The nature of capital and income . e v for ;, this may be written, 1 -(-1 -y + W^ + v^ + ad. inf., or v (l+v^-i-ad. inf.). Since the series evidently converges, the parenthesis is equal to , which may be seen by simply dividing 1 by 1 — v. Hence the value of the annuity is, v 1-v which reduces to — if we substitute for v its original value. * 1 +1 This sum, - dollars, is, therefore, the capital-value of anannuity of f 1. By proportion, the capital-value of any other annuity a is § 4 (to Ch. XIII, § 3) Formulae and Diagrams for Capital-value of Annuities payable Annually,Semi-annually, Quar
The nature of capital and income . e v for ;, this may be written, 1 -(-1 -y + W^ + v^ + ad. inf., or v (l+v^-i-ad. inf.). Since the series evidently converges, the parenthesis is equal to , which may be seen by simply dividing 1 by 1 — v. Hence the value of the annuity is, v 1-v which reduces to — if we substitute for v its original value. * 1 +1 This sum, - dollars, is, therefore, the capital-value of anannuity of f 1. By proportion, the capital-value of any other annuity a is § 4 (to Ch. XIII, § 3) Formulae and Diagrams for Capital-value of Annuities payable Annually,Semi-annually, Quarterly, Continuously In case the annuity accrues semi-annually, the teeth will befiner, but twice as frequent. In Figure 34 we see the behavior2b 370 NATURE OF CAPITAL AND INCOME of the capitalized annuity if the annuity is payable annually,semi-annually, or quarterly. If the annuity be $4 a year, theteeth drop $4 if payable annually, $2, if semi-annually, and $1,if quarterly. If the frequency of the installments of income be. 12^^ 12,^^ [S^-^ 121 2 Semi-annual Payments Quarterly PaymentsFig. 34. indefinitely increased, we reach the limiting case of a continuousincome, when the teeth disappear entirely and the value of theannuity remains at a constant level. The value of the annuityof $4 in all these cases, just after any installment of in-come is received, will be $100, if the rate of interestis 4%, provided the rate is respectively reckoned annually,semi-annually, and quarterly in the various cases. In thecase of continuous income, the value of the annuity of $4will always be f 100 if the rate of interest be 4% reckonedcontinuously. The same remarks apply, of course, to anannuity of any number of dollars. Its value after each install-ment is equal to the annual income divided by the rate of in-terest, or the annual income multiplied by the purchase order to obtain the formulae for the value of a perpetuitypayable semi-annually, quarterly, or continuously, in te
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