Assurances, Single Premiums   29

We have seen that (x), according to the life table, must fail during one and obviously can fail in only one of the next w+1 —x years where w is the highest age for which a value appears in the lx column of the life table. We have also seen that the terms of the expansion

1= x+    x+l +   +dx+tt-1+   +

lx   lx   lx   lx

represent the respective probabilities that (x) will fail in the first, second, ... , nth, . . . and (0)+1 —x)th year from now. Further the value of one payable at the end of each of these years for certain is: v at the end of the first year, v' at the end of the second year, etc., up to v"+1' at the end of the (w+1—x)th year.

Hence

A _7, dx +v2dx+1 + . . -1-iJndx+-1 +... +71"-x+1 .d,o . lx   lx   lx   lx

3. Again, consider lx lives all aged exactly x and suppose one unit is to be paid at the end of the year of death of each of them, then

IX d,   will be required at the end of the first year.

1X cl,+1 will be required at the end of the second year. .................................................. 1 Xdx+n_1 will be required at the end of the nth year. ..................................................

1 Xdx,   will be required at the end of the (w+1—x)th year.

The total present value of all the payments to be made as the lives fail is

vdx+v2dx+1 + . . . +zr". dx +n-1 + . . . +v'+1. d,,, and hence the average value required for a life now aged x is

Ax= - [vdx+v2dx+1+ .   +2'ndx+n-1+ . . . +v"—x+1dm].

Ax is the value or single premium for a whole life assurance where the sum assured is one payable at the end of the year in which a life now aged exactly x fails.