CHAPTER VII

COMMUTATION SYMBOLS

1. Up to this point we have been developing a theory in terms of fundamental symbols without making any attempt to ascertain arithmetical values. If we are to apply our theories we must be able to compute the numerical values of the symbols we have been using.

2. Let us attempt to find the numerical value of A25 which is the value now of an assurance of 1 payable at the end of the year of the death of a life now aged exactly 25. We have

1

A25 = — [vd25+v2d26+v3d27+ . . . ].

125

The values of d and the values of v are tabulated and we can per-form the series of multiplications, add our results and divide by the value of 125. This process is a tedious one, but it can be done. Before doing it we should do well to look ahead a little. We shall next wish to find the value A26 and we have

1

A26 = — [vd26+v2d27+v3d2s+ . . . ]. 126

Not one of our former multiplications, and there were many of them, is going to be of use again.

If we do obtain one value of Ax we might use the fact that (1+i) Ax =4x+pxAx+1 to obtain, one by one, the values at younger or older ages.

But we can do better, for in general

1   LL

   A, =   [vdx +v2dx+1 +v3dx+2 + . . . ]

   =   [vx+ldx +vx+2dx+1 +vx+3dx+2 + . . vxlx