Value & Capital, CHAPTER XI, Section 5

This section continues the discussion of long lending in the context of a "spot economy."  Earlier in this chapter, the analysis explored a model in which a long-period loan could be built up out of a sequence of one-week loans, with the interest rate for the long loan being the arithmetic average between the current short rate and the forward short rates for the periods comprising the loan duration.  Section 5 describes a different way of simplifying the analysis of interest rates;  this approach views all loans as having indefinite (i.e. infinite) duration.  Another way of viewing this (my own interpretation, not the author's exact words) is that a loan gives the lender the right to an ongoing series of payments from the borrower.  When the borrower wishes to pay off the loan, he or she must buy back that right from the lender (at a price that reflects both the payment size, and the interest rate prevailing at the time of the buy-back).

Once a loan is made, the lender holds an asset -- the right to be paid a fixed amount "in perpetuity, at regular intervals, as interest on the loan."  The value of this asset will change according to the interest rate at any particular time.  If the interest rate goes down, a new loan that generates the same payment as the existing loan would require a greater loan amount (principal), so the capital value of the existing loan would increase with a decrease in interest rate.  Conversely, the capital value would decrease for an increase in interest rates.

Suppose R is the current week's interest rate, and R' is the rate a borrower expects for the following week (the author continues to base his illustrations on the use of one week as a loan period).  The author points out that a loan's capital value "will change in the course of the week in the proportion R/R'."  He then states that the effective rate a lender will have to pay is

R + (R / R') - 1

This expression can be understood as follows:  the first term represents that portion of the return that results from the existing interest rate;  the second and third terms represent the change in capital value expressed as an interest rate.

An individual who wants to borrow can issue new securities (thus becoming obligated to make the associated payments); or he could sell existing securities he already possessed (which has the effect of increasing his net indebtedness).  An individual who wants to lend can do so by buying old or new securities, thus becoming entitled to the associated stream of payments.  A buyer is indifferent between old and new securities (assuming there is "an equal degree of default risk"), so there must be an equivalence between these securities' prices if they generate the same income per period.  We may view this equivalence either as the interest rate on new securities adjusting to the prices of existing securities, or as the prices of those securities adjusting to the new interest rate.  Either way, the new prices of the old securities are completely determined by the new interest rate, which is the only market rate of interest in the system (in the words of the author, there is a "purely arithmetical relation between the prices of old securities and the rate of interest.")