30   Life Contingencies

4. The value or single premium for a term assurance where the sum assured is one to be paid at the end of the year in which (x) fails, should (x) die within n years, is written A.

Axe =Z'ax +v2dx+1 +...+vndx+n-1 = E, vdx+r-1

lx   lx   lx   lxr=l

= L [vdx+v"dx+1+ ... +v`"-x+ldj

1 [vn+l

—   dx+n+vn+2dx+n+1 + . . . +vw-x+ldm]

lx

=Ax —vn. lx+n. 1 [vdx+n+v2dx+n+1+ . . . +v"—x—n+1dJ Ix lx+n

n

=Ax =Ax—Axl .Ax+,=.

5. Similarly, the value or single premium for a deferred life assurance where the sum assured is one to be paid at the end of the year in which (x) dies, provided (x) dies after attaining at least age x+n, is written n Ax =Ax„~

—x = Z [vn+l dx+,z +vn+2dx+n+1 + . . . + vw-x+ld„,l = vnnpx . Ax+n. x

Also Axn =Ax—Al ;

so that Ax =A +n I Ax =AxTil

6. Again, A =v dx +v 1x+1 CZ~dx+1 +v2dx+2 + • • .] =vqx+vpx•Ax+1• lx   lx   lx+ 1

This can be written

A.x. (1+i) =qxX 1+pxXAx+1

=Ax+1+qx. (1—Ax+1)

=1—Px(1 —Ax+1)

and further, as a means of obtaining Ax+1 from Ax, we have

1+i qx

px   px

=Ax . ux— kx, (Fackler's accumulation formula)

Ax+1 =Ax.