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 annui- 
ties, 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