48   Life Contingencies

(m) 1 —{1 Aim)

a,il   — 1 A

x —_ .~(m)   J (m)   x   :lc.)

xni

axx

1

=   = — .

S   S   b

Ax~.

12. It will be noticed, in the case of whole life and endowment assurance, that there is a very simple linear relation between the value of the assurance and the value of the corresponding annuity.

For example, we have 1=d(1-Fax)+Ax

and 1=d(l+axn_11)+Axni•

That is, if a represents the value of the assurance and (3 the value of the corresponding annuity, we have

1=d(l+0)+a.

Since the range of 0 is limited to values lying between 0 and 1 we can prepare tables, one for each rate of interest desired, showing the value of a corresponding to values of (3 beginning at 0 and rising, by differences of .01 for example, up to 1 . Such a table is called a Single Premium Conversion Table.

If we enter such a conversion table with the value ax at a given rate of interest we arrive at the value Ax at the same rate of interest and conversely.

Also entering the same conversion table with the value ax,-F--_f. we arrive at the value Ax nl .

13. Consider, again, the difference in value between al') and ax

Due to the earlier payments made under azm), the difference each year amounts to interest on 1 for m -1 of a year

m   m

plus interest on 1 for m—2 of a year

m   m