Salomon Brothers Understanding the Yield Curve Part 7

Salomon Brothers Antti Ilmanen (212) 783-5833 Raymond Iwanowski (212) 783-6127 riwanowski@sbi.com The Dynamics of the Shape of the Yield Curve: Empirical Evidence, Economic Interpretations and Theoretical Foundations Understanding the Yield Curve: Part 7 Salomon BrothersFebruary 1996 T A B L E O F C O N T E N T S P A G E Introduction 1 How Should We Interpret the Yield Curve Steepness? 2 • Empirical Evidence 3 • Interpretations 5 • Investment Implications 8 How Should We Interpret the Yield Curve Curvature? 8 • Empirical Evidence 9 • Interpretations 13 • Investment Implications 16 How Does the Yield Curve Evolve Over Time? 17 • Time-Series Evidence 17 • Cross-Sectional Evidence 20 Appendix A. Survey of Term Structure Models 22 • Factor-Model Approach 22 • Arbitrage-Free Restriction 25 • One Example: The Vasicek Model 26 • Comparisons of Various Models 27 Appendix B. Term Structure Models and More General Asset Pricing Models 31 References 33 F I G U R E S 1. Evaluating the Implied Treasury Forward Yield Curve’s Ability to Predict Actual Rate Changes, 1968-95 3 2. 60-Month Rolling Correlations Between the Implied Forward Rate Changes and Subsequent 4 Spot Rate Changes, 1968-95 3. Evaluating the Implied Eurodeposit and Treasury Forward Yield Curve’s Ability to Predict 5 Actual Rate Changes, 1987-95 4. Average Business Cycle Pattern of U.S. Realized Bond Risk Premium and Curve Steepness, 1968-95 6 5. Treasury Spot Yield Curves in Three Environments 9 6. Correlation Matrix of Yield Curve Level, Steepness and Curvature, 1968-95 9 7. Curvature and Steepness of the Treasury Curve, 1968-95 10 8. Curvature and Volatility in the Treasury Market, 1982-95 11 9. Average Yield Curve Shape, 1968-95 12 10. Evaluating the Implied Forward Yield Curve’s Ability to Predict Actual Changes in the Spot Yield Curve’s Steepness, 1968-95 12 11. Average Treasury Maturity-Subsector Returns as a Function of Return Volatility 14 12. Mean Reversion and Autocorrelation of U.S. Yield Levels and Curve Steepness, 1968-95 18 13. 24-Month Rolling Spot Rate Volatilities in the United States 19 14. Term Structure of Spot Rate Volatilities in the United States 20 15. Basis-Point Yield Volatilities and Return Volatilities for Various Models 21 February 1996Salomon Brothers 1 I N T R O D U C T I O N How can we interpret the shape (steepness and curvature) of the yield curve on a given day? And how does the yield curve evolve over time? In this report, we examine these two broad questions about the yield curve behavior. We have shown in earlier reports that the market’s rate expectations, required bond risk premia and convexity bias determine the yield curve shape. Now we discuss various economic hypotheses and empirical evidence about the relative roles of these three determinants in influencing the curve steepness and curvature. We also discuss term structure models that describe the evolution of the yield curve over time and summarize relevant empirical evidence. The key determinants of the curve steepness, or slope, are the market’s rate expectations and the required bond risk premia. The pure expectations hypothesis assumes that all changes in steepness reflect the market’s shifting rate expectations, while the risk premium hypothesis assumes that the changes in steepness only reflect changing bond risk premia. In reality, rate expectations and required risk premia influence the curve slope. Historical evidence suggests that above-average bond returns, and not rising long rates, are likely to follow abnormally steep yield curves. Such evidence is inconsistent with the pure expectations hypothesis and may reflect time-varying bond risk premia. Alternatively, the evidence may represent irrational investor behavior and the long rates’ sluggish reaction to news about inflation or monetary policy. The determinants of the yield curve’s curvature have received less attention in earlier research. It appears that the curvature varies primarily with the market’s curve reshaping expectations. Flattening expectations make the yield curve more concave (humped), and steepening expectations make it less concave or even convex (inversely humped). It seems unlikely, however, that the average concave shape of the yield curve results from systematic flattening expectations. More likely, it reflects the convexity bias and the apparent required return differential between barbells and bullets. If convexity bias were the only reason for the concave average yield curve shape, one would expect a barbell’s convexity advantage to exactly offset a bullet’s yield advantage, in which case duration-matched barbells and bullets would have the same expected returns. Historical evidence suggests otherwise: In the long run, bullets have earned slightly higher returns than duration-matched barbells. That is, the risk premium curve appears to be concave rather than linear in duration. We discuss plausible explanations for the fact that investors, in the aggregate, accept lower expected returns for barbells than for bullets: the barbell’s lower return volatility (for the same duration); the tendency of a flattening position to outperform in a bearish environment; and the insurance characteristic of a positively convex position. Turning to the second question, we describe some empirical characteristics of the yield curve behavior that are relevant for evaluating various term structure models. The models differ in their assumptions regarding the expected path of short rates (degree of mean reversion), the role of a risk premium, the behavior of the unexpected rate component (whether yield volatility varies over time, across maturities or with the rate level), and the number and identity of factors influencing interest rates. For example, the simple model of parallel yield curve shifts 2 Salomon BrothersFebruary 1996 is consistent with no mean reversion in interest rates and with constant bond risk premia over time. Across bonds, the assumption of parallel shifts implies that the term structure of yield volatilities is flat and that rate shifts are perfectly correlated (and, thus, driven by one factor). Empirical evidence suggests that short rates exhibit quite slow mean reversion, that required risk premia vary over time, that yield volatility varies over time (partly related to the yield level), that the term structure of basis-point yield volatilities is typically inverted or humped, and that rate changes are not perfectly correlated — but two or three factors can explain 95%-99% of the fluctuations in the yield curve. In Appendix A, we survey the broad literature on term structure models and relate it to the framework described in this series. It turns out that many popular term structure models allow the decomposition of yields to a rate expectation component, a risk premium component and a convexity component. However, the term structure models are more consistent in their analysis of relations across bonds because they specify exactly how a small number of systematic factors influences the whole yield curve. In contrast, our approach analyzes expected returns, yields and yield volatilities separately for each bond. In Appendix B, we discuss the theoretical determinants of risk premia in multi-factor term structure models and in modern asset pricing models. H O W S H O U L D W E I N T E R P R E T T H E Y I E L D C U R V E S T E E P N E S S ? The steepness of yield curve primarily reflects the market’s rate expectations and required bond risk premia because the third determinant, convexity bias, is only important at the long end of the curve. A particularly steep yield curve may be a sign of prevalent expectations for rising rates, abnormally high bond risk premia, or some combination of the two. Conversely, an inverted yield curve may be a sign of expectations for declining rates, negative bond risk premia, or a combination of declining rate expectations and low bond risk premia. We can map statements about the curve shape to statements about the forward rates. When the yield curve is upward sloping, longer bonds have a yield advantage over the risk-free short bond, and the forwards "imply" rising rates. The implied forward yield curves show the break-even levels of future yields that would exactly offset the longer bonds’ yield advantage with capital losses and that would make all bonds earn the same holding-period return. Because expectations are not observable, we do not know with certainty the relative roles of rate expectations and risk premia. It may be useful to examine two extreme hypotheses that claim that the forwards reflect only the market’s rate expectations or only the required risk premia. If the pure expectations hypothesis holds, the forwards reflect the market’s rate expectations, and the implied yield curve changes are likely to be realized (that is, rising rates tend to follow upward-sloping curves and declining rates tend to succeed inverted curves). In contrast, if the risk premium hypothesis holds, the implied yield curve changes are not likely to be realized, and higher-yielding bonds earn their rolling-yield advantages, on average (that is, high excess bond returns tend to follow upward-sloping curves and low excess bond returns tend to succeed inverted curves). February 1996Salomon Brothers 3 1 Another way to get around the problem that the market’s rate expectations are unobservable is to assume that a survey consensus view can proxy for these expectations. Comparing the forward rates with survey-based rate expectations indicates that changing rate expectations and changing bond risk premia induce changes in the curve steepness (see Figure 9 in Part 2 of this series and Figure 4 in Part 6). 2 The deviations from the pure expectations hypothesis are statistically significant when we regress excess bond returns on the steepness of the forward rate curve. Moreover, as long as the correlations in Figure 1 are zero or below, long bonds tend to earn at least their rolling yields. Empirical Evidence To evaluate the above hypotheses, we compare implied forward yield changes (which are proportional to the steepness of the forward rate curve) to subsequent average realizations of yield changes and excess bond returns.1 In Figure 1, we report (i) the average spot yield curve shape, (ii) the average of the yield changes that the forwards imply for various constant-maturity spot rates over a three-month horizon, (iii) the average of realized yield changes over the subsequent three-month horizon, (iv) the difference between (ii) and (iii), or the average "forecast error" of the forwards, and (v) the estimated correlation coefficient between the implied yield changes and the realized yield changes over three-month horizons. We use overlapping monthly data between January 1968 and December 1995 — deliberately selecting a long neutral period in which the beginning and ending yield curves are very similar. Figure 1. Evaluating the Implied Treasury Forward Yield Curve’s Ability to Predict Actual Rate Changes, 1968-95 Notes: Data source for all figures is Salomon Brothers (although Figures 3 and 11 have additional sources). The spot yield curves are estimated based on Treasury on-the-run bill and bond data using a relatively simple interpolation technique. (Given that the use of such synthetic bond yields may induce some noise to our analysis, we have ensured that our main results also hold for yield curves and returns of actually traded bonds, such as on-the-run coupon bonds and maturity-subsector portfolios.) The implied rate change is the difference between the constant-maturity spot rate that the forwards imply in a three-month period and the current spot rate. The implied and realized spot rate changes are computed over a three-month horizon using (overlapping) monthly data. The forecast error is their difference. 3 Mo. 6 Mo. 9 Mo. 1 Yr. 2 Yr. 3 Yr. 4 Yr. 5 Yr. 6 Yr. Mean Spot Rate 7.04 7.37 7.47 7.57 7.86 8.00 8.12 8.25 8.32 Mean Implied Rate Change 0.65 0.32 0.27 0.23 0.14 0.12 0.11 0.08 0.07 Mean Realized Rate Change 0.003 0.001 0.000 0.000 0.001 0.001 0.001 0.002 0.002 Mean Forecast Error 0.65 0.32 0.27 0.23 0.14 0.12 0.11 0.08 0.07 Correlation Between Implied and Realized Rate Changes -0.04 -0.08 -0.10 -0.08 -0.10 -0.13 -0.13 -0.12 -0.13 Figure 1 shows that, on average, the forwards imply rising rates, especially at short maturities — simply because the yield curve tends to be upward sloping. However, the rate changes that would offset the yield advantage of longer bonds have not materialized, on average, leading to positive forecast errors. Our unpublished analysis shows that this conclusion holds over longer horizons than three months and over various subsamples, including flat and steep yield curve environments. The fact that the forwards tend to imply too high rate increases is probably caused by positive bond risk premia. The last row in Figure 1 shows that the estimated correlations of the implied forward yield changes (or the steepness of the forward rate curve) with subsequent yield changes are negative. These estimates suggest that, if anything, yields tend to move in the opposite direction than that which the forwards imply. Intuitively, small declines in long rates have followed upward-sloping curves, on average, thus augmenting the yield advantage of longer bonds (rather than offsetting it). Conversely, small yield increases have succeeded inverted curves, on average. The big bull markets of the 1980s and 1990s occurred when the yield curve was upward sloping, while the big bear markets in the 1970s occurred when the curve was inverted. We stress, however, that the negative correlations in Figure 1 are quite weak; they are not statistically significant.2 4 Salomon BrothersFebruary 1996 3 Figure 7 in Part 2 shows that the forwards have predicted future excess bond returns better than they have anticipated future yield changes. Figures 2-4 in Part 4 show more general evidence of the forecastability of excess bond returns. In particular, combining yield curve information with other predictors can enhance the forecasts. The references in the cited reports provide formal evidence about the statistical significance of the predictability findings. Many market participants believe that the bond risk premia are constant over time and that changes in the curve steepness, therefore, reflect shifts in the market’s rate expectations. However, the empirical evidence in Figure 1 and in many earlier studies contradicts this conventional wisdom. Historically, steep yield curves have been associated more with high subsequent excess bond returns than with ensuing bond yield increases.3 One may argue that the historical evidence in Figure 1 is no longer relevant. Perhaps investors forecast yield movements better nowadays, partly because they can express their views more efficiently with easily tradable tools, such as the Eurodeposit futures. Some anecdotal evidence supports this view: Unlike the earlier yield curve inversions, the most recent inversions (1989 and 1995) were quickly followed by declining rates. If market participants actually are becoming better forecasters, subperiod analysis should indicate that the implied forward rate changes have become better predictors of the subsequent rate changes; that is, the rolling correlations between implied and realized rate changes should be higher in recent samples than earlier. In Figure 2, we plot such rolling correlations, demonstrating that the estimated correlations have increased somewhat over the past decade. Figure 2. 60-Month Rolling Correlations Between the Implied Forward Rate Changes and Subsequent Spot Rate Changes, 1968-95 Notes: The Treasury spot yield curves are estimated based on on-the-run bill and bond data. The implied rate change is the difference between the constant-maturity spot rate that the forwards imply in a three-month period and the current spot rate. The implied and realized spot rate changes are computed over a three-month horizon using (overlapping) monthly data. The rolling correlations are based on the previous 60 months’ data. C or re la tio n C oe ffi ci en t Correlation C oefficient -0.4 -0.2 0.0 0.2 0.4 0.6 0.8 -0.4 -0.2 0.0 0.2 0.4 0.6 0.8 Five-Year Spot Rate Two-Year Spot Rate Three-Month Spot Rate 72 12 31 77 12 31 82 12 31 87 12 31 92 12 31 In Figure 3, we compare the forecasting ability of Eurodollar futures and Treasury bills/notes in the 1987-95 period. The average forecast errors are smaller in the Eurodeposit futures market than in the Treasury market, reflecting the flatter shape of the Eurodeposit spot curve (and perhaps the systematic "richness" of the shortest Treasury bills). In contrast, the correlations between implied and realized rate changes suggest that the Treasury forwards predict future rate changes slightly better than the February 1996Salomon Brothers 5 4 However, some other evidence is more consistent with the expectations hypothesis than the short-run behavior of long rates. Namely, long rates often are reasonable estimates of the average level of the short rate over the life of the long bond (see John Campbell and Robert Shiller: "Yield Spreads and Interest Rate Movements: A Bird’s Eye View," Review of Economic Studies, 1991). Eurodeposit futures do. A comparison with the correlations in Figure 1 (the long sample period) shows that the front-end Treasury forwards, in particular, have become much better predictors over time. For the three-month rates, this correlation rises from -0.04 to 0.45, while for the three-year rates, this correlation rises from -0.13 to 0.01. Thus, recent evidence is more consistent with the pure expectations hypothesis than the data in Figure 1, but these relations are so weak that it is too early to tell whether the underlying relation actually has changed. Anyway, even the recent correlations suggest that bonds longer than a year tend to earn their rolling yields. Figure 3. Evaluating the Implied Eurodeposit and Treasury Forward Yield Curve’s Ability to Predict Actual Rate Changes, 1987-95 Notes: Data sources are Salomon Brothers and Chicago Mercantile Exchange. The Eurodeposit spot yield curves are estimated based on monthly Eurodeposit futures prices between 1987 and 1995. The Treasury spot yield curves are estimated based on on-the-run bill and bond data. (Note that the price-yield curve of Eurodeposit futures is linear; thus, the convexity bias does not influence the futures-based spot curve. However, convexity bias is worth only a couple of basis points for the two-year zeros.) For further details, see Figure 1. 3 Mo. 6 Mo. 9 Mo. 1 Yr. 2 Yr. 3 Yr. 4 Yr. 5 Yr. 6 Yr. Eurodeposits Mean Spot Rate 6.32 6.40 6.48 6.58 6.98 — — — — Mean Implied Rate Change 0.16 0.18 0.19 0.20 0.20 Mean Realized Rate Change -0.02 -0.02 -0.02 -0.02 -0.03 Mean Forecast Error 0.18 0.20 0.21 0.22 0.23 Correlation Between Implied and Realized Rate Changes 0.39 0.18 0.11 0.06 0.02 Treasuries Mean Spot Rate 5.67 5.90 6.01 6.13 6.64 6.86 7.07 7.29 7.41 Mean Implied Rate Change 0.47 0.30 0.27 0.28 0.19 0.16 0.15 0.12 0.11 Mean Realized Rate Change -0.01 -0.01 -0.01 -0.01 -0.02 -0.02 -0.03 -0.03 -0.03 Mean Forecast Error 0.47 0.30 0.28 0.29 0.21 0.18 0.18 0.14 0.14 Correlation Between Implied and Realized Rate Changes 0.45 0.32 0.28 0.17 0.04 0.01 -0.01 0.01 0.01 Interpretations The empirical evidence in Figure 1 is clearly inconsistent with the pure expectations hypothesis.4 One possible explanation is that curve steepness mainly reflects time-varying risk premia, and this effect is variable enough to offset the otherwise positive relation between curve steepness and rate expectations. That is, if the market requires high risk premia, the current long rate will become higher and the curve steeper than what the rate expectations alone would imply — t