The Term Structure of Interest Rates

THERE IS a tendency to regard Friederich Lutz’s classic article, “The Structure of Interest Rates,”1 as the “synthesis of ‘received doctrine’,” 2 explaining the relation between the yields of securities which are basically similar but mature at different dates. Yet, in reality, there are two main threads running through the literature, whose separate identities are distinguished only rarely.3 Both of these threads are to be found in Hicks’ exposition in Value and Capital.4 The first view, which is widely accepted as “the Hicks view,” is that “the long rate is the arithmetic average between the current short rate and the relevant forward short rates.” 5 That is, changes in the long-term rate of interest are primarily related to actual and expected changes in short-term rates. The second view is that “in stable conditions, when the long rate is expected to remain steady, 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. . . .” 6 That is, changes in the short-term rate of interest are primarily related to actual and expected changes in long-term rates. In Hicks’ opinion, these observations were not at variance with each other. In fact, he regarded them as “perfectly consistent.” 7 It will be argued later that both influences are at work, determining the structure of interest rates, and that they provide fundamentally inconsistent explanations of the term structure of bond yields. The problem is to reconcile these inconsistencies.

Abstract

THERE IS a tendency to regard Friederich Lutz’s classic article, “The Structure of Interest Rates,”1 as the “synthesis of ‘received doctrine’,” 2 explaining the relation between the yields of securities which are basically similar but mature at different dates. Yet, in reality, there are two main threads running through the literature, whose separate identities are distinguished only rarely.3 Both of these threads are to be found in Hicks’ exposition in Value and Capital.4 The first view, which is widely accepted as “the Hicks view,” is that “the long rate is the arithmetic average between the current short rate and the relevant forward short rates.” 5 That is, changes in the long-term rate of interest are primarily related to actual and expected changes in short-term rates. The second view is that “in stable conditions, when the long rate is expected to remain steady, 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. . . .” 6 That is, changes in the short-term rate of interest are primarily related to actual and expected changes in long-term rates. In Hicks’ opinion, these observations were not at variance with each other. In fact, he regarded them as “perfectly consistent.” 7 It will be argued later that both influences are at work, determining the structure of interest rates, and that they provide fundamentally inconsistent explanations of the term structure of bond yields. The problem is to reconcile these inconsistencies.

THERE IS a tendency to regard Friederich Lutz’s classic article, “The Structure of Interest Rates,”1 as the “synthesis of ‘received doctrine’,” 2 explaining the relation between the yields of securities which are basically similar but mature at different dates. Yet, in reality, there are two main threads running through the literature, whose separate identities are distinguished only rarely.3 Both of these threads are to be found in Hicks’ exposition in Value and Capital.4 The first view, which is widely accepted as “the Hicks view,” is that “the long rate is the arithmetic average between the current short rate and the relevant forward short rates.” 5 That is, changes in the long-term rate of interest are primarily related to actual and expected changes in short-term rates. The second view is that “in stable conditions, when the long rate is expected to remain steady, 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. . . .” 6 That is, changes in the short-term rate of interest are primarily related to actual and expected changes in long-term rates. In Hicks’ opinion, these observations were not at variance with each other. In fact, he regarded them as “perfectly consistent.” 7 It will be argued later that both influences are at work, determining the structure of interest rates, and that they provide fundamentally inconsistent explanations of the term structure of bond yields. The problem is to reconcile these inconsistencies.

The first Hicksian view has been presented most precisely by Lutz who states that it is possible “to look upon the long rate as being fundamentally the average of the expected future short rates.”8 In the following discussion, this position will be referred to as the “Hicks/Lutz” 9 view. The second view has been presented most precisely by Sayers, who argued that the expected long-term rate is a unique concept to which the market adjusts. In his opinion “the effect of a change in short-term rates upon the long-term market is narrowly limited by what people expect the long-term rate to be in the near future.” 10 In the following discussion, this position will be referred to as the “Sayers” view.

Both the “Sayers” and “Hicks/Lutz” views fail to explain two aspects of the interrelation between long-term and short-term interest rates. In the first place, they both suggest that, over any short period, long- and short-term rates will usually move in the same direction. (Sayers even states that “a pronounced movement in short-term rates is bound to be followed by a movement, in the same direction, in long-term rates.”)11 It cannot be denied that, over short periods, long- and short-term rates move in the same direction more frequently than they move in the opposite direction. However, divergent movements of these two series are not exceptional. Thus, in the period between January 1948 and July 1962 there were, for the United States, 155 months when both the long-term rate and the short-term rate12 moved, and for 65 of these months the movements in the two rates were in opposite directions. For the United Kingdom, there were 123 months with movements in both rates, of which 46 were movements in opposite directions. For Belgium, the number of months with such movements was 112, with 51 in opposite directions; for Switzerland, 64 months with 14 showing opposite movements; for France, 56 months with 31 opposite; for Canada, 54 with 11 opposite; for Germany, 4413 with 20 opposite; and for Belgium, 39 with 15 opposite. Almost 40 per cent of these 647 observations are contrary to that which generally accepted theory would lead one to expect.

In the second place, both views would suggest that, unless the financial markets formed very complex views regarding the future course of long-term or short-term interest rates, any yield curve, relating long-term and short-term interest rates with the rates on securities of intermediate maturities, would rise or fall consistently from one end of the range to the other, that is, it would be monotonic.

Yield curves are frequently consistent, in this sense. However, they frequently are not monotonic and contain humps or troughs.14 Further, curves with simple humps or troughs are not the only alternatives.15 Any theory of the relation between long-term and short-term rates of interest must comprehend yield curves with Bactrian as well as Arabian humps, and W as well as U shapes.

Chart 1, which describes changes in the market for U.S. Government securities during 1961, presents a not untypical story. Each curve is the yield curve at the end of the month indicated on the left half of the lower axis for securities of up to 18 years’ maturity, with the yields on different maturities recorded on the right axis. For all the curves, equal distances from the origin to points on the curve represent the same periods to maturity; the scale of these distances for the December curve is recorded on the right half of the lower axis. Several of the curves rise to a maximum and then fall. On several occasions, the yields on long-term and short-term securities moved in different directions. These movements cannot be explained solely by shifts in the composition of the outstanding debt held outside the Federal Reserve System and U.S. Government accounts (indicated by the bar charts for the end of each month indicated on the lower axis, with the percentage distribution of debt by maturity recorded on the left axis).

Chart 1.

United States: Term Structure of Interest Rates and Government Debt Maturities, 1961

Citation: IMF Staff Papers 1963, 002; 10.5089/9781451947151.024.A003

The Theory

The structure of interest rates is determined by the operations of the capital market. The market for government securities may be considered as being subject to two major influences: (1) the activities of the monetary authorities;16 (2) the reactions of private purchasers and sellers of securities.

It may be assumed that all the reactions in the capital markets are consistent with the decisions made in the market for government securities. Hence, to simplify the present argument, it may be assumed that the only market which is significant is the one for these issues, and that the only significant structure of rates, for the present analysis, is that on government securities of differing maturity. These assumptions are consistent with the view, which is fundamental to the argument, that the entire capital market is a unified system. (Any complications arising from the effects of income and capital gains taxes have been consciously ignored for simplicity.)

The basic nature of interest is outside the scope of this discussion. If the result of these considerations is to suggest that our theory “seems to leave interest hanging by its own bootstraps” 17 or that it describes an Alice-in-Wonderland world and is “a grin without a cat,” 18 that result is irrelevant to the present discussion. All that need be assumed, for the present purpose, is that there is a stated long-term rate, or short-term rate, or that both exist. The problem is to develop satisfactory conclusions regarding the structure of rates from the existence of a given rate or rates. Partly for purposes of exposition it will be assumed that the central bank has unlimited power to set and maintain any long-term rate and any short-term rate which it desires. By open market operations, it can make its desires effective.19

Finally, it should be emphasized that the ensuing analysis is applicable only to the security markets in those countries where a significant part of the community’s liquid assets is held in the form of short-term financial claims other than money, where there is a significant body of investors who are willing to hold long-term securities, and where there is an adequate number of lenders and borrowers willing to shift the maturity structures of their balance sheets in reaction to market stimuli. Most importantly, members of the community must be able and willing to make these decisions. Less importantly, the market institutions must be developed so that not only may the decisions be made but the stimuli may be exerted on both lenders and borrowers. Although a major part of the world’s financial transactions takes place in countries where these conditions exist, it should be recognized that these countries are relatively few in number.

The market

At the risk of caricaturing the activities of private purchasers of government securities, and of misrepresenting some distinguished authors, it is possible to envisage the securities market as divided into three overlapping segments, each one dominated by an identifiable group. The first group comprises those who may be called the “Sayers” type of bondholder. They are almost exclusively interested in the market for long-term government securities. They may be exemplified by the members of the British Insurance Association, who gave evidence before the Radcliffe Committee to the effect that they were really interested only in holding long-term securities, and for some periods would choose between holding cash or securities, depending on their view regarding the current and prospective levels of long-term interest rates.20 Similar bondholders in the United States were interviewed in 1943 in the course of a survey for the Mutual Life Insurance Company. One interpreter of this survey concluded that it provided “little or no evidence that the course of short-term rates was regarded as determining the prospect for the long rate . . . .”21 The second group comprises those who will be called the “Hicks/Lutz” type. They are interested almost exclusively in the market for short-term government securities. They may be exemplified by the “money managers” of commercial banks (who are responsible for insuring that a predetermined volume of liquid assets is held by the banks), and the financial officers of companies (entrusted with the short-term investment of funds which are to be kept in liquid form). Finally, there is a third group. It must be emphasized that, in the terms of the Radcliffe Committee Report, “the market for credit is a single market,” 22 or, in Riefler’s terms, “the markets for United States Government securities are usually characterized by a high degree of fluidity as between the various maturity sectors.” 23 Hence, this third group comprises all those bondholders who do not clearly fall into the “Sayers” group or the “Hicks/Lutz” group. The members of this group, who spread their holdings between the long and short ends of the range of available securities and provide continuity in the yield curve, the limits of which are determined by the two preceding groups, are referred to as the “Radcliffe/Riefler” type.

It must be remembered that there are no clear divisions between the different sectors of the market. For example, in the United Kingdom, the discount houses invest primarily in Treasury bills and other short-term assets; yet, in recent years, bonds (admittedly mostly short-dated) have accounted for approximately 30 per cent of their assets.24 The decisions of these institutions may be assumed to be dominated by expectations regarding future levels of short-term interest rates. On the other hand, many of the “Sayers” type investors, perhaps acting personally in a non-Sayersian capacity, also hold short-term securities.

The essential unity of financial markets arises from the willingness of many lenders and borrowers to alter the maturity structure of their assets and liabilities, within relatively narrow ranges, as the structure of interest rates alters. Borrowers, who would otherwise prefer to obtain funds for fairly long periods, will be willing to shorten their commitments, provided long-term interest rates rise in relation to short-term rates. Lenders, who would otherwise prefer to obtain the liquidity attributes of relatively short-dated instruments, will be willing to lengthen the maturity of their assets, as a result of such a shift in relative rates. These reactions will tend to lower long rates and raise short rates. Conversely, a rise in short rates relative to long rates will elicit offsetting reactions by borrowers and lenders. These shifts will not involve the movement into the very short-term market by borrowers or lenders who are essentially interested in the very long-term market, or vice versa. Rather, changes in the structure of rates will encourage small shifts in the maturity structure of debts desired by borrowers and lenders interested in the maturity ranges affected by the changes in the rate structure. These movements will guarantee that, probably with the assistance of professional operators, there will be an inherent smoothness in the yield curve appropriate for differing maturities. However, they will not guarantee that both ends of the yield curve will move in the same direction over time.

Further, the shifts in asset and liability structures do not necessarily require purchases and sales of securities. A holder of long-term bonds may wish to build up his cash or to move into shorter bonds. If there is a market consensus in this direction, the prices of long-term bonds will fall, and these shifts will be discouraged. However, bondholders may be assumed to have a flow of funds accruing to them from a wide variety of sources: current income for investment, insurance premiums to be held in trust, redemption of prior holdings, etc. These accruals may be used to acquire assets, and the disparity between asset holders’ existing asset structures and their desired structures will determine the disposition of these accruals. The impact of this flow of accruals on the market will strongly influence the structure of bond prices, and may dominate the pattern of financial flows.

In the exposition which follows, attention is limited to security holders only. The considerations which limit the terms acceptable to each of the first two groups (essentially long-term and short-term investors) will provide limits to the yield curve acceptable to the three groups. Any terms better or worse than these at any point on the curve will be assumed to encourage reactions which will drive the curve toward its limits. It may be assumed that the reactions of both borrowers and lenders will be complementary so that the yield curve will be pushed by both purchasers and sellers of securities to its theoretical limits. That is, the theory of the structure of interest rates can be set forth solely in terms of lender reactions. The reactions of borrowers may be assumed to strengthen the forces set in motion by the reactions of lenders.

The Nature of Expectations

Most of the literature dealing with the term structure of interest rates is based on the premise that market expectations play a predominant role in determining the relation between long-term and short-term rates. A few writers have questioned the validity of any theory based on expectations.25 However, most of the discussion has centered on the nature of market expectations.

It seems reasonable to recognize that market expectations exist and that they may be given a finite value. To assume that the market expects bond yields to be different in the future, and to suggest that a value may be assigned to this expectation, does not imply that the market has perfect foresight, nor that all members in the market hold the same views, nor that these views are held with complete certainty.

To maintain that there is a market consensus regarding the movement of rates is not to suggest that members of the market “reckon the expected change in bond prices over the further future, and so on to Kingdom Come.” 26 All that is required is an acceptance of the point that members of the market expect bond prices to rise or fall in the future, and that they hold some view regarding the degree of price movement in the not too distant future. The exceptional horizon is not till Kingdom Come; it need only be a few months, as Hawtrey has suggested.27 If members of the market believe that bond prices (i.e., bond yields) are going to be different in a few months, they can make decisions which take account of these expectations. An investor in mid-1939 deciding to buy long-term securities, short-term securities, or to hold cash, and reaching the conclusion that he should buy a 20-year bond at the prevailing rate, because he believes that the yield on a 20-year bond will fall in the near future, does not treat us “to the spectacle of an individual standing on the eve of World War II, atomic bombs, the cold war, the postwar inflations, the Employment Act of 1946, et hoc genus omne, trying to decide what short-term interest rates will be fifteen to twenty years in the future.” 28 He treats us to the spectacle of an individual faced with the probability or improbability of a German invasion of Poland, the mid-1939 price of 20-year bonds, and an opinion regarding the price of 20-year bonds prior to Christmas 1939.

To accept that there is a market consensus regarding the future course of interest rates does not even imply that one believes that there is a uniform market view regarding the direction or degree of change. A market consensus implies that the market operates so that relative prices change in directions, and to the extent necessary, to reconcile the conflicting views of the market. The shifts in asset and liability structures, brought about by discrepancies between the market’s views of the future and the current price structure, proceed to the point where relative prices are altered, so that further shifts are discouraged.

Again, the preceding argument does not imply that the views of the future are held with certainty, but only that they are held with sufficient conviction to influence members of the market either to shift their position or to accept alterations in the price structure. By no means does a theory based on such expectations require that the market has perfect foresight; it only requires that there be “a market view.”

The assignment of a numerical value to such a consensus raises more difficult problems. However, it seems reasonable to consider that the conflicting views in the market regarding the expected changes in interest rates, and the strength with which these views are held, may be combined into something like a weighted average, which can be given a series of numerical values measuring the interest rates that are expected to prevail in the future. It is in this sense that the concept of a “market consensus” will be used here.

Holders of Long-term Securities

Long-term lenders of the “Sayers” type may be considered as individuals who decide to accept the current level of long-term rates or to hold cash. They are investors reacting like the British insurance companies, whose representatives responded to the query, “even with big amounts accruing for investment you may be holding off the long market for a matter of weeks?” by, “Yes, or even months.” 29 The current long-term rate, together with the market expectations of these investors, will determine the distribution of their assets between cash and long-term securities. The influence of those investors who decide to hold cash rather than securities may be ignored in considering the shape of the yield curve. Their actions will determine the level of the liquidity preference function at the current interest rate. Only the actions of those willing to purchase long-term securities will be effective determinants of the yield curve.

It may be assumed that any holder of a bond prefers to retain it rather than to sell it and hold money. However, it need not follow that the market accepts the current long-term rate as a measure of the expected long-term rate. A holder of a long-term bond may be assumed to accept the current long-term rate as an approximation to his expectations regarding the upper limit to the long-term rate of interest expected in the not too distant future. To others, the current rate is one of the limits to the choices available. If they expect long-term rates to fall in the future, they can sell securities or, at least, refrain from buying.

With a given current long-term rate and given market expectations, it is possible to derive a yield pattern, which is similar to an indifference curve. This derivation is indicated in Chart 2a. In this chart, it is assumed that the central bank engages in open market purchases and sales of 20-year bonds so as to maintain their yield at 4 per cent. Also, it is assumed that there is a market consensus that, in the future, 20-year bonds will yield 6 per cent.30 Potential bond purchasers are faced with the alternatives of buying 20-year bonds, holding cash, or buying other bonds. Some investors will not be prepared to accept 4 per cent for 20 years. With the expectation that yields on 20-year bonds will rise to 6 per cent, they will wish to buy shorter-term securities, so that at some date in the future they may enjoy the 6 per cent yields. Hence, they will bid up the prices of shorter-term securities, i.e., bid down their yields. This process is represented graphically for bonds maturing in AiBi years by the line AiBiCiD. An income stream of 1 per cent extending for AiBi years, and then rising to 6 per cent for CiD years (on the assumption that AiBi plus CiDi equals 20 years) has the same present discounted value as an income stream of 4 per cent for 20 years.31 Thus, if the prices of long-term bonds are determined solely by “Sayers” type investors, market pressures will reduce the yield on bonds of AiBi years maturity to 1 per cent. Similarly, income streams represented by the lines AiiBiiCiiD, and AiiBiiCiiD have the same present value as a 4 per cent income stream for 20 years, represented by AivDiv. Hence, the prices of bonds with AiiBii years to maturity will be bid to the point where their yield to maturity is 2 per cent, and the yields of bonds with AiiiBiii, years to maturity will settle at 3 per cent. That is, points Bi, Bii, Biii, Div, and consistent intermediate points may be joined to plot a curve of the yields on securities with different maturities, regarding which, with a given market consensus, investors may be indifferent as to the yield and maturity that they will accept. Such a curve may be considered to be the type of yield curve which would be produced by the actions of “Sayers” type investors, when the central bank fixes the yield on 20-year bonds at 4 per cent, and there is a market consensus that the yield on 20-year bonds will move to 6 per cent in the not too distant future.

Chart 2A.
Chart 2A.

Yield

(percent per annum)

Citation: IMF Staff Papers 1963, 002; 10.5089/9781451947151.024.A003

On the basis of this logic, it is possible to draw a complete family of yield curves, on the assumption that there is a market consensus that the yield on 20-year bonds will approximate 6 per cent in the future, and that the central bank is currently engaging in open market operations, designed to set the yield on 20-year bonds at the levels indicated by the ends of the curves. This is done in Chart 2b.32 Alternatively, it is possible, on the basis of similar logic, to draw a family of yield curves, on the assumption that the long-term rate is fixed at 4 per cent, but showing the shapes of the curves with different market consensuses regarding the future level of long-term interest rates. This is done in Chart 2c, where the values of the market consensuses are indicated by numbers above each curve.33

Chart 2B.
Chart 2B.

Yield

(percent per annum)

Citation: IMF Staff Papers 1963, 002; 10.5089/9781451947151.024.A003

Holders of Short-term Securities

Short-term lenders of the “Hicks/Lutz” type may be considered as individuals who must accept the current level of short-term rates. Security holders who are interested above all in maintaining the liquidity of their asset positions are, in fact, faced with few alternatives. In countries with highly developed capital markets, they must, in effect, accept the rates set by central bank discount policies. The only alternative is to hold financial assets or noninterest bearing cash. As long as the rate of interest on short-term assets is positive (or greater than the cost of investment), short-term assets will be preferable to cash.

Chart 2C.
Chart 2C.

Yield

(percent per annum)

Citation: IMF Staff Papers 1963, 002; 10.5089/9781451947151.024.A003

However, if the “Hicks/Lutz” logic be accepted, some investors can be persuaded to move out of short-term investments and into longer-term (or vice versa) if there is sufficient divergence between short-term and long-term interest rates. With a given short-term rate, and a given market consensus regarding the future course of short-term rates, there will be a structure of rates for bonds with different maturities which will not involve incentives for investors to sell bonds of any maturity in order that they may buy bonds of a different maturity. The derivation of such a structure of rates is indicated in Chart 3a. In this chart, it is assumed that there is a market consensus that short-term rates will reach 6 per cent at some time in the future. A very simple form of this expectation is accepted. It is assumed that the market believes that one year from the present the short-term rate of interest will be half-way between its current level and its expected future level; and that each 12 months thereafter the difference between the rate and its expected level will be halved.34 In Chart 3a, it is assumed that the present short-term rate is 2 per cent. The expected course of the short-term rate, on these assumptions, is given by the curve HE. If these rates were to prevail, the capital sum indicated by curve VV (plotted against the right-hand scale) would accumulate over time from the investment of 100 and its quarterly compounded reinvestment. For any accumulation through reinvestment at prevailing rates, there is an alternative single interest rate which would yield the same capital sum at compound interest. Thus if 100 were invested and reinvested quarterly, together with accumulated interest for period OTi, at the rates indicated by curve HE, it would accumulate to AAi Alternatively, 100 invested at a compounded rate of OYi would accumulate to AAi over a period OTi Similarly, 100 invested at OYi, compounded quarterly for OTii, would accumulate to the same amount (AAii) expected to accumulate from the investment and reinvestment at the expected short-term rates over the period OTii. Curve HL is the locus of all points (Pi, Pii,. etc.) so described. This curve describes the “arithmetic average between the current short rate and the relevant forward short rates” 35 for the periods OTi, OTii, etc. It is a hypothetical yield curve based on “Hicks/Lutz” logic.

A family of curves, similar to that presented in Chart 2b, may be drawn for a range of levels of the current short-term rates, with a long-term expectation of 6 per cent. This is done in Chart 3b Alternatively, on the basis of similar logic, it is possible to draw a family of yield curves, on the assumption that the short-term rate is fixed at 2 per cent, but to show the shapes of the curves with different market consensuses regarding the future level of short-term rates. This is done in Chart 3c.

Chart 3B.
Chart 3B.

Yield

(percent per annum)

Citation: IMF Staff Papers 1963, 002; 10.5089/9781451947151.024.A003

Chart 3C.
Chart 3C.

Yield

(percent per annum)

Citation: IMF Staff Papers 1963, 002; 10.5089/9781451947151.024.A003

Holders of Intermediate-term Securities

In the preceding argument, complete yield curves have been derived on the assumption that the shape of a yield curve was determined either solely by the “Sayers” type of investor or solely by the “Hicks/Lutz” type. It was recognized earlier that any such assumption is invalid. Most competent observers agree that “Sayers” and “Hicks/Lutz” types are caricatures of two relatively small segments of the market. The remaining members of the market form the “Radcliffe/Riefler” group. To derive a composite yield curve, it is necessary to reconcile the forces exercised on the yield curve by the three groups of investors. It may be postulated that the shape of the yield curve relating to long-term securities will be determined predominantly by the “Sayers” type investors, and its shape relating to short-term securities will be determined predominantly by the “Hicks/Lutz” type. However, if the market is “a single market for credit,” the yield curve must be a smooth progression. The “Radcliffe/Riefler” purchasers of securities may be expected to adjust their security holdings between long-term and short-term securities, so that the “Sayers”-determined long end of the curve is smoothly joined to the “Hicks/Lutz”-determined short end.

It should be emphasized that both the “Sayers” curves, described in Charts 2a, b, and c, and the “Hicks/Lutz” curves, in Charts 3a, b, and c, are not actual yield curves. Rather, they are limits to the possible yield curves. The actual yield curves will lie between their relevant “Sayers” and “Hicks/Lutz” curves. If it is assumed that the central bank takes action to determine the long-term and short-term rates, it may be assumed that the long end of the actual yield curve will lie on the “Sayers” curve, and the short end will lie on the “Hicks/Lutz” curve, and that the forces described in both the “Sayers” and “Hicks/Lutz” curves will be effective influences on the actions of the “Radcliffe/Riefler” investors.

The forces determining the shape of the yield curve are suggested graphically in Chart 4a. It is assumed that the market has reached a consensus that, in the pertinent future, the average short rate and the normal level of long rates will both be 6 per cent. It is further assumed that the central bank fixes the 3-month rate at 2 per cent, and the 20-year rate at 4 per cent. (These assumptions are consistent with those underlying Charts 2a and 3a.) Hence, the “Sayers” yield curve in Chart 2a can be transferred to Chart 4a to give one limit to the actual yield curve (line SR). Similarly, the “Hicks/Lutz” curve in Chart 3a can be transferred to provide the other limit to the yield curve in Chart 4a (line HL). Consequently, the “Radcliffe/ Riefler” yield curve will lie in the shaded area between the two curves. On the assumptions given regarding central bank determination of interest rates, it will be a smooth curve joining points H and R. Further, at the short end, it will not diverge far from the HL curve, and at the long end it will not diverge far from the SR curve. In Chart 4a, curve HR has been derived by computing a series of averages between the HL and SR values with moving weights. At the 20-year point, the SR values have a weight of 80, and the HL values a weight of zero. As the calculation moves to earlier periods, the weights assigned to the SR values are reduced, and the weights assigned to the HL values are increased by 4 a year. Curve HR results from this computation as a suggested possible yield curve.

Chart 4A.
Chart 4A.

Yield

(percent per annum)

Citation: IMF Staff Papers 1963, 002; 10.5089/9781451947151.024.A003

It must be emphasized that curve HR is presented as nothing more than a possible curve. It is argued that, on the assumptions outlined above, the yield curve will lie between the HL and the SR curves. It is not suggested that this system of weighting is a precise measure of the strength of the reactions of the “Radcliffe/Riefler” investors relative to the actions of the “Sayers” and “Hicks/Lutz” investors.

In particular, the assumption of complete independence between long and short rates is an abstraction. They are essentially, rather than completely, independent. Moreover, their interdependence is asymmetrical. On the one hand, open market operations in short-term securities will alter the yields on short-term securities and the total money in the economy. Changes in the capital values of short-term financial assets and in money will alter the liquidity of the economy, hence provoking a movement along the community’s liquidity preference function. This movement will induce a change in long-term rates. On the other hand, it can be argued that the liquidity preference function is essentially independent of the short-term interest rate. If so, the liquidity effects of a change in long-term rates should not have marked repercussions on short-term rates.

On the basis of this logic, a wide variety of shapes may be derived for possible yield curves. Given the possible variations in the relations between actual long-term and short-term interest rates, and between actual yields and market consensuses, there is a very large number of families of curves, similar to those presented in Charts 2b and 3b (or 2c and 3c). A few of these families have been sketched in Charts 4b,4c, Chart 4d.1, and Chart 4d.2. Charts 4b and 4c are based on a 6 per cent market consensus. In Charts 4b, the short-term rate is 2 percent, and the yield curves for alternative long-term rates are presented.

Chart 4B.
Chart 4B.

Yield

(percent per annum)

Citation: IMF Staff Papers 1963, 002; 10.5089/9781451947151.024.A003

Chart 4C.
Chart 4C.

Yield

(percent per annum)

Citation: IMF Staff Papers 1963, 002; 10.5089/9781451947151.024.A003

Chart 4D.1.
Chart 4D.1.

Yield

(percent per annum)

Citation: IMF Staff Papers 1963, 002; 10.5089/9781451947151.024.A003

Chart 4D.2.
Chart 4D.2.

Yield

(percent per annum)

Citation: IMF Staff Papers 1963, 002; 10.5089/9781451947151.024.A003

In Charts 4c, the long-term rate is 4 percent, and the yield curves for alternative short-term rates are presented. In Chart 4d.1, the short-term rate is 2 per cent and the long-term rate is 4 per cent in all cases. The curves are those appropriate for different market consensuses. As these curves lie close together and intersect, the lower left hand quadrant of this chart is shown in Chart 4d.2, with the scales doubled and with the market consensuses identified by numerals.

Examples of practically all the shapes for yield curves which have been observed when market data are plotted are to be found in these charts.

In this presentation, it is assumed that the market consensuses regarding long-term and short-term rates are the same. This assumption is consistent with most of the literature. However, it is not necessarily valid. The market consensus regarding short-term rates may differ from that regarding long-term rates. If this should happen, it would not create any inherent difficulty for the derivation of yield curves; the appropriate “Sayers” curve, based on the given long-term rate and long-term consensus, could be averaged with the appropriate “Hicks/Lutz” curve, based on the given short-term rate and a different market consensus.

Empirical Evidence

As yet, this argument has been presented solely in theoretical terms. Two recent investigations of observed data lend support to an “expectations” theory of the term structure of interest rates, but not necessarily to a pure “Hicks/Lutz” or to a “Sayers” explanation of this structure. Rather sketchy observations lend support to the hypothesis outlined above.

Meiselman has analyzed movements of the yield curve for U.S. securities during this century.36 He concludes that “year to year movements of components of the yield curve and its implied forward rates during the 1900-1954 period were, in fact, systematically related to the difference between futures and realized short-term rates, and hence to each other. The systematic covariation of segments of the yield curve contradicts the widely held view that the market is a ‘segmented’ one.” 37 His observations support the view that the yield curve is a smooth progression based on a market consensus regarding expected rates. He finds that the shape of this curve alters with changes in this consensus. Unfortunately, the Meiselman analysis is limited to changes in expected short rates. It might well be that, if he had made similar comparisons based on changes in expected long rates, he would have reached similar conclusions. In brief, Meisel-man’s results are consistent with the theory presented here.

Wood has examined the U.S. data for the postwar period.38 On the basis of monthly data for the period January 1947-December 1961, he observed that “both the elastic expectations hypothesis and the segmented markets version of the hedging theory are generally inconsistent with” 39 the data. However, he found that a measure of expectations regarding the movements in the short-term rate over the near future made some contribution to an explanation of the ratio of short-term to long-term rates. Again, unfortunately, Wood did not examine expected movements in the long-term rate. Yet his results are consistent with the view that the long-term and short-term securities markets are essentially independent, but that arbitrage in the market for intermediate-term securities results in a smooth yield curve based on a long-run expected level of interest rates.

Finally, a comparison of three actual yield curves with possible curves derived on the basis of the analysis presented here suggests that this analysis may approximate to a valid description of market behavior. It should be recognized that a few statistical comparisons cannot be accepted as verification of any hypothesis. Even so, it is proper to accept such comparisons as providing some support for theoretical propositions, if the observed data correspond to those which might be expected on the basis of the theoretical postulates. In Chart 5, the observed yield curve (A) for U.S. Government securities at the end of January 196040 is compared with a theoretical yield curve (T) derived in the same way as curve HR in Chart 4a, with a market consensus of 6 per cent, a Treasury bill rate of 4 per cent, and a 20-year bond yield of 4½ per cent. Similar comparisons are made between the observed yield curve on U.K. Government securities at the end of March 196141 and a theoretical curve based on a 7 per cent market consensus, a 4½ per cent Treasury bill rate, and a 6 per cent 20-year bond yield, and between the observed yield curve on Canadian Government securities on February 14, 195842 and a theoretical curve based on a 5½ per cent market consensus, a 3 per cent Treasury bill rate, and a 4 per cent 20-year bond yield.

Chart 5.

Comparison of Three Actual and Theoretical Yield Structures Yield

Citation: IMF Staff Papers 1963, 002; 10.5089/9781451947151.024.A003

Structure des taux d’intérêt d’après l’échéance

Résumé

I1 existe deux théories largement acceptées pour expliquer la structure des taux d’intérêt d’après l’échéance. L’une prétend que, foncièrement, le taux à long terme est égal à la moyenne des taux à court terme prévus pour l’avenir, l’autre, que la structure des taux est fonc-tion de l’évolution attendue des taux à long terme dans l’avenir.

II ressort de l’observation qu’aucune de ces deux théories n’offre une explication parfaitement satisfaisante de la structure des taux d’intérêt d’après l’échéance. Le marché des valeurs à revenu fixe est formé de deux marchés essentiellement indépendants. I1 est probable que c’est à partir de la première théorie que l’on peut le mieux expliquer la structure des taux du marché à court terme; tandis que la deuxième semble la plus appropriée pour décrire la structure des taux du marché à long terme.

Toutefois, du fait que pour beaucoup de prêteurs et d’emprunteurs les titres sont trés facilement interchangeables, les divers secteurs du marché constituent un ensemble aux liens assez lêches. Par conséquent la courbe des revenus sera probablement régulière, l’extrémité représentant le court terme étant déterminee par les emprunteurs et les prêteurs qui s’intéressent principalement aux valeurs à court terme, et l’extremité représentant le long terme, par les emprunteurs et les prêteurs qui s’intéressent principalement aux valeurs à long terme. Par le moyen de l’arbitrage, les deux extrémités de la courbe se rejoindront selon un processus régulier.

On a pu constater, grâce à quelques observations, que les courbes ainsi obtenues sont en accord avec les faits tels qu’ils se sont produits récemment.

Estructura de las tasas de interés según los plazos de vencimiento

Resumen

Existen dos interpretaciones teóricas muy difundidas sobre la estructura de las tasas de interés según los plazos de vencimiento. La una sostiene que la tasa de interés a largo plazo es en esencial el promedio de las tasas de interés a corto plazo previsibles; la otra mantiene que la estructura de las tasas de interés depende de las perspectivas sobre la evolución de las tasas de interés a largo plazo.

La experiencia demuestra que ninguna de estas interpretaciones encierra una explicación completamente satisfactoria de la estructura de las tasas de interés. El mercado de valores de renta fija está dividido en dos mercados esencialmente independientes: el mercado a corto plazo, en donde la estructura de los intereses puede probable-mente explicarse mejor basándose en la primera de las teorías antes señaladas; y el mercado a largo plazo, que posiblemente se ajuste más exactamente a la segunda.

No obstante, el alto grado de substituibilidad de valores que existe para muchos prestamistas y prestatarios enlaza los varios sectores de dichos mercados en un conjunto más o menos integral. En consecuencia, la curva de rendimiento acusaría posiblemente una tendencia bastante regular en la que uno de sus extremos se determinaría por los prestatarios y prestamistas cuyo interés primordial está en los valores a corto plazo, y el otro, por los prestatarios y prestamistas interesados más que todo en valores a largo plazo. Mediante el arbitraje se unen ambos extremos y se imprime regularidad a la curva.

Existen algunas indicaciones de que las curvas así obtenidas están de acuerdo con la experiencia recientemente observada.

*

Mr. Dorrance, Chief of the Finance Division, has been a lecturer at the London School of Economics and a member of the staff of the Bank of Canada.

1

Quarterly Journal of Economics, LX (1940-41), pp. 36-63 (reprinted in American Economic Association, Readings in the Theory of Income Distribution [Philadelphia and Toronto, 1951], from which the citations made here are taken).

2

Joseph W. Conard, An Introduction to the Theory of Interest (Berkeley and Los Angeles, 1959), p. 290.

3

There are other opinions regarding the term structure of interest rates. Thus Dudley G. Luckett maintains that any theory of the term structure of rates based on expectations is “incorrect” because it is “in fact, reasoned from an unwarranted assumption to a questionable conclusion” (“Professor Lutz and the Structure of Interest Rates,” Quarterly Journal of Economics, LXIII [1959], p. 131). J. M. Culbertson maintained that there was a basic relation between the rates for different maturities determined by the relative liquidity of short-term and long-term debts, but that the actual structure of rates would be influenced by the relative maturity structure of existing debts (“The Term Structure of Interest Rates,” Quarterly Journal of Economics, LXXI [1957], pp. 485-517).

4

John Richard Hicks, Value and Capital (Oxford, 2nd ed., 1946).

5

Ibid., p. 145.

6

Ibid., pp. 151-52.

7

Ibid., p. 152.

8

Op. cit., p. 512.

9

The designation of one of his two views as the “Hicks” view is partly justified by his reliance on this explanation of the term structure of interest rates in one discussion of the problem (see “Mr. Hawtrey on Bank Rate and the Long-Term Rate of Interest,” The Manchester School, Vol. X, No. 1).

10

R. S. Sayers, Modern Banking (Oxford, 5th ed., 1960), p. 143.

11

Ibid., p. 139.

12

As identified in International Monetary Fund, International Financial Statistics.

13

Starting with February 1956.

14

One of the more mystifying statements of the Radcliffe Committee, given the readily available statistical data, is the Committee’s view that “It is sometimes alleged that there is such a hump, but we can discern no pronounced hump for the range of maturities in question…” See Committee on the Working of the Monetary System [Radcliffe Committee], Report (Cmnd. 827, London, August 1959), p. 201.

15

For example, some of the yield curves reproduced in Bank of Canada, Annual Report of the Governor to the Minister of Finance for the Year 1954 (at p. 8), and for the Year 1957 (at p. 31), are complex curves with more than one hump.

16

Throughout this presentation, it is assumed that the central bank is the only monetary authority. This assumption is irrelevant to the argument, provided that all the monetary authorities in a country follow consistent policies.

17

J. R. Hicks, op. cit., p. 164.

18

D. H. Robertson, “Alternative Theories of the Rate of Interest,” Economic Journal, XLYU (1937), p. 433.

19

It is recognized that the implementation of certain interest-rate policies may have undesired repercussions leading to a change in the interest levels desired by the authorities (vide the United Kingdom experience in the summer of 1947). However, the possibility that the central bank may change its policy is not of direct relevance to a discussion of the effect of specific decisions determining the level of long and short rates.

20

Committee on the Working of the Monetary System [Radcliffe Committee], Minutes of Evidence (London, 1960, referred to hereafter as Radcliffe Evidence), pp. 469-78.

21

H. C. Wallich, “The Current Significance of Liquidity Preference,” Quarterly Journal of Economics, LX (1945-46), p. 493.

22

Op. cit., p. 42.

23

“Open Market Operations in Long-Term Securities,” Federal Reserve Bulletin, November 1958, p. 1264.

24

Committee on the Working of the Monetary System [Radcliffe Committee], Principal Memoranda of Evidence (London, I960), Vol. 2, p. 214.

25

E.g., Culbertson, op. cit.

26

Joan Robinson, “The Rate of Interest,” Econometrica, Vol. 19 (1951), p. 102, fn. 20.

27

“A Rejoinder,” The Manchester School, October 1939, p. 156.

28

Luckett, op. cit., p. 141.

29

Radcliffe Evidence, p. 476.

30

It is recognized that a spread of two points between the current yield and the market consensus is most unlikely to emerge. However, in order to clarify the graphic presentation, the basic argument throughout this exposition is outlined in terms of the rather wide spreads of a 2 per cent short-term rate, a 4 per cent long-term rate, and a 6 per cent market consensus.

31

Throughout this exposition, all the computations assume that interest is paid and compounded quarterly.

32

The preceding exposition explained the positive slope of the 4 per cent line. The negative slope of the yield curves, when the central bank sets the yield on 20-year bonds above 6 per cent, can be similarly explained. If the market expects long-term rates to be at the 6 per cent level in the future, and if purchasers now have the opportunity to obtain a yield of more than 6 per cent for 20 years, purchasers will acquire a bond with a maturity of less than 20 years only on condition that they are compensated for the difference in yields over the period from the maturity of the shorter bond to 20 years from the present.

33

These graphic presentations take the market consensus as a datum. Any effect which the central bank’s policy may have on the consensus is assumed to be already reflected in the given consensus.

34

No particular significance may be attached to the form of this expectation. It is one of several simple, asymptotic assumptions which might be made. It has been used here primarily because it involves a simple type of expectation.

35

See above, page 275.

36

David Meiselman, The Term Structure of Interest Rates (Englewood Cliffs, N. J., 1962).

37

Ibid., p. 60.

38

John H. Wood, “An Econometric Model of the Term Structure of Interest Rates,” an as yet unpublished paper, presented to the Econometric Society, Pittsburgh, December 27, 1962.

39

Ibid.

40

From Treasury Bulletin, March 1960, p. 58.

41

Derived from Bank of England, Quarterly Bulletin, December 1961, pp. 58-60.

42

From Bank of Canada, Annual Report of the Governor to the Minister of Finance for the Year 1957, p. 31.

IMF Staff Papers: Volume 10, No. 2
Author: International Monetary Fund. Research Dept.