Annuities   41

The total future life time of the lx persons is J lx+.t.dt years.

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The complete expectancy of life is   Jlx+t.dt= tpx.dl years.

lx o   o

The rectangle OL = the area under the curve and therefore the area A = the area B.

The value of ax, an annuity on (x) of 1 per annum payable mo-

mently, is represented by   v`tpxdt, where the payment at the

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end of time t is dt and the probability of survival to receive it is tpx.

Therefore lxax = J vtlx+t. dt =the sum of all the elements of the

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area under the curve such as lx+t. dt each multiplied by its own power of v.

But lxaexl   o

=lx.Jvt.dt=the sum of all the elements of the area of the rectangle OL such as lx.dt each multiplied by its own power of v.

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Comparing lx. J v`tpx. dt and lx. ~e vtdt, we see that so far as the o   o part of the area which is common to both is concerned, they are

identical. But although the area of A is equal to the area of B, the discount factor v has a smaller value with each element of B than it has with any element of A.

Therefore ax must be less than a_ °exl.

12. Again, we have seen that

ex =px+2px+3px+ . . . +npx+ . .

Consider the following expressions:

(1) ax =vpx+v22px+v3px+ . . . +vkPx+   ,hh

. . . +vm mx xt'x,

where w is the maximum age of the table, and k is the nearest integer to the value of ex.

2. c4—xi =v+v2+v3+ . . . +vk, very nearly.

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