APPENDIX TO CHAPTER XIII 369

§ 3 (ro Cu. XIII, § 3)
Formula for the Present Value of a Perpetual Annuity
The proposition that the present value of a perpetuity is
% was proved in the preceding chapter. An alternative proof,
i
and the one which is usually given in treatises on annuities,
 is as follows: Consider a perpetual annuity of $1
per annum; it is required to find the capital value. It is

evident from what has preceded that the first payment,
$1, being due at the end of a year, has a discounted

; the second has a present

 

value at the present time of 1
1 2 i

a : 5 ,; the third, ax iin and so on indefinitely.
Therefore, the present value 2 the entire series will be, —
Sha
1+7 art (1 is
If, for brevity, we substitute » for 1 L ., this may be written,
v

worth of

 

+ ete., ad inf.

 

 

v+v? + v* + ad. inf,
or v (142+ ad. inf.).
Since the series evidently converges, the parenthesis is equal

 

to i 1 , which may be seen by simply dividing 1 by 1—2.

 

Hence the value of the annuity is, v (3 1 5) ’

 

which reduces to 1 if we substitute for v its original value, 7
1

This sum, 1 dollars, is, therefore, the capital-value of an
i

 

annuity of $1. By proportion, the capital-value of any other
annuity a is 2.
1

§ 4 (ro Cu. XIII, § 3)

Formule and Diagrams for Capital-value of Annuities payable Annually,
Semi-annually, Quarterly, Continuously

In case the annuity accrues semi- -annually, the teeth will be
finer, but twice as frequent. In Figure 34 we see the behavior

2B