44   Life Contingencies

3. Consider,

,hh   n   n—1

Ax =vnnt'x = E vrrhh t'x E vrrpx =ax

   r=1   r=1

That is, the value now of one payable at the end of n years, if (x) should be alive at the end of n years, or the value of a pure endowment of 1 to (x) due at the end of n years, can be expressed as the difference in value between two temporary annuities of 1 each on the life (x). The first annuity to provide 1 per annum for a possible n years while the second annuity will withdraw 1 per annum for a possible n—1 years.

dx+r—1

4. Again,   Az = E, yr   

r=1 (lx+r—1 r   —lx+r)

r=1   lx   //

=v E yr—1 lx+r—1 — E yr lx+r
   r=1   lx   r=1   lx

or   Az =vax~—ax~.

In words, we may say that the value now of one payable at the end of the year in which (x) dies, should (x) die within the next n years, is the difference in value between two annuities. The first annuity will provide 1 at the end of each year on which (x) enters for a possible n years, while the second annuity will withdraw 1 at the end of each year that he completes for a possible n years: the difference being the value of the 1 payable at the end of that year within the period which he may begin and not complete.

Also,   Ax =`4x l Ax nl

=ax —ax   +vax —ax
=vax n~ —ax n~ .

This relation may also be stated in words in a similar manner so as to confirm its truth.