THE THEORY OF INTEREST

for the separate years constituting that long term. Theproposition affirming the existence of separate rates forseparate years amounts to this: that normally there shouldbe a difference between the rates for short term and longterm loans, sometimes one being the larger and sometimesthe other, according to the whole income situation.

impossible to give a concrete example of an average of a rate of inter-est for a long term loan as an average of the year to year rates, becauseas already noted, the year to year rates have only a hypotheticalexistence. The nearest approach to the concrete existence of separateyear to year rates is to be found in the Allied debt settlements, bywhich the United States agreed that Italy, France, Belgium , and othercountries should repay the United States through 62 years, with spe-cific rates of interest changing from time to time. These are equivalentto a uniform rate for the whole period according to theory. Forinstance, the proposed French debt settlement provided for annualpayments extending over 62 years, beginning with 1926 at interest ratesvarying from 0 per cent for the first 5 years, 1 per cent for the next10 years, 2 per cent for another 10 years, 2 Y 2 per cent for 8 years,3 per cent for 7 years, and 3 Vz per cent for the last 22 years. Theproblem is to find an average rate of interest for the whole period whichwhen applied in discounting the various payments provided for in thedebt settlement, will give a present worth, as of 1925, equal to the prin-cipal of the debt fixed in that year, namely, $4,025,000,000. Clearly noform of arithmetic mean, weighted or unweighted, will give the desiredrate. A rough computation indicates that the rate probably falls between1% per cent and 1% per cent. Discounting the annual payments bycompound discount gives a total worth in 1925 at 1% per cent of$4,197,990,000; at 1% per cent of $3,893,610,000. These results show thatIY 2 per cent is too low, since the present worth obtained by discountingat this rate is greater than the principal sum which was fixed at$4,025,000,000. The rate 1% per cent is too high because the discountedpresent worth is less than the principal.

Discounting the annual payments at 1.6 per cent we obtain $4,072,-630,000. We can now locate three points on a curve showing the inter-est rates corresponding to different present values. By projecting aparabolic curve through the three determining points, we find theordinate of the point on the curve which has the abscissa of $4,025,000,000is 1.64. Hence the average rate of interest for the whole period, withina very narrow margin of error, is 1.64 per cent.

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