寿险精算公式汇总

1.(x)=1-F ()=P (X>x)

>=0x X r S r x x 生存函数： 2.我们约定：x (0)=0,S (0)=1;x F 3.r ()(X>y )=

()X X S y P X x S x > 4.

=Pr(T(x)>t)=Pr(X>x+t )

(+)=()t x X

X p X x S x t S x > 5.

++q =Pr[t

=-=-x t u t u x t x t x t u x q q p p ≤

6. 0

=exp[-(y)dy]=exp[-(x+s)ds]x t

t t x x p μμ+⎰⎰ 7.()f (t)=p x+t t ;T x t x μ

⋅≥（） 0 8. ) 1( x x dl x l dx

μ=- 9. +1x s s x x x

l p s q l ==-⋅ 10. ()1-11s 1111x x x x s x s s s x x s s s q p

q q p q q ++--⋅--⋅=-==-⋅-⋅

11.常值死力假定：x+=-ln s x p μ 12. 1111111(1)1(1)x s x s

x x x x x s x x x l p l s p s p s q l l l l +-++++⎛⎫⎧⎫==+-=+-=-- ⎪⎨⎬ ⎪⎩⎭⎝⎭ 13.1

:0

=(t)dt n t x n t x x A v p μ⋅⋅⎰ 14. n 年定期死亡保险： ()()2211::x n x n

Var Z A A =- 15. n 年期生存保险：()()2211::x x n n Var Z A A =-

16.延期保险：=

(t)dt t m x t x x m A v p μ∞⋅⋅⎰ 17.递推公式：1121

2221:+1:-1:+1:-1=+;

=+x x x x x n x n x n x n A vq vp A A v q v p A （）。

18.1:x n x x n x n

A A E A +=+ 19. m x x m k x x k m k k X m A E q p v A ++∞

=+==∑1

| 20.11:++:()=(IA)+x x x x n x n n n

IA A A +⋅（（IA ）n ） 21.-1

1

1::=0

(IA)k+n k x n x n k A =∑（1） 22. x i /A Ax δ

=

23./x x x A M D = 24.1+x :A =-/x x n x n （M M ）D 25.1:+n =/x x x n A D D 26.1++:--(IA)=x x n x n x n

x R R n M D ⋅ 27.0=x t x a E dt ∞⎰ 28.1=+x x a A δ 29. ()

()()22221δδδx x T T T A A Var Var a Var v v -==⎪⎪⎭⎫ ⎝⎛-= 30.ar(+)=0T T V a δυ 31.:0n t t x x n a v p dt =⎰ 32.22:::2

()()()x n x n x n Var Y a a a δ=-- 33. ::=1x n x n A a δ+ 34.:=x n x x n a a a +

35.11x x x a vp a +=+ 36.:1:11x x n x n a vp a +-=+

37.1(1)x x ia i A =++

38.x +1x a a = 39. +:-=x x n x n x N N a D 40.x x

()=x A P A a 41.x :()=

h x x h A P A a 42.x 1-()=x x a P A a δ 43.x ()=1-x x A P A A δ 44.11:::x n x n x n

A P a = 45.1:+:P =P (1)x n x x x n n n P A +- 46.1~:..:1P =x n x n n P s +

47.++()=-()x x t x t x t V A A P A a 48.+++-()()=

x t x x t x t x t M P A N V A D 49.111::-:+:-()=[()-()]x n x t n t x n t x t n t V A P A P A a + 50.++k +=x k x x k x x k M P N V D - 51...111:k

:+k:-k :-k 1

k :=A -<;=0=.x n x n x n x k n x n V P a k n V k n +