Value & Capital, CHAPTER XII -- THE DETERMINATION OF THE RATE OF INTEREST

The author, Sir John Hicks, begins Chapter XII of Value and Capital by raising the question of what determines the rate of interest.  Among economists, the traditional answer to this question had been the demand and supply of "capital," but Hicks notes that the definition of capital in this context had been somewhat imprecise.

Does capital mean 'real capital' in the sense of concrete goods and the power to dispose over a given quantity of them? ... Or does 'capital' mean 'money capital' in the sense of loanable funds -- power to dispose over a given quantity of money?  It makes a great deal of difference which interpretation we take.

Before addressing this particular question, Hicks notes another apparent controversy among "those who adhere to the monetary approach," namely whether the interest rate is "determined by the supply and demand of loanable funds (that is to say, by borrowing and lending)" versus being "determined by the supply and demand for money itself."

Hicks notes that the latter view is the one proposed by John Maynard Keynes in his General Theory, and Hicks explains that he will endeavor to show that these two views lead to the same results.

Value & Capital, CHAPTER XI, Section 6

In this section, the final one of the chapter, the author summarizes his findings on the behavior of interest rates in an example economy.  In earlier sections he had defined two separate ways of constructing a simple economy having only one market rate of interest.  He argues that each of them has its advantages, so he proceeds to discuss them both in this section ("We shall therefore try to drive them for a while in double harness.")

In the first approach, there is a one-period interest rate (the "short rate") that is used as the unit from which the whole system of interest rates is built.  The author argues that "a system of nothing but short lending would break down in practice because many borrowers would desire the additional security that comes of borrowing for longer periods, and lenders would be prepared to grant them this security in return for a concession of rather higher rates of interest."

In the second example, interest rate is for funds loaned indefinitely.  The author notes that borrowers wanting to borrow for an extended length of time would be content with such a system.  In addition, those lenders "whose object is simply to derive a regular income from their capital, and have no thought of anything else" would likely find indefinite lending to be satisfactory.  In practice, the consideration by a lender of wanting his capital back for other uses will tend to expose the problems with indefinitely long lending.

As we have seen, the rate of interest which can be earned on a loan of any finite duration, by investing in undated debentures, is highly conjectural.  If there is a serious rise in the long-term rate of interest, the effective yield may be completely wiped out. But this is much less likely to happen if the security acquired has a definite maturity, even if it is disposed of at a different date from that at which it falls due.
    Thus lenders will always tend to reduce the risks to which they are subject if they can substitute shorter lending for longer lending ... In general we may suppose that they will be willing to make some sacrifice of interest (which may be great or small) in order to achieve greater security.

So short and medium-term interest rates will tend to be below the prevailing yield on indefinite loans;  the difference will correspond to a risk premium determined by "the estimate put upon the gain in security."  The prevailing yield on loans of indefinite duration "will lie below the current (long-term) market rate when that rate is expected to rise in the future, above it in the contrary case."  When the long-term rate is expected to be stable, "the short rate will lie below it to the extent of the normal risk-premium; when the long rate is expected to rise, the short rate will lie below it still further;  it is only when the long rate is expected to fall that the short rate may lie above the long rate."

These conclusions, based on the analysis of long-term rates, are consistent with those derived from the analysis of short rates.  The only difference is which of the rates (short or long) is the focus of expectations regarding future rate changes.  In both cases, the analysis of this section describes how a significant portion of the borrowing and lending will happen at rates different from the single market rate of interest, in order to accommodate the desire for greater security.  The author concludes the section (and hence the chapter) by noting that "there is a tendency for short and long rates to move in the same direction, but for the movement of short rates to have the larger amplitude."

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.")

Value & Capital, CHAPTER XI, Section 4

This section continues the discussion of interest rates on loans of various durations, based on the view of a long-duration loan as a series of single-period loans (the text uses a week as the length of a single period).  The text notes that interest rates for loans beginning in a future week "are strictly analogous to the futures prices" discussed in the previous chapter "and are determined in almost exactly the same way."

The previous chapter's discussion of forward trading discusses the role of hedging and speculation in bringing about the coordination of plans in a private enterprise economy. A similar distinction of hedgers and speculators is relevant in discussing the market for long-terms, although the author concedes that "it is not usual" to think of the loan market in this way.  The text notes that, all other things being equal, someone who "[enters into] a long-term loan contract puts himself into a more risky position than he would be in if he refrained from making it;  but there are some persons (and concerns) for whom this will not be true."  For a person or firm already committed to a large project that will require capital over a large number of future time periods, entering into a long-term loan will amount to hedging their future supplies of loan capital.  This opportunity to reduce their risk will give them a strong incentive to borrow long.

The text notes that there does not appear to be a similar incentive on the other side of the market, but it does mention "an important circumstance which demands attention."  The discussion goes on to explain that, essentially, there are some fixed costs of making a loan, regardless of the size of the loan.  Thus "the difficulty of short lending may sometimes have the effect of driving lenders into the long market."  Although this effect may be real, the text implies that it is relatively weak, and therefore, "If no extra return is offered for long lending, most people (and institutions) would prefer to lend short."

The result of these effects on the demand and supply for loan capital will tend toward "a large excess of demands to borrow long which would not be met.  Borrowers would thus need to offer better terms in order to persuade lenders to switch over into the long market (that is to say, enter the forward market)."  As with speculators in the forward commodity market, lenders in the long loan market would only participate if by so doing, they "gain sufficiently to offset the risk incurred."  To balance supply and demand for loanable funds, the interest rate for any particular future week must be high enough to induce a sufficient quantity of loan capital to be available for the loan contracts.  This forward short rate will have to be higher than the short rate the lenders expect to see in that future period, in order to compensate the lenders for the risk they incur.  The excess above the expected rate corresponds to a risk premium analogous to that in forward commodity markets.  If there is an expectation that future rates will rise, the excess will be even greater;  if rates are expected to fall, the premium will be reduced.

The section closes by noting that these rules "must also apply to the long rates themselves, which, as we saw in the last section, are effectively an average of the forward rates."

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.

Value & Capital, CHAPTER XI, Section 2

This section examines factors that determine the rate of interest on loans.  Given the conclusions of the previous section, the focus is on money rates of interest.  Loans -- even those made at the same time -- may have different interest rates.  There are two main reasons for these differences:  (1) differences in the length of the loan period and schedule of repayment; and (2) differences in the risk of the borrower defaulting on the loan.  According to the author, the second of these reasons "is responsible for the element of 'risk premium' in interest rates as generally understood."

A borrower judged to have poor credit will have to pay a higher interest rate than a borrower with good credit.  This is because of the extra risk that loaning to such a borrower imposes on a lender.

[E]ven if the supposedly untrustworthy borrower does discharge his obligations, he will not pay more than he is obliged; that sets a maximum to the receipts which can be expected by the lender; all the possible variations from it are in one direction.

Therefore a lender will only be induced to lend to a less sound borrower if he is offered better terms than for a loan to a sound borrower.

A borrower's creditworthiness is a matter that individual lenders must subjectively assess.  These types of judgments are likely to vary among lenders.  For a small loan, a business may find it possible to raise the desired amount of funding "by appealing only to that inner circle of potential lenders with which it has good standing, and who thus may be expected to be willing to lend to it on relatively favourable terms."  To raise larger amounts, the business must either appeal to other lenders, who will want a higher rate, or else persuade lenders in the "inner circle" to lend more.  The amount that a particular lender will loan is limited by that lender's willingness to incur risk from "putting all his eggs in one basket."  Offering better terms (such as a higher rate of interest) not only persuades an individual lender to lend more, but it also induces additional lenders to be willing to lend.  The result of this effect is that

Each particular borrower thus finds himself confronted with a sort of 'supply curve for loan capital', analogous to the supply curves of other factors of production which confront a producer when he is in a 'monopsonistic' (or monopoly buyer) position.

The section closes by noting that there is no reason to suppose this supply curve would be perfectly elastic.  Although the author does not go into detail, he notes in passing that "this consideration introduces into the theory of interest questions analogous to those which have been discussed by writers on Imperfect Competition," which a complete theory of interest ought to take into account.

Value & Capital, CHAPTER XI -- INTEREST

In this first section of Chapter XI, the author, Sir John Hicks, references the discussions of the preceding sections to introduce the concept of a loan transaction, as well as that of interest.  Previously, he had introduced the concept of a "Spot Economy" in which all transactions were made for immediate delivery, as well as that of a "Futures Economy" in which "everything was fixed up in advance" for some considerable amount of lead time.  These concepts were helpful exploring potential sources of disequilibrium in an economy.  In this section he notes that "there is no reason why the two sides of a bargain should be due to be executed at the same date."  Thus we have another kind of transaction -- loan transactions -- where the transaction is divided in time;  one side is executed immediately, and the other at some future date or series of future dates.

Any exchange of this sort is technically a loan, but by far the most common in practice is an exchange of money now for money later.  It is uncommon to exchange goods of one sort now for goods of another sort later, for the same reason that barter is not more common:  the inconvenience of searching for someone who wants exactly the goods you have and can offer you goods that you want in exchange.  Transactions involving goods now for money later, or goods later for money now, occur frequently, but Hicks notes that "they are naturally thought of as reducible to a money loan plus a spot transaction (or a forward transaction)."  To illustrate, he imagines a pure-barter case of exchanging "coffee now for coffee a year hence" and notes that it can be reduced to a spot transaction, a forward transaction, and a money loan.  He then goes into a discussion of how a forward market implicitly determines a rate of interest in terms of whatever commodity is being traded.  Using an example in which the price of coffee for delivery in one year is trading at 3 percent above the spot price, and the interest rate on money is 5 percent, he determines the coffee rate of interest to be 105/103, or approximately 2 percent.  The reasoning is as follows:  a person who wants to "lend coffee for one year" could, in this example, sell the coffee in the spot market, loan the money proceeds at 5 percent, and cover the sale of the coffee by purchasing on the forward market.  To put it another way, a person who has coffee now should be able to obtain an increased amount of coffee in a year (even though the price of coffee is expected to increase) because, in this example, money can grow slightly faster.  Hicks concludes the discussion by noting that the coffee rate of interest will be the same as the money rate of interest only if the forward price of coffee is the same as the spot price (i.e. if the denominator of the fraction above is 100 instead of 103).

After concluding the discussion on commodity rates of interest, Hicks summarizes by saying that these rates of interest are of relatively "little direct importance," just as a rate of exchange between two commodities is of relatively little importance "when neither of the two commodities is the standard of value" (i.e. when neither one is being used as money).  With no further assumptions about the properties of money, then, 

[W]e are entitled to assume that all loans are in money terms; for any loan transaction which takes place otherwise is always capable of being reduced to a money loan combined with a spot transaction and a forward transaction.