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The nature of capital and income . qual to one half, the altitudes will nolonger represent the individual payments, but double those semi-annual payments, i.e. the per annum rate. Thus, if the annu-ity is $4 per annum payable semi-annually, the rectangle OTmeans $2, its base is one half, and its altitude, TT, will not be2, but 4, the rate per annum. Similarly, quarterly payments are represented by rectanglesOS, XT, TU, etc., whose altitudes will again represent the rateper annum of each quarterly payment. Finally, for continuous payments, we shall have an infinitenumber of infinitesimal rectan

The nature of capital and income . qual to one half, the altitudes will nolonger represent the individual payments, but double those semi-annual payments, i.e. the per annum rate. Thus, if the annu-ity is $4 per annum payable semi-annually, the rectangle OTmeans $2, its base is one half, and its altitude, TT, will not be2, but 4, the rate per annum. Similarly, quarterly payments are represented by rectanglesOS, XT, TU, etc., whose altitudes will again represent the rateper annum of each quarterly payment. Finally, for continuous payments, we shall have an infinitenumber of infinitesimal rectan Stock Photo
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The Reading Room / Alamy Stock Photo

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2AN9PR3

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7.2 MB (113.9 KB Compressed download)

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2727 x 917 px | 23.1 x 7.8 cm | 9.1 x 3.1 inches | 300dpi

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The nature of capital and income . qual to one half, the altitudes will nolonger represent the individual payments, but double those semi-annual payments, i.e. the per annum rate. Thus, if the annu-ity is $4 per annum payable semi-annually, the rectangle OTmeans $2, its base is one half, and its altitude, TT, will not be2, but 4, the rate per annum. Similarly, quarterly payments are represented by rectanglesOS, XT, TU, etc., whose altitudes will again represent the rateper annum of each quarterly payment. Finally, for continuous payments, we shall have an infinitenumber of infinitesimal rectangles, forming in the aggregatethe whole figure represented, the altitude of which at any pointwill be the rate per annum at which income is flowing at that^oint. APPENDIX TO CHAPTER XIII 373 By limits we may pass from income which flows at a uniformrate to any income stream. Evidently, therefore, any con-tinuous income-stream may be represented by a curv© (Fig. 37)of which the ordinate represents the per annum rate of flow at any. point of time, and the area EC between any two ordinates BEand CF represents the total income which flows within the timeintercepted between those ordinates. For the case of uniform flow, the continuous income streamis represented in Figure 38 by the area OB. OA represents the Fig. .38. rate of income, and OC represents the capital-value of this in-come. This capital-value remains constant, as shown by the 374 NATURE OF CAPITAL AND INCOME horizontal line CD, and the rate of interest (reckoned continu- § 6 (to Ch. XIII, § 5) Formula for Capital-value of a Terminable Annuity Let a represent the annual payment of the annuity, t itsduration or term, and V its present value. We are requiredto find V in terms of a, t, and i, the rate of interest. Wehave observed that a man who owns such a terminableannuity owns the difference between a perpetual annuity be-ginning at present and another perpetual annuity deferred tyears. Consequently, the value of his pr

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