The nature of capital and income . values of these two; that is, it is equal to thevalue of a perpetual annuity beginning now, less the presentvalue of a perpetual annuity beginning t years hence. Thedeferred annuity which begins at the end of t years will, we know, be worth then the sum of -, and will be worth now what- i ever is the present value of this -. This present value is of i course found simply by discounting the - just obtained, and is arr-TT^t This expression should therefore be subtracted fromthe value of the other perpetual annuity which begins now, ofwhich the present value is
The nature of capital and income . values of these two; that is, it is equal to thevalue of a perpetual annuity beginning now, less the presentvalue of a perpetual annuity beginning t years hence. Thedeferred annuity which begins at the end of t years will, we know, be worth then the sum of -, and will be worth now what- i ever is the present value of this -. This present value is of i course found simply by discounting the - just obtained, and is arr-TT^t This expression should therefore be subtracted fromthe value of the other perpetual annuity which begins now, ofwhich the present value is -7. This subtraction gives the formula, § 7 (TO Ch. XIII, § 5) Discussion of Formulae for Terminable Annuity by Diagrams. TotalDiscount. Total Interest. Depreciation. In Figure 39 let AB represent the term t of the annuity, ADthe value of a perpetual annuity beginning at the point of timeA, and BE the equal value, taken at the end of the term, of adeferred perpetual annuity beginning at that time. Now the APPENDIX TO CHAPTER XIU. Fig. 39. present value at time A of the value BE, at time B, is evi-dently AC, found by drawing the discount curve CE. There-fore the value of the terminable annuity is equal to AD - AC, 376 NATURE OF CAPITAL AND INCOME or DC, which is the total discount on BE; the amount bywhich it is, as Bohm-Bawerk says, diminished in time per-spective. Similarly, the capital-value of the annuity, taken at any timelater A (just after an installment of income), is equal to the
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