36   Life Contingencies

dx dx+1 dx+2

— ,   ,   ... respectively

lx   lx   lx

d

ax =oXx+al Xdlzl+a,Xdx+2+.

x   lx

= ll [vd,+1 + (v+v2)dx+2+(v +v2+v3)dx+3+ ... 'J = Ix [vlx+l +v2lx+2 +v31x+3+ . . . ].

If the first payment were to be made at once the value of the annuity would be 1+ax=ax. This is called an annuity-due.

If i = 0, so that vn =1 for all values of n, we have

   ex =Px+2px+3px+    

4. If an annuity on (x) is not to run for more than n years, so that the maximum number of payments that can be made is n, we have a temporary annuity

ax=vpx+v` •2px+v3•3px+ . . . +vnnpx'

=[vpx+v22px+v33px+ . . . to end of table]

—[vn n+1px+vn n+2px+ . . . to end of table]

=ax—vnnpx[vpx+n+v22px+n+ • • ]

n

=ax —v npzax+n '

Also axnj =ax —vnnpxax-}.n = +axn li

5. If no payments of an annuity on (x) are to be made for the first n years, so that the annual payments of one each will begin when the man is aged x+n+1, we have a deferred annuity

n+1   +2
n ax =v 'n+lpx +vn 'n+2px+

=vnnpx[vpx+n+v2'2px+n+ . . . ] =vnnpxax+n

=ax—axn, =ax—axn+11•