38   Life Contingencies

(Ia) = E, nvnnpx =vpx+2v2.2px+3v3 ' 3Px + .

n=1

= ax +vpx ax+1 +v22pxax+2 + • •

CO

-1

_ E V

n n—lpxax+n—1 n=1

or   = vpx ax+1 +v22px ax+2 +v33px ' ax+3 + . . .

co

=   vnnpxax+n.

n=1

8. Suppose that a life annuity on (x) of 1 per annum is to be paid twice a year: the symbol for it would be written ax2). The following diagram will illustrate the payments:

year of death

A   B

I   I   I   I

1   2

2   1   1   i   i   i   i   1?

2   2   •••••• 2   2   2   2 obviously axe) is greater than ax by the value of

1. a possible i that may be paid in the year of death

2. the interest on 2 for half a year during each completed year of life.

On the assumption of a uniform distribution of deaths in each year, the annuitant is equally likely to die in either half of the last year, therefore the value at B of the possible last payment

3

of 2 is 1X =IA

The value each year of the half year's interest on the   is 2(1+i)'—2 =4i—1 2+—etc.

A year's interest on the final ; vl is 4 (1 +i) -I• i = 4 i — sit+ .. .

These are nearly equal and we can say that the sum of the values of (i) and (ii) above is approximately the value of an ultimate upon which interest will be paid until the 4 is paid, which is the value of in cash.

Therefore axe) =ax+4, approximately.

9. Similarly if a life annuity on (x) of 1 per annum is to be paid 4 times a year, its value, written ax4), will exceed the value of ax by the value of