Zero-Coupon Yields and the Cross-Section of Bond Prices

Transcription

1 Zero-Coupon Yields and the Cross-Section of Bond Prices N. Aaron Pancost First version: April 9, 2012 This version: November 20, 2012 Abstract I estimate the risk-neutral parameters of a three-factor affine term structure model and use it to price individual cross-sections of Treasury bond prices from 1990 to I find that the cross-sectional fit is worse during recessions; in particular the model has trouble pricing old, high-coupon bonds during the recent financial crisis. In December 2008 the average spread between over- and under-ten-year-old bonds was almost 40 basis points; by the end of 2011 it had narrowed to about basis points. This age effect explains roughly half of the variation in the model s yield residuals since The model underprices on-the-run bonds during the early 1990s but this effect is quite small in the last twenty years relative to the age effect. Keywords: bond data, term structure model, non-linear filtering, Great Recession JEL Classifications: G12, G14, C33 I would like to especially thank Michelle L. Barnes for her support and encouragement in the early stages of this project. I also thank my advisers, John H. Cochrane and Ralph S.J. Koijen, as well as Lars Peter Hansen, Francis Longstaff, Monika Piazzesi, Che-Lin Su, Harald Uhlig, Pietro Veronesi, Alexander Zentefis, and seminar participants at the University of Chicago Economic Dynamics working group. All remaining errors are my own responsibility. 1

2 1 Introduction Gaussian affine term structure models (ATSMs) have in recent years become an important mechanism for understanding and forecasting the yield curve. The no-arbitrage restrictions in these models can in principle use information from the cross-section of yields to help forecast future yields, going beyond the well-known failures of the pure expectations hypothesis. A solid statistical understanding of both the cross-section and time-series behavior of yields could then help explain the dynamic interactions between interest rates and the macroeconomy. While much progress has been made along this dimension, one aspect of these models has so far been overlooked: estimating them on actual bond prices. Ang and Piazzesi (2003), Wachter (2006), and Ang, Bekaert and Wei (2008) take as primitive yield data the Fama-Bliss interpolated yield curve, which uses zero-coupon bills to back out implied short-maturity forward rates, then computes longer-maturity forward rates by assuming that these are flat between observed maturities. One problem with this approach is that Fama and Bliss (1987) originally limited themselves to a maximum maturity of five years, because at longer maturities the distance between bonds in the data grows and the quality of the interpolation assumption deteriorates. Nevertheless Diebold, Rudebusch and Aruoba (2006), Diebold and Li (2006), Rudebusch and Wu (2008), Christensen, Diebold and Rudebusch (2011), and Joslin, Singleton and Zhu (2011) use the Fama-Bliss data, or the same interpolation strategy, to go out further than five years along the yield curve. Rather than rest on a forward-rate interpolation assumption, many papers instead use fitted yield curve data. A popular choice are the zero-coupon yields estimated by Gürkaynak, Sack and Wright (2007), which are updated at regular intervals and available for free on the Federal Reserve Board s website. These yields are estimated using a method proposed by Svensson (1995), which itself is an extension to longer maturities of the approach of Nelson and Siegel (1987). Kim and Orphanides (2005), D Amico, Kim and Wei (2010), and Hamilton and Wu (2010a) all use these yields to estimate their own yield curve models. In principle the estimated parameters from the Fed s website could be used to generate time paths for yields of any maturity. In practice researchers use only a handful of maturities, because there would be no extra explanatory power from using more: the yields that come out of this model reflect movement in only six underlying factors. Recognizing that it may not be appropriate to use fitted data from one model to estimate another, Cochrane and Piazzesi (2008), Duffee (2010), and Orphanides and Wei (2010) have tried to combine Fama-Bliss and Gürkaynak-Sack-Wright yields. Most obviously, data generated from a Svensson model cannot be used to estimate another Svensson model; this is presumably why Nelson-Siegel models such as Diebold, Rudebusch and Aruoba (2006) and Christensen, Diebold and Rudebusch (2011) rely on Fama-Bliss yields, and the only models that use fitted Gürkaynak-Sack-Wright estimates are ATSMs. 2

3 There is another option: go straight to the actual bond price data that Fama and Bliss (1987) and Gürkaynak, Sack and Wright (2007) used in the first place. For example, Hu, Pan and Wang (2010) examine in great detail the residuals from estimating a Nelson-Siegel model on the underlying CRSP prices. The main difference between their analysis and mine, besides the choice of model, is one of detail; while they document business-cycle variation in the model fit, as I do, they do not link it explicitly back to the characteristics of the individual bonds, such as the bond s age or coupon rate. Linking a model s failures directly to the data on which it was estimated can provide important clues as to what model simplifications are at odds with the data, and how the model might be improved. Using the prices of individual Treasury securities directly can also make estimation easier and, potentially, more efficient. Typical ATSMs can have more than a dozen parameters, depending on how prices of risk are parameterized, and maximizing likelihoods in such high-dimensional spaces can be numerically difficult. However, Andreasen and Christensen (2011) show that consistent estimates of hidden states and model parameters can be achieved by simply minimizing a sum of squared price or yield residuals, so long as the number of prices in each cross-section tends to infinity; a typical cross-section of Treasuries has over a hundred bonds, compared to (for example) the six degrees of freedom in the Gürkaynak, Sack and Wright (2007) yield curve or the seventeen maturities used by Diebold, Rudebusch and Aruoba (2006). In addition, Andreasen and Christensen (2011) show that the parameters that govern the risk-neutral dynamics can be estimated separately, yet still consistently, from the price-of-risk parameters; I exploit this separability to estimate only the risk-neutral part of a canonical ATSM. Andreasen and Christensen (2011) also present Monte Carlo evidence suggesting that large cross-sections with a non-optimal estimator can have lower root mean-squared error than optimal estimators using small cross-sections. Two recent papers that also look at the prices of individual Treasury securities are Krishnamurthy (2002) and Gürkaynak, Sack and Wright (2007). Both papers argue that on-the-run bonds those most recently issued by the Treasury enjoy higher liquidity, and thus lower yields, than other bonds of similar maturity. I also find significant spreads between bonds of different ages; however the age difference that I find most severely mispriced is much larger. While on-the-run bonds are around six months older than the closest corresponding off-the-run bonds, I find that an age difference of about ten to fifteen years best describes the model s price residuals, at least in the last twenty years. In this sense I advocate replacing the bond/old bond spread of Krishnamurthy (2002) with a bond/really old bond spread. To my knowledge this is the first paper to bring a no-arbitrage affine term structure model into contact with the actual CRSP Treasury data. One reason most of the ATSM literature may have avoided doing this is that it presents complications on the data side of the paper that have little to do with the paper s main motivation, such as incorporating survey data (Kim and Orphanides 2005), regime shifts (Ang, Bekaert and 3

4 Wei 2008), a return-forecasting factor (Cochrane and Piazzesi 2008), or macroeconomic factors generally (Ang and Piazzesi 2003). While the methodological contributions these papers have made are important, their demonstrated effectiveness and empirical plausibility are only as good as the data on which they are estimated. In particular, looking at how well one model fits another model s estimates overstates the degree to which it fits the true underlying data the only reason the fit isn t perfect, is that the two models are slightly different. Figure 1 illustrates such a situation. In the top panel I graph the Gürkaynak, Sack and Wright (2007) henceforth GSW zero-coupon yields in December 2008, along with a yield curve fitted to these data using the risk-neutral parameters given in Table 2 below. There is nothing remarkable about this picture. In the bottom panel I plot the actual yields for this month from the CRSP database against the GSW zeroes. Notice that on this date there were two separate yield curves, operating at the long and short end though with about a five-year overlap. The GSW zeroes navigate a middle ground between the two, and a model estimated on them would return an excellent fit, despite the strange behavior of the yield curve on this date. Importantly, this phenomenon is not an isolated incident; it begins roughly in mid-2008 and lasts until about the middle of Of particular interest is that the longer-maturity, higher-yield yield curve in the figure consists entirely of older, higher-coupon bonds each was issued over ten years ago, and has a coupon higher than 6%. The lower yield curve consists entirely of bonds issued within the last ten years, and except for a single security at the far left, all have coupons less than 6%. The two dotted blue lines in Figure 1 are yield curves estimated on the bonds older and younger than ten years, respectively. The situation graphed in Figure 1 may seem overly dramatic, coming as it does so close to the recent financial crisis and the fall of Lehman Brothers. But it illustrates well the fact that, as pointed out by Hu, Pan and Wang (2010), there is interesting time variation in the cross-sectional fit of yield-curve models. Figure 2 plots the sum of squared log-price residuals (divided by the number of bonds in each cross-section) for an ATSM fitted to the actual CRSP bond price data. As might be guessed from Figure 1, the model fit deteriorates drastically during the recent financial crisis; in addition it seems to fit relatively worse during the last two recessions. Notice also that the poor fit persists long after the official end of all three recessions, as dated by the NBER. In this paper I estimate individual cross-sections of bond prices using the risk-neutral part of a canonical affine term structure model. The main contribution of the paper is to show how this can be done consistently, on actual Treasury bond prices, without estimating the market prices of risk or, equivalently, the physical dynamics of the factors. I restrict myself to risk-neutral pricing in this paper for expositional reasons; there is a lively debate in the literature on the best way to restrict prices of risk in these models see for example Cochrane and Piazzesi (2008), Duffee (2010), Bauer (2011), or Joslin, Singleton and Zhu (2011) and my 4

5 4 GSW Zeroes and Fit on Estimated on 17Sep , re run on 22Sep yield Years to Maturity (a) GSW Zero-Coupon Yields in December Yield Curve on Estimated on 17Sep , re run on 22Sep yield CRSP data GSW zeroes duration (years) (b) Actual CRSP Yields in December 2008 Figure 1: Estimated Yield Curves In Late

6 12 x 10 5 Sum of Squared Log Price Residuals over Time Estimated on 17Sep Figure 2: ATSM Cross-Sectional Fit Over Time approach has nothing to add to it. Moreover, as should be evident from Figure 1(b) and as I document quantitatively below in Section 4, these models will not fit the underlying bond data regardless of how one specifies the prices of risk. If these models cannot even fit the cross-section of bond prices, then they certainly cannot forecast them. A second contribution of the paper is to show how a standard ATSM can be used to price coupon bonds, which make up all the Treasury securities with a maturity greater than one year. Correctly pricing such bonds requires dealing with annoying minutiae of the data, such as the day of the month and month of the year in which the bond pays its semi-annual coupon, and the accrued interest on the first coupon payment. Having to deal with prices of coupon bonds is a challenge of this paper s approach to estimation, but it is a strength rather than a weakness. After all, if the yield data at medium to long maturities consist entirely of coupon bonds, shouldn t we have an empirical model that can handle them? Although I restrict myself to risk-neutral pricing in this paper, the approach can be readily extended to include prices of risk. Given any set of risk prices (or equivalently, a time-series representation of the factors) and the already-estimated risk-neutral parameters, cross-sections can be priced to incorporate time-series 6

7 information using the particle or unscented Kalman filter. Filtering optimally resolves the tension between the time-series estimates of the factors (based on the factors estimated last period) and the factors that best fit today s cross-section (from the risk-neutral loadings). As I outline below, a particle or unscented Kalman filter is necessary because the measurement equation when using coupon bonds is nonlinear. Finally, while the Andreasen and Christensen (2011) SR approach applied to actual bond prices can be used to estimate any Nelson-Siegel or affine no-arbitrage model more complicated than the simple threefactor model below, it does rule out principal-components analysis, for example as done by Cochrane and Piazzesi (2008). This method is popular because it allows an ostensibly model-free way to compute the factor loadings. However, it requires an observation on each date of the same maturities in every cross-section, while the maturity structure in the actual Treasury data changes on each date: a 120-month bond this month becomes a 119-month bond next month, and the Treasury does not issue 120-month bonds every month. Without a single time series for the yield at each maturity, there is no way to create a correlation matrix whose eigenvectors represent the factor loadings. This is not really a problem, because one expects the 119-month bond to behave very similarly to the 120-month bond; how similarly, and in what ways, is exactly what the model is designed to tell us. 2 Model In the standard affine term-structure model, the factors that price zero-coupon bonds evolve over time as a vector autoregression: X t+1 = µ + φx t + Σε t+1, (1) where ε t+1 is an iid standard normal random variable. The stochastic discount factor M t+1 is assumed to be M t+1 = exp ( δ 0 δ 1X t 12 λ tλ t λ tε ) t+1 (2) λ t = λ 0 + λ 1 X t, where (δ 0, δ 1, λ 0, λ 1 ) are parameters and the time-varying market prices of risk λ t are an affine function of the factors. Let p (n) t be the log price at time t of a zero-coupon bond that pays $1 in n periods. The standard 7

8 asset-pricing relation is p (1) t = log E t [M t+1 ] p (n) t = log E t [M t+1 exp ( p (n 1) t+1 )] which, together with equations 1 and 2, implies the well-known recursion for log prices in affine models: p (n) t = A n + B nx t B n+1 = δ 1 + B nφ (3) A n+1 = δ 0 + A n + B nµ B nσσ B n where the parameters (µ, φ ) govern the risk-neutral dynamics of the factors, µ = µ Σλ 0 (4) φ = φ Σλ 1, and A 0 and B 0 are both zero. The recursions in equation 3 give the factor loadings; that is, the precise affine function of the state that prices any zero-coupon bond. However, because the model can price any cross-section of n riskless bonds with k < n factors, it exhibits a stochastic singularity: once I ve used k bond prices to determine the k factors on any given date, the prices of the other n k bonds are already determined, and these will almost surely be different than the actual observed prices. Thus estimating the model on actual yield data requires allowances for measurement error. While there are real measurement errors involved in calling the CRSP data the true bond prices unobserved bid-ask spreads, the fact that these are dealer quotes and not actual trades, etc a more sophisticated view of the model residuals is that they are specification errors. This places the burden of proof on the theory, rather than the data. Recognizing that all yields are priced with (specification) error allows me to relax the assumption that the data consist of individual yields at a handful of fixed, convenient maturities. In fact, there is no reason that the maturity structure of the data I use for estimation must be time invariant. If the vector of maturities at time t is an n t 1 vector τ t = (τ 1,t, τ 2,t,...τ nt,t), then equation 3 defines a time-varying n t (dim X t + 1) 8

9 measurement matrix Z t Z(τ t, θ) at each date that can be used to price zero-coupon bonds as p (τ1,t) t p (τ2,t) t... p (τn t,t) t } {{ } p t A τ1,t B τ 1,t A τ2,t B τ 2,t = } A τnt B,t τ nt,t {{ } Z t 1 X t + Σ M t η t (5) where Σ M t is the specification-error covariance matrix and η t is an n t 1 vector of standard independent normal random variables. The matrix Z t depends only on the vector τ t of maturities in the cross-section at time t and the risk-neutral parameters θ = {δ 0, δ 1, µ, φ, Σ}; thus the risk-neutral factors can be estimated from any cross-section, for given parameters, directly via OLS (GLS if Σ M is not spherical). Importantly, these estimates for the time-t factors are consistent for large cross-sections, regardless of the time-series behavior of the yield curve or the time-series parameters. Notice also that in using equation 5 there is no need for the maturity structure τ t to be constant over time, for there to be only one security of each maturity, or for the maturities to be spaced in any particular way. I do not literally believe that an environment sufficient for the existence of a representative agent, with an SDF of the specific functional form given in equation 2, was the data-generating process for these data. In applying equation 5 to the CRSP data I am instead assuming that (i) the specification errors η t are small, and (ii) that they are not correlated with objects or events of interest to us or the market participants. However, I argue below in Section 4 that both assertions are violated in these data; to the extent that (i) and (ii) fail, they can point us towards ways of improving the model. With price of risk parameters {λ 0, λ 1, Σ} or, equivalently, the time-series parameters {µ, φ, Σ} governing the factors, given the specification-error covariance matrices Σ M t, and assuming that the η t are i.i.d standard normal random variables, equations 1 and 5 define a state-space system on which one can apply the Kalman filter to estimate the state X t and price bonds. The system is straightforward except for the fact that the measurement matrix Z t and specification-error variance Σ M t vary over time as the number and maturity structure of the bonds in the data vary. In the case of coupon bonds things become more complicated. Pricing a coupon bond is no different than pricing a portfolio of zero-coupon bonds, and the price of a sum should be the sum of the prices. This is fine except that ATSMs give the log prices of bonds as an affine function of the state; the nonlinearity means that estimating the factors on a given cross-section is not as convenient as OLS. In addition, identifying the yieldto-maturity of a coupon-paying bond with the yield on a zero-coupon bond of the same maturity is incorrect, 9

10 because much of the value of a long-maturity coupon-bearing bond lies in the upcoming coupon payments. Thus the CRSP yields plotted in Figure 1 are plotted against their duration, rather than the maturity of their principal payment. However, this approximation is not necessary: for a given set of parameters and factors, the model returns the value of the principal and each coupon payment in the bond. For example, a two-year bond with a four-percent coupon has a payment of $1 in two years, and payments of 4 in six, twelve, eighteen, and twenty-four months. Each of these payments is individually a zero-coupon bond, and the complete portfolio can be priced using the loadings from equation 3 as P (24,4) t = P (24,0) t = e A24+B 24 Xt + ( ) P (6,0) t + P (12,0) t + P (18,0) t + P (24,0) t (e A6+B 6 Xt + e A12+B 12 Xt + e A18+B 18 Xt + e A24+B 24 Xt ). (6) Equation 6 cannot be used immediately to compute level prices of coupon bonds given the factors X t, for two reasons. First, it ignores accrued interest: Treasury bonds in my sample period pay coupons semi-annually, but purchasers are only entitled to a fraction of the first coupon payment proportional to the time span between purchase and that coupon payment. Second, many securities do not mature at the end of the month, but in the middle; for pricing coupon payments that occur in only one or two months, plus or minus two weeks can make a big difference. Neither refinement of equation 6 is difficult to handle. Accrued interest can easily be calculated and subtracted from the estimated price, and payments made within the month can be calculated by interpolating the factor loadings. The price of a Treasury bond that matures in τ months, with semi-annual coupon rate c, and that pays coupons and interest at fraction of the month d, when the factors are X, is P (X, τ, c, d) = P Z (X, τ + d 1) }{{} principal + c ( (τ b)/6 j=0 P Z (X, 6j + b + d 1) } {{ } coupons ) (1 f)p Z (X, b + d 1) }{{} accrued interest adjustment, (7) where b is the time to the next coupon payment in months, f = (b + d 1)/6 is the fraction of the first coupon that the purchaser earns, and P Z (X, τ) is the discount function the level price of a zero-coupon bond maturing in τ months given the factors X, using the loadings from equation 3: P Z (X, τ) exp(a τ + B τ X). (8) 10

11 Notice that the limit of summation (τ b)/6 in equation 7 will always be an integer, because all Treasury securities pay coupons in the same month of the year in which they eventually mature. Given a set of risk-neutral parameters, I estimate a cross-section of log bond prices p t by stacking equation 7 as p t = ( ) log P (X t, τ 1,t, c 1,t, d 1,t ) ( ) log P (X t, τ 2,t, c 2,t, d 2,t )... ( ) log P (X t, τ nt,t, c nt,t, d nt,t) } {{ } Z(X t,τ t,c t,d t) +Σ M t η t. (9) Given the physical dynamics of the factors and values for Σ M t, equations 1 and 9 define a state space system on which a filter can again be estimated. Because the function Z(, τ t, c t, d t ) in equation 9 is nonlinear, however, one cannot use the basic Kalman filter; a nonlinear filter such as the unscented Kalman or particle filter is required. Rather than use the filter, which requires estimates of the physical dynamics of the factors, in this paper I estimate equation 9 on each cross-section in the data, and choose to the risk-neutral parameters minimize the total sum (across cross-sections) of squared log-price residuals. 3 Data and Parameter Estimation Details of my sample of 31,023 CRSP bond prices from January 1990 to December 2011 are given in Table 1. I exclude tax-exempt, callable, flower, and consol bonds, as well as any bond with less than one year to maturity. I do not exclude on-the-run securities, which I define as the most-recently issued bonds at the 2-, 3-, 4-, 5-, 7-, and 10-year maturities, in order to better understand when, and how badly, the model misprices them. Each row of Table 1 gives details for particular sub-samples; the second column reports the average number of bonds in each cross-section for that sub-sample. As noted bu Hu, Pan and Wang (2010), there is a noticeable dip in the late 1990s and early 2000s in the number of Treasury bonds in the cross-section. The next column reports the average number of on-the-run bonds in each cross-section. The subsequent columns plot sub-sample means and standard deviations (in parentheses) of maturity, duration, coupon rates, and yield to maturity. There is a declining trend over my sample in yields and, with them, coupon rates, reflecting the Treasury s desire to issue bonds close to par. I estimate model parameters by minimizing the sum of squared log-price residuals; this is equivalent to 11

12 Date Range # Bonds # On-the-Run Maturity Duration Coupon Yield to Maturity Full Sample (3.18) (2.37) (2.92) (2.09) (3.14) (2.21) (2.38) (1.45) (2.34) (1.82) (2.34) (0.61) (3.6) (2.65) (2.74) (1.59) (3.43) (2.59) (2.29) (1.57) Table 1: Sample Summary Statistics maximum likelihood under the assumption of homoskedastic normal iid pricing errors, a common assumption in the literature. I choose the initial point for the parameter search to closely fit an eigenvector decomposition of fifteen GSW yields; I describe the details below in Appendix A. The standard errors reported in Table 2 are the asymptotic standard errors, whose calculation I describe in Appendix B. For simplicity and ease of exposition in this paper I focus on risk-neutral pricing; that is, I estimate equation 9 on each cross-section individually, taking Σ M t to be proportional to an identity matrix, without in any way using the time-series relation between cross-sections. In particular this means I do not apply the unscented Kalman filter on equatons 1 and 9 together; re-arranging the temporal order of my crosssections would have no effect on the estimates. For given risk-neutral parameters, these estimates of the unobservable factors are consistent for the same reason that non-linear least-squares is consistent; consistency of the estimates of the risk-neutral parameters {δ 0, δ 1, µ, φ } which apply to all cross-sections is proven by Andreasen and Christensen (2011). A crucial assumption for these estimators to be consistent is that the number of observed bond prices in each cross-section tends to infinity; this estimation strategy is not appropriate for the standard approach in the ATSM literature where only a fixed number of yields are observed in each cross-section. This is another strength of using the actual CRSP data, which (from Table 1) typically features more than a hundred prices in each cross-section. Not only does ignoring the time-series dimension make the calculation more transparent, it reduces the number of parameters I need to estimate from twenty-one (including λ 1, or equivalently, φ) to just thirteen, in {δ 0, δ 1, µ, φ }. To ensure identification I restrict φ to be lower triangular, and the last element of δ 1 to be positive. I don t need to estimate Σ because unobserved factors can always be pre-multiplied by Σ 1 to have an identity VAR covariance matrix technically speaking this uses time-series information from the sample, but only to normalize the factors. Multiplying the factors and loadings by a time-invariant matrix beforehand has no effect on the estimates. I estimate risk-neutral parameters that minimize the sum of squared residuals from the raw CRSP prices 12

13 µ φ curvature(t-1) slope(t-1) level(t-1) curvature(t) (0.046) ( ) slope(t) (0.024) ( ) ( ) level(t) (0.064) (0.0016) ( ) (2e-05) δ 0 δ 1 constant curvature slope level ( ) 0 (0.0026) ( ) 7e-05 (7e-05) Table 2: Estimated Risk-Neutral Parameters using the MATLAB function lsqnonlin, which is specially designed for minimizing sums of squares. My initial point is calibrated to fit the fifteen GSW yields from January 1990 to November 2010; other than this calibration, I use only CRSP price data, from January 1990 to December My estimated parameters, along with their asymptotic standard errors, are given in Table 2; the forward loadings which they define are plotted in Figure 3. These loadings have the traditional level, slope, and curvature shapes, even though the loadings defined by the initial point (plotted in the bottom right panel of Figure 8) do not. These are the parameters I use to create Figures 1 and 2 and in the regressions below. 4 Deviations from the Law of One Price A common charge against affine term structure models is that they overfit the data; without restrictions, fully-flexible ATSMs may fit themselves so well to sample-specific patterns that they imply ridiculous risk prices or Sharpe ratios (Duffee 2010), or fail to use the no-arbitrage restrictions to help with forecasting, instead using the VAR(1) forecasts to estimate the risk prices (Joslin, Singleton and Zhu 2011). Figures 1 and 2 belie this conclusion. These plots are generated from the best cross-sectional fits possible from a maximally-flexible three-factor ATSM, because in calculating them I have not specified the prices of risk a fully-specified model, even one with prices of risk derived from an unconstrained VAR(1) of the factors, can only do worse by trading off some of the cross-sectional fit in order to better fit the time series. Moreover, there are readily-quantifiable patterns in the mistakes that the model makes patterns that suggest either deviations from the law of one price, or (more likely) model misspecification. In this sense the 13

14 6 x Iteration: 1: Forward Factor Loadings Estimated on 17Sep curvature slope level Years to Maturity Figure 3: Estimated Forward Loadings common perception that maximally-flexible ATSMs fit the data too well is incorrect. For example, I estimate below a spread of about 40 basis points between older, higher-coupon bonds and more-recently-issued, lowercoupon bonds during the height of the crisis. In principle an arbitrageur could short the younger bonds and to finance buying the older bonds, netting an immediate profit, and in addition earn the coupon spread as long as she held the position. This has nothing to do with the model over-fitting, and moreover 40 bps is economically very large, regardless of its statistical significance. Age and Coupon Effects As mentioned in the introduction, the two yield curves evident in the CRSP data from late 2008 through 2009 are split by age coupon rate: the bonds in the upper yield curve were all issued over fifteen years prior and have coupon rates above 6%, and those in the lower curve were all issued less than ten years ago and have coupons below 6% (one bond has a coupon equal to 6%). To quantify how systematic this pricing error is over time, I regress each cross-section s yield residuals on a constant and a dummy variable equal to 14

15 one if the bond in question was issued over ten years ago, and zero otherwise. 1 The value of this regression coefficient over time an age factor is plotted in the top panel of Figure 4. The two-standard-deviation lines plotted in the figure come from the standard OLS variance estimates. 2 Notice that the age factor is especially high during the recent financial crisis this is clear from Figure 1 but that it begins to surpass its previous sample maximum value as early as May 2008, well before the Lehman event. I have normalized the values plotted in the top panel of Figure 4 so that they are interpretable in basis points; the value in December 2008 is almost 40 bps, well in accord with an eyeball measurement using Figure 1. While I have yet to determine formally whether this value is statistically significant, an average 40-bps spread between bonds that hitherto had close to identical yields (less than 10 bps for much of the sample) is certainly economically significant. Moreover the age factor explains roughly half of the variation in the model s pricing errors, at least since 2005, as can be seen in the bottom panel of Figure 4. The distance of the bond-age coefficients from zero is an indication of the degree to which bonds issued over ten years ago have higher yields (lower prices) than predicted by the term structure model. If one truly believes in the single model-implied discount function of equation 8, then a level of this age factor far from zero represents a deviation from the law of one price, because regardless of its age any coupon bond is just a portfolio of zero-coupon bonds whose price derives from the discount function. One possibility, that raised by Hu, Pan and Wang (2010), is that during times of market stress arbitrageurs are unable to raise enough capital to smooth away all such arbitrage opportunities, which arise because some market participants have preferences for some habitats for example, more recently-issued bonds. The age factor plotted in Figure 4 is a useful statistical description of the systematic pricing errors of the maximally-flexible ATSM; however it was picked particularly to fit the observed abnormal behavior of Treasury securities in December A natural question to ask is, how much of the model residuals can be explained with a more natural set of bond characteristics? To answer this question, in Figure 5 I plot the results from regressing yield residuals on a constant and either each bond s age in months (top two panels) or each bond s coupon rate in percent (bottom two panels). The left panel plots the estimated coefficients and two-standard-deviation bands, while the right panel plots the regression R 2 over time. Steadily declining interest rates over the sample period, combined with the Treasury s desire for newly-issued bonds to price close to par, mean that the age of bonds and their coupon rate are highly correlated, and thus the plots are almost identical (except for the scale of the y-axes). In fact, I m not sure any empirical evidence from 1 The figures using the log-price residuals look virtually identical, but the magnitudes are not readily interpretable in basis points. 2 These estimates are not formally correct, as they do not account for serial correlation. However, correcting for serial correlation requires keeping track of over a thousand individual securities across cross-sections. The plotted standard errors, while incorrect, are nevertheless a useful gauge of the strength of the correlation between the model s residuals and bond-specific characteristics. I leave careful and precise estimation of the correct standard errors for these regressions to interested future researchers. 15

16 60 Yield Residual Coefficients over Time Variable: Old Bond Dummy Regressors: Constant / Old Bond Dummy Estimated on 17Sep , re run on 22Sep coefficient (bps) month (a) Old-Bond Dummy Coefficients over Time 0.9 R Squared over Time Regressors: Constant / Old Bond Dummy Estimated on 17Sep , re run on 22Sep R Squared month (b) Old-Bond Dummy R 2 over Time Figure 4: OLS Regressions of Model Residuals on a Constant and Old-Bond Dummy 16

17 0.25 Yield Residual Coefficients over Time Variable: Bond Age (months) Regressors: Constant / Bond Age (months) Estimated on 17Sep , re run on 22Sep R Squared over Time Regressors: Constant / Bond Age (months) Estimated on 17Sep , re run on 22Sep coefficient (bps) R Squared month month (a) Age (months) Coefficients over Time (b) Age (months) R 2 over Time 10 Yield Residual Coefficients over Time Variable: Coupon Rate (%) Regressors: Constant / Coupon Rate (%) Estimated on 17Sep , re run on 22Sep R Squared over Time Regressors: Constant / Coupon Rate (%) Estimated on 17Sep , re run on 22Sep coefficient (bps) 4 R Squared month (c) Coupon Rate (%) Coefficients over Time month (d) Coupon Rate (%) R 2 over Time Figure 5: Bond Age and Coupon Rate as Regressors 17

18 this sample, at least could disentangle whether the pricing errors come from the age of the bonds, or their coupon rates. Whether the model s systematic pricing errors stem from age or coupon effects matters greatly for attempts to improve the theoretical model. An age factor suggests a liquidity story, but a coupon factor has a (potentially) even simpler explanation: it s possible that the market is perfectly liquid, but couponbearing securities are just not always well modeled with a single discount function that prices payments of $1 at all maturities in particular, perhaps the size of the payment (the coupon rate!) matters in ways that do not scale linearly. One issue with this interpretation is that it fails to address why the model fit appears to fluctuate with the state of the business cycle, as in Figure 2. Nevertheless, the fact that in times of macroeconomic stress the single discount-function model does not accurately price coupon bonds is highly suggestive. On- vs. Off-the-Run Bonds Krishnamurthy (2002) shows that newly-issued Treasury bonds typically have lower yields than other bonds of comparable maturity; indeed, Gürkaynak, Sack and Wright (2007) exclude these on-the-run securities from their estimation, under the impression that they are considerably more liquid than other bonds. To investigate whether these effects are present in my own sample, in the bottom panel of Figure 6 I again plot the yield curve in December 2008, with bonds issued over ten years ago plotted as blue circles, bonds issued less than ten years ago but off-the run plotted as black dots, and the on-the-run bonds plotted as red X s. The top panel of Figure 6 plots the same data in March 1991, for comparison. On the earlier date the on-the-run bonds have lower yields, most dramatically at the long end; bonds that have been around for more than ten years are neither systematically over- or under-priced relative to newer bonds with the same duration. In the 2008 data, on the other hand, the largest spreads seem to be between old and really old bonds of comparable maturity. Figure 7 quantifies these conclusions over time; the top two panels show the estimated coefficients and R 2 from regressing model residuals on a constant and an on-the-run dummy, while the lower two panels plot the coefficients from regressing the same residuals on a constant, an on-the-run dummy, and each bond s age in months. It appears that although on-the-run bonds in the early 1990s had higher prices (lower yields) than those predicted by the model, in the later part of the sample this effect is quite small and in no way explains the age effects described above, which are essentially unchanged. Thus the qualitative fact in the bottom panel of Figure 6 that on/off-the-run spreads are small, while old/really-old spreads are large, appears to be in force mainly in the last twenty years. 18

19 8.5 Yield Curve on Estimated on 17Sep , re run on 22Sep yield Over 10 years old Under 10 years old Under 10 years old, on the run duration (years) 3.5 Yield Curve on Estimated on 17Sep , re run on 22Sep yield Over 10 years old Under 10 years old Under 10 years old, on the run duration (years) Figure 6: Yield Curves in 1991 and

20 30 Yield Residual Coefficients over Time Variable: OTR Dummy Regressors: Constant / OTR Dummy Estimated on 17Sep , re run on 22Sep R Squared over Time Regressors: Constant / OTR Dummy Estimated on 17Sep , re run on 22Sep coefficient (bps) 0 10 R Squared month (a) On-The-Run Dummy Coefficients over Time month (b) On-The-Run Dummy R 2 over Time 30 Yield Residual Coefficients over Time Variable: OTR Dummy Regressors: Constant / Bond Age (months) / OTR Dummy Estimated on 17Sep , re run on 22Sep Yield Residual Coefficients over Time Variable: Bond Age (months) Regressors: Constant / Bond Age (months) / OTR Dummy Estimated on 17Sep , re run on 22Sep coefficient (bps) coefficient (bps) month (c) On-The-Run Dummy Coefficients over Time (Age included in regression) month (d) Bond Age (months) Coefficients over Time ( on-the-run dummy included in regression) Figure 7: On-The-Run and Bond Age as Regressors 20

21 5 Conclusion I propose a framework for taking Gaussian no-arbitrage affine term structure models directly to raw price data. Contrary to the usual concern in the literature that these models overfit the data, I find that model residuals are large, have interesting business-cycle variation, and are correlated with bond-specific characteristics in particular, old high-coupon bonds have higher yields than those predicted by the model, especially during the last few years. On-the-run bonds seem to have lower yields than other bonds during the early 1990s, but this effect is dwarfed in the recent sample by the age/coupon effect. The largest pricing errors are during the most recent financial crisis and correlate with the time since issuance or, equivalently, the coupon rate: older, higher-coupon bonds trade at a roughly 40-bps discount at the height of the crisis. While the model estimated in this paper is the simplest workhorse three-factor ATSM without any observable factors, and although I have not included any risk pricing, the method can be easily extended to include both using the unscented Kalman filter. Another direction, which needs no risk prices or timeseries estimation at all, would be to use simulated cross-sections to analyze how the Fama-Bliss interpolation procedure would fare against a high or low age factor (or any other indicator of poor model fit); it may be that the specific ways in which the procedure seemed to break down during the crisis can be understood in terms of such a factor, which would allow a better understanding of how the approximation works, and when it doesn t. Another avenue of research along the lines proposed in this paper is to examine the model s pricing residuals even more closely. For example, how tightly do bond age and coupon correlate with a bond s ability to be used as collateral in the repo market, i.e. with specialness as examined by Duffie (1996) and Krishnamurthy (2002)? Another possibility is to examine the pricing residuals statistically; it would be interesting to examine heteroskedasticity, serial correlation, or even bond fixed effects. The simple maximumlikelihood framework used here would allow simple likelihood-ratio tests for such effects, although their interpretation would be difficult. It would also take us further from an economic interpretation of the pricing errors. Now that we know that the standard model over-prices old and high-coupon bonds, the next step is to understand why. Are these bonds simply less liquid than other bonds, or does it stem directly from their high coupon rates? If the former, can the no-arbitrage ATSM be extended to incorporate these liquidity effects? If the latter, what accounts for the fact that in good times a coupon-bearing security prices close to the sum of its parts, but in bad times prices much lower? Armed with a simple and transparent method for estimating yield-curve models directly on Treasury securities, we can begin to answer such questions. 21

22 A Parameter Calibration Estimating the parameters of ATSMs can be a tricky business. Ang and Piazzesi (2003) use a grid search to maximize the likelihood function, but only manage to find a local maximum (Hamilton and Wu 2010b). Hamilton and Wu (2010b) and Joslin, Singleton and Zhu (2011) propose a less risky method; unfortunately it requires some yields to be priced without error, so that the k factors are an invertible linear combination of k yields. This method is not feasible when the maturity structure of observed yields changes in each cross-section, as it does in the CRSP data. Notice that this also excludes the possibility of an eigenvector decomposition. To start my parameter search, I go back to a simpler dataset on which an eigenfactor decomposition is possible: the GSW yields. I take an eigenvector decomposition of the 1-year forwards from 1 to 15 years on this dataset, which gives me a set of factor loadings as well as the three factors on each date. Following Cochrane and Piazzesi (2008), I find risk-neutral model parameters to match these loadings as closely as possible. Cochrane and Piazzesi (2008) perform a grid search over risk-neutral parameters to minimize a sum of squared residuals; I propose a method that is significantly easier and faster, and (at least in my sample) performs just as well. Calibration on Annual Data My calibration method is simplest when the frequency of observation and the shortest observed maturity are the same. Assume T years of annual data of zero-coupon yields at maturities from 1 to N years. For reasons that will become apparent it is easier to calibrate risk-neutral ATSM parameters from forwards; one-year forward rates can be computed from yields as f (1) t y (1) t f (n) t ny (n) t (n 1)y (n 1) t. Because this panel of yields is strongly balanced, they permit an eigenvector decomposition; specifically, for a de-meaned T N forward matrix F, I calculate Q and a diagonal matrix D such that F F = QDQ. Taking the columns of Q that correspond to the three largest elements of D (eigenvalues of F F ) gives the loadings on the three principal components; call this matrix Q. The time-series evolution of the unobserved 22

23 factors corresponding to these loadings is given by X = F (T 3) (T N) Q (N 3) Because an eigenvector decomposition gives the loadings Q and factors X t such that each cross-section is, roughly, F t + a Q X t + a where the vector a is the vector of forward means, the eigenvector decomposition gives us the best-fitting forward loadings of the data. Denote these eigenvector loadings as {â n, ˆb n }, so that â n is the average n-year forward rate and ˆb n is the nth row of Q. I would like to find risk-neutral parameters that give me forward loadings {A f n, Bn} f that are close to the eigenvector loadings; forward loadings are related to the risk-neutral parameters through the formulas A f n = δ 0 B n 1µ 1 2 B n 1ΣΣ B n 1 (10) B f n = δ 1φ n 1, (11) where the {B n } come from equation 3. The linear structure of these formulas allows a simple method to calibrate the risk-neutral parameters {δ 0, δ 1, µ, φ } I pre-multiply the X t by an estimate of Σ 1 (from an unconstrained VAR) so that the Σ in these formulas is the identity matrix. I start with the slope loadings {Bn}. f Because the shortest (one-year) yield is included in the data, I can estimate δ 1 immediately as the one-year regression coefficient: ˆb 1 B f 1 = δ 1φ 0 = δ 1. Equation 11 implies a linear relationship between the other loadings and φ : B f 2 = δ 1φ = B f 1 φ B f 3 = δ 1φ φ = B f 2 φ... B f n = δ 1φ n 2 φ = B f n 1 φ, 23

24 which means that, because I want {B f n } {ˆb n}, I m looking for a single matrix φ which satisfies ˆb 2 ˆb 3... ˆb 1 ˆb 2... φ. ˆb N ˆb N 1 I estimate such a φ through linear regression: each column of φ contains the regression coefficients of that factor s forward loadings, from 2 to N, on the same loadings from 1 to N 1. I calibrate δ 0 and µ with a similar trick. δ 0 can be estimated immediately as the mean of the 1-year yield: â 1 δ 0 Given estimates of {δ 1, φ }, I can calculate the price slope loadings {B n} which satisfy B n = δ 1 + B n 1φ. From there, everything on the right side of equation 10 is known except for µ. This means that to set A f n â n I m searching across 15 maturities for the three coefficients µ that satisfy â 2 δ B 1B 1 â 3 δ B 2B 2... B 1 B 2... µ. â N δ B N 1 B N 1 B N 1 Thus µ can also be estimated as the three coefficients in a single 15- observation linear regression. Calibration on Monthly Data Because the CRSP price data I use is monthly, the above procedure needs to be modified so that the frequency of observation is not the same as the shortest observed forward rate. Now assume monthly data of zero-coupon yields at maturities from 1 to N years. In this case, I can t compute one-month forwards, 24

25 because I don t have yields of adjacent maturities. Instead I compute year-long forwards as f (0) t 0 f (n) t ny (n) t (n 12)y (n 12) t. n = 12, 24,...12N As before, I compute an eigenvector decomposition of these year-long forwards to get eigenvector loadings {â 12i, ˆb 12i }N i=1. I map the year-long forward loadings into the log-price loadings as follows. From equation 3, B n = δ 1 + B n 1φ = δ 1(I + φ + φ φ n 1 ) = δ 1(I φ ) 1 (I φ n ). Because a year-long forward rate is just the difference between two log-prices with maturities twelve months apart, the eigenvector loadings relate to the {B n} as ˆb n B n 12 B n = δ 1(I φ ) 1 (φ n φ n 12 ) = δ 1(I φ ) 1 (φ 12 I)φ n 12 = γ φ n 12, where the last equality defines γ. Thus I use the same regression trick as for the annual data, only this time it returns φ 12 : ˆb 12 B 12 = γ ˆb 24 B 12 B 24 = γ φ 12 = ˆb 12φ 12 ˆb 36 B 24 B 36 = γ φ 12 φ 12 = ˆb 24φ 12...etc, so ˆb 24 ˆb ˆb 12N ˆb 12 ˆb ˆb 12(N 1) φ 12. (12) I then estimate φ as the matrix twelfth root of φ