\stackrel{o}{e}}_x$

$${\mathring{e}}_x\approx e_x+\frac{1}{2}$$

本页文档最后更新于：2020年8月1日