34   Life Contingencies

and Ax> v`x , since Ax =Ax(1+i)l and ex =1 +ex approximately.

11. Suppose that we have two lives, one aged x and the other aged y.

The value or single premium for a joint life assurance where the sum assured is one to be paid at the end of the year in which the pair is broken is written A,,„.

oo

   Axy=   vn(n—lpxy—npxy n1

[V(lxy—lx+l : y+l) + v2 (lx+1 : y+1 -1x+2 :y+2) 7xY

1

   =   [vdxY +v2dx+1 : y+1 +    ] .
xy

12. The value or single premium for a joint life term assurance where the sum assured is one to be paid at the end of the year in which the pair is broken, if the breakage occurs during the term, is written Azy , where the term is n years

1   n

   AL   n• =— [vdxy+v2dx+l :y+1+   +v dx+n—1 :y+n—1]

xY

n

=Axy—v npxyAx+n:y+n

13. The value or single premium for a survivorship life assurance where the sum assured is one to be paid at the end of the year_in which the last survivor of (x) and (y) dies is written Ax,, ,

oo

   Axy —   v (n—lpxy ,epxy n=1

   =   vn (n—lpx+n—lpy n—lpxy)—(npx+npy npxy) n=1

_ E [vn(n—lpx—npx)+vn(n—lPy npy)—vn(n—lpxy—npxy) n=1

=Ax+Ay —Axy.