374 NATURE OF CAPITAL AND INCOME 
horizontal line CD, and the rate of interest (reckoned continu- 
. 04 
1 a 
ously) is 00 
§ 6 (ro Cu. XIII, § 5) 
Formula for Capital-value of a Terminable Annuity 
Let a represent the annual payment of the annuity, ¢ its 
duration or term, and V its present value. We are required 
to find V in terms of a, t, and i, the rate of interest. We 
have observed that a man who owns such a terminable 
annuity owns the difference between a perpetual annuity be- 
ginning at present and another perpetual annuity deferred ¢ 
years. Consequently, the value of his property is the differ- 
ence between the values of these two; that is, it is equal to the 
value of a perpetual annuity beginning now, less the present 
value of a perpetual annuity beginning ¢ years hence. The 
deferred annuity which begins at the end of ¢ years will, we 
know, be worth then the sum of 2 and will be worth now what- 
i 
ever is the present value of this 9 This present value is of 
7 
course found simply by discounting the 2 just obtained, and is 
a 
aTy This expression should therefore be subtracted from 
the value of the other perpetual annuity which begins now, of 
which the present value is 2 This subtraction gives the 
1 
a a 
formula, T= at 
§ 7 (ro Cum. XIII, § 5) 
Discussion of Formule for Terminable Annuity by Diagrams. * Total 
Discount.’ ‘Total Interest.” Depreciation. 
In Figure 39 let AB represent the term ¢ of the annuity, AD 
the value of a perpetual annuity beginning at the point of time 
A, and BE the equal value, taken at the end of the term, of a 
deferred perpetual annuity beginning at that time. Now the