54   Life Contingencies

Px.ax =the value of an annuity-due of Px per annum on the life of x,

the annual premium for a whole life assurance of ax. or, again

dAx   the annual interest due in advance each year on a loan of Ax

the single premium for a whole life assurance of d, and so on.

The above links may be modified and meanings can generally be given to the modifications.

For example, Ax =1—dax, which means that if the unit payable under the policy on (x) were payable at once its value would be 1, but as it is not payable until the end of the year of death we must deduct the value of the interest throughout the lifetime of (x).

Also, for example, Ax =v(1+ax) —ax, which means that if (x) was entitled to a payment of 1 at the end of each year he began which is equivalent to v at the beginning of each such year, and (x) was obliged to pay 1 at the end of each year he completed, his property in regard to these annuities would be only the value of that 1 due at the end of the year which he will ultimately begin but not complete.

5. For a term shorter than the whole of life, we have 1=(Pin, +d )ax

but note 1= (Pz +d)ax n +Ax -4

and   1= (Px ni +d)ax n +Ax;, .

Also   dAx i1 =Px -i; (1 —Ax j )

but note dAzn~+Px; .Axe =Pxn (1—A4)

and   dAx iij-1 +Pint. Ax =Px,i (1—Ax4).

6. It will be noticed, in the case of whole life and endowment assurances, that there is a very simple relation between the annual premium for the assurance and the value of the corresponding annuity.