The nature of capital and income . curve DC. In other words, instead of having a uniformdiscount curve from B to D, we have a broken discount curveBCD, with a different percentage rate of rise for the twoyears considered. In this way we may obtain the present value of any series ofincome items precisely as before, with the exception that the dis-count curves are now somewhat irregular. Thus, if the seriesof income items at different points of time are of the magnitude 392 NATURE OF CAPITAL AND INCOME represented in Figure 48 by AB, CD, EF, and GH (read in theorder of futurity), the remotest it
The nature of capital and income . curve DC. In other words, instead of having a uniformdiscount curve from B to D, we have a broken discount curveBCD, with a different percentage rate of rise for the twoyears considered. In this way we may obtain the present value of any series ofincome items precisely as before, with the exception that the dis-count curves are now somewhat irregular. Thus, if the seriesof income items at different points of time are of the magnitude 392 NATURE OF CAPITAL AND INCOME represented in Figure 48 by AB, CD, EF, and GH (read in theorder of futurity), the remotest item, AB, is discounted bymeans of the discount curve BC, and the next to the last itemis added to the capital-value at C, bringing the capital-value tothe point D, from which the next discount curve DE is drawn,and so on until we reach the point I. IJ is thus the capital-value of the given series of income items. In this way it is possible to review all the special casesof capitalizing income which were considered in Chapter XIII,. Fig. 48. and correct them for the general case of a rate of interestwhich is variable but foreknown. Such a calculation, how-ever, is of very little practical consequence, inasmuch as thevariations in the rate of interest are seldom if ever it worth while to pursue the subject, it wouldbe convenient to simplify the calculations by substituting,where possible, for the series of rates of interest i^, i^,?3, etc., an average, ./, such that if the given series of in-come items were discounted uniformly according to the rateof interest j, we should obtain exactly the same presentvalue as when the several separate rates i^, tj, H, etc., areemployed. The formula for the average rate of interest j of APPENDIX TO CHAPTER XIII 393 any particular number of individual rates, as i^ i^, h, etc.,used for discounting individual items of income, 01,03,0^would evidently be, -r /H . -so ~r /^ . •Na4- • • 5= 1+1 ^ (TkkiT^ ^ (1 + ^0(1+ 12X1 + 13) ?•
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