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 beginning
 at present and another perpetual annuity deferred ¢
years. Consequently, the value of his property is the difference
 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 whati

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