52   Life Contingencies

Similarly   Pxn -,axn

,7 =Ax nl

Pxnlax, =Axe

Px ,1ax-,71 =Axn.

The student should note from the above statements the exact meanings of the symbols for the different annual premiums.

2. From Px.ax=Ax and 1=dax-l-Ax,we can, by algebra, deduce x = 1 —d =   dA

P       x

ax   1—Ax

If a man aged exactly x were to borrow 1 for the period of his life and agree to pay Px+d in advance each year so long as he lived, the lender would receive interest for each year including the year of death together with the annual payment Px which is by definition equivalent in value to the return of the capital at the end of the year of death, or 1 = (Px+d)ax.

If a man aged exactly x were to borrow Ax, use it as a single premium for an assurance of 1, agree to pay interest on the loan each year in advance, and charge his policy with the repayment of the loan at the end of the year of his death, he would be paying dAx each year in advance for a policy of 1—Ax or dAx =Px(1-Ax).

3. Schedule on opposite page.

4. Any two of these fundamental links 1=dax+Ax

1= (Px+d) (1+ax)

dAx =1),(1 —A x)

Ax=Px.ax

may be deduced by algebra from the other two. The student should make sure that he can do this.

In some cases two different explanations may be given of any one connecting link, since the product of any two ordinary symbols may be translated into two different statements; for instance,