The Relationship Between Central-Bank Interest Rates and Market Yields

Eric Dowty
Shape Software
521 Hidden Valley Road
Kingsport, TN 37663
dowtz@shapesoftware.com
Copyright 2009, 2011

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Contents

Abstract

The relationship between central-bank rates (discount and federal funds in the U.S.) and yields for government and corporate securities between 1919 and the present can be described with an empirical linear regression equation:

       Yield_M,t = k_1(M) + k_2(M) * B_t + k_3(M) * <B>_{t-1 to t-p}  + k_4(M) * Risk

where the average yield for a given maturity and rating M over time interval t (year, month, day) is a linear function of the current value of the central-bank rate B at time t; the average of that rate over past time intervals in a period p of about 4 years; and a risk parameter, which can be a difference in yields of medium- versus low-risk securities (Baa-rated corporate versus 10+-year Treasuries), unemployment rate or difference between actual and trend GDP. For 44 monthly and yearly time series of U.S. securities the weighted averages are R² = 0.954 (range 0.91-0.98) and standard error = 0.501% (range 0.22-0.77%).  Results almost as good can be obtained when the third term uses a single past value of central-bank rate about three years previous. The risk term is important only at certain times when general confidence is low, for example in the U.S. during the Great Depression or in the credit crisis of 2008-2009. Best agreement for the complete time range 1919-present requires linear time variation of at least one of the coefficients, but for periods of up to 75 years (1934-present) this is not necessary. The magnitude of deviations from the relationship is correlated over time with the general activity of the Fed in changing interest rates, not with variation of macroeconomic parameters such as inflation or deficits.

Adding inflation rate, government deficits or debt, or foreign investment as independent variables to the equation gives no meaningful improvement. Inspection of historical data confirms that neither inflation nor deficits have major direct influence on yields in the U.S. Preliminary analysis of data from more than 23 other countries (205 time series) gives similar results.

The shape of yield curves as well as their variation in slope over time is described to a very good approximation by the equation. The supposed prophetical aspect of yield curves is explained by the fact that a flat or negative yield curve has often been induced by the Fed raising discount or federal funds rate prior to recessions, usually in response to inflation. Investment, GDP growth and employment are correlated with the difference between long- and short-term rates, not with the absolute value of long-term rates.

Yields can be interpreted as an expected future central-bank rate, which is estimated by the second and third terms of the equation, plus a maturity premium function given by the regression constant k₁ plus the risk term. This is consistent with Hicks' theory of term structure, in which long-term rates are determined by expectation of future short-term rate. A special and important case of the class of "speculators" in Hick's theory is banks and other re-lenders whose provision of money at long term will be calibrated with reference to the expected future central-bank rate plus a profit margin, and will thus tend to make long-term yields converge on this value.

1. Introduction - The Conventional Wisdom

This paper explores the influence of various factors on yields of market-traded securities and some other debt, primarily through a simple regression equation.  There is a great variety of theories about what determines interest rates, and in some of the more important ones the central bank is considered to be a major influence through its assumed control of the money supply. A few aspects of such theories will be discussed in Section 12, although the main objective of this paper is empirical. But despite such theories, even among their adherents there seems to be a great reluctance on the part of many authorities to admit that the central bank controls long-term interest rates, and in some cases there has even been even an outright denial that central banks have any influence at all:

"The Federal Open Market Committee (FOMC) implements monetary policy by setting an intended, or target, federal funds rate, and then engaging in open-market transactions to keep the actual fed funds rate close to the intended rate. This institutional fact has the unfortunate side effect of leading some to believe that the Federal Reserve has more control over interest rates than it in fact does." (Poole, 2003).

"Where, however...open-market operations have been limited to the purchase of very short-dated securities, the effect may, of course, be mainly confined to the very short-term rate of interest and have but little reaction on the much more important long-term rates of interest." (Keynes, 1936, Ch. 15, p. 197; but see Section 12.2.1.)

"... the Fed only controls short-term interest rates, while investment spending depends on long-term rates." (Krugman, 2007)

“...all of us recognize that the influence that the Federal Reserve has on long-term interest rates is negligible.” (Burns, 1975)

"Look, I can't affect interest yields." (Volcker, 1982)

This paper will demonstrate quantitatively that operations by the central bank to maintain short-term yields are the main determinant of all interest rates, apparently under all economic conditions. In addition, three subsidiary questions will be addressed and answered. The current assumptions about these matters, the conventional wisdom, appear to have arisen without very close reference to the data. The questions are:

1. How do Supply and Demand in Credit Markets Influence Rates. The supply of Treasury securities as a function of national budget deficits and demand for securities on the part of foreign as well as domestic investors are presumed by many to be major determinants of yield rates:

"In order to make lenders willing to continue to finance us at reasonable rates, we do have to persuade them that we are serious about returning to a more balanced fiscal situation going forward." (Bernanke, 2009a)

"We have thus been able to finance larger deficits for longer and at lower interest rates, as foreign demand has kept Treasury yields low." (Roubini, 2009)

2. What is the Effect of Inflation. Yields are commonly thought to be influenced by lenders' expectation of inflation - lenders supposedly should demand higher yields if they expect the principal to be devalued by inflation.

"Nominal interest yields are highly sensitive to inflation and to inflation expectations." (Dewald, 1998).

"Typically S&L customers would deposit money in passbook savings accounts, which paid only 3 percent interest but were federally insured; then the S&L would lend out these funds in the form of thirty-year mortgages at 6 percent interest. ... But inflation spelled doom for this tidy state of affairs. It drove both short-term and long-term interest yields sharply higher, putting the S&Ls in a terrible squeeze." (Greenspan, 2007 - referring to the Savings and Loan industry during the inflation of the 1970's and early 1980's).

"[T]he large decline in interest rates after 1981 was due to a large decline in inflation expectations." (Poole, 2003).

"A perceived loss of monetary policy independence could raise fears about future inflation, leading to higher long-term interest rates..." (Bernanke, 2009b)

3. What Determines the Slope of the Yield Curve. The conventional wisdom is that the market determines the slope of the yield curve, and the yield curve is thought by many to have prophetic properties - a flat or negative slope has often preceded recessions over the last 50 years or so, and even in 1929. Various explanations for this correlation have been offered, for example:

"The reason for the historical relationship between the slope of the yield curve and the economy's performance is that the long-term rate is, in effect, a prediction of future short-term rates. If investors expect the economy to contract, they also expect the Fed to cut rates, which tends to make the yield curve negatively sloped. If they expect the economy to expand, they expect the Fed to raise rates, making the yield curve positively sloped." (Krugman, 2008).

From some of the quotations above (by Fed personnel) and many other statements which could be cited, the operating assumptions of the Federal Reserve seem to be basically the same as the conventional wisdom.

These questions will be answered in a straightforward empirical way by reference to the interest-rate data. The main part of the paper will deal with data for the United States since these records are the most complete, but available data for other countries will also be considered.

2. The Regression Equation

The data are available as average values (or sometimes mid- or end-period daily-average values) over different time units; days, weeks, months and years. Sources of data are in the section Data Sources and are referred to in the text and tables with numbers in brackets. The yield for each maturity and rating (M) averaged over each time interval (t) as a day, week, month or year, has been fitted by least-squares to linear equations of the type:

Equation (1):            Yield_M,t = k_1(M) + k_2(M) * B_t + f(B_t-n1,B_t-n2, ...) + k_4(M) * Risk

where B_t is the effective or target central-bank policy rate for the given time interval and B_t-n1 is the average central-bank rate for the interval at n1 time units previous and so on. The values of n1, n2 etc. will be called the time offsets. Terms for other independent variables will also be tested, for example inflation rate and deficits.

The third term or group thereof, f(B_t-n1,B_t-n2, ...) is a simple linear function of the central-bank rates at previous times, with one term:

Equation (2):            f(B_t-n1,B_t-n2 , ...) =  k_3(M) * B_t-n1

or several terms with individual offsets:

Equation (3):            f(B_t-n1,B_t-n2 , ...) =  k_3a(M) * B_t-n1 + k_3b(M) * B_t-n2 +  ...

or an average of the central-bank rate at all time offsets up to a limit p:

Equation (4):            f(B_t-n1,B_t-n2 , ...) =  k_3(M) * <B>_{t-1 to t-p}

The regression constant and the terms in central-bank rate at time t and the various time offsets - that is the first three terms - account for 91 to 97% of the variance of yields of Treasury securities in monthly time series since 1919. The different forms of the function of time offsets in equations (2)-(4) will be discussed in Section 3. The final term in "Risk" in equation (1), which becomes important at certain times for other types of securities, will be discussed in Section 4. That section will also discuss variation in the regression coefficients over long periods.

The equations were developed in a straightforward and strictly empirical way; taking first the direct correlation between the dependent variable (market yield) and the main independent variable (central-bank rate), then taking the first derivative of the independent variable. This is equation (2), with terms rearranged. Equations (3) and (4) are elaborations on this basic relationship. So far a second derivative of the independent variable has not been found to be useful. The risk term is also empirical, inspired by observed yield spreads during the credit crisis of 2008-9 (Krugman, 2008-2009).

The independent variable - central-bank rate. In the U.S., yields are influenced by the Fed in two main ways: by the arbitrary setting of the discount rate (or rates; each regional bank set its own rate up to 1970); and by open-market operations to control the federal funds interbank lending rate, or at certain times in the past to control other short-term rates. Since 1954 federal funds rate has normally been the main instrument of Fed interest-rate policy. The discount rate was replaced in January 2003 by the primary credit rate, which was 1.5% higher than the discount rate at that time. In this paper, the main independent variable B_t for monthly data will be the discount rate (New York) up to July 1954 and effective federal funds rate thereafter, and for yearly data the discount rate through 1954 and federal funds rate thereafter. It is possible to use a splice of discount rate and primary credit rate instead of federal funds, and if time variation of coefficients is allowed (Section 4), the results are only slightly inferior to those using federal funds, usually by less than 0.01 in R². In this paper the simplest possible representation of central-bank rate has been chosen, but it should be realized that multiple rates have often been simultaneously operative in the U.S. and other countries. In principle, the rate chosen is the minimum rate at which money can generally be borrowed at short term.  The central-bank rate used throughout this paper was taken from data sources [2/115], [3/12.1] and [1].

From 1942 to 1947 the Fed conducted open-market operations to keep the yield on 3-month Treasuries at 3/8% (see Section 7.1) and this might be considered the policy rate for that time, but for simplicity the discount rate (1%) through the interval has been used; using the 3-month T-Bill rate makes little difference in overall statistics.

Through the 1970's and at least part of the 1980's a main objective of Fed policy was to control monetary aggregates and in 1979 it was announced that attempts to control the money supply would be "directly" through reserves rather than through interest rates. Nevertheless target federal funds rates were set through this period (Federal Reserve, 2009), and monthly averages of effective federal funds generally were quite close to the targets, or midpoints of the limits. The discount rate was varied sympathetically with federal funds target rate, though its variation was usually not as great. The policy of "direct" control seems to have been abandoned perhaps as early as mid-1980.

Compilations of data averaged on a weekly or daily basis are not as extensive as those on a monthly or yearly basis. Effective federal funds rate has at times been quite variable on the time scales of weeks or days, so agreement is generally poorer for calculations on these scales. So far no advantage has been found to using time intervals shorter than a month.

3. Results for Regression on Central-Bank Rate Only

This section will establish the importance of central-bank rate as the main determinant of yields of all types of securities of all maturities. Only terms in central-bank rate will be used, no term in risk, with coefficients constant over time. Inflation rate was also tested as an alternate or additional independent variable.

3.1. Results - U.S. Treasuries, 1953-present (monthly)

This is the most complete set of actual yield averages, and it is simpler than other groups of data because of the minimal risk of the securities and apparent lack of time variation in the derived coefficients. The data are monthly-average yields from April 1953 to the present as published by the Federal Reserve. The series for maturities of less than one year are Treasury Bills. Averages for securities of less than one year maturity referred to by the Fed as "constant maturity" are also available, but are less complete. The group of time series of Treasuries of 4 weeks to 30 years maturity as given in Table 3.1.2 were the primary data used to evaluate which of equations (2), (3) or (4) gives best results, and what the values of time offsets should be, though other series were also considered (below in Section 3).

For equations (2) or (4) the value of the single time offset or maximum offset was varied from one to 92, the regression was done for all time series in Table 3.1.2, and that value was chosen for Table 3.1.1 which gave the maximum weighted average of R² for all the series in the group. Minimizing the weighted average standard error almost always gives the same result. For equation (3) the value of each of the offsets had to be varied independently, a much more time-consuming operation.

Table 3.1.1. Overall results for regression of yields on 11 series of U.S. Treasury securities of varying maturities 1953-present (monthly) on various combinations of independent variables. The R2 and standard error values for each maturity (as given in Table 3.1.2) are weighted by the number of observations. Offsets are as defined above for equations (1)-(4). The standard error of the regression is in absolute yield percentage points. Independent Variables Weighted Average R2 Weighted Average Standard Error  Inflation 0.460 2.135  Central-bank rate 0.881 0.934  Central-bank rate + inflation 0.882 0.928  Central-bank rate + 1 offset, 33 months (eq. 2) 0.941 0.687  Central-bank rate + 1 offset, 33 months + inflation 0.943 0.679  Central-bank rate + 2 offsets, 21 and 42 months (eq. 3) 0.950 0.637  Central-bank rate + 3 offsets, 12, 33 and 44 months 0.952 0.628  Central-bank rate + average 1-48 months (eq. 4) 0.951 0.631  Central-bank rate + average 1-48 months + inflation 0.953 0.624

At these high levels of explanation, changes in R² should probably be considered in terms of the residuals - for example increasing R² from 0.88 to 0.94 means that the unexplained variance is reduced from 12% to 6%.

Central-bank rate plus one offset gives good overall results, accounting for about 94% of overall variance, and adding offsets beyond that improves overall R² by small amounts. Adding offsets does have greater effect on the longer maturities, for which agreement tends to be poorer than for short maturities.

Inflation by itself explains some of the variance over this period, but it is much worse than central-bank rate alone, and adding inflation as an independent variable when central-bank rate is present improves the fit by only a very small amount. Adding an offset or other minor tweaks to the equation usually gives more improvement than adding inflation. The coefficients for inflation in any of the cases in which central-bank rate is also present are small, for example for equation (4) varying from -0.12 for 4-week maturity to +0.13 for 30-year maturity. If inflation were actually a cause of the overall variance we would expect all positive coefficients. The results when inflation is an independent variable will be discussed further in Section 7.

Figure 3.1.1. Dependence of the multiple correlation coefficient R2 on the number of months averaged (p) in the third term of equation (5). Data as in Table 3.1.2. The curves are in strict order top to bottom at 20 months

The simple average of offsets 1 through 48 months gives results almost as good as using three offsets but with two fewer adjustable coefficients. As shown in Figure 3.1.1 the optimal number of averaged offsets for each maturity is fairly constant near 54 months except for the shortest maturities, and for these the coefficients for the third term are small (see Table 3.1.2). For equation (2),  R² similarly peaks for most maturities when the single offset is around 33 months. For simplicity standard numbers of offsets in equation (4) equal to 4 years will be used hereafter. Thus the equation used for the remainder of Section 3 is equation (5) where p is 4 for yearly data and 48 for monthly data.

Equation (5):           Yield_M,t = k_1(M) + k_2(M) * B_t +  k_3(M) * <B>_{t-1 to t-p}

In all the tables showing regression results in this paper, k₁ is the regression constant, k₂ is the coefficient for central-bank rate and k₃ is the coefficient for the average value of central-bank rate over the preceding 48 months or 4 years. The Mean and standard deviation (StDev) are those of the raw yield data. R² is the squared multiple correlation coefficient for the above equation, StdErr is the standard error of the regression, and Nobs is the number of observations (months or years in most cases). The R² and standard error of each series are weighted by the number of observations to derive the group averages at the head of the tables.

3.2. Results - Other U.S. Security Series

Details and data sources for each group are given in Section 5. Regression on central-bank rate only (current and past rates) usually accounts for over 90% of the variance, although not as much for other securities as for Treasuries. The group of Treasury Inflation-Protected Securities (Section 5.8) is not included because of the necessity for extra terms and the short time they have been available.

The regression coefficients as a function of maturity fall into reasonably smooth patterns as shown in Figures 3.2.1 through 3.2.4. The series in group 1 for less than one year maturity are Treasury bills, to get the greatest range in time, so perhaps some discontinuity is to be expected between these and longer-maturity series, which are those designated "constant maturity" by the Fed. Also, the 2-year, 7-year and 30-year series start about 1977 rather than 1953. Currently the "average" values for "constant maturity" securities, presumably notes and bonds, are derived daily from a single fitted curve, while those for T-Bills are apparently derived in a different way, independently from the derivation of "constant maturity" rates.

Figure 3.2.1. Coefficients for equation (5), regression of Group 1, U.S. Treasuries 1953-present. k1 is the regression constant, k2 is the coefficient for central-bank rate and k3 is the coefficient for the average of central-bank rate for 1-48 months previous. k1 has vertical dimension of rate (percent), but k2 and k3 are dimensionless.

Figure 3.2.2. Coefficients for equation (5), regression of Group 1, U.S. Treasuries 1982-present. Constant-maturity series were used instead of T-Bills for 3- and 6-month, and regression started in 1982.

Figure 3.2.3. Coefficients for equation (5), regression of Group 2, U.S. Treasuries 1919-present. Approximate average maturities are assigned to the 3-5 year and 10+-year Treasury series.

Figure 3.2.4. Coefficients for equation (5), regression of Group 3, basic yields for corporate bonds 1919-1970.

When the results for series which span all or most of the complete time range, 1919 to present or 1919-1970, are viewed, systematic deviations emerge. This is illustrated with a group of such series (Group 8) selected from among the monthly groups (2), (4) and (5). The results for each series are given in Table 3.2.2 and Figure 3.2.5.

Deviations of the calculated from observed values are of three types.

1) In long-term series in which there is considered to be some degree of default risk, there are certain times when the observed values are distinctly higher than calculated, especially peaking at about 1933, 1938 and 2008. Lower bond ratings give larger discrepancies.

2) In long-term series which span most or all of the entire range from 1919 on the calculated values before about 1930 are distinctly too high and those over the last 30 years or so are too low. This is most apparent when the type 1 deviations have been corrected with the risk term.

3) There are relatively short-term deviations (usually 2 years or less) in the results for all securities especially in the period 1953 to present which are consistent for most series. For example, observed yields reach a peak about 1995 which is distinctly underestimated, and conversely yields are overestimated about 1998-99 and 2001. The consistency of these deviations among series is shown statistically by the fact that using the 1-year series as the main independent variable instead of central-bank rate raises R² for the data in Table 3.1.1 from 0.951 to 0.976, or in other words halves the unexplained variance. These deviations between observed and calculated increase rapidly as maturity is increased as shown in Figure 3.2.6. Such deviations are present but less important before about 1934, and least important between 1934 and 1955.

Figure 3.2.6. Difference between observed and calculated for Group 1 (Treasuries 1953-present) for December 1994 as a function of maturity (red), with the values for the k1 coefficient as in Figure 3.2.2.

The first two types of deviation can be largely removed with the risk term and time variation, which will be addressed in Section 4.

4. The Risk Term and Time Variation

This section will discuss the addition of the risk term and time variation of the regression coefficients. The order in which they are introduced is arbitrary.

The risk factor as well as time variation will be evaluated mostly with Group 8, the monthly series which cover all or most of the interval 1919-present, taken from the data in Sections 5.2, 5.4 and 5.5. Results for this group using central-bank rate only are in Table 3.2.2.

4.1. The Risk Term

The usual yield curve, a plot of yield versus maturity, may sometimes be affected by risk (see Sections 6.6 and 12.1.1). But there is also another type of yield curve, yield versus rating, which clearly reflects default risk. It seems, however, that investors at least at times perceive default risk differently than the rating agencies, and the yields for a given rating can vary markedly in a way not accounted for by equation (5). This curve of yield versus rating definitely steepens when there is general uncertainty about the economy and/or specific problems in credit markets themselves. Although the qualitative effect of perceived risk on yields is fairly well known, its importance with respect to equation (1) became clear only during the 2008 credit crisis (Krugman, 2008-2009).

The affect of this differential default risk perception is addressed with the risk term in equation (1), and the risk factor or parameter can be approximated in several ways. The first is simply the difference between the yields for Baa-rated corporate bonds (Section 5.4) and the average for long-term (10+year) Treasuries (Section 5.2). Another is the unemployment rate and a third is the difference between actual GDP and potential or trend GDP. Potentially another is various polls of investor or general confidence.

When added to the regression as an independent variable either the yield difference, unemployment or difference between trend and actual GDP give statistical improvement for Group 8 as shown in Table 4.1.1 but especially in the agreement shown visually in the calculated-observed plots for the critical time periods, that is during the Depression and 2008-2009 (Section 5).  Where only yearly values are available for unemployment or GDP monthly values are derived by interpolation.

Table 4.1.1. Results for adding independent variables representing perceived risk to the regression for Group 8, Selected monthly 1919-1970 or 1919-present (12 series). No time variation of coefficients is used. Independent Variables Added to Equation (5) Weighted Average R2 Weighted Average Standard Error  No extra independent variables 0.909 0.734  Baa-10+year 0.932 0.656  Unemployment 0.930 0.668   Trend minus actual GDP 0.927 0.666

Adding the University of Michigan consumer sentiment index to the Group 8 regression improves R² by 0.004 compared to 0.014 for adding Baa-10+year spread over the life of the index (from 1952).

The improvement is quite meaningful and noticeable for those monthly series in Group 8 which are considered to have some risk. Improvement for Group 3, the somewhat hypothetical corporate lowest-risk series, is slightly better for unemployment than for the yield difference but in neither case is it large (Table 4.2.1). The Treasuries in Group 1 give mostly negative coefficients for this term and though the overall improvement is slight there are noticeable effects in the 1930's and 2008-2009. The 3-month and 10+year Treasuries in Group 8 also give negative coefficients (Table 4.2.1) and show the effect visibly during the Depression, that is their yields go down as the yields of more risky securities go up.

The correlation of these parameters with economic cycles as defined by the National Bureau of Economic Research is shown in Figure 4.1.1. The deviation of GDP from a smoothed long-term trend also correlates with the parameters in Figure 4.1.1, but further assumptions are required to determine the trend and GDP data are available only on a yearly basis up to 1947 and quarterly thereafter. It is quite possible that in the future other economic measures or combinations thereof could give an equal or better measure of risk perception which is independent of security yields themselves - certainly there are now other indices - but for the entire period 1915-present no appropriate data have so far been found other than some kind of yield difference, unemployment or GDP difference.

Figure 4.1.1. The risk parameter (black curve): yield for Baa corporate securities minus yield for long-term Treasuries, and other measures. Periods of recession, as determined by the NBER, are shown by vertical gray bars. The vertical scale applies to the risk parameter; the values for unemployment have been divided by 3, and the values of the University of Michigan consumer sentiment index were subtracted from 120 and divided by 30. Data: [1], [14], [20], [28/D85], [47].

Employment is considered to be a trailing indicator and using unemployment from several months after the calculation date gives better agreement, but the statistical improvement is only very slight, in the third decimal place in both R² and standard error. In view of this and the fact that this amount of delay could not be used for yearly data, results are given for the calculations using unemployment for the calculation date. Prior to 1948 there are no official monthly unemployment data - the values used were obtained by linear interpolation using the official yearly average value at the midpoint of the year. The methods of reporting unemployment have changed over time and there is dispute over whether the official yearly values for the time of the Depression are consistent with later values.

Since the yield difference between Baa and long-term Treasuries is presumably a better measure of risk perception in the securities markets themselves and unemployment rates are not as detailed and subject to some controversy, the yield difference will generally be used instead of unemployment for calculations in this paper, in the form used hereafter, called the complete equation:

Equation (6):           Yield_M,t = k_1(M) + k_2(M) * B_t + k_3(M) * <B>_{t-1 to t-p} + k_4(M) * Risk

It is not significant from a statistical point of view that the fits for the Baa corporate series and for the long-term Treasury series are improved by the addition of the risk factor as a yield difference, since this factor is adding information directly from these series. It is meaningful, however, that fits are improved for all series in a consistent way: the coefficients for corporate and state/local bonds are increasingly positive with lower rating, and the coefficients for Treasuries are usually negative. The obvious correlation of this parameter with other macroeconomic indicators, and the fact that unemployment or GDP difference give nearly equal or better improvement in some series (especially yearly series), also demonstrate that it is not just a statistical artifact.

The risk parameter could well be influenced by things other than differential default risk; especially, as the general perception of uncertainty increases, bond investors may be less tolerant of risk and shift to higher-rated securities, but this would appear to have a similar effect on yields. Different types of securities are affected differently by risk, as illustrated further in Section 6.7.

There is presumably a matter of supply and demand in the response of yields to risk in that in times of low confidence demand apparently decreases for the less highly-rated bonds and increases for others (the negative coefficients for Treasuries indicate that their yields actually decrease with respect to times when general confidence is high). But such supply and demand are different from the overall supply and demand factors discussed in Section 7.

The risk parameter is definitely not directly correlated with either U.S. Government deficits (Figure 7.1.1) or inflation (Figure 8.1.2), except insofar as inflation was frequently followed by recession, which causes increased risk perception.

Differential international risk is discussed in Section 12.3.7.

4.2. Time Variation

It is clear from the type 2 deviations that in terms of the basic empirical regression relationship the coefficients have changed over time. This could have been a fairly continuous change, or the change could have been confined to certain restricted periods. During the Great Depression there were certainly major changes in banking and securities markets - many banks failed and there was a great deal of regulatory and insurance legislation at this time. Monetary standards were changed in the U.S. and in many other countries. The policies of the Federal Reserve also changed at various times - with respect to changing central-bank rates it was moderately active up to the early 1930's, lay dormant for 30 years or so, and became very active in the second half of the century.

Two main approaches have been tried. Simply splitting the regression into two time periods does reduce the type 2 deviations and greatly improves the fit for the earlier years. Appendix A gives some results for fitting two parts of the interval 1915-2010 separately. However, this method doubles the numbers of coefficients and complicates presentation of results. It does not clearly indicate a time or short range of time during which there was a sharp change in the relationships. The calculation for Group 3 can be extended back to 1900, using commercial-paper rates instead of central-bank rate as the independent variable (see Appendix A), and while the behavior is very different at the two ends of the 1900-1970 time range the fits show no obvious break.

The other approach is to apply  linear time variation to the coefficients k₁, k₂, k₃ and k₄ of equation (6). Let

Equation (7):         T = (t - 1915) / (2010-1915) 

or other start and end dates as appropriate for the data. Each coefficient k₂ and higher can be replaced by two:

Equation (8):         k_2(M) * B_t = k_2a(M) * T * B_t  + k_2b(M) * [1-T] * B_t ... etc.

In the actual regression, each independent variable must be split into two by multiplying the original by T and [1-T] respectively. The regression always gives a constant k₁, so time variation of k₁ is introduced with an additional term,  k_1b(M)  * T.

Thus there could be a complete set of coefficients for the year 1915 and another for the year 2010, with any date in between using a linear combination of these coefficients. Varying either k₁ alone or k₃ alone improves the statistics distinctly while varying k₂ gives less improvement. For the longer-term series, k₁ tends to become smaller and k₃ larger over time. The effect of adding time variation to a second and third coefficient is relatively small, so the provisional decision was made to vary only k₃ since its values are generally the most different in the two time segments (above). This removes most of the type 2 deviations, while improving R² values in some cases by several percent. No judgment about the exact nature of the time variation is implied by this decision; it should just be noted that there is definitely some time variation over the time span of the existence of the Federal Reserve, and this can be accounted for rather well using linear models adding a small number of adjustable parameters, in this case one for each series. This gives a total of five terms and five adjustable coefficients, but three of the four independent variables are derived from one data series, central-bank rate.

Using central-bank rate only for Group 8 gave average R² 0.915 and average standard error 0.715.  Adding linear time variation of the k₃ coefficient gives R² 0.924 and average standard error 0.638.  Adding both linear time variation of the k3 coefficient and the risk term gives R² 0.961 and average standard error 0.471. 

All results in this paper use the raw values of current and past central-bank rates, but the values of any other variables, including risk, unemployment, inflation, and others in Section 7, are normalized by subtracting the average of that variable over the complete time span. This normalization choice affects only the value of the k₁ coefficient, not those of the other coefficients or the statistics; for reasons discussed in Section 12.1.1 it seems best to avoid biasing the absolute values of that coefficient.

Table 4.2.1 gives the results of stepwise addition of different independent variables to each of the groups. Addition of a term for inflation has negligible effect, at most 0.003 improvement in R². Addition of a term for GDP difference usually has about the same effect as a term for unemployment. Complete results for all series of U.S. securities are given in the next section.

5. Regression Results for the Complete Equation

All results include the risk term (k₄) using the yield difference and time variation of the k₃ coefficient, although one or the other of these additions may not give significant improvement in some cases.

For all 44 monthly and yearly time series of U.S. securities considered in subsections 5.1 through 5.7 - all those which could be obtained covering more than a few years while avoiding redundancy - the averages, weighted by time span of each series, are R² = 0.954 (range 0.91-0.98) and standard error = 0.501% (range 0.22-0.77%).

5.1. Complete Equation Results - U.S. Treasuries, 1953-present (monthly)

Neither risk nor time dependence has much importance in this group - compare with Table 3.1.2. Distinct negative risk spikes are visible in 2008-9 which are mirror images of the positive spikes in lower-rated corporate bonds (see Sections 5.4, 5.5 and 11), but these have little effect on overall statistics.

5.2. Complete Equation Results, U.S. Treasuries, 1919-present (monthly)

All the series in this group are pieced together from more than one source, and at least at times involve averages over a wider range of maturities than Group 1. The 10+year series, which is used in deriving the risk factor, is available from the Fed [1] from 1925 to 2000 and from the Treasury [44] after that.  For 1919 to 1924 [2/128] the yields are actually for 8 year or longer maturities. The 3-month series actually includes 6-month before 1933, the 9-12 month series is 1-year only after 1953, and the 3-5 year series includes tax-exempt notes 1932-1941.

The risk term (k₄) has little importance in this group - omitting it reduces average R² for the group by only 0.005 and does not change R² for the 10+year series at all (to the third decimal place).  The most noticeable effect of risk appears to have been to reduce the yields of the 3-month series during the Depression.

5.3. Complete Equation Results - Basic Yields for Corporate Bonds 1915-1970 (Yearly)

The values in these series are not actual averages of yields; they represent the estimated limit of low risk or high quality at each year (Durand, 1942).

5.4. Complete Equation Results - Corporate Bonds and Mortgages 1919-present (monthly)

The Industrial, Railroad and Utility series end before the time of highly variable central-bank rate, which causes the R² values for at least the Railroad and Utility series to be slightly inferior because the total variance is less. The standard error for these series is better than that for the others for the same reason. The monthly mortgage series, on the other hand, lacks the earlier period of lesser variation.

5.5. Complete Equation Results - State and Local Bonds, 1919-present (monthly)

The yields for these bonds vary with federal income tax rates; the equation takes no account of this.

5.6. Complete Equation Results - Stock-market and other short-term loans 1919-1978 (yearly)

This group is included mainly to show the rates for stock market loans in comparison to other short-term rates (see Section 7.2 for discussion). These are all low-risk securities, and risk coefficients (k₄) are all negative. In this and some other respects the banker's acceptances and 4-6-month commercial paper might be considered a short-term continuation of the Group 3 corporate bonds.

5.7. Complete Equation Results - Mortgages 1919-present (yearly).

 The monthly mortgage series in Table 5.4.1 apparently is based on the same data as the HUD yearly series.

5.8. Complete Equation Results - Treasury Inflation-Protected Securities 2003-present (monthly).

Both the principal and interest paid on these securities are adjusted by the current Consumer Price Index. If the only buyers of TIPS were bond investors who are determined to get a constant real yield, that is nominal yield minus inflation, then nominal TIPS yields should be insensitive to inflation and central-bank rate. Actually the yields of TIPS tend to closely parallel those of the equivalent conventional Treasury, as shown in Figure 5.8.1, and the spreads up until late 2008 were approximately the same as an average value of inflation over the last few years. The yields of the shorter-term securities were quite variable during this time (Section 5.1). It may have appeared before late 2008 that the spreads are insensitive to current inflation rate, but as Figure 5.8.1 shows, the spreads were apparently affected by the very large and rapid movements of inflation rate from late 2008 to early 2009. This comparison is best made on the basis of month/month inflation numbers because the usual year/year numbers reflect events of the past 12 months and peaks and valleys can be displaced in time. The effect was greater the shorter the term.

Figure 5.8.1 Spreads between TIPS and the equivalent constant-maturity Treasuries, with monthly inflation rate. Data: [1], [48]

Clearly TIPS investors have not been obtaining a constant real return, they have been basing TIPS returns on the current real return on conventional Treasuries. The calibration to conventional security yield requires an estimate of future inflation over the life of the bond. There are several ways to deal with these bonds; terms for inflation could be added to equation (1) or the regression could be on the spreads. There are several choices for adding the influence of inflation; month-to-month values, or the standard year-over-year values; or a longer-term average or integral, as used for central-bank rate. Equation (9), which regresses on the actual nominal TIPS yields using the yield of the corresponding conventional security as an independent variable, gives good results with a reasonable number of parameters. Both the month-to-month and year-to-year inflation values are used.

Equation (9):        Yield_M,t = k_5(M) + k_6(M) * Yield_conventional + k_7(M) *  Risk

                                       +  k_8(M) * I_mm + k_9(M) * I_yy

The regression constant k₅ is different from the regression constant in equation (1). In equation (9) it consists mostly of the "constant" part of expected future inflation as well as any other influences which are constant over this period. Thus there are three coefficients, k₅, k₈ and k₉ which are influenced by current and past values of inflation, and the risk term may also be in part a proxy for inflation (see below). Results for this equation are given in Table 5.8.1 and Figure 5.8.2. Since the variation of inflation over this period was rather limited, a different equation may be required in the future. The spreads between TIPS and conventional securities probably also include a premium for protection from future increases in inflation, and this itself may be variable; it seems likely to be correlated with the current level of inflation.

Figure 5.8.2. Observed and calculated yields for TIPS 2003-present (monthly) using equation (9). The green curve gives the yield for the equivalent conventional security, which is the main independent variable.

A risk term is required for the best results using any equation. Using equation (9) the risk coefficients (k₇) are positive, but since the main independent variables in this equation are conventional Treasuries, the risk coefficients for which are usually negative (Table 5.1.1), the effect is largely to cancel out the effects of risk. Several interpretations of this dependence are possible.

One simple interpretation is that TIPS yields are more neutral with respect to perceived risk than those of conventional Treasuries on the one hand, which decrease in times of risk, or those of lower-rated corporate securities, which increase in times of risk (Table 5.4.1). However, this does not seem to account for the much greater effect on TIPS the shorter the maturity (Figure 5.8.1); this seems more consistent with inflation expectation. Figure 5.8.2 compares the risk parameter with inflation as possible influences on the spreads.  Risk is not really a good fit to the shape of the spreads. TIPS spreads reached extreme negative values as oil price and inflation fell through the second half of 2008, but there was no reaction of TIPS when Baa yields and the risk parameter again went high in March 2009.  The sharp, single-valued spike in the inflation curve agrees better with the shape of the spread curves, but the magnitude of the spike in comparison to monthly fluctuation at other times is not sufficient to account quantitatively for the main dip in spreads in late 2008 if the response is assumed to be linear (an arbitrary convention in this paper).  The risk parameter appears to supply this extra magnitude.

Figure 5.8.2. 5-year TIPS spread compared to monthly inflation and the negative of the Risk parameter (Baa-10+year Treasury yields). Data: [1], [48]

Thus it seems likely that TIPS were reacting mostly to inflation rather than risk perception, but because there were major movements of federal funds rate, inflation and risk during a short time in 2008-2009 the situation is ambiguous - clarification of this may await a time when movements of risk and inflation are not so closely correlated. For several reasons the spreads themselves cannot be taken as straightforward estimates of future inflation and nothing else, though this is certainly the dominant contribution.

5.9. Evaluation of the Regression Results

It should be clearly understood that the regression analysis is not simply an exercise in curve-fitting, which would involve expressing yields as a function of time. The calculated market yield for each maturity at any month or year is independent of time itself and of past, present and future yields for that maturity and of any future values whatsoever; the yield is derived strictly from the central-bank rate for that and previous months or years (not future years) and the current risk parameter, with an overall linear time dependence of one coefficient for each series which extends over the entire time range. The regression equations give a good approximation to the observed yields, and most importantly, there are no obvious large and consistent correlations of the differences (red areas in the figures showing observed and calculated yields) with some of the macroeconomic variables previously thought to be important in yields, though correlations with other economic variables should be sought.

The risk factor and time variation of k₃ largely eliminate the type 1 and 2 deviations, leaving only the deviations of type 3.

Although up to 5 coefficients for each series are used for best fits of the data which extend over the years 1919-present, quite good results for low-risk securities can be obtained with 3 coefficients for periods up to nearly 80 years (1933-present - see Appendix A), and the coefficients vary in systematic ways (Figures 3.2.1-4). For the most recent and complete data extending over nearly 60 years, Group 1 monthly Treasury series 1953-present, time variation and the risk term are not very important in terms of overall statistics and good fits are obtained without them (Table 3.1.2).

Results of multivariate analysis and modeling must always be viewed with caution because adding or removing variables can change things drastically. However, if any other variables are to replace central-bank rate in these calculations, such variables or combinations thereof would have to be very closely correlated with the central-bank rates over the entire period examined, that is the life of the Federal Reserve. Some factors which have been suggested or assumed to affect market yields will be discussed in Sections 7 and 8.

Most of the remaining deviations (type 3) are concentrated in periods in which the Federal Reserve was most active and the central-bank rates varied most widely - that is in the early period 1915-1933 and especially in the later period 1956-present. When central-bank rate was varied only very slowly or not at all between about 1934 and 1955 market yields also varied only slowly and within a limited range. Variation of economic parameters - inflation, unemployment, GDP growth, deficits, etc. - during the 1934-1955 period was at least as great as during earlier or later periods (see Figures 8.1.2 and 8.2.1). The relatively short time scale of the type 3 deviations is more consistent with that of variation in central-bank rate than that of some economic parameters, especially government debt (Section 7). This suggests that the cause of the type 3 deviations would best be sought in limitations of the equation with respect to the exact treatment of bank rate, attempts by issuers and investors to anticipate central-bank rates, or differences in strategy or other aspects of central-bank operations, rather than in factors not directly involved with central-bank rate.

Time limits. As far as U.S. securities are concerned, this study is necessarily limited to the years 1915 and after, since there was no official central bank between 1833 and 1915, though perhaps it would be useful to study the rates of state banks during that time, if possible. But apart from this, finance in general and the objectives of central banks (for example in Britain) in the 19th century were distinct in several ways from those after World War I (Appendix B: Homer and Sylla, 2005). This study is thus not intended to apply to interest rates before 1915.

Predicting. Equation (1) is not intended for near-term prediction. The predictive power can be improved somewhat by adding information on current and past values of the dependent variable - that is the yield values of the series itself. Adding linear terms for current and past values in a similar way to equations (2) and (3) has so far succeeded in reducing the standard error for a 6-month advance prediction of the 10-year Treasury series 1953-present by only about 0.1% from that given in Table 3.1.2. However, not all possible combinations have been tried. The predictive value of the equation lies more in giving approximate values (standard error 0.6%  average for recent Treasuries, but larger for more risky securities) for any time in the future if central-bank rates can be specified, rather than giving exact values for near dates. Such predictions could be quite profitable for those who have reasonably exclusive knowledge of the future intentions of the central bank. Over some 2-year periods between 1975 and 1985 the yield on 10-year Treasuries increased and decreased by as much as 6%. Such profiteering could be forestalled either by holding rates constant or making the exact course of action public well in advance.

The discussion in Sections 6-11, evaluating other possible influences on yield rates, will assume that the relationship between central-bank rate and yield in equation (1), demonstrated and illustrated by the results in Section 5, is fundamental and causative, but will rely on no theory.

6. Yield Curves

6.1. Yield Curves, U.S. Treasuries, 1953-present (monthly)

Figure 6.1.1 shows the calculated and observed yield curves for each month. Start the animation by clicking on the Play button.

Figure 6.1.2 shows the observed and calculated yield-curve slope parameters (20-year minus 3-month yield). For the period when 20-year data are missing (1987-1993) the observed values were obtained from the 30-year series.

Figure 6.1.2. Observed and calculated yield-curve slope parameter (20-year yield minus 3-month yield) for U.S. Treasury Securities, 1953-present (monthly).

The calculated values for Figure 6.1.2 were obtained using no information from the future, only current and past values of central-bank rate and the risk parameter. The risk parameter is not essential; very similar results are obtained without this term. Variations in slope of the yield curve are primarily determined by the balance between the second and third terms of equation (1); without the third term the shape of the yield curve would be essentially constant except for any variations due to changing risk as a function of maturity.

6.2. Yield Curves, Basic Yields of Corporate Bonds, 1900-1970 (yearly)

Figure 6.2.2. Observed and calculated yield-curve slope parameter (20-year yield minus 1-year yield) for the basic yields for corporate bonds, 1919-1970, yearly.

6.3. Yield Curve Slope Parameters - U.S. Treasuries 1919-present (monthly)

Figure 6.3.1. Observed and calculated yield-curve slope parameter (10+-year yield minus 3-6-month yield) for U.S. Treasury Securities, 1919-present (monthly). Data as in Section 5.2.

6.4. Equilibrium Yield Curves

The equilibrium yield curves, that is the curves obtained when central-bank rate is held constant for 4 years, for the basic-yield series 1919-1942 and the Treasury series 1953-present are shown in Figure 6.4.1. In order to show possible changes over time, this group of basic-yield series includes all the maturities reported up to 1942, while the series in Section 5.3 above included only the smaller number of  maturities which carried through to 1970.

Part of the differences between these two groups may be due to differences between corporate and Treasury securities, and the curves for the highest central-bank rates are less reliable because these rates were not actually held for any great length of time. The basic-yield series for corporate bonds were in part derived by drawing a yield curve for each year at what was perceived to be the high-quality, low-yield limit (Durand, 1942), and the "constant maturity" treasuries yields are also based on a fitted curve, so it is conceivable that operator prejudice had some influence on the shape of these curves.

With these caveats in mind, the differences are consistent with the type 2 deviations (Section 3.2) observed in the series which span the complete time range (Sections 5.2, 5.4 and 5.5). The general behavior shown for the period 1915-1942 is consistent with that shown throughout the 19th and early 20th century (observed data starting in 1900 are shown in Figure 6.2.1), that is relatively rapid and sometimes extreme variation in central-bank and short-term market rates and much less variation in long-term rates (Appendix B; Homer and Sylla, 2005). For the 1919-1942 period the yield curve for the average discount rate (4%) is flat - thus inconsistent with the positive-sloping curve for after 1953. When the entire range is considered together, the yields calculated for longer terms tend to be too high for the period before about 1930 and to a lesser extent too low for the later period (the later period after 1953 outweighs the period of the 1920's in the regression). During the transition period of about 1933 to 1960 or so, the central-bank rate was low and both sets of curves are at least in qualitative agreement in having a positive slope. Thus it is hard to determine where the transition occurred and how sudden it was.

Figure 6.4.1a. Equilibrium yield curves for basic yields for corporate bonds, 1919-1942. Each graph shows the equilibrium yield curves, from bottom to top, when the central-bank rate is held constant at 0, 1, 2, 3 4, 5 and 6 percent, respectively. The red curve, 4%, is the approximate long-term average for both discount and federal funds rates. The value of the risk factor used is the average for each period.

Figure 6.4.1b. Equilibrium yield curves for treasuries, 1953-present. See Figure 6.4.1a for explanation

6.5. Correlations - Yield Curve vs. Central-bank rate and Yield Curve vs. 10+-year Treasuries.

Figure 6.5.1a shows the correlation of change in the yield-curve slope parameter (10+-year minus 3-month Treasury yields) with change in central-bank rate for the 1919-present Treasury data (Section 5.2) and Figure 6.5.1b shows change in the yield-curve slope parameter versus 10-year yield over one year, in both cases with linear regression line. The resulting parameters are:

for y = a + bx,
and y is change in yield-curve slope parameter (10+-year minus 3-month Treasury yields);
when x is change in central-bank rate, a = 0.00, b = -0.52 and R = -0.80 (Figure 6.5.1a);
when x is change in 10+-year Treasury yield, a = 0.03, b = -0.14 and R = -0.14 (Figure 6.5.1b).

The data viewed this way again show that change in the slope of the yield curve is a result of change in central-bank rate and short-term yields, not long-term yields; the yield curve is controlled by central-bank rate.

Figure 6.5.1a. Change in Treasury yield-curve slope parameter (10+-year yield minus 3-month yield) over 12 months as a function of change in central-bank rate.

Figure 6.5.1b. Change in Treasury yield-curve slope parameter (10+-year yield minus 3-month yield) over 12 months as a function of change in 10+-year Treasury yield.

6.6. Correlations - Yield Curve vs. Inflation

The slope of the Treasury yield curve as related to inflation has varied over time.  The data were divided into two periods according to the general activity level of the Fed in varying central-bank rate  In the earlier period 1920 to 1966 the correlation is insignificant (R = -0.06) despite greater variation in inflation, and in the later period 1967-present it is moderate (R = -0.60).  These results are discussed further in Section 8.2.

Figure 6.6.1a.  Correlation of the yield-curve slope parameter with inflation for the period 1920 through 1966. R = -0.06.

Figure 6.6.1b.  Correlation of the yield-curve slope parameter with inflation for the period 1967 to present. R = -0.60.

7. Supply and Demand in Credit Markets

Supply and demand in this section is to be understood in terms of the overall amount of debt to be financed, and the overall amount of money going into a credit market. Shifting of demand between securities of different ratings, which is involved in the risk term, and shifting between different maturities, which is discussed in Section 12, are different matters.

7.1. Foreign Investment and Government Debt

The U.S. Government debt held by foreign entities increased from about $1.2 trillion at the end of 2002 to over $4.2 trillion by the end of 2010. Many people have expressed the belief or assumption that this and foreign purchases of non-Treasury securities have driven down long-term yields. Over the same time interval the gross U.S. National Debt increased from $6.4 trillion to over $13.5 trillion, and others have suggested that this has raised yields - and some have suggested that yields have not varied more widely because these influences cancelled out (Introduction). As the deficits projected forward from 2009 have increased to over a trillion per year, many have positively predicted that this will raise interest rates, and/or that foreign purchase of U.S. debt will be necessary to keep rates down. Deficits and foreign investment will be considered together since one popular view is that their effects have been offsetting during the last 10-30 years.

Figures 7.1.1 and 7.1.2 show gross debt, gross deficit and total foreign investment as percentages of GDP.

Figure 7.1.1. Gross Debt and Gross Deficit as percentages of GDP, with the 10+-year Treasury series for reference. Data: [21].

Figure 7.1.2. Federal debt held by foreigners and yearly inflow of foreign investment as percentages of GDP, with the long-term Treasury series for reference. Data: [17].

All the variables shown in Figures 7.1.1 and 7.1.2 were tested as independent variables. Several other measures of foreign investment were tested as well, with similar results. Since deficits are only released yearly and the foreign investment data are quarterly, the data from Table 3.2.2 were transformed to yearly averages. Also, since major variation of the parameters in question occurred during World War II or later and data are incomplete for the World War I period, results are given for starting the series in 1933, which allows neglect of the time variation of coefficients. The yearly averaging smoothes out some of the variation involved in the type 3 deviations and the starting statistics are better than those in Table 3.2.2. The complete time range was also examined with time variation and the conclusions are the same. Here are the results before adding the supply/demand variables:

Except for the 3-month Treasury series all these are long-term securities. Using debt, deficit or foreign investment as independent variables without including central-bank rate and risk may give results which are statistically significant; for example debt plus total foreign investment gives R² of 0.378 (standard error 2.351). However, in all cases the coefficients for debt or deficit are negative, because the highest values of these variables came when interest rates were generally low, and the coefficients for foreign investment are positive - the opposite of the effects which are generally assumed. Results when adding the supply/demand variables as percentages of GDP as in Figures 7.1.1 and 7.1.2 to the regression when central-bank rate and risk are present are given in Tables 7.1.2 and 7.1.3.

Table 7.1.3. Results for adding supply and demand variables, 1953-present. Independent Variables Weighted Average R2 Weighted Average Standard Error  No extra independent variables - k1, k2 and k3 only 0.934 0.675  Add Risk only - k1, k2, k3 and k4 0.953 0.593  Add debt plus foreign investment yearly increment 0.970 0.489  Add Risk and time variation only 0.970 0.478

For the full time interval 1933-present, no combination of the supply and demand variables improves R² more than adding simple time variation. Table 7.1.3 gives the results for the period when federal funds rate was the main instrument of Fed policy, to address the question of whether federal funds as opposed to discount rate could make any difference. It does not. Regressing on debt or deficit alone always gives negative coefficients, because the period of greatest deficits and debt coincides at least partly with the period of lowest yields.

We should consider the actual contribution of the debt and foreign investment terms as well as the overall statistics. Figure 7.1.3 shows the results for adding both debt and foreign investment inflow for the 10+year Treasury series. The debt term reaches as much as 0.7 percentage points above its baseline at the peak of debt in 1946, but the calculated value is 0.4 above observed at that time, and over the interval 1960 to present, when the debt has been presumed to be influential on yields, the variation in this term is less than 0.4. Although foreign investment gives the greatest statistical improvement, the contribution from the foreign investment term is positive - that is if using this term is justified, increasing investment would seem to have raised yields, just the opposite of what has been assumed. Coefficients for any of the foreign investment parameters are always positive for all series. If time variation of the k₃ coefficient is allowed, the statistical benefits of adding foreign investment vanish and the coefficients are tiny.

Figure 7.1.3. Observed and calculated yields for the 10+year Treasury series (yearly) with terms for national debt and foreign investment inflow divided by GDP, showing the contributions from individual terms in the equation. A single set of coefficients was used for the entire time interval (no time variation). k1 is 1.460%.

The mathematical methods of this paper indicate that the influence of debt or deficit and foreign investment could only be slight. While these methods are not the only way the question can be addressed, these methods are successful in accounting for yields over the entire period of existence of the Federal Reserve.

In the case of debt and deficit, no mathematical analysis is necessary to see that they did not affect yields appreciably in the period of about 1930 to 1965 as shown in Figure 7.1.4. The deficits and total debt during World War II reached values far higher with respect to GDP than at any other time during the 20th century, and there was no meaningful change in yields - the figure shows the basic yields for corporate bonds, but other series are very similar in lack of correlation with deficits. Yields did not begin to increase until after the war when the Fed began raising the discount rate - by this time deficits had dropped to zero and the debt as a fraction of GDP was steadily decreasing. No obvious effect of the huge wartime deficits and debt on the yield curve is apparent either (Figure 6.2.1). The slope of the yield curve was decreasing during the war, but this process started before the war and went on long after it.

Figure 7.1.4. Basic yields for corporate bonds (observed) over the period 1916 to 1970,  compared with U.S. gross deficit  as a percentage of GDP. Data: [2/131], [3/12,14], [21]

The period of the War and for some time afterwards was different in many ways from other times during the life of the Federal Reserve. The War was financed mostly with savings bonds which were not negotiable, and in the case of the E series which made up the bulk of the bonds, available only to private persons. It was the patriotic duty of citizens to buy bonds, and lack of production of consumer goods and rationing forced high savings rates.

The declared policy of the Fed and Treasury through the War was to maintain yields of all maturities at low rates. According to Homer and Sylla (2005, p. 354) the Fed bought most of the 3-month Treasury Bills which were issued, and their yields were maintained at 0.375% (Figure 5.2.1). But it was not necessary for the Fed to buy long-term Treasuries to keep their rates down; for example at the end of 1945, 74% of Fed Treasury holdings had maturities of 90 days or less, and 99% had maturities of one year or less (Data: [3/9.5]). After the policy of maintaining low rates ended in late 1947, more long-term securities were bought (Figure 7.1.5), but the total did not change greatly and long-term market rates were little affected (Figures 5.2.1 and 5.3.1). Open-market operations 1942-1947, apparently directed at the 3-month T-Bill rate, seem to have been little different from the time after 1953 when they were directed at federal funds rate. Non-federal securities did not behave differently from Treasuries during this time, and there was no discontinuity in long-term rates when the formal policy of keeping rates low was terminated.

Figure 7.1.5. Holdings of all Federal Reserve Banks, by maturity, between 1940 and 1952. Data: [3/9.5]

In the absence of data from other countries it could conceivably be argued that all the special policies and factors somehow balanced or negated the effect of the deficits (and inflation - see  Section 8) which according to the conventional wisdom are supposed to cause rates to increase, though this would seem to require a remarkable degree of coincidence. Note also that if the government (the Treasury in the U.S.) has the power to enforce low rates by fiat or persuasion in wartime, it should be considered whether that power also operates in peacetime. But evidence from other countries is available and reduces the possibility of such fortuitous balance to insignificance. The lack of variation of yields in the United States through the World War II and post-war period shown in Figure 7.1.4 is duplicated in all other countries examined so far for which the data are available, while in many cases debts were even larger than in the U.S. - see Section 9.

7.2. Stock-Market Loans

The volume of loans secured by stocks in customer's accounts at brokerages and banks is dependent on price performance of stocks, as shown in Figure 7.2.1 - the major peaks coincide with peaks in stock prices. Although these are usually short-term loans, the high interest rates on margin loans during certain periods, especially the late 1920's, has sometimes been attributed to increased demand - or more accurately this has been assumed to be the reason for the high rates.

Figure 7.2.1. Loans to brokerages and banks for stock purchases as a percentage of GDP, with call money rate (Section 5.6) for reference. From 1922 to 1963 the series is for total brokerage borrowing, which is mostly for purchases on margin. From 1966 to present the series is for total borrowing by banks and brokerages for margin purchases. Compare with Figure 5.6.1. Data: [2/142], [3/12.23, 12.27], [4/23], [5/22], [6/19], [7/14], [8/1.36].

The yields on call money since 1915, that is money loaned by banks to brokerages for margin accounts, do not follow the trend of loan volume in Figure 7.2.1, but are fully consistent with equation (1), as shown in Section 5.6. The yields on these loans follow the movements of central-bank rate closely, but at distinctly higher rates than most other short-term instruments, though lower than bank prime loan rate. The loan rate was higher in 1920 than in 1929, despite the enormously higher volume in 1929. It can clearly be seen in Figure 7.2.1 that the volume of margin credit was going down from about 1968 through 1976, while rates were generally going up. Complete data on call money rates for after 1976 have not been obtained - in later years they have tended to be pegged to federal funds rate, currently at 1.75% above. Of course these are the rates for loans by banks and others to brokers, and do not rule out higher rates charged to customers by brokers in times of high demand.

7.3. Mortgages

Mortgages are not traded individually on markets but the general assumption seems to be that their rates are influenced by supply and demand. Demand for residential mortgages is dependent on a number of factors, such as constriction of purchases during wars and release afterwards, and mortgage rates themselves. Figure 7.3.1 shows total mortgage debt in the U.S. per capita. Housing and mortgage demand boomed during the 1920's, dropped during the Depression and World War II and boomed after the War. High rates depressed demand from the late 1960's through the early 1980's and a post-1947 low was reached in 1981 when interest rates were near all-time highs. Home ownership rate actually dropped during the 1980's whereas there was rapid growth of home ownership during the inflation of the late 1940's and early 1950's when rates were not raised (Census, 2008).

Mortgage rates are not correlated with demand, but are well accounted for by equation (1) as shown in Sections 5.4 and 5.7.

Figure 7.3.1. Total mortgage debt in the U.S. per capita, in thousands of 2009 dollars, with HUD mortgage rate (Section 5.7) for reference. Compare with Figure 5.7.1. Data: [28/N262], [31].

7.4. Summary of Supply and Demand Evidence

The analysis in this paper of some supposedly important supply and demand variables fails to find any evidence that these variables directly affect interest yields. This follows on three grounds; the success of equation (1) in explaining yield rates using central-bank rates and the risk factor alone; the failure of these supply and demand variables to have meaningful influence in the regression; and inspection of the historical record of these variables, yield rates and central-bank rates. Although data for every conceivable type of security and debt have not been examined, there is a sufficient number of different types in the various groups in Section 5 to justify a provisional conclusion that the lack of influence of supply and demand is general in credit markets.

Market yields vary over the short term - or have so varied during periods when the Fed was active in changing central-bank rate. Average prices for Treasuries and other securities are reported on a daily basis and carefully watched. These variations are glibly interpreted by media analysts in terms of the latest financial news, but this news and the interpretation thereof seldom have to do with supply and demand; they are usually about general economic performance and how the Fed will react to this performance and change central-bank rate. Part of the market reaction to economic and financial news, and a great deal at certain times, must be classified under the risk factor. But predictions or expectations of Fed policy certainly play a role; the variability of these expectations may account to some degree for the type 3 deviations as defined in Section 3.2, and why the prevalence of such deviations is correlated with the general activity level of the Fed (as in the figures of Section 5).

Transactions at the discount window and open-market operations which maintain the chosen federal-funds rate affect the monetary aggregates, so changes in supply and demand presumably have consequences to the money supply as measured by the aggregates. Currency exchange rates are also presumed to be affected by supply and demand. These are large subjects, much more complicated than the simple empirical relationships among interest rates discussed in this paper.

8. The Effects of Inflation

8.1. The Effect of Inflation on the Regression

It has been shown in Section 3 that for Treasuries 1953-present inflation is not a meaningful contributor to the variance when the factors in equation (1) are present, and the results are similar for all series examined (Table 4.2.1): inflation rate alone is poorly correlated with yields at best, and not at all at some times, while central-bank rate alone is well correlated; adding one or more terms for past central-bank rates improves the fit greatly; and adding inflation rate to the regression when any function of central-bank rate is present improves it only slightly at best. As in the case of supply-and-demand variables (Section 7), it makes no difference whether the period in which the main Fed policy instrument was federal funds rate is considered separately or together with the period in which the main instrument was discount rate.

The overall weighted average R² for the regression of the Group 8 selected monthly 1919-present data (Table 3.2.2) on inflation rate alone is 0.078. Weighted average R² for regression of this group on current central-bank rate alone without time dependence is 0.76 (0.79 with time dependence). For the basic yields for corporate bonds (Group 3) 1919-1970, R² for regression on inflation alone is 0.002.

Figure 8.1.1 shows the components in the regression of 10+year Treasuries and Baa corporate bonds with inflation as an independent variable; the contribution from inflation (red line) is obviously negligible.

Figure 8.1.1a. Components from the regression of yields of 10+year Treasury bonds including inflation as an independent variable, otherwise as in Table 5.2.1.

Figure 8.1.1b. Components from the regression of yields of Baa-rated corporate bonds including inflation as an independent variable, otherwise as in Table 5.4.1.

Another way to evaluate the influence of inflation is to measure its correlation with the residuals from the calculations using only central-bank rate and risk (red curves in the figures of Section 5). For the 10-year Treasury series 1953-present (Section 5.1), R² is 0.023. For the 10+year Treasury series 1919-present (Section 5.2), R² is 0.00005. The latter period is more meaningful since it includes periods of inflation and deflation in which the Fed did not deliberately cause central-bank rate to track inflation.

The lack of dependence of market yields on inflation or deflation is also shown graphically and conclusively by the data from 1915 to about 1970 (Figure 8.1.2). No mathematical analysis is necessary for this, though again for the group shown here weighted average R² for regression on inflation alone is 0.002. Data on other corporate, state and local and Treasury bonds tell the same story (Section 5). Time shifts obviously cannot reconcile inflation and yields in Figure 8.1.2, nor can they do so in the regression in any period.

Figure 8.1.2. Basic yields for corporate bonds (observed) over the period 1915 to 1970, with inflation rate. Data: [2/131], [3/12,14].

Did the attitude of the market drastically change some time after 1955, so that inflation became important in yields? Regression of yields on inflation during this later time does not support this idea (Section 3). Certainly the attitude of the Fed changed - it began to actively track inflation rate with central-bank rate, at least until about 1989. In fact the attitude and strategy of the Fed have changed a number of times; the Fed was fairly active before 1929, changed rates only very slowly 1934-1955, actively raised and lowered rates in response to inflation from about 1955 to 1989, and since then its activity has been mainly directed at attempting to alleviate recession. Equation (1) accounts well for all types of high-quality security yields through all these changes and through booms, depressions and recessions, unemployment ranging from 1.2% to over 24%, deficits and surpluses, war and peace, changes in exchange rates and monetary standards as well as inflation and deflation. The accuracy of the equation does correlate negatively, however, with the general activity level of the Fed.

8.2 Inflation Expectation

The conventional wisdom is clearly that long-term bond yields are determined at least partly on the basis of expectation of inflation (Poole, 2003). Some people claim or assume that yields themselves can give an estimate of this expectation (for example Dewald, 1998). These ideas are in severe conflict with several types of data.

Slope of yield curves. The normal state of yield curves is considered to be positively sloping, that is yields increasing with maturity, and this is often explained as at least partly a result of the demand of long-term lenders for increased yields because of the risk of inflation. It would seem to be a straightforward prediction from this that the slope of the yield curve should become more positive when inflation is increasing, if investors base their expectations on recent past experience. The opposite correlation is observed when the Fed was active after about 1967 in changing central-bank rate, and there is no correlation before that - see Section 6.6. The yield curve slope parameter rapidly increased from 1929 through 1935 (Figure 6.2.2), a time of severe deflation (Figure 8.1.2). At that time it reached a maximum which was not exceeded until the drastic movements of central-bank rates during the 1980's, and the yield curve slope parameter decreased slowly during the 1930's and 1940's as inflation rate increased erratically. During the period from about 1965 through 1989 when the policy of the Fed was directed primarily toward fighting inflation and federal funds rate was correlated strongly with inflation rate, the yield curve flattened when the Fed was raising rates and became positive when the Fed was cutting (Figures 6.1.1 and 6.1.2).

Equation (1) reproduces all these changes in yields and in the yield curve (Figure 6.3.1), with no dependence whatsoever on either inflation or expectation thereof.

TIPS yields. It was shown in Section 5.8 that yields of Treasury Inflation-Protected Securities are dependent on both central-bank rate (as it influences conventional yields) and inflation rate. If investors demanded and were able to enforce positive real rates they could keep the yields of TIPS constant, but instead the yields are closely tied to the nominal yields on conventional Treasuries. Because the yields of TIPS are dependent on future inflation, buyers must estimate future inflation in order to compare their yields to those of conventional Treasuries, but the dependence of TIPS yields on inflation can be accounted for quite well as a linear function of current and past inflation rates with some contribution from risk - there is no evidence for estimation of future inflation by any other means. Thus the only type of security for which inflation expectation is demonstrable is just the type for which the conventional wisdom predicts insensitivity to inflation.

Poll Results.  Figure 8.2.1 shows actual consumer expectation of inflation for one year in the future from the University of Michigan Poll. Consumers' expectations are closely tied to current inflation, though extreme movements are discounted or damped. The following equation for the polled expectation of year-over-year inflation in percent gives R² = 0.86:

           Expected Inflation  = 1.43 + 0.58 * Iyy

where Iyy is current year-over-year inflation. Federal funds rate and 10-year Treasury rate have negligible influence if added to the equation.

There is no evidence that consumers can see into the future to predict inflation. The huge discrepancy between the polled expectation and the long-term rate is completely inconsistent with the idea that long-term rates, such as the 10-year rate shown, in some way represent expectations of inflation. The fact that TIPS rates can also be explained in terms of current and recent inflation (Section 5.8) indicates that the expectations of bond investors and consumers are not completely different.

Figure 8.2.1. Inflation, 10-year Treasury bond yield and consumer expectation of inflation from the University of Michigan Poll. Data: [1] [38], [48]

If long-term rates are considered to be a prediction of future inflation by the bond market it has been a terrible prediction, far worse than that of consumers as represented by the poll; after 1981 the 10-year rate was always far above inflation for any number of years in the future. Of course before that when both rates were usually rising the discrepancy was in the opposite direction. The conventional wisdom would have it that investors were grossly misestimating inflation in this way while simultaneously correctly predicting future recessions through the slope of the yield curve. A far more satisfactory explanation for the generally high long-term rates after 1981 is that the Fed determined to keep federal funds rate much higher with respect to inflation than before and that long-term rates followed according to equation (1) as shown in Section 5.1; and a more plausible explanation for the correlation between the slope of the yield curve and recessions is given in Section 10.

Many theorists claim that the Fed can affect consumers' or investors' expectations of inflation through declarations or demonstrations of policy. There is no evidence of this in either the University of Michigan poll or TIPS yields. Expectations of inflation are certainly determined primarily by current and past values of inflation itself.

8.3. The Importance of Real Rates.

Many economists believe that investors have some power to maintain a reasonably favorable real yield, that is nominal yield minus inflation (Poole, 2003). This real yield is shown for 10+-year Treasuries in Figure 8.3.1 (the more complex form of real yield, the Fischer equation, gives qualitatively similar numbers but blows up when inflation is near -1.0%, as happened several times in this interval).

Figure 8.3.1 .Real yield for 10+-year Treasury securities, with inflation rate. Data: [1], [28/E135], [48]

Real yields for long-term securities throughout the period 1919-present have been essentially a mirror image of inflation - clearly during this period there was either no desire or no ability of investors to maintain positive real yields. Since about 1965 when the Fed raised all rates higher than ever before, real rates have tended to be more consistently positive, but the mirror-image relationship has still held most of the time. Since this paper has demonstrated a simple relationship of yields to central-bank rate during this entire time, it is obvious that what changed around 1965 was the policy of the Fed, not the attitude of investors toward maintaining a real rate of return. Of course this is not to say that real rates do not affect economic behavior, just that the need or desire of investors for a given return does not directly affect real rates. Real rates are determined primarily by inflation and only secondarily by central-bank policy. This is apart from influence of the risk factor as accounted for in equation (1) and beyond at most the type 3 deviations which are on average about 4.5% of the variance (of course the latter variations may be very important to certain people such as speculators).

There is no consistent correlation of real rates with either good or bad overall economic performance.  Real rates were strongly negative during World War I, World War II and the Korean War and cold-war rearmament, when GDP growth was high and unemployment was low. High inflation at these times was driven by shortage of consumer goods and of civilian labor.  Real rates were also low in the late 1940's and during the inflation spikes of 1975 and 1980, when there were recessions. Real rates were very high during the down-cycle of the Depression 1930-1933, but have also been positive during many times of good economic growth. As discussed in Section 10.2, the influence of interest rates on growth seems to be a matter of the difference between long- and short-term rates, not absolute long-term rates, either nominal or real.

As discussed in Section 8.2, nominal TIPS yields (Section 5.8) are not constant or even always positive, as would be expected if bond investors were able to enforce positive real rates, but vary with the real yields on conventional Treasury securities.

8.4. Summary - the Effects of Inflation

To summarize, nominal yields for conventional securities are not directly affected by either inflation itself or by expectation of inflation, and there is no evidence that investors are able to maintain positive real yields, or even try to do so. Real rates for short-term securities could be kept reasonably constant by the Fed, and in this sense its expectation of inflation is indirectly relevant if it follows such a policy. But even since 1965 real yields have not been kept constant or even always positive. Investors currently keep an eye on inflation, but this is most likely for the purpose of guessing future policy-rate changes by the Fed - when the Fed was not changing policy rate in the 1940's and early 1950's market yields did not react to inflation, nor did they react to deflation in the 1930's.

The conventional wisdom would have it that both inflation and government deficits should increase yields, but the combination of massive deficits and significant inflation during and after both World War I and World War II apparently had no such effect in the U.S. (Figure 8.4.1) or any country (Section 9). The situation was similar in the U.S. during the Civil War (Appendix B). Considerable deflation in the 1920's and 1930's caused no decrease in yields. It is difficult to imagine, in the real world, more definitive tests of the set of beliefs in the conventional wisdom or a more complete failure of its predictions.

Figure 8.4.1. Discount rate, basic yield rate on 30-year corporate bonds, deficit as a percentage of GDP and inflation rate. [1], [3/12.14], [21], [28/E135]

Data on yields for some short-and long-term bond series (Homer and Sylla, 2005) as well as inflation rates are available for the 19th as well as the 20th century and there was no correlation between inflation and yields before 1915 in the U.S. or any other country (Appendix B).

9. Other Countries

Data for 23 other countries - all those for which a reasonable amount of data could be obtained - have also been examined by regression in a preliminary way, in some cases going back to 1800. Regression results with figures are given in Appendix C. So far, the following conclusions can be made.

1)  Visually and qualitatively the pattern of central-bank rates versus market yields during the 20th century is very similar to that in the U.S. (Figure 9.1). Especially, in the period about 1933 to 1965 central-bank rates in every country for which data are available were held low, rising only gradually until the late 1960's. Market yields tracked central-bank rates during this period, and several significant periods of deflation and inflation had no major or consistent effect on market yields. National deficits are not shown in the figures, but almost all European and North American countries as well as Japan incurred very large debts during this period, and that also had no discernible effect on market yields.

2)  Regression results on monthly and yearly data (Appendix C) are generally reasonably good but somewhat erratic. For the 23 countries for which there are monthly series (167 total series), the weighted average R² is 0.92 and the weighted average standard error is 0.71%. Generally these are the most recent and best data. For yearly data generally extending over longer intervals (38 series),  the weighted average R² is 0.86 and the weighted average standard error is 0.86%. Many of these yearly series are combinations using spliced data from different sources. Regression on inflation alone did not give significant values of R² if the period of about 1960-2000 was omitted. In a few countries only, adding inflation as an independent variable to the regression after 1960 improved agreement, but for several reasons the meaning of this is not clear - see Appendix C.

There are several reasons why regression is not expected to be as successful as on U.S. securities; sample numbers for yield averaging may be much smaller; the official policy rate, usually discount rate, may not have been the lowest short-term rate available; risk data have not been obtained yet in most cases while perceived risk may have been much higher at times than in the U.S. and may have had quite different effects on yields (see point 3); the smaller economies are more subject to external influence than the U.S.; there were major changes in governments in many countries; central bank policies often underwent more and/or more drastic changes than in the U.S.; and consolidation into European economic unions obviously required major adjustments in some countries. In some cases to get any continuity through the latter part of the 20th century yearly series had to be patched together from the compilation of Homer and Sylla (2005) and from the archives of central banks or national statistical offices.

3) Credit markets in other countries have been disrupted at times by endogenous banking crises and other events. Examples include Norway in 1940 and Norway, Sweden and Denmark in 1992. The response of yields can be quite different from that in the U.S., depending on actions by the central banks. In the Scandinavian crisis of late 1992 discount rates were not raised but overnight interbank rates were briefly raised very high, and  yields on short-term government securities also went very high. Data on non-government securities have not been obtained yet for these countries. The credit crisis of 2008 did not cause any increase in short-term government or interbank rates in Denmark, Norway or Sweden.

4) Since the credit crisis of 2008-2009 the European Common Bank and also the central banks of those European countries which do not belong to the monetary union have been keeping their policy rates low, interbank rates have been very low, and long-term government-bond rates in prosperous countries have been generally low, despite high deficits and total debt in some countries (e.g. France, Britain). In some countries (e.g. Greece, Ireland, Spain, Portugal) which have experienced very severe difficulties rates have been elevated. While there is some correlation of long-term rates with deficits and debt the very high rates in some countries are clearly associated with increased perception of default risk in the affected countries.

10. The Meaning and Importance of the Slope of Yield Curves

10.1. The Prophetic Power of the Yield Curve.

The conventional wisdom about the supposedly prophetic aspect of the yield curve apparently assumes that the slope is determined by long-term rates, which in turn are supposedly determined by investors, but it is quite clear that it is the movement of central-bank rate which directly determines the slope, as discussed in Section 6.5. Krugman's (2008) explanation of the relation between the yield curve and recession (see Introduction) assumes, following Hicks (1946), that long-term rates are at least partly determined by investors' expectation of future central-bank rate, which is consistent with  the data (see Section 12.1), but the data do not support the idea that such expectations are based to any great extent on forecasts of recession; the dominant influence by far must be current and past central-bank rates, which account for over 90% of the variance of all yields in the U.S.; and the long-term end of the curve is anyway the non-active end.

Some historical aspects of likely investor attitudes are also inconsistent with the conventional wisdom. The slope of the curve for basic corporate yields and for Treasuries went from negative in 1929, when the Fed had been raising the discount rate for some time, to very positive in 1932-1935 (Figures 6.2.2 and 6.3.1). In direct contradiction to Krugman's (2008) explanation, during most of this time investors hardly expected the economy to expand, nor was it a time when investors expected the Fed to raise rates. The slope decreased slowly through the 1960's, an exceptional boom time.

The regression results of this paper reproduce the main features of yield-curve variation over the entire period of the existence of the Federal Reserve (Figures 6.1.1, 6.2.2 and 6.3.1). Again, the regression equation uses only the current and past values of federal funds or discount rate and current value of a risk parameter - the equation for yield at any given time does not include any information from the future. Changes in the yield curve are mostly imposed by the Fed - these changes usually do not have directly to do with any expectations on the part of investors about the general state of the economy, prophetic or not. The Fed's expectations are relevant, but during the main period when the yield curve was supposed to have prophetic properties the policy of the Fed was dominated by its attempts to control inflation. Actually, as shown in Sections 8 and 9, during that period central banks in all countries simply tended to vary their policy rate in parallel with the current inflation rate, without demonstrating any ability to foresee future rates. There were several recessions in the period from the middle 1930's to the middle 1950's during which the Fed did not change central-bank rate much, and these recessions were not predicted by a flat or negative yield curve (Figures 6.2.2 and 6.3.1).

Figure 10.1.1. Slope of the yield curve (10-year minus 3-month Treasury rates) compared with the risk parameter (Baa minus 10+year Treasury rates) before and during the credit crisis of 2008-9. Data: [1], [20]

One might consider that the general inertia of long-term yields with respect to central-bank interest-rate changes, which is in some sense what determines the slope of the yield curve, is a kind of expectation, but this would appear to be the expectation that the central bank interest-rate changes will not be permanent, and not an expectation in one way or the other about the future state of the economy or about inflation. In addition to taking account of current and past central-bank rates, long-term investors may well try to anticipate future moves by the central bank by other means, but there is no reason to think that they are usually successful at it (see Section 12.1.2, last paragraph).

Why then has a flat or negative yield curve often preceded recession?  Friedman and Schwarz (1963) pointed out that up until 1960 every raise in rate or other constrictive action by the Fed had been followed by recession, and there have been even more examples since then. As shown in this paper, a flat or negative yield curve has occurred when the Fed has been raising central-bank rate for a significant period of time. Rates have been raised in this way under three conditions: 1) when the Fed was trying to combat inflation, mostly in the period about 1955-1989;  2) when the Fed was trying to restrain over-expansion as in 1929 and perhaps in 1999-2000;  and 3) when the Fed was restoring rates back to "normal" after a recession during which it had deeply lowered rates. The third condition really only applies to the periods after the recessions of 1991 and 2001, periods in which combating inflation was not a major objective of the Fed.

Thus recessions have frequently followed sharp increases in central-bank rate to combat inflation or over-expansion and it is these increases which caused the slope of the yield curve to decrease. The cause of the recessions could in any given instance be either a result of inflation itself; a collapse of the overexpansion; the constrictive action of the rise of interest rates; or some combination of these things.  For the purposes of this subsection, it is enough to know empirically that interest-rate raises have in many but not all cases been followed by recession.  At times when the Fed did not raise rates in response to inflation or overexpansion, such as during the 1930's through early 1950's, recessions were not preceded by flat or negative yield curves. The empirical evidence on the possible direct effect of interest rates on economic growth will be discussed in Section 10.2.

This explanation for the correlation of flat or negative yield curves with subsequent recession relies on simple causative relationships which do not require prophetic insight on the part of investors or anyone else. If investors were not in the habit of failing to recognize overexpansion and to predict its results, such overexpansion and subsequent recession might not occur at all. Bond investors might be justified in expecting recession after periods of inflation, but their response would depend on predicting the actions of the Fed in changing interest rates, which have greatly varied over time. There is no reason to think that investors, any more than consumers or the Fed (Section 8), can predict future inflation, nor is there any reason to think that their expectations of inflation directly affect yield rates. See the last paragraph of Section 12.1.2 for the question of whether the market can accurately predict future short-term rates

10.2. The Real Importance of the Slope of the Yield Curve

It is axiomatic in economics that investment is strongly affected by long-term interest rates and that investment is a major determinant of economic production or growth.  However, a correlation of investment or growth rates with long-term interest rates is not observed. Through the 19th and most of the 20th centuries long-term rates simply varied too little and too slowly to correlate with anything. Real long-term rates have varied much more - they are primarily a function of inflation - but as discussed in Section 8.3 the variation has no consistent relation to economic growth. During the period from the late 1960's to the early 1990's when short-term rates were often changed rapidly and long-term rates did rise to unprecedented levels there was no particular effect of the high long-term rates, or of the absolute level of short-term rates either. Figure 10.2.1 shows that high levels of both long- and short-term rates (the figure shows their negatives) during this period did not themselves severely inhibit growth. For example GDP growth reached a peak of 8.5% in the first quarter of 1984 when the 10+year rate was over 11%. What does show a good correlation with GDP growth is the slope of the yield curve, shown in Figure 10.2.1 as the difference between the 10+year Treasury rate and federal funds rate. The valleys of GDP growth correspond with the those of the slope of the yield curve with a delay of somewhat over a year, as for unemployment (Figure 10.2.3).  There is also a negative correlation of GDP growth with rate of change of short-term rate. As shown in Figure 6.5.1a, change in short-term rate is strongly anti-correlated with change in the slope of the yield curve.

Figure 10.2.1. Real GDP growth rate and interest rates; 10+year Treasury and federal funds rates, shown as negatives; and the difference between the two rates. Data: [1], [18]

A plausible cause of this correlation appears to be the expected profitability of loans. Banks and other re-lenders essentially borrow at short-term at rates which are close to the central-bank policy rate, and loan at higher long-term rates.  The greater the difference between these rates (strictly speaking between expected future short-term rates and offered long-term rate; see Section 12.1), the more incentive there is for banks to loan at a given level of risk.  Figure 10.2.2 shows that the rate of bank loans and leases was generally consistent with this hypothesis during the period of high rates. Corporate investment (data: [46]) followed a similar pattern.

Figure 10.2.2. Real GDP growth rate with the difference between 10+year Treasury and federal funds rates, and the real growth rate of loans and leases by Federal Reserve member banks. Data: [1],  [11], [18]

Figure 10.2.3 shows that there was a very close correspondence of federal funds rate with unemployment rate, which is an indicator of recessions, during the period of about 1955 to 1989, when the Fed was most active in attempting to control inflation with interest rates.  The peaks correspond best with a delay of about 1 1/4 year for unemployment. In 1989 the Fed began to decrease federal funds rate sharply in response to signs of difficulty because of the savings and loan crisis, and this marked the end of the period when interest-rate changes were primarily directed at controlling inflation.

Figure 10.2.3. Federal funds rate and unemployment rate in the period 1955-1995.  The horizontal (time) scale for unemployment is displaced 1.25 years to the left from that for federal funds. Data: [1], [47]

It is not suggested that short-term rates directly controlled unemployment rate, as might appear from Figure 10.2.3.  The stair-step increase in unemployment up to around 1982 was probably due to lack of sufficient time between recessions for employment to fully recover, so unemployment peaked at the time of maximum rates. The presumed chain of causation is; restricted difference between long- and short-term rates causes reduction of lending, which causes contraction, which causes reduced employment.

Although central banks certainly control the slope of the yield curve through their cumulative actions the correlations during the restricted time in Figures 10.2.1 through 10.2.3 do not prove that central banks always control economies, whatever the mechanism. At previous times when both short- and long-term rates were fairly constant, there were even greater variations in GDP growth and unemployment, so clearly other factors can be far more important than interest rates.  The Fed raised rates in 1929 and also in 2000, causing negative yield curves, but the following recessions are more plausibly ascribed to collapse of overexpansion than to the direct effect of interest rates. During the inflation spikes of 1975 and 1980 the greatly increased price of oil itself must have had restrictive effects in countries like the U.S. which import a large fraction of their oil, reducing the amount available to companies to invest and to consumers to spend on other than energy and transportation. Thus it is not completely clear how much of the downturns at this time were due to direct effects of commodity prices and how much to deliberate efforts by central banks to control inflation.  The fact that the significant recession of  July 1981 to November 1982 occurred well after inflation had begun to decrease rapidly in early 1980 suggests that the highly negative yield-curve slope (a historical extreme in 1981) was important at this time anyway.  The negative values of yield-curve slope during the time of inflation in the 1970's and 1980's were much greater than at any other time. But if and when interest rates do influence economies, the evidence is consistent with some effect due to the difference between short- and long-term rates, and not with the direct influence of long-term rates.

11. The Credit Crisis of 2008-9 and Some Predictions

[The essential predictions in this section were present in the version of the paper first submitted for copyright in 2009, using data as of May, 2009 - see Section 12 in that version]

Starting in 2007 the financial world was disrupted by a major credit crisis and puncture of a housing bubble. In late 2008 yields of less-highly-rated securities rose sharply and those of Treasuries decreased. The risk parameter as the difference between Baa corporate bonds and 10+-year Treasuries reached a monthly high of 5.7% in November 2008, much higher than at any time since the Great Depression. Unemployment later reached levels exceeded since the Depression only by the high of 10.8% set in 1982. Federal funds target rate was set at an all-time low of 0.0 to 0.25% (limits) in December 2008.

Figure 11.1 shows the recent monthly data as of August 2011 for some representative series. The calculated values use the data starting from 1953 (Section 5.1). Recent trends are well accounted for, but it should be kept in mind that the long-term Treasury and Baa series are themselves used to derive the risk parameter.

Figure 11.1a. Observed and calculated yields for 5-year Treasury securities. Monthly data current through August 2012. After that date the calculation assumes a constant federal funds rate of 0.15. The assumed risk parameter remains constant at 2.2%. ----  Observed ----  Calculated ----  Federal Funds

Figure 11.1b. Observed and calculated yields for 30-year Treasury securities. Parameters as in Figure 11.1a. ----  Observed ----  Calculated ----  Federal Funds

Figure 11.1c. Observed and calculated yields for Baa-rated corporate securities. Parameters as in Figure 11.1a. ----  Observed ----  Calculated ----  Federal Funds

The figure also shows projected yields after August 2012, assuming that the federal funds rate of 0.15% will hold indefinitely, and that the risk parameter remains constant at the value for the last date reported. The average of the risk parameter over the entire period 1953-2010 is 1.75%. Yields are calculated to decrease up to four years from December 2008 when federal funds reached minimum.

The results of this paper allow a confident prediction that long-term Treasury yields will not increase greatly unless and until federal funds rate is increased. If perceived risk increases again due to another financial crisis or other factors, Treasury yields will be depressed, as they were in late 2008 to early 2009, after which they could rebound, as they did in 2009. On the other hand, the effect of perceived risk on non-Treasuries with less than highest ratings is to increase yields, so the calculated decreasing trend of Baa securities holds only if general confidence does not decrease drastically again. Of course there are relatively short-term deviations from the equation, mostly classified as type 3 deviations, and the standard errors in the tables of Section 5 must be kept in mind, but so must the fact that large deviations are correlated with drastic movements of central-bank rate - a prediction of what might happen if central-bank rate is held constant is probably more secure than a prediction of what might happen after some given sequence of drastic rate changes.

While the spread between Baa-rated corporates and 10+-year Treasuries reached a peak in November 2008 and decreased rapidly through 2009, unemployment did not peak at 10.1% until October 2009, 11 months later (Figure 4.1.1), and thus using unemployment as the risk parameter gives distinctly poorer regression results for the period after 2007. However, if unemployment continues to decrease from its peak of late 2009, the predictions from using unemployment as the risk parameter are qualitatively the same as those from using yield spread; Treasury yields will continue to go down as long as central-bank rate is held low and so will those of lower-rated securities unless unemployment increases again.

Figure 11.2 shows the calculated yield for the 10-year Treasury on the assumption that federal funds rate is raised starting in January 2014, at the same pace as it was raised in 2004-2006 and to the same value, 5.25%. By 2015 the Treasury yield curve would be flat, as it was in 2006.

Figure 11.2. Observed and calculated yields for 10-year Treasury securities, on the assumption that federal funds rate is increased to 5.25% from January 2014 to 2016 at 0.16% per month. Other parameters as in Figure 11.1a. ----  Observed ----  Calculated ----  Federal Funds

Deficits, foreign investment and inflation should not directly affect yields, but may cause the Fed to change central-bank rates. Again, while supply-and-demand factors presumably do not affect yields as long as central-bank rate is held constant, the operations by the Fed to maintain central-bank rate change the quantity of money in short-term accounts which contribute to the monetary aggregates, and this in turn is supposed to affect inflation rate and other aspects of the economy.

These predictions assume that the relationships observed in the past will continue to hold. It is possible that correct knowledge of the factors which influence interest rates could cause changes in the bases for the relationships, for example in the way that borrowers and lenders estimate future central-bank rates (see Section 12).

The Fed followed a program starting in 2009 of buying long-term securities in open-market operations to try to bring down long-term yields directly rather than simply bringing down the federal funds rate with operations on short-term Treasuries. As the credit crisis of 2008 evolved, the Fed expanded its holdings from about $860B to almost $2.2T (Data: [8], [34]). This rapid expansion was exceeded historically only by that in World War II. Since there is no precedent in the US for this type of open-market operations, except for a much smaller program of buying long-term Treasuries after late 1947 (Figure 7.1.5), they could change the empirical relationships and the predictions, but so far there is no indication of such changes. The question of the effects of massive purchase by the Fed of long-term securities is discussed further in Section 12.3.3. That long-term yields have come down and will continue to do so is clearly not a proof that these operations were successful in affecting yields because these decreases are predicted on the basis of previous central-bank-rate cuts only. Figure 6.1.2 shows that the slope of the yield curve for Treasuries remained steep through 2009 and 2010; its behavior in this respect was similar to that in the recessions of 1991 and 2001, in which there was no purchase of long-term securities.

In early 2009 an increase in yields for long-term Treasury securities over a few months was attributed by many commentators and authorities (e.g. Bernanke, 2009a) to large increases in government borrowing, and some predicted and continue to predict much higher increases in the future. Such attribution and predictions are very improbable, both on the basis of the calculated future yields and on the basis of historical observations during times of very high world-wide government debt (Section 7 and Section 9). Rather the analysis in this paper indicates that this rise was the upward leg of one of the two sharp (negative) risk spikes in Treasuries. This confirms the interpretation previously given by Krugman (2009).

Some people argue that drastic increases in federal debt in the U.S. will cause yields of Treasury securities to go up because of fear of default - this would be an effect on yield rates due to some sort of risk factor rather than supply and demand. If considered in terms of comparative risk of domestic investment, this would actually amount to a reversal of the effect of increased general risk perception on Treasury yields as it has operated in the U.S. since 1919, including through the credit crisis of 2008-2009. This argument also comes up sharply against historical facts; there is no evidence for such downgrading during and after World War II when debt was over 120% of GDP, nor during the large proportionate increase in debt from about 1981 to 1995 (Figure 7.1.1). In fact the risk parameter decreased continuously from its Depression high as the World War II debt was being run up (Figure 4.1.1) and generally remained low for many years afterwards. As a rule, U.S. government deficits increase in peacetime when tax revenue falls off and there are increased expenditures for unemployment relief, etc., and that happens - and is happening currently - because private enterprise is not doing well.

In mid 2011 there was much talk in the news media and in political discourse of possible default on U.S. bonds because Congress for a while delayed passing the usually automatic extension of the statutory debt ceiling. One rating agency, S&P, has actually formally downrated U.S. Treasuries because of this. However, this certainly had no noticeable effect on bond yields, which tended to decrease during this time, perhaps in part because of international risk effects.

Through 2010 and 2011 there were significant fluctuations in the yields of the bonds of several European countries as they have been considered at possible risk of default. At least at some of these times, it appeared that the yields of U.S. Treasuries decreased, presumably as investors switched to them. Even in those European countries and in general there certainly is not a direct and invariant correlation of government debt with yields on the respective government bonds; rather yields sometimes go high when the overall soundness of the country's government or economy seems dubious or there are particular events such as banking crises. At times governments have incurred even higher debts than levels projected for the U.S., such as during World War II, and yields on their bonds do not seem to have been drastically affected (Figure 9.1). Some of the governments concerned came to an end, though not necessarily for economic reasons. Evaluating the future effects of relative international risk perception would require accurate prediction of the relative soundness and prosperity of all the countries involved, as well as the comparative importance of international versus domestic debt.

12. Theoretical Aspects of the Determination of Interest Rates.

Equation (1) was developed with no theory in mind, and is scarcely more than an empirical description of the data. It relies on no theory or assumptions, although insofar as it describes the data adequately it falsifies several prevalent beliefs. The equation itself is very minimal and any number of improvements can be imagined. Essentially all conclusions made up to this point are empirical; there have been only a few remarks pointing to possible theoretical interpretations.

12.1.  The Theory of the Determination of Interest Rates

This section will give first a rudimentary interpretation of the results in terms of expected future rates, which was derived from equation (1) in a simple way with no direct reference to any particular existing theory; then an account of how the results and this interpretation are in accord with Hicks' theory of the term structure of interest rates. The influence of systematic re-lenders, for example banks, must be an important influence in the mechanism outlined by Hicks.

12.1.1. Yields as Expected Central-Bank Rate Plus a Maturity Premium Function

The actual yield for a given maturity may be regarded as determined by reference of the market or market participants to a maturity premium function (curve) which gives an increment over an expected future central-bank rate. This expected central-bank rate may perhaps be more strictly an integration of the expected central-bank rates over the term of the security, or in some cases the expected rate at a projected date of sale. The maturity premium curve is in effect an ideal yield curve. Thus we postulate:

Equation (11)           Yield_{M, t} =  B_exp,M + f(M) 

where B_exp,M is the expected future central-bank rate. We can take the expected central-bank rate to be estimated by the second and third terms of equation (1) or equation (5) or (6):

Equation (12)         B_exp,M   =  k_2(M) * B_t + k_3(M) * <B>_{t-1 to t-p}

Then from equations (11) and (6) we have

Equation (13)        f(M) = k_1(M) + k_4(M) * Risk

The risk-invariant part of the maturity premium function in question can thus be equated with the values of the k₁ constant in equation (1) as dependent on maturity in Figures 3.2.1 to 3.2.4 and all the tables of results. The risk term gives the dependence on differential risk perception, which varies over time. As in equation (1) the subscript M refers to a given rating as well as maturity, so equation (13) is for a combined maturity-risk premium function and f(M) is really a surface. But usually when considering maturity premium we would be including only securities of a constant rating, so f(M) is a curve at a given level of perceived risk.

The curves for k₂ and k₃ in Figures 3.2.1 to 3.2.4 are the factors for the market's estimation of future central-bank rate for a given maturity. The longer the term, presumably the further in the future for which the estimate is made and the greater the reliance on past values (k₃) instead of current value (k₂) of central-bank rate.

This estimation of a future central-bank rate by the market should probably not be regarded as the result of conscious and deliberate calculation; some hypotheses about mental processes are discussed below. While the results of this paper can be interpreted in terms of predictions on the part of market participants, this does not imply that there is any sort of prophetic insight by anyone, simply that participants attempt to predict future central-bank rates based on available information, mostly current and past rates. There is no implication that such predictions are generally correct, and the validity of the central-bank-rate prediction part of equation (1) is not dependent on their correctness, only on whether factors influencing the predictions are correctly included.

Thus the data show that it is possible for the market as a whole or for market participants to expect yields conforming to a maturity premium curve of  fairly constant shape, while the actual yield curve which results at any time is not of an ideal shape, and does not itself represent the market's idea of maturity premiums. This does not say why the market would place primary importance on an estimate of future central-bank rate, or why it should then determine yields by reference to the maturity premium function; likely answers to these questions are discussed in succeeding subsections.

Figure 12.1.1.1 illustrates how investors in bonds of different maturities can expect different future central-bank rates, to which a constant maturity premium function is added to determine the yield for that maturity. This diagram is for Treasury securities in January 2007. Federal funds rate had been low from 2002 through early 2004, then was raised to 5.25% by late 2006. Thus in effect the investors in longer-term bonds were still expecting federal funds in the future to be relatively low, which resulted in a virtually flat yield curve.

Figure 12.1.1.1. Calculated Treasury security yields in January 2007. The black curves show maturity-premium functions for respectively 1-, 3-, 5-, 10- and 30-year securities going downward (some maturities are omitted for clarity). Investors are hypothesized to start from an expected future central-bank rate, near where each curve intersects zero maturity, and add to that the maturity premium function. Where that curve reaches the desired maturity (circled dots) is the market yield. This figure uses data for constant-maturity Treasuries starting in 1982, as in Figure 3.2.2.

The figures in Section 3.2 showing the values of k₁, k₂ and  k₃ are from calculations which do not include time variation. Including time variation for series which span the entire interval 1919-present typically improves R² by 2% or less, while periods of up to about 75 years may be improved to a negligible extent. As discussed in Section 4.2, it was somewhat arbitrarily decided to include time variation of k₃ , but this does not rule out some time variation of other coefficients.

A master equation could be constructed for each rating or type of security, such as Groups 1 and 2 - which would hopefully be similar - and (3), by fitting the curves of the constants k₁, k₂ and k₃ to polynomials or other functions in maturity. At present the limited data on maturities and the uncertainty about the best form of the equation do not really justify this.

As mentioned in Section 4.2, all results in this paper use the raw values of current and past central-bank rates, but the values of any other variables, including yield spreads, unemployment, inflation, and others in Section 7, are corrected by subtracting the average of that variable over the complete time span. If this is not done the values of the k₁ coefficient are biased by the average values of those other variables. An alternative to using the average would be to chose some date to give the base value for the other variables.

Obviously the maturity premium function is similar to an ordinary yield curve and in some sense is an ideal yield curve but it is not the same as actual calculated or observed yield curves, and a distinct term is warranted. Even the equilibrium yield curves in Figure 6.4.1 are mostly not maturity premium curves, because the expected central-bank rate is itself dependent on maturity. In the case of Treasury securities since 1953 in Figure 6.4.1b the k₂ and  k₃ constants are such that when central-bank rate is held constant for four years the shape of the yield curve is not highly sensitive to central-bank rate, but this is not the case for the basic yields for corporate bonds from an earlier period in Figure 6.4.1a. Calculated yield curves coincide exactly with the maturity premium function only when central-bank rate has been zero for four years.

Other terms could be added to equation (12) corresponding to different factors the market uses in predicting future central-bank rates besides current and past central-bank rates. It may readily be imagined that at times the market attempts to guess at the future actions of the central bank based on the latest economic and financial news, or perhaps some individuals are influenced by this rather than past rates.  A reasonable hypothesis is that the general idea of a base rate or "normal" rate in Hicks' terminology (see Section 12.1.2) is formed from current and past central-bank rates, and market participants then try to guess which direction the central bank will move from that according to different supposed influences.  In some cases participants' expectations about the actions of others in the market, for example speculators, may come into play.

One obvious thing which might influence the market's expectation of future central-bank rate is inflation, but this influence would depend not only on predicting inflation but on knowing how the central-bank will react to inflation, and this reaction has varied greatly over time. The Fed's dual mandate requires it to minimize unemployment as well as try to maintain constant prices, and these objectives are often in conflict. The Fed did certainly react to inflation during and after World War I, although this was not nearly as drastic as its actions during the 1970's and 1980's. From 1934 to around 1955 the Fed did not react at all in terms of its policy rates to inflation. After that it began to track inflation with federal funds rate. In early 1980, inflation rate began to fall steadily as the price of oil stabilized and the Fed dropped federal funds rate extremely rapidly, or allowed it to drop. Then in late 1980, the Fed suddenly raised federal funds drastically, although inflation was still falling. There were further sharp changes in federal funds rate through the 1980's which were not closely related to major changes in inflation rate. In 1989, the Fed began dropping rates again as the recession of 1990 loomed, and from then until the present its actions have been related more to counteracting recession than inflation. The Fed began dropping Federal funds in 2007 as the credit crisis developed, but in early 2008 there was a sharp increase in inflation because of very high oil prices and the drop in federal funds was temporarily halted. Then oil price collapsed, inflation turned to deflation, and the drop in federal funds was resumed. It would certainly seem if not necessarily rational at least conservative to base expectations of future central-bank rate more on current and past rates than on trying to predict the actions of the Fed in reaction to changes in the inflation rate.

In a few countries, adding inflation to the regression after about 1960 - but not before - improves the fit meaningfully (Appendix C, Section C.24).  This could be a result of unrecognized risk factors, but may be a result of a real secondary influence of inflation - that is assumption by the market that elevated inflation will mean higher short-term rates in the future because of central-bank action in response.  There is certainly no reason why response must be uniform in all countries, and as with many economic mass actions may be a result of temporary or cyclic group psychology

The empirical evidence strongly indicates that economic factors not connected with either central-bank rate or perceived default risk are not important in determining market interest yields. When central-bank rate is held nearly constant, as it was in the period about 1934-1955, rates are generally constant and show little or no influence of deficits, inflation or apparently any other external factor aside from the perception of risk. In periods when the central bank changes its base rate frequently, the market must guess at the future base rate, and while a simple relationship like equation (1) can account for much of the variation of such consensus guesses in terms of current and past central-bank rates we would not expect to be able to specify exactly all of the psychological factors involved.

Yet the results of the empirical analysis in this paper still seem to be in conflict with the law of supply and demand; how can the price and yield of securities traded in an open market be insensitive to what appear to be obvious supply and demand forces?  But actually while the overall credit market is "open" in some respects it is hardly the same as markets for commodities which have a limited supply and a limited demand at any given time. The central bank is after all a significant part of this market; its objectives are very different from those of other participants and it is able to create supply and demand at will. The open-market and discount-window transactions involved in the process of maintaining a base rate - the rate, in effect, at which money originates in modern systems - evidently are tied more closely to markets for long-term securities than previously thought. Even in the market, demand for securities is not such a simple thing as demand for basic commodities like food or fuel and depends on other factors besides current price (or interest rate).  The value of a long-term security depends heavily on the expected level of short-term interest rates over the term of the security. To assume that long-term rates are completely independent of short term rates - as several of the authorities quoted in the Introduction do - is to ignore this dependency or to assume that expectation of future short-term rates is completely independent of current and past rates.  

The theoretical bases for interpreting these relationships have actually existed for some time and the remainder of this section will explore them from various angles.

12.1.2. Hicks' Theory of the Term Structure of Interest Rates

Hicks (Originally 1939; Second Edition 1946, Ch. 11) systematically analyzed the term structure of interest rates, which he considered to be a result of the actions of what he called "hedgers and speculators", market participants who on the one hand want to minimize their risk in obtaining funds for the long term, and on the other hope to make a profit from some perceived imbalance in long- versus short-term rates. Both of these groups must act on the basis of their expectations of future rates. He wrote in summary:

"If short rates are not expected to change, the long rate will exceed the short rate by a normal risk-premium; if the current short rate is regarded as abnormally low, the long rate will lie decidedly above it; the short rate can exceed the long rate if the current short rate is regarded as abnormally high."  (Hicks, 1946, Ch. 11, p 147).

This account is very close to the empirical results of this paper and their interpretation in rudimentary terms in Section 12.1.1. Hicks also wrote "...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". (Ch. 11, p. 152). This of course is generally observed, subject to the perturbations of the type 3 deviations.  The general mechanisms which underlie Hick's conclusions are illustrated in Section 12.1.3.

The "normal risk-premium" as a function of term or maturity corresponds at least roughly to the maturity-premium function defined empirically in Section 12.1.1. More strictly, the "normal risk-premium" for long-term bonds of a given term is the amount by which the point for that term lies above the point for the short-term bond in question on one of the equilibrium yield curves in Section 6.4. The maturity-premium function may be considered to consist of various kinds of risk, but given the existence of banks and other institutional re-lenders, must also reflect their costs and profits - see Section 12.1.3.

What Hicks refers to indirectly as the "normal" short-term rate is the expected future central-bank rate of equation (12), which to a close approximation is a mixture of current and past rates. To the extent that the regression using equation (1) is successful, it shows that the "normal" condition is what has existed, not a guess about the future based on parameters which do not appear in equation (1). Such guessing may occur, but its influence appears to be limited to the type 3 deviations or the residual variance, which is on average about 4.5% of the total. Note that what is considered to be "normal" is dependent on the term of the long-term rate; the shorter this term, the closer "normal" is to the current short-term rate, so the "normal" short-term or central-bank rate which applies for this particular long bond is definitely not the overall average. The predicted short-term rate is also definitely not an extrapolation using the rate of change; if the rate has been going up, the predicted rate is lower, not higher than the current rate.

The conditions under which short rates are not expected to change are those of the equilibrium yield curves in Section 6.4. Some real-life examples approximating these conditions are the U.S. from about 1934 to about 1955, or indeed almost any country during that time, Japan since about 1998, and likely the U.S. from about 2009 going forward, although long-term rates have not reached minima (Section 11). Short-term rates were very low in all these cases, so the relevant collection of "normal risk premiums" as well as the actual yield curve are close to the maturity-premium function.

The case in which the current short rate is regarded as abnormally low is that in which the expected future central-bank rate or short-term market rate is above the current rate, which usually occurs when the central bank has been dropping its policy rate.

The case in which the current short rate is regarded as abnormally high is in practice that in which the central bank has been raising its policy rate so that it is well above the market's expectation. That expectation in equation (1) and equation (12) is always based on past rates. When the central bank has been raising for some time, its current rate is above the average of past rates, and this is when the slope of the yield curve is flat or negative.

Hicks analyzed the problem of term structure on the alternate assumptions that either the short-term rate or the long-term rate could be varied while the other is held constant. Since this ambiguity about which end of the rate structure is fixed carries through Hicks' more general analysis of interest rate, investment and income, it seems possible that Hicks and perhaps also Keynes did not fully realize the degree to which modern central banks can control or are willing to control short-term rates by open-market operations. Perhaps the constraints of a gold standard were still on their minds. Hicks may have had it in mind that the overall level of interest rates is at least partly determined by market forces including supply and demand and the influence of inflation in addition to the quantity of money. But the reality is that since early in the 20th century, certainly since the founding of the Federal Reserve in the U.S., central banks have controlled short term rates as a matter of course. The Fed was able to control 3-month T-bill rates at exactly 0.375% from 1942 to 1947 (Figure 5.2.1) with massive purchases (Section 7, Figure 7.1.5), and while federal funds rate may show quite large excursions from the target on a daily basis (more commonly in the past than at present), the control is complete on a monthly time scale.

Does the fixing of the short-term end of the rate spectrum invalidate Hicks' analysis of term structure?  Since the determination of term structure in Hicks' theory relies on the activity of "hedgers and speculators" - although it will be shown in Section 12.1.3 that banks must be included amongst them, and in Section 12.2 that the influence of the quantity of money must usually work in the same direction - it might be asked whether extremes of supply and demand or some condition of inflation could sometimes overwhelm their activities with respect to long-term rates. This would seem to be a question ultimately to be decided empirically, and for U.S. credit markets at least the answer from Sections 7 and 8 is that it has never happened. It is of course possible that the ability of the central bank to modify short-term rates could be limited, for example by exhaustion of short-term securities to buy or sell. There could also be restrictions in the availability of money through the discount window or interbank loan. Some likely reasons why supply and demand and inflation have little or no influence will be discussed in following subsections.

The results of this paper supply quantitatively the main part of the determination of the expectation of future short-term rates, or the "normal" rate in Hicks' terminology, and that is the mixture of current and past central-bank rates in the second and third terms of equation (1); and these results also determine the "normal risk-premium", which is derived from the maturity premium function. The fact that the central bank can under most circumstances control a chosen short-term rate precisely causes the control of all interest rates to be subject to the dominance of the central bank, through its cumulative actions. Of course the control of long-term rates is less precise, and the central-bank actions take time to be effective on long rates. The actions of the central bank are modified by changing perceptions of default risk, but that influence amounts to only a few percent of the variance over the long term, as compared to well over 90% for that of the Fed in the U.S.

The two general elements of the prediction of future short-term rates and risk-perception effects which appear in Equation (1) and the interpretation thereof in Section 12.1.1 may in principle give an almost complete theoretical account of the determination of interest rates. What is missing from Equation (1) are the criteria that are used by the market to predict future short-term or central-bank rates in addition to current and past rates, and possible international risk effects (Section 12.3.7). These missing things, in addition to inadequacy of the equation with respect to the two main elements, are presumably what accounts for the type 3 deviations or the remaining 4.5% of the variance for U.S. securities. Of course these latter variations can be important in the short term for some purposes, such as speculation. Other more direct influences cannot be ruled out, but testing of a large number of possibly relevant economic variables has so far not shown any consistently meaningful correlations.

Equation (12) uses central-bank rate, while the rate which applies in Hicks' theory is the short-term rate which is available to the "speculator". If the "speculator" is a bank (see next subsection), the two may be generally identical, although there may be a choice between federal funds rate and discount rate, or amongst other central-bank or inter-bank rates in other systems, and the rate paid on deposits also enters. Given this range of possibilities and the fact that the process is a complex psychological one, as recognized by Hicks and Keynes (1936), there is obviously a limit to the precision of equations like those used in this paper. 

Keynes (1936) had discussed the term structure of interest only rather sparingly (for example Ch. 13, pp 168-170), but certainly recognized that expectation of future rates is critical and that speculation could be significant: "...the individual, who believes that future rates of interest will be above the rates assumed by the market, has a reason for keeping actual liquid cash, whilst the individual who differs from the market in the other direction will have a motive for borrowing money for short periods in order to purchase debts of longer term." (Ch. 13, p 170).

Can the market predict future short-term rates?  Long-term rates are based on what amounts to a prediction of future short-term rates, but is that prediction accurate? Krugman's (2008) explanation of the relation between the yield curve and recession (see Introduction and Section 10) implies that prediction tends to be correct, based on divination of future recession, whereas the explanation in this section does not.  There are various ways such predictions can be tested mathematically, but simple inspection shows that the idea that long-term rates actually predict short-term rates is completely false. Compare the green curve for federal funds rate with the black curve for actual 10-year Treasury rate in Figure 5.1.1 (for example - many other figures show the same thing). The peaks and valleys for the two curves tend to be fairly nearly coincident, but where there is any displacement it is the peaks and valleys in the long-term curve which are displaced to later times - one might say that short-term rates tend to predict long-term rates (with a considerable damping effect), but certainly not the other way around. If the market were correctly predicting short-term rates for the future, displacing the long-term curve to the right should lead to near coincidence, but if this is done any coincidence is worsened and peaks tend to coincide with valleys. Of course the actual relationship is exactly as predicted by Hicks' theory if the estimate of future rate is based on past rate, and the blue curve for predicted rate according to equation (1) does give the correct positions for the major peaks, keeping in mind the standard errors and the fact that there may be excursions due to other influences on the prediction of future short-term rate.

12.1.3. The Effects of Banking and Other Institutional Re-lending

Much of banking depends on borrowing money at short term (including demand deposits) and lending it out at a higher rate, frequently at long term. The need to cover costs and make a profit on such transactions - or at least not to sustain a loss - is a powerful factor which ties long-term rates offered by banks and other re-lenders to the expected future central-bank rate, which currently is the minimum rate at which banks can borrow, or near to it. Consider the simple case of a commercial bank or S&L which takes in deposits paying interest very near the current central-bank rate and may also borrow from other banks at that rate or from the Fed at the discount rate, and loans the money out in long-term mortgages. If the commercial bank expects central-bank rate over the next 10 years, say, to be 6%, there is no incentive whatever for it to offer mortgages or other loans which extend over this period at 6% or below because it would make no profit; thus there would be no supply of money from banks at such rates. The demand for mortgages becomes irrelevant as the rate reaches the limit of expected future short-term rate plus re-lending profit. To put it another way, the willingness of banks and other re-lenders to make money available in long-term markets will be calibrated with reference to the expected central-bank rate plus a minimum profit, including operating costs, demanded or required to stay in business, and thus the influence of re-lenders will tend to make long-term rates converge at an average or consensus value of this nature. Clearly, borrowing at short-term and relending at long both requires an estimate of future central-bank rate, and tends to impose a positive slope to the maturity premium function.

Inflation is irrelevant to the first order in re-lending since the bank's obligations to its depositors, or to others from whom it borrows money including the central bank, are devalued as much as the obligations to the bank in the form of mortgages and other loans. If inflation is expected to be very high, however, this would affect the value of minimum required future profits. To put it another way, bank profits are not strongly dependent on real rate, just on the spread between long and short-term rates, and to adjust this it is necessary to predict future short-term rates, not inflation.

Obviously banks and other re-lenders compete with each other and this tends to keep the spread between the rates which they expect to pay on short-term loans and what they offer for long-term loans at a minimum.  Bankers in particular, being typically conservative (at least in the past), will be unlikely to make a drastic change in offered rates every time the central bank changes its policy rate, since economic developments may cause the policy-rate change to be reversed over the course of the loan; but as incremental policy-rate changes accumulate re-lenders will be forced to change long-term rates.

If we consider that banks and other institutional re-lenders are part of credit markets, then they must fit into Hicks' schema, presumably as "speculators", while bank borrowers are "hedgers". This is a very special case of "speculation", however, because it is not an occasional matter, dependent on special conditions arising, nor are bankers considered to be especially venturesome. In modern finance, there is a tendency for others besides banks to be re-lenders and some of these others have considerably more of the character of opportunistic speculators.

Mortgagors and other borrowers from banks in theory also must consider expected future short term rates in comparison to long-term rates offered, although for mortgages financing by short-term may be restricted to variable-rate mortgages, which have not always been available.

Until fairly recently mortgages have not been traded on the open market like bonds. Nevertheless they have obviously behaved very much like long-term bonds with respect to response to central-bank rate (Sections 5.1 and 5.7), and their rates have shown essentially no response to supply and demand in the mortgage market itself (Section 7.3).

Banks are also important because they are generally considered indispensible and must impose their costs and profit requirements on Hicks' "normal risk premium" or the maturity premium function. Their operations affect the money supply, so that changing the term structure of interest rates can modify the intentions of the central bank in that respect - see Section 12.3.3. Central banks create money through the lending of commercial banks, and it is impossible to understand that lending without considering how it is affected by the expectation of the rates at which commercial banks will need to borrow themselves in the future.  This is one important reason why short- and long-term rates could not possibly be completely independent. The case of banks should also illustrate why inflation is not important in Hicks' theory of the structure of interest rates.

Until the 1970's rates paid on checking accounts were very low or zero, but central-bank rates were also relatively low during most of this time. Rates paid on savings deposits tended to vary with central-bank rate, being sometimes higher and sometimes lower and sometimes limited by law (Data: [27]). Discount or federal-funds rates may be more important than deposit rates with respect to bank lending policy, because these are the rates which must be paid if funds are required on short notice. Reserve requirements should affect the relative impact of bank re-lending as compared to other sources of credit, and might have a secondary effect on rates as increased loan volume could allow somewhat lower rates.

If banks do not consider it to be profitable to lend at long term, they would presumably shift at least partly to short-term lending, where the profit margin is more predictable. When allowed, lenders can promote variable-rate mortgages. Although rates for prime business loans are currently fixed at federal funds rate plus 3%, loan volume could be increased by changing loan criteria. The effect of this would be to overall reduce demand for long-term securities, while increasing demand for short-term securities. The latter change is absorbed by the central bank.

12.2. The Influence of the Money Supply

While the influences discussed in Sections 12.1.2 and 12.1.3 are not directly tied to the quantity of money, it is apparent that the influence of the quantity of money in Keynes' General Theory and some other theories require a connection between short- and long-term rates, though the quantitative nature of this relation may not be specified.

12.2.1 Keynes

Keynes (1936, chapter 13) concluded that the money supply strongly influences interest rates, for example: "If this explanation is correct, the quantity of money is the other factor, which, in conjunction with liquidity-preference, determines the actual rate of interest in given circumstances" (p. 167). Keynes defines "the rate of interest" specifically as "the complex of the various rates of interest current for different periods of time, i.e. for debts of different maturities." (p. 167, footnote). The entire discussion would appear to be superfluous if the rates referred to in the complex included only the discount rate and/or the short-term rates affected by normal open-market operations - that is to say those rates considered to be directly set by the central bank - and thus if Keynes' arguments are correct, increasing the money supply by reducing the discount rate or by central-bank purchase of short-term securities must tend to cause the rates for long-term securities to decrease as well. An increased supply of money made available to banks through the discount window or purchase from them of short-term securities is available for long-term loans and must therefore tend to decrease their rates, though this will be tempered by apprehension that the supply will decrease in the future - or that the short-term rates will go back up.

Keynes' statement that "...at a level above the rate which corresponds to full employment, the long-term market-rate of interest will depend, not only on the current policy of the monetary authority, but also on market expectations concerning its future policy" (p. 202; italics his) is obviously consistent with the empirical results of this paper and the rudimentary theoretical explanation thereof in Section 12.1.1, as well as Hicks' theory of term structure. This paper further shows that expectations of the future policy of the monetary authority are mostly anchored in its current and past policies, and that there do not appear to be deviations from the relationship regardless of the state of employment; the qualification in Keynes statement appears to pertain to the conditions under which inflation can occur.

The relationship between employment and interest rates is empirically quite well defined once the influence of the central bank has been accounted for. As discussed in Section 4.1, at times when unemployment has been high, the yields on low-risk securities, especially Treasuries, have gone down somewhat, but yields on moderate- and high-risk securities have gone up markedly. There is no evidence that the state of employment itself has a direct effect on yields. In the period during and after World War II when unemployment was very low all yields remained quite low, but in the period of high inflation in the 1970's and 1980's both unemployment and yields were often simultaneously high. High unemployment is strongly correlated with greater differential default risk, although that risk is better indicated by yield spreads, and not with reduced interest yields through any sort of supply-and-demand effect, and conversely low unemployment does not cause interest rates to rise, rather long-term corporate bond rates tend to decrease somewhat because risk is perceived to be low. 

Inflation appears to cause conflicting influences. The quantity of money usually increases during general inflation, and this by itself according to Keynes' analysis would tend to decrease interest rates, but liquidity-preference, the other factor which determines interest rates in Keynes' theory, is not constant either. Demand for credit increases, or in effect there is an increased supply of securities, but there must also be increased desire of those who have money to hold bonds in preference to cash, provided nominal interest rates are not zero. If inflation is 6%, bond buyers may like to have a rate greater than this, but they will take 3% in preference to zero. Since there are other options for avoiding holding cash, such as stocks, commodities and real property and the prices of these things are variable for different reasons including the expectation of inflation, it does not seem possible to relate these opposing tendencies quantitatively, but the empirical evidence (Section 8) suggests that they at least roughly cancel, if they actually have any importance as compared to the factors discussed in Sections 12.1.2 and 12.1.3, which are basically insensitive to inflation.

Keynes versus Hicks.  To a large extent Keynes' analysis of the effect of the quantity of money reinforces Hicks' analysis of the market factors influencing term structure and some of the critical elements are present in both analyses.  As the central bank increases the quantity of money by reducing the discount rate and/or buying short-term securities, according to Keynes long-term rates must also tend to come down.  But the predictions of the two theories are not the same, and Keynes' views about the effects of employment and liquidity-preference on interest rates are not confirmed empirically.  Also the relationship between interest rates and the money supply is not as strict as Keynes and many others seem to have thought. Keynes wrote "As a rule, we can suppose that the schedule of liquidity-preference relating the quantity of money to the rate of interest is given by a smooth curve which shows the rate of interest falling as the quantity of money is increased" (1936, p. 171). If the quantity of money is at all represented by the usual aggregates, this is of little value if not exactly false; for example, the rate of growth of the M2 aggregate was strongly negative 1930-1933, then very strongly positive in the following years and even more positive during World War II and the 10 or so years after, while long-term interest rates remained fairly constant and closely related to central-bank rates (Section 5). While the central bank controls interest rates within defined limits, its ability to control the money supply is in practice not strong. 

12.2.2. IS-LM and Liquidity Traps

Hicks (1937) developed the schematic method of illustrating some of the basic economic relations involving interest rates now known as the IS-LM method. The IS-LM diagram is often used to illustrate the condition known as a liquidity trap (or by some other names). His theory of the term structure of interest rates discussed in Section 12.1.2 is almost entirely independent of the more complete theory of economic principles in Hicks' (1946) book and from IS-LM analysis and there are many assumptions and derivations in the latter which are irrelevant to the present paper, but a few points are relevant.

Firstly, the use of the IS-LM diagram when dealing with the influence of the central bank assumes that the interest rate, at the intersection of the IS and LM curves, can in most cases be changed if the central bank displaces the LM curve, that is changes the quantity of money. Since the interest rate in this diagram is that rate or complex thereof which influences investment and this is normally assumed to be long-term rates, and since the central bank usually adjusts the money supply by controlling short-term rates, either by setting the discount rate or by open-market operations on short-term bonds (other methods are rarely used), it follows that long-term rates must be generally correlated with short-term rates. Empirically as shown in this paper the overall or effective long-term interest rate is set within fairly close limits by the cumulative actions of the central bank and the current risk perception.

The use of IS-LM diagrams assumes that the absolute values of interest rates - especially long-term rates - have important influence on investment and on economic growth. Empirical evidence of this is lacking and during the period of the late 1960's to the early 1990's when interest rates varied most and reached the highest values ever in the U.S., GDP growth and employment were correlated with the difference between long- and short-term rates, not the absolute value of long-term rates (Section 10.2). Keynes had warned that the actual quantitative influence of the rate of interest was unknown, and had to be determined empirically (1936, Ch. 12, p 164).

While short-term rates are set immediately, this is normally during a gradual process of adjustment; for example the Fed began reducing federal funds rate from 5.25% in August 2007 and did not reach the ultimate target of 0-0.25% until December 2008. Although the central-bank actions which affect short-term rates may quickly affect the money supply as measured by the aggregates, long-term rates are not immediately reset; their value depends on central-bank action over several years in the past. Investment is presumed to be dependent mostly on long-term rates, and if this is true investment will only affect national income after a time lag. On the other hand if the most important aspect of interest rates is the difference between long- and short-term rates the effect may diminish as the yield curve stabilizes over time. The IS-LM diagram and many other economic relationships purport to describe a state of equilibrium, but in view of the fact that the various time lags may approach the time scale of a normal business cycle it should be realized that a state of equilibrium is mostly imaginary.

Krugman (2009b) and some others have argued that the US has been in a liquidity trap from about the end of 2008 to the present (2013) and also that Japan has been in one since the late 1990's. Since the Fed has been going to great lengths to expand the money supply, buying long-term bonds and taking other measures to the point that banks have huge excess reserves, while investment has generally been low, it would seem that there should currently be a liquidity trap as defined in the IS-LM diagram if there ever is one. Krugman has at times attributed the low long-term interest rates since about 2009 (to 2013) to the existence of a liquidity trap, and at other times to the constraints of Hicks' theory of the term structure - these are two separate things.

However appropriate and useful the description of current conditions in the US and many other countries may be in the original sense of a liquidity trap as defined by Keynes or by any other definition, the low long-term rates in the U.S. (Section 11) and Japan (Appendix C, Section C.13) are not themselves proof of a liquidity trap, because they are predicted by equation (1), which is completely general and not dependent at all on any special conditions. Nor is a liquidity trap required as an explanation for the lack of influence of high debt in the U.S., for the same reason, and specifically because much higher debt after World War II did not cause higher rates, as discussed in Section 7. In the period of about 1934-1955 there was extreme variation of all kinds of economic parameters, but central-bank rate was held low and long-term rates were also low (e.g. Figure 8.4.1). Except for some brief recessions, notably that of 1937-1938, GDP growth was quite high during this time and liquidity-trap or "depression-like" conditions did not generally exist, if such conditions refer to lack of investment and/or slow growth in GDP or employment. Unemployment was high during the early part of the period 1934-1955, but when there is rapid GDP growth and increase in employment description as a "trap" is not appropriate. Long-term interest rates were low during the period 1934-1955 because the Fed held its policy rates low, not because of any liquidity trap.  Long-term rates in the U.S. and in many other countries which are not considered to be in danger of default are currently low because the market estimation of future short-term rates is low, not because of a liquidity trap, although current conditions are more like those of a liquidity trap than those in the post-WW-II years.  The market estimation of future short-term rates is low primarily because central banks have established a low base or "normal" level by holding their rates low for the last four years.  This estimate may also be affected to a lesser extent by current events which are thought to influence future central-bank policy; and rates in more stable countries such as the U.S. and Germany may be depressed by international risk effects (see Sections 9 and 12.3.7).  In those countries in Europe and elsewhere which are currently (2012-2013) most prosperous such as Germany, there certainly is no upward pressure on interest rates - in fact rates in those countries are lower than predicted by equation (1) presumably because of the international risk effect.

12.3. Other Considerations and Points of View

12.3.1. Absorption of Short-Term Supply and Demand

It is impossible to argue that the shortest-term yields are not effectively set, within limits, by central-bank policy in the U.S. and most other countries with strong central banks. It follows that variations in supply of and demand for short-term securities which might cause deviation from the chosen rate are absorbed by the central bank and the banking system - any actual variations in supply of and demand for short-term securities must usually cause variations in bank reserves and monetary aggregates, not yields. About half of current U.S. government debt is in securities with maturity of two years or less. Thus a large part of the variation in supply and demand, or any other economic variable which could change yields of Treasury securities, is ipso facto absorbed by the banking system at present, though long-term securities were more important in the past.

Judging at least from the quotes from economic authorities in the Introduction, at least some authorities consider that there is a transition from this nearly complete control of yields by the central bank for the shortest-term securities towards complete lack of control of yields for the longest-term securities - towards a rate of interest controlled by supply and demand and expectation of inflation (this assumes that the remark by former Fed Chairman Volcker applied only to long-term rates - it would be complete nonsense if applied to short-term rates). But equation (1) accounts well for yields as a function of central-bank rates and risk perception only, and the residual does not correlate with inflation or several commonly-assumed types of supply and demand. The transition in going from short- to long-term rates is mostly one of decreasing immediate response and increasing dependence on past history of central-bank rates.

12.3.2 Expectations and Choices in Credit Markets

This section considers credit markets, focusing on security issuers who wish simply to raise money, and those who buy and sell according to their own needs and outlook, without attempting to re-lend or speculate. Such people are generally concerned with current rates as compared to returns on other investments such as stocks or real property and whether those rates will go up or down in the future; and relative rates of short- versus long-term bonds. Investors aim to maximize return and issuers to minimize cost.  In many cases it is clear that there must be some deliberate effort to predict future long-term rates.

Market response to central-bank rate. If it is thought that long-term rates will be going up, while short-term rates remain constant or are not considered, bond buyers will tend to avoid long-term securities and will at least in some cases buy short-term securities instead, while issuers and sellers will tend to offer long- over short-term bonds. These changes in supply and demand may cause long-term rates to go up. Conversely the consensus that long-term rates will be going down may itself cause long-term rates to decrease; this has been realized by most theorists, and is one reason for considering that rates are to a large extent psychological.

Figure 12.3.2.1. Yearly change in federal funds rate, compared with the results of the University of Michigan poll as to which direction interest rates are expected to move over the next year. The poll results are shown as the difference between the percentage of respondents expecting rates to move up and those expecting rates to move down. Both quantities are yearly averages. Data: [1], [38]

Market and central-bank response to supply/demand and inflation. If the law of supply and demand is not to be abandoned entirely for credit markets, the empirical results seem to indicate that changes in supply and demand are counteracted by processes such as inflow from and outflow to other types of investment or hoarding  and/or are transmitted down the range of maturities, to be ultimately absorbed by the central bank. Both of these processes must operate to some extent. If an increase in supply of bonds causes interest rates of long-term bonds to increase enough, money will tend to be drawn from the stock market, real property, etc. It should also tend to cause those who would otherwise be investing in short-term bonds to buy long-term bonds. The consequent decrease in private demand for short-term bonds is of course made up by the central bank, at present in the U.S. through open market operations, as it holds the yields of those bonds at the chosen value.

The devaluation of principal because of inflation is the same for money loaned continuously at short term as for money loaned at long term. If there is no prospect of rates being raised by the central bank the devaluation factor gives no reason for investors to shun long-term securities in favor of short-term securities and thus no reason for the maturity premium function to steepen because of inflation. In times of inflation if long-term rates do not go up - and this was generally the case before the 1950's to 1960's - investors may shift from bonds of all maturities to other investments such as real property or stocks. Of course if there is general expectation of inflation and if the central bank does not raise interest rates the prices of those other investments will go up, anticipating their increase in money value, which will obviously dampen a shift. If there is a decrease in overall demand for bonds, explaining the empirical finding that inflation does not directly influence interest rates is not fundamentally a different problem from the overall problem of the influence of supply and demand. In this case any increase in long-term rates will tend in principle to be counteracted by shifting of investors from short- to long-term bonds, issuers perhaps curtailing the supply of long-term bonds, and the central bank again buying short-term bonds to keep their rates constant. The theoretical problem of the response of interest rates to inflation is apparently a complex one, and it should be a better approach to start from the empirical observation that there is little or no net response, rather than the fictitious assumption that lenders can somehow enforce positive real rates. Again, the factors discussed in Section 12.1.2 and 12.1.3 may overwhelm any other influences on interest rates.

12.3.3. Implications for Central-Bank Strategy

Evidence in this paper clearly shows that central banks effectively control long-term interest rates - within limits - so if long-term rates are a major determinant of investment and thus economic growth central banks would presumably have even more influence than previously thought, but only with a considerable time delay.  However, other empirical evidence mentioned in Section 10.2 suggests that the main influence of interest rates on investment and growth may not be the absolute level of long-term rates, but the difference between long- and short-term rates, or in other words the slope of the yield curve, which is also controlled by central banks.  The influence of central-bank control of short-term rates on yield curves is immediate, although it is usual for major changes in policy rate to be carried out over many months, but is not lasting over time, as they adjust to equilibrium states.

Can central banks modify the equilibrium yield curves, or possibly speed up the attainment of an equilibrium state by buying and selling long-term securities instead of confining open-market operations to short-term? In the absence of empirical evidence such trades would appear to be similar in most respect to trades by anyone else. Suppose the rates are initially high and the Fed decides to reduce short term rates to around 2% and long-term rates to around 3%. Setting the short-term yields to 2% would be a routine matter, but it would also be normal for long-term yields to be considerably higher than 3% initially. A marginal reduction of long-term yields because of Fed purchases of them could cause investors to shift to shorter term - thus at least partially negating the effect of Fed purchases on long-term rates - and to maintain the 2% rate the Fed would have to sell short-term securities, or at least buy fewer than it would otherwise.  Figure 12.5.3.1 shows that as the Fed began to buy long-term securities in early 2009, holdings of short-term securities were reduced, and after 2009 they were considerably lower than before the crisis. Something similar happened in late 1947 when the Fed suddenly began buying longer-term bonds and cut back on purchase of short-term bonds. (Figure 7.1.5). This caused no change in yield curves (Figures 6.2.1 and 6.3.1).

Figure 12.3.3.1. Federal Reserve holdings by maturity, 2008-2011. Data:  [34], [39]

If the Fed holds federal funds and discount rates constant and is able to set long-term rates so low that expected bank profits are squeezed (assuming bank lending rates follow market rates), apparently long-term bank lending will dry up. It was hypothesized in Section 12.1.3 that expected bank profits are in effect calculated with respect to expected future central-bank rate, and earlier in Section 12.1.1 that the expected future rate is estimated mostly by reference to current and past rates. Then at times when past central-bank rates were higher than the current rate, the estimate of future central-bank rate will be higher than the current rate and bank lending may dry up even though the current spread between short- and long-term rates seems normal or exceptionally high. We see from this why banks may not stop long-term lending immediately when the yield curve becomes flat or negative as the central bank is raising short-term rates, and conversely why they may not begin to increase lending immediately when the central bank begins to cut rates and the current yield curve goes strongly positive. However, as discussed in Section 10.2, when the yield curve has gone flat or negative there have usually been contractions in lending.

Thus attempting to maintain too flat a yield curve may not be successful on the long-term end, unless the Fed entirely takes over the market, and if successful could essentially set the Fed at cross-purposes with itself and with private banks with respect to modifying the money supply.  The empirical evidence mentioned in Section 10.2 as well as the theoretical considerations from Section 12.1.3 suggest that too small a difference between long- and short-term rates could actually be harmful, as reducing the incentive for re-lending. The results of this paper suggest that the main effect of central-bank operations may be in immediate modification of the short-term end of the yield curve.  Of course if perception of demand is low, that is so-called liquidity-trap conditions exist, interest rates and/or re-lending may not be the limiting factors in investment.

If market participants can be completely convinced that the Fed intends to reset the entire complex of rates at some new set of levels consistent with the maturity premium function, the process of readjustment might be speeded up; but such conviction would have to apply to a period concomitant with the term of a given security. For someone to be convinced  that when the Fed or other central bank changes its rate it will hold that rate constant for 30 years would be absurd. There appear to be no reasons why bond investors would value the current promises of monetary authorities over past experience when it comes to predicting future central-bank rate over a long term; central banks rarely hold to a given policy for more than a few years, unless it is the policy of holding the rate low and constant as in most countries 1934-1955. Assuming that the second and third terms of equation (1) are a prediction of future central-bank rate as proposed in Section 12.1.1, the longer the term of the bond the less investors place faith in the current rate and the more they use past rates for this prediction.

The type 3 deviations (Section 3.2) involve yield movements of up to two percent over periods generally less than about 2 years. The possibility that these movements are due to some kind of supply/demand changes cannot be completely ruled out, though the type of changes which might be responsible have not been identified. Thus it is possible though unlikely that similar movements could be induced directly by Fed open-market operations on long-term securities. On the other hand, if these yield movements are due to attempts by the market to guess at future central-bank rate they are influenced by the perceived intentions of the Fed, so a program of buying long-term securities may reassure the market about the intention of the central bank to keep rates low. Keeping the market guessing about the long-term intentions of the central bank is an inducement to instability of yields, as recognized by Keynes and other theorists. Of course a real understanding of the factors which control rates might be beneficial but still would not tell investors what the central bank will do in the future.

There is certainly no empirical evidence as yet that central banks can control long-term rates by direct open-market operations on them rather than through fixing the minimum lending rate as in equation (1), or that they can control long-term rates as closely as they control short-term rates The Bank of Japan attempted such a policy starting in early 2001 and there were no detectable effects on long-term bond rates, which were already low (Appendix C, Section 13).The formal announcement of the first and second programs of buying long-term securities in the U.S. in 2009 and 2010 and the actual commencement of these programs were not followed by immediate decrease in yield rates, but by increase. Such overall decreases as did occur over the time interval in question are accounted for by equation (1) within its determined standard error, as shown in Section 11. Most central banks throughout the world cut short-term rates severely during the 2008-2009 crisis but did not generally buy long-term bonds, and in most cases long-term rates followed as in the US.  In Europe some countries such as Greece, Spain and Ireland experienced distinctly higher rates, but this was clearly a result of an increased in perceived default risk. In some other countries with better outlooks, such as Germany, rates dropped even more than predicted by equation (1) (see Appendix C), likely because their default risk was perceived as comparatively less.

The shape of the observed Treasury yield curve in early 2013 may seem anomalous, with a rather low slope at low maturities, but this type of curve is predicted at near-zero central-bank rate - see Figures 6.1.1 and 6.4.1b. There is no reason to conclude that it results from the Fed's buying of mid-maturity securities.

As mentioned in Section 3, using the 1-year Treasury rate instead of federal funds as the independent variable in the regression improves the fit considerably for group (1), essentially halving the unexplained variance. This suggests that targeting the 1-year instead of shorter rates might give the central bank a tighter control of long-term rates, although it is not clear that this would be possible if the 1-year rate is set by the market entirely according to expectations of shorter rates.  Doing this might also require changes in the volume of issuance of government securities, since the very short-term securities play a special role in finance.

12.3.4. The Nature of Maturity Premiums

The empirical results indicate that the maturity premium function for a given type of security and rating is actually fairly constant over time, not subject to change because of inflation or supply-and-demand influences though sensitive to perception of risk, and that it may be critically important in determining interest yields. What factors contribute to maturity-premium functions? Several things can be identified or postulated, but it is hard to draw firm conclusions about their actual importance. Actual reasons for maturity premiums do not have to be based on correct knowledge and reasoning nor do they even have to be rational. Following are comments on - not a complete review of - some supposed or likely contributors to maturity premiums.

Re-lending. The influence of the necessary costs and profits of re-lending is inescapable (Section 12.1.3); the only question is how much of the actual maturity premium function it accounts for. These costs and profits must balance the advantage to the borrower of long-term rates in terms of hedging his risk, as described by Hicks (1946, Ch. 11).

Inflation. If the risk of inflation were largely responsible for variation in interest rates this should have certain effects as discussed in Section 8, and these are not observed. The belief that yields are affected by changing rates of inflation may have arisen only after central banks began to adjust rates systematically in parallel with inflation. However, it is conceivable that if there is a moderate, essentially constant long-term expectation of inflation this could contribute to the maturity premium function. Such an expectation would be entirely rational since prices have generally moved upward over the years, though not in any regular fashion.

Risk of future rate changes. The value of securities offered for resale on the market depends on the current rates, which may change, and long-term securities involve more potential for loss or gain than short-term. This is considered to introduce a risk or uncertainty premium. Part of the justification for this seems to have been that rates were supposed to be directly susceptible to variations in inflation and supply and demand. While these suppositions have been shown in this paper to be incorrect, rates have nevertheless been subject to variation at the discretion of the central bank. In any case, such a risk premium would be expected to decrease if central-bank and other rates hold reasonably constant for a long period, as they did for about 30 years after 1934 (Figures 6.2.2 and 6.3.1). The slopes of the observed yield curves did decrease slowly over this time, but this decrease is reproduced by the equation, which has no time dependence of the k₁ coefficient, although the decision of which coefficients to vary with time was somewhat arbitrary. If the data are broken into periods, the slope of the maturity premium function, k₁ for 10+year minus k₁ for 3-6-month Treasuries, is actually greater for the 1919-1955 period (1.81%) compared to the later 1956-present period (1.20%) which had much greater rate variability. The period 1934-1955, which has the least rate variability, has even greater slope (1.90%). However, these two Treasury series are the only series obtained so far which differ only in maturity and are complete over the entire time range (and then only by splicing several compilations).

Default risk.  Default risk itself is normally incorporated into ratings and yields, but if the perception of default risk of an issuer increases, whether or not securities are re-rated, its outstanding long-term issues lose more value than its short-term issues. On the other hand, when general perception of risk increases, Treasuries and the highest-rated of other securities may gain value. The amounts involved were not trivial in the 2008-2009 credit crisis. Thus there should be a maturity premium for possible changes in perceived default risk.

Panic risk. Risk associated with liquidity differences has entered into the financial system in different ways at different times. Before the establishment of federal deposit insurance in 1933, banks were subject to the risk of panic runs by depositors or other short-term lenders. The lack of liquidity on the long-term side arose largely because mortgages were not generally traded. The difference between interest rates paid on deposits and interest rates charged for long-term loans presumably included a premium for this. Deposit insurance greatly reduced the risk to both depositors and lending institutions but did not eliminate the premium, simply changed it into an actual insurance premium that has to be paid by banks and S&Ls to the FDIC. More recently, implicit government guarantees, which are a sort of taxpayer-supplied insurance, have come into play - mainly in the bailout of the S&L insurance corporation after the failures of 1982-1992 and in the bailout in 2008-2009 of investment banks, commercial banks, the government-sponsored mortgage institutions (FNMA and FHLMC) and insurance companies, but also in sundry other individual rescues and rescue attempts over the years. In principle this should reduce the maturity premium. On the other hand banking has become increasingly oligopolistic as the number of banks has shrunk through failures and mergers, which has probably raised profit margins. Mortgages had become increasingly marketable in packages, which in principle could reduce maturity premiums, but this process went awry when their risk (rating) was not assessed correctly in the face of a housing and real-estate asset bubble. How these financial changes will affect maturity-premium functions remains to be seen. Panic risk may be considered to be an independent influence, or simply part of the costs of re-lending which influence the maturity premium function as in Section 12.1.1.

An important constraint on hypotheses about maturity premiums is that the relationship between short- and long-term rates appears to have been quite different in the 19th century, even in countries which had central banks (Appendix B).

12.3.5. Comprehensive Theories Involving Interest Rates

12.3.6 Monetary Standards

The basic monetary standard of the U.S. changed at least twice over the existence of the Federal Reserve, but no discrete, direct effect of these changes on the relationships has so far been conclusively identified. In 1933 the valuation of gold versus the dollar was changed, ownership of gold was made illegal and all gold in the U.S. was confiscated. The U.S. formally retained the gold standard and the currencies of most industrialized countries were pegged to that of the U.S. in 1946, and finally in 1971 all vestiges of the gold standard were abandoned, as was the Bretton Woods agreement to keep international monetary ratios usually constant.

Britain went off the gold standard in 1914, back on in 1925 and off again in 1931. The data on long-term rates for this time are meager and yearly only (Appendix C, Section C.4) but certainly any effects of these changes are no larger than fluctuations at many other times.

The lack of discontinuities does not mean that standards had no indirect influence on interest rates, through the central bank. There have been varying legal limitations to how much the Fed has been able to exercises its basic tools. The raising of rates in late 1931 was a response to the abandonment of the gold standard by Britain and some other countries, and the U.S. revaluation of gold in 1933 probably allowed the Fed to decrease rates. It seems very likely that the end of the U.S. gold standard and the Bretton Woods agreement in 1971 not only gave the Fed and other central banks increased freedom, but perhaps also the perception of responsibility to vary interest rates for one reason or another.

12.3.7 International Risk Effects

Credit and banking crises and other events can affect the perception of risk in the debt securities of one country relative to those of others and this obviously affects yields. Through 2010, 2011 and 2012 the default risk of government bonds in some countries such as Greece, Ireland and Spain was intermittently perceived to be elevated and the yields rose while yields on the bonds of less-risky countries such as Germany may have fallen. It is possible that yields of U.S. bonds also fell to some extent at times for this reason. There does not appear, however, to be a consistent pattern to these variations; at times, such as late 2010, yields in all countries rose (despite announcement of the second program of quantitative easing in the US), while early in 2011 yields fell in some countries and rose in others. Variation in yields because of such events is not explicitly accounted for in equation (1).

How should international effects be considered - as changes in overall supply and demand as capital moves from more risky to less risky countries, or as quasi-permanent changes in the maturity premium functions of the respective countries? It seems that in the affected countries themselves the affect is on the maturity premium function, as the yield increases persist until the risk situation is resolved in one way or another. The affect on other countries is harder to assess as changes in supply and demand and/or risk are dispersed. So far no international effects have been conclusively identified in the yields of US securities, but they have not yet been sought in any detail. As yet no parameters have been identified to consistently correlate with the obvious default-risk effects after 2009 in some countries (Section 9).

The European Monetary Union probably presents additional complications because countries may run different deficits and issue greatly different amounts of national bonds. Assuming supply and demand can travel up and down the range of maturities, maintaining reasonably similar interest rates means that the single central bank must buy greatly different quantities of the different national bonds on the open market. This may conflict with perceptions of national interest, and if the central bank does not buy enough of a country's bonds to keep rates down, this in turn may feed into perceptions of default risk.

13. Conclusions

13.1. Answers to the Questions.

Correlation does not necessarily prove causation, but lack of correlation can disprove causation. Yield rates are correlated to some extent with inflation, but only during certain periods when central banks deliberately raised their policy rates in counter to inflation. When central banks did not pursue this policy, the correlation is completely absent. There is no significant correlation of yield rates with government deficits or debt at any time in the U.S. or apparently in any other country when there was no imminent threat of default.

For U.S. Treasuries since 1953 the simple regression on central-bank rate alone, that is one term for current rate and one term for past rates, accounts on average for almost 95% of the variance. With the addition of a risk term and linear time variation of one coefficient, results as good or better are obtained for all types of securities from 1919 to the present. In the case of simple correlation there may be uncertainty about the direction of causation, but in the case of the relationship between central-bank rate and market yields the temporal relationship is unambiguous; the term or terms in past central-bank rate are essential to a good fit, and this applies to the entire period over which both central-bank rate and market yield are available (1919-present). The dependence on past central-bank rate is what accounts for variation in the slope of yield curves, which has hitherto been explained only by invocation of clairvoyance on the part of the bond market. The simple equation only predicts long-term interest rates within certain limits - a standard error of about 0.6-0.7% yield under current conditions - but the limitations are understandable in the context of Hicks' theory because many as-yet unpredictable factors may influence the expectation of future short-term rates.

The observed relationships among interest rates, inflation and various aspects of supply and demand lead inescapably to conclusions about causation and the questions posed in the Introduction have been answered quantitatively. Regarding the main issue of the influence of the central bank, yields even for long-term securities are mostly determined by central-bank action. The Fed cannot set the yields of all Treasury securities to anything it wants at a given time (at least not without massive and unprecedented open-market operations, which may present unexpected complications; see Section 12.3.3); when it changes the federal funds and/or discount rates, yields respond less rapidly the longer the maturity, so the market does play a role. But this graduated response is largely of a consistent, predictable nature, which this paper shows has been almost constant over long periods of time through all sorts of economic conditions.

Differential perception of risk can change yields considerably at times. A further presumed influence is the attempts of issuers and investors to guess future central-bank actions on criteria other than present and past rates. What is most important is that yields are set within the stated standard errors by the cumulative actions of central banks and by differential perception of risk or other factors not directly related to overall supply of and demand for credit or securities.

The subsidiary questions are also answered:

1) Overall supply of and demand for securities have no demonstrable effect on yields as long as central banks operate to control short-term yields. That central banks furnish supply and demand to maintain short-term rates at the chosen levels is obvious; mechanisms which in effect transfer supply and demand from long- to short-term exist and are implicit in theories of Keynes and Hicks. The operation of banks and other re-lenders must embody such mechanisms.

2) There is no valid evidence for any meaningful direct effect of inflation or expectation thereof on market yields or on the yield curve, except in the case of inflation-protected securities. Investors have no power to keep yields positive with respect to inflation.

3) The actual yield curve is also set mostly by the cumulative actions of central banks and not directly by inflation, supply and demand, or any prediction on the part of investors about the economy. The regression results can be viewed in terms of a maturity premium function which seems to be largely invariant over time and insensitive to supply-and-demand or inflation. The correlation of a flat actual yield curve with subsequent recession in the U.S. is due to raising of interest rates by the Fed to counter inflation or overexpansion; the aftereffects of such inflation or overexpansion, combined probably with effects of the elevated rates, cause recession.

The conventional wisdom on these points, as represented by the quotes in the Introduction, is incorrect. The assumptions of the conventional wisdom are simply not consistent with the actual data on yield rates. This is not a model-dependent conclusion; see, for example Figures 7.1.1, 7.1.4, 7.2.1, 7.3.1, 8.1.2 and 8.4.1. To some extent the major empirical conclusions in the paper can be asserted by inspection of historical data without mathematical analysis, but the simple regression equation is important in showing that the same relationships have prevailed over the entire existence of the Federal Reserve, and for long periods of time - to up to 75 years - with no change in coefficients.

A point not directly related to the determination of interest rates is that there is no good evidence that long-term interest rates have a strong direct effect on investment or economic growth; rather the main effect of interest rates appears to be in the difference between long- and short-rates. Most of the evidence is from the period of the late 1960's to early 1990's, since long-term rates at least did not vary greatly at other times.

A somewhat more general though tentative point is that consensus estimates of the future value of economic parameters such as central-bank short-term policy rate or inflation rate tend to be based on current and recent-past values of that parameter, not on extrapolations using recent rate of change or other complex techniques of prediction or on policy statements of authorities.

Interest-rate literature. A great deal has been written about interest rates, and no attempt has been made in this paper to review all of this work. But it appears that if the basic empirical relationships had been understood the conventional wisdom as described in the Introduction would not be what it is. How should conflicts between the conclusions of this paper and previous work be resolved?

Any calculation or model which purports to demonstrate or measure the effects of inflation, government deficits and/or debt, any other supply and demand variable or independent variable of any kind in the United States, should be applied to the entire period of existence of the Federal Reserve and have success at least as good as equation (1) as given in the tables and figures of Section 5. That is, such calculation should account for comparable fractions of the variance over the entire time range. Calculations or models based on correlations over a selected term, or which deal with only a small fraction of the variance, may be essentially meaningless.

13.2. Final Remarks

Many powers are typically attributed to the Federal Reserve and other central banks; they are supposed to be able to counter inflation, control unemployment, stimulate investment, prevent or ameliorate recession, control international exchange rates and perhaps do other things. That the Fed does have influence in most of these areas is indisputable, but a given action by the Fed can be quite complex in its effects and whether or not such action has the intended or assumed outcome is often highly questionable. Yet ironically the powers which the Fed unequivocally has on the basis of straightforward empirical evidence, that is the powers to determine yields of all maturities and to modify the slope of actual yield curves, have been routinely overlooked, denied, misunderstood or undervalued. Past actions of central banks appear in a different light when the nature of the relationship between central-bank rates and market yields is understood.

References.

Data Sources. (referred to in the tables and figures in brackets)