Value & Capital, CHAPTER XI, Section 3

This section revisits the concept of reducing a long-term loan into a combination of spot and forward transactions.  In particular, the discussion examines a case of a loan made for two weeks, which is equivalent to a loan for one week, plus a forward transaction renewing the loan for the succeeding week.

Looked at in this way, the rate of interest for loans of two weeks ... is compounded out of the 'spot' rate of interest for loans of one week and the 'forward' rate of interest, also for one-week loans, but for loans to be executed in the second week.  If no interest is to be paid until the conclusion of the whole transaction, then the same capital sum must be arrived at by accumulating for two weeks at the two-weeks rate of interest, or alternatively by accumulating for one week at the one-week rate, and then accumulating for a second week at the 'forward' rate."  The two transactions are ultimately identical.

Using the notation R1, R2 and R3 for the current one-, two-, and three-week rates, respectively, and the notation r1, r2, and r3 for the single-week rates for "forward" transactions in weeks one, two, and three, respectively, we have the following equations for the costs of paying back one-, two-, and three-week loans, respectively:

1 + R1 = 1 + r1

(1 + R2)^2 = (1 + r1)(1 + r2)

(1 + R3)^3 = (1 + r1)(1 + r2)(1 + r3)

These relationships are less complicated if we assume simple interest.  In that case the following equations express the costs of interest for the one-, two-, and three-week loans, respectively:

R1 =  r1

 2 R2 = r1 + r2

3 R3 = r1 + r2 + r3

Thus the long rate for a loan of a given length is the arithmetic average between the current short rate and the forward short rates for the periods comprising the loan duration.