壽險精算公式匯總

1、1.生存函數(shù)：Sx (x)=1-F X r ( x)=Pr (Xx)x=02.我們約定：Fx (0)=0,Sx (0)=1;3. Pr (XyXx )= SX ( y)SX (x)4.t px =Pr(T(x)t)=Pr(Xx+tXx )= SX (x+t) SX ( x)5.t u q x = PrtT(x)t+u= t +u qx - t qx = t px -t +u pxt px = exp-xtt(x+s)ds6.x(y)dy=exp-07. f T ( x) (t)= t px（ x+t ） t0;8.(x)1 dl xl xdx9.lx +s1s q xs pxl x10.1-s

2、q x s1px1s qxpx1s qx1 ss1s qx1s qx11.常值死力假定：x+ s =-lnpx12.1 s px sl x1l x 11s11pxs(1 px )1 (1 s) q xl xsl xl x 1l x1nvtpxx (t)dt13. Ax:n =t0Var Z211214. n 年定期死亡保險：A x: nA x: n15. n 年期生存保險： VarZ2 Ax:1nAx:1n216.延期保險： mAx =vttpxx (t)dtm17.遞推公式： A1x:n=vqx +vpx A1x+1: n-1 ; 2 A1x: n =v2qx +v2 p（x2 A1x+1:

3、n -1）。18. AxA1n Ex Ax nx:n19.m| AXvk 1k px qx km Ex Axmk m11IA ）x+ n +nAx+n）20.(IA)x =(IA) x:nAx:（n1n -1A121.(IA)（ k+1）kx:nk=0x:n22.Axi Ax /23. AxM x/ D x24. A 1x:n =（ Mx -Mx+ n）/Dx25. Ax:1n =Dx+n /Dx26. (IA) 1x:n= Rx -Rx +n -n M x+ nDx27. ax =0t Exdt28. 1= ax + AxTT22T1vVar vAx Ax29. Var aVar2230.V

4、ar( aT +T )=0n31. ax:nvttpxdt032.Var (Y)2(ax:n2ax:n ) (ax:n )233. Ax:n + ax: n =134. ax = n ax +ax:n&1&35. axvpx ax 1&36. ax:n1 vpxax 1:n 137.1ia x(1 i ) Ax&ax +138. ax&N x -N x +n39. ax:n=D x40. P( Ax )= Ax ax41.h P( Ax )=Axax:h42. P( Ax )=1-axax43.P( Ax )=Ax1- Ax1A1x:n44.P x: n&ax:n45. Px:n = n PxPx:1n (1 Ax +n ) 1146.Px:n = P x: n.sn47.t V ( Ax )= Ax+t -P( Ax )ax+ tM x+ t - P( Ax ) N x+ t48. t V ( Ax )=Dx+ t111) a49. V ( Ax:n)= P( A)- P( Ax:ntx t:n -tx +t: n-t50. kVx = M x+