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,Tx表示(x)的余命的随机变量(Future lifetime rv.),T0 为新生儿的余命的随机变量。Kx表示 (x) 的取整余命的随机变量(Curtate future lifetime rv.)。

T0的分布

cdf of T0:

F0(x)=P(T0x)

sf of T0:

S0(x)=P(T0>x)=1F0(x)

pdf of T0:

f0(x)=ddxF0(x)=ddxS0(x)

Relationship between T0 and Tx

Events that are equivalent (important relationship between T0 and Tx): TxtTx=T0xt|T0>x

Tx的分布

cdf of Tx:

Fx(t)=P(Txt)=F0(x+t)F0(x)1F0(x)

sf of Tx:

Sx(t)=P(Tx>t)=1Fx(t)=S0(x+t)S0(x)

pdf of Tx:

fx(t)=ddtFx(t)=f0(x+t)S0(x)

Chaining Survival

S0(x+t)=S0(x)Sx(t) Sx(t+u)=Sx(t)Sx+t(u)

Conditions on Sx(t)

Sx(0)=P(Tx>0)=1 limxSx(t)=limxP(Tx>t)=0 Sx(t) is non-increasing in t

Assumptions on Sx(t)

Sx(t) is differentiable t>0

E[Tx]< Var[Tx]<

The force of mortality

μx,称为force of mortality, hazard rate, transition intensity, 定义为: μx=limdx0+P(T0x+dx|T0>x)dx=limdx0+P(Txdx)dx 需要掌握的几个公式: μx=f0(x)S0(x)=xlogS0(x)

μx+t=fx(t)Sx(t)=xlogSx(t)

Sx(t)=exp(0tμx+rdr)

Typical μx curve for humans

μx on log10 scale (ELT15 (Males) Mortality Table) The main features of μx 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: TxUni(0,ωx)
  • Compertz Model
  • Generalized (or Modified) De Moivre Model

Actuarial Notation

Survival:

ptpx=P(Tx>t)=Sx(t)

Mortality:

qtqx=P(Txt)=Fx(t)

Deferred Mortality:

qt|uqx=P(t<Txt+u)

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

p1px=ppx, q1qx=qqx, qt|1qx=qt|qx.

3 ways to calculate qt|uqx

qt|uqx=Fx(t+u)Fx(t)=qt+uqxqtqx qt|uqx=Sx(t)Sx(t+u)=ptpxpt+upx qt|uqx=ptpxqx+tqu

Relationship between F and f

qtqx=0tpupxμx+udu

Approximate qx with μx+u

For small qx:

qx011μx+uduμx

qx can also be approximated by μx+12.

Distributional Quantities of Tx

Mean of Tx

Complete expectation of life: e˚x=E[Tx]=0tfx(t)dt=0tptpxμμx+tdt=0ptpxdt

Median of Tx

This is the value m s.t. P(Tx>m)=0.5. i.e. pmpx=0.5

Variance of Tx

VarTx=E[Tx2](ETx)2

Distribution of Kx

pmf:

P(Kx=k)=P(kTxk+1)=qk|1qx=qk|qx

cdf

P(Kxk)=1pk+1px=qk+1qx

sf:

P(Kx>k)=pk+1px

Mean of Kx

ex=E[Kx]=k=0kP(Kx=k)=k=1pkpx

Variance of Kx

Var[Kx]=2k=1kpkpxex(ex)2

Relationship between ex and eox

e˚xex+12

精算后花园

精算后花园