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