APPENDIX TO CHAPTER XIII . 379 
The sum of these expressions is the value ¥, which we are 
seeking. In other words,— : 
  
i 
{Tas tae 
pg 
or ~ L 
Vi *GT 
Some special cases may be considered. First, if the annual 
income a is the interest on the “ principal” P,—i.e. if a= Pi 
a . ‘ ’ 
(or P=), the second term vanishes, as its numerator is 
1 
evidently zero, and since the first term, is by present hy- 
pothesis P, the equation then becomes V'=P. 
Secondly, if a is greater than iP, it may be readily shown that 
V will be greater than P; and if a is less than iP, that V is 
less than P. 
The formula given is of practical importance, as it enables 
us to compute the price at which a bond must sell in order to 
yield a certain rate of interest. 
To apply the formula numerically we need only to assign 
particular values for the magnitudes involved. Let us take the 
numerical case already considered, where P= $100, a= $5, 
i=.04, and ¢t=10. In this case the formula becomes, — 
ed 
ne 3 Li ny! 
= donT 
which reduces to 108, as we found before. 
Similarly, it may be shown that if bonds are sold on a 69 
basis, the price of the bond in question would be $92}. 
We have derived the value of a bond, V, just after a 
payment of “interest.” In this case the bond is said by 
brokers to be sold “ex-interest.” If, on the contrary, it is 
sold “flat,” that is, with interest, its value will evidently be 
increased by the “interest” a, and will be ¥'+a. The price 
at any time between installments will evidently be > A