Value & Capital, Notes to Chapter XIV -- B. INTEREST AND THE CALCULATION OF INCOME, part 3

In this section, the author considers a "common sense" analysis of a person expecting to receive funds "derived from the exploitation of a wasting asset, liable to give out at some future date."  In this case, the author explains, "we should say that his receipts are in excess of his income, the difference between them being reckoned as an allowance for depreciation."  To avoid consuming more than his income, such a person must invest part of his receipts (or, in the language of the text, "re-lend" them) so that his income from these investments (or the interest he earns on re-lending) will compensate "for the expected failure of receipts from his wasting asset in the future."  In this case, with receipts expected to decline, income will decrease if the interest rate decreases.  If we change the assumption of a wasting asset and assume that expected receipts will increase (for whatever reason), income will be higher for a lower rate of interest (because the person consuming as much as his income will have to borrow during early periods to have enough funds to consume so much).

The author then recalls the previous section's interpretation of the capital value of a stream of receipts as the weighted average period of the receipts (weighted by discounted values of receipts), and the comparison of this average period of receipts with the period of a standard stream of receipts to test whether a rise in the interest rate would increase or decrease income.  The author asks whether this test can be reinterpreted so as to agree with the common-sense case described above.

The author gives his answer for the case in which prices and interest rates are expected to remain constant.  In this case all three approximations of income give the same results, with the standard stream of receipts having the same constant value in all periods.  If the average period of the given stream of receipts is greater than that of the standard stream, then the given stream must have lower value initially.  But because the two streams must have the same capitalized value, the given stream must catch up later.  (In the language of the text, there must a "crescendo.")  The author concludes that "The average period turns out to be nothing else but an exact method of measuring the crescendo (or diminuendo) of a stream of values."  In the case of a stream of identical quantities, continuing indefinitely, and "discounted throughout by the same rate of interest" the author shows that the average period works out to be the reciprocal of the rate of interest, the calculations being as follows:


Finally, the author gives a formula for the crescendo of a stream of values, with each period's value expanding by the same proportion.  The formula for the crescendo c is

c = i – 1 / P

where i is the interest rate, and P is the average period of the stream.