07-082

Copyright © 2005–2007 by Luis M. Viceira.

Working papers are in draft form. This working paper is distributed for purposes of comment and discussion only. It may not be reproduced without permission of the copyright holder. Copies of working papers are available from the author.

Bond Risk, Bond Return Volatility, and the Term Structure of Interest Rates Luis M. Viceira

Bond Risk, Bond Return Volatility, and the TermStructure of Interest Rates

Luis M. Viceira1

First draft: May 2005This draft: February 2007

Abstract

This paper explores time variation in bond risk, as measured by the covariation ofbond returns with stock returns and with consumption growth, and in the volatilityof bond returns. A robust stylized fact in empirical finance is that the spread betweenthe yield on long-term bonds and short-term bonds forecasts positively future excessreturns on bonds at varying horizons, and that the short-term nominal interest rateforecasts positively stock return volatility and exchange rate volatility. This paperpresents evidence that movements in both the short-term nominal interest rate andthe yield spread are positively related to changes in subsequent realized bond risk andbond return volatility. The yield spread appears to proxy for business conditions,while the short rate appears to proxy for inflation and economic uncertainty. Adecomposition of bond betas into a real cash flow risk component, and a discount raterisk component shows that yield spreads have offsetting effects in each component.A widening yield spread is correlated with reduced cash-flow (or inflationary) risk forbonds, but it is also correlated with larger discount rate risk for bonds. The shortrate forecasts only the discount rate component of bond beta.

JEL classification: G12.

1Graduate School of Business Administration, Baker Libray 367, Harvard Univer-sity, Boston MA 02163, USA, CEPR, and NBER. Email lviceira@hbs.edu. Websitehttp://www.people.hbs.edu/lviceira/. I am grateful to John Campbell, Jakub Jurek, and AndréPerold for comments and suggestions, and to the Division of Research of the Harvard BusinessSchool for generous financial support.

1 Introduction

There is strong empirical evidence that the expected return on long-term bonds inexcess of the return on short-term bonds is time-varying. Linear combinations ofbond yields or, equivalently, forward rates, forecast future bond returns (Fama andBliss 1987, Campbell 1987, Campbell and Shiller 1991, Cochrane and Piazzesi 2005,Fama 2005). For example, the spread between the yield on long-term bonds and theshort-term interest rate forecasts positively future bond excess returns.

Standard asset pricing models predict that in equilibrium the expected excessreturn on an asset is equal to the product of the systematic risk of the asset timesthe price of risk. According to these models, time variation in expected bond excessreturns must be the result of time variation in the aggregate price of risk, or in thequantity of bond risk, or in both.

There is ample empirical evidence documenting the existence of common com-ponents in the time variation of expected excess returns on bonds and on stocks(Fama and French, 1989). This evidence suggests that time variation in expectedbond excess returns is explained, at least partially, by a time-varying aggregate priceof risk, as in the external habit formation models of Campbell and Cochrane (1999).Indeed, Wachter (2006) shows that external habit preferences can generate a positiveforecasting relation between the yield spread and future bond excess returns.

This paper adds to this research by investigating whether the quantity of bondrisk (to which I will refer from now on simply as “bond risk”) is also time varying,and whether this time variation is correlated with variables which have been shownto explain time variation in bond excess returns. If expected bond excess returnschange in response to changes in bond risk, one might expect that variables thatforecast bond excess returns also explain changes in bond risk.

I study two intuitive measures of bond risk. The first one is the covariance–or anormalized measure of this covariance–of bond returns with stock returns. This isthe measure of bond risk implied by the standard CAPM. The CAPM implies thatthe risk of an asset is measured as the covariance of the returns on that asset withthe returns on the aggregate wealth portfolio, which is typically proxied by the stockmarket. From an asset allocation perspective, the covariance of stock returns andbond returns plays a central role in the asset allocation decisions of investors. Thesecond measure of risk I consider is the covariance of bond returns with per capita

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consumption growth. This is the measure of bond risk implied by the standardConsumption CAPM.

The research conducted in this paper builds on previous work by Campbell (1986),Barsky (1989), Campbell and Ammer (1993), Shiller and Beltratti (1992) and oth-ers which has explored the determinants of the comovement of stock and bond re-turns. The focus of this work has typically been the unconditional moments describingthis comovement. More recently, Boyd, Hu, and Jagannathan (2005) and Andersen,Bollerslev, Diebold and Vega (2005) have shown that stock returns, interest rates andexchange rates appear to respond to macroeconomic news differently over the businesscycle, suggesting the existence of a business cycle component in the variation of thesecond moments of returns. This paper looks explicitly at the relation between timevariation in the realized second moments of bond returns–volatility and covariancewith stock returns and with consumption growth–and time variation in variableswhich proxy for business conditions.

The use of realized second moments of returns to study time variation in riskhas a long tradition in Finance that dates back at least to the work on stock returnvolatility of Officer (1973), Merton (1980), French, Schwert and Stambaugh (1987),and Schwert (1989). This tradition has been reinvigorated recently with work byAndersen, Bollerslev, Diebold and Labys (2003), Andersen, Bollerslev, and Meddahi(2005), Barndorff-Nielsen and Shephard (2004) and others. This work provides thetheoretical foundation for the use of time series of realized secondmoments as the basisfor modeling the dynamics of volatility and covariance.2 The empirical applicationsof this work have typically focused on the study of time variation in the volatility ofstock returns (both aggregate stock market returns and returns on stock portfolios)and in the volatility of exchange rates. This paper focuses instead on the study oftime variation in bond risk and, secondarily, bond return volatility.

Figure 1 illustrates why it might be interesting to study time variation in bondrisk. This figure plots 3-month rolling estimates of bond CAPM betas for the period1963 through 2003, computed using daily returns on stocks and bonds. The covarianceof long term bond returns with stocks is typically estimated to be positive, but small.For example, Table 1 shows that the full sample estimate of the CAPM beta of bondsduring the sample period shown in Figure 1 is 0.03. Figure 1 shows that this lowfull-sample estimate of bond beta hides considerable variation over time. It also

2Andersen, Bollerslev, Christoffersen and Diebold (2006a, 2006b) provide an excellent summaryof recent findings in this literature and their practical implications.

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suggests that some of this variation might be systematic: Changes in beta appear tobe persistent and mean reverting; periods of low bond betas are followed by periodsof large bond betas. Average bond betas were very close to zero in the 60’s and70’s–with the exception of the inflationary period of the late 60’s and early 70’s–,significantly positive from the late 70’s through the mid-90’s–with the exception ofthe recessionary period of the early 90’s–, and negative in the last part of our sample(1997 through 2003). Of course, there is considerable short-run volatility around thoseaverages, some of which might be driven by transitory “flight to quality” events fromstocks to bonds–as during the crash of October of 1987 or the Asian crisis of 1998,when bond betas fell dramatically–, or from long-term assets (bonds and stocks) tocash–as in the early 80’s, when bond betas increased dramatically.

The empirical analysis conducted in this paper confirms the visual impressiongiven by Figure 1 that there is systematic variation in bond risk. I find that thecovariance of stock returns with bond returns and bond return volatility both followmean-reverting, persistent processes. I also find that the short-term nominal interestrate forecasts positively realized bond CAPM betas, bond consumption betas, andbond return volatility. Additionally, the spread between the yield on long-term bondsand the short rate forecasts positively bond CAPM betas and bond consumptionbetas.

The finding that the short rate also forecasts positively bond CAPM betas, bondconsumption betas and bond return volatility adds to previous research showing sim-ilar forecasting ability of the short rate for stock return volatility and exchange ratevolatility.3 Recent research (David and Veronesi 2004) has also shown that inflationuncertainty, as measured by a time series of cross-sectional dispersion in professionalforecasts of inflation, is positively related to subsequent realized stock-bond returncovariance. This paper shows evidence that this variable is no longer a statisticallysignificant predictor of the covariance of stock and bond returns once we control forthe effect of the short rate in the same regression.

The empirical results in this paper about the relation between time variation in theshort rate and in the second moments of bond returns lend support to the hypothesisin Glosten et al. (1993) that the short rate forecasts stock market volatility because

3Campbell (1987), Breen, Glosten and Jagannathan (1989), Shanken (1990), Glosten, Jagan-nathan and Runkle (1993), Scruggs (1998), and others have documented a positive relation betweenthe short nominal rate and stock market volatility. Giovannini and Jorion (1989) document a positiverelation between the short nominal rate and exchange rate volatility.

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it reflects inflation uncertainty, which in turn is likely to be correlated with aggregateeconomic uncertainty.4 If the short rate proxies for inflation uncertainty and economicuncertainty, one might expect that it should also forecast bond return volatility andthe covariance of stocks and bonds, in addition to stock return volatility.

Furthermore, consistent with this hypothesis, this paper also presents strong em-pirical evidence that the effect of the short rate on the covariance of stock and bondreturns, bond return volatility, and stock return volatility is much weaker in the pe-riod following the implementation of strong anti-inflationary policies by the FederalReserve under Paul Volcker in the early 1980’s, than in the period that preceded it.There is considerable empirical evidence that the last two decades of the twentiethcentury have been characterized by a significant and persistent reduction in the volatil-ity of a broad array of macroeconomic variables, including inflation, aggregate andsectoral output growth, consumption growth, employment growth, and investmentgrowth (Kim and Nelson 1999, McConnell and Perez-Quiros 2000, Stock and Watson2002). Lettau, Ludvigson, and Wachter (2006) argue that this sustained decline inthe volatility of macroeconomic activity can account for a substantial fraction of therun-up in aggregate stock prices during the 1990’s.

This paper also finds that the yield spread forecasts positively the bond-stockreturn covariance at short and long horizons, and bond return volatility at shorthorizons. This result suggests that the ability of the yield spread to forecast bondexcess returns is related not only to changes in aggregate risk aversion, but alsopossibly to changes in bond risk. Moreover, in contrast to the effect of the short-rateon the second moments of bonds and stocks, the effect of the yield spread on bondbeta and bond return volatility appears to be stable over time.

Nominal bond returns reflect changes in expected future inflation–which deter-mine nominal bond real cash flows–, changes in expectations of future real interest

4The argument in Glosten et al. (1993) builds on the Fisher hypothesis, which establishes adirect link between expected inflation and the short rate, and on the fact that empirically there isa positive relation between inflation volatility and its level (Fischer 1981, Brandt and Wang 2003).Also, Fama and Schwert (1977), Fama (1981), Huizinga and Mishkin (1984), Pennacchi (1991),Boudoukh and Richardson (1993), Boudoukh, Richardson and Whitelaw (1994) and others havedocumented a negative correlation between inflation and real activity, and between inflation andstock returns. Levi and Makin (1980), Mullineaux (1980), Hafer (1986), and Holland (1988) andothers have documented a positive relation between inflation uncertainty and economic growth.This would suggest that times of higher inflation uncertainty are also times of higher aggregateuncertainty.

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rates, and changes in expected future bond excess returns (or bond risk premia). Thefirst effect is a real cash flow effect, while the last two effects are discount rate effects.Therefore, changes in bond risk must be related to either changes in the real cashflow risk of bonds, or in the discount rate risk of bonds, or both. I use the vectorautoregressive-log linearization approach of Campbell and Shiller (1988), Campbell(1991), and Campbell and Ammer (1993) to conduct an empirical decomposition ofunexpected bond excess returns into real cash flow news and discount rate news com-ponents, and explore systematic time variation in the covariation of these componentswith unexpected stock returns and with unexpected consumption growth.

Of course, this statistical decomposition can be sensitive to the specification of thevector autoregressive system (Campbell 1991, Campbell and Ammer 1993, Campbell,Lo, and MacKinlay 1997, Campbell and Vuolteenaho 2004, Chen and Zhao 2006). Tominimize this issue, the system includes variables that plausibly help capture thecomponents of bond returns, and changes in expected inflation, ex-ante real interestrates, and expected excess returns on bonds and stock. Because many of these vari-able are not observed at high frequencies, the analysis of systematic variation in thecomovement of the components of bond returns with stock returns and consumptiongrowth is based on monthly and quarterly observations.

The structure of the paper is as follows. Section 2 describes the empirical measuresof realized second moments of bond return and stock returns used in the paper, andpresents an analysis of their univariate time series properties. Section 3 presentsthe main empirical results about in the paper, based on a multivariate time seriesanalysis of the second moments of bond returns. Section 4 examines an empiricaldecomposition of bond CAPM and consumption betas into cash flow and discountrate betas. Section 5 concludes.

2 Empirical measures of realized second momentsof bond returns

The basic empirical analysis in this paper is based on realized second moments ofbond and stock returns measured at a daily frequency, in the spirit of Officer (1973),Merton (1980), French, Schwert and Stambaugh (1987), Schwert (1989), and others.Specifically, I consider the realized covariance of stock and bond returns, the realized

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volatility of bond returns, the realized volatility of stock returns, and normalizedmeasures of the covariance of stocks and bonds.

The realized covariance of stock and bond returns between the end of month t andthe end month t+n (a proxy for integrated instantaneous covariance) is measured as

σS,B (t, n) =1

22n

tDPd=t1

rS,d × rB,d, (1)

where [t1, tD] denotes the sample of available daily returns between the end of montht and the end of month t+n, and rS,d and rB,d are the stock and bond log returns onday d. Following standard practice in the literature on realized second moments, thenumber of days in a month is normalized to 22, and the mean correction is omitted.(Considering demeaned returns does not change the conclusions from the empiricalanalysis.)

Similarly, the realized volatility of stock and bond returns between the end ofmonth t and the end month t+n, a measure of integrated instantaneous volatility, ismeasured as

σ2i (t, n) =1

22n

tDPd=t1

r2i,d, (2)

where i = S and B for stocks and bonds respectively.

I also consider normalized measures of the stock-bond return covariance. One suchmeasure is correlation. Another is beta. My analysis focuses on the realized CAPMbeta of bonds, because of its more natural interpretation as bond risk. Realized bondCAPM beta between the end of month t and the end month t+ n is measured as

βS,B (t, n) =σS,B (t, n)

σ2S (t, n). (3)

These measures of realized second moments have been previously considered in thecontext of stock returns and exchange rates. Andersen, Bollerslev, Diebold and Labys(2003), Andersen, Bollerslev, and Meddahi (2005), Barndorff-Nielsen and Shephard(2004) and others have shown that the theory of quadratic variation implies that thesemeasures converge uniformly to integrated instantaneous volatility and covarianceunder weak regularity conditions.

I build time series of these realized second moments using daily returns on stocksand bonds from the Center of Research in Security Prices (CRSP) at the University of

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Chicago. The CRSP Fixed Term Indices File provides daily total returns and yieldson constant maturity coupon bonds, with maturities ranging from 1 year to 30 yearsbetween June 14, 1961, and December 31, 2003. The CRSP Daily Stock Indices Fileprovides daily total stock returns from July 2, 1962 through December 31, 2003. Thissample is about one year shorter than the bond return sample, and thus determinesthe common sample period. The empirical analysis is focused on the return on the5-year constant maturity bond, and on the return on the value weighted portfolio ofall stocks traded in the NYSE, the AMEX, and NASDAQ.

Table 1 presents full-sample statistics of daily bond and stock returns for thecommon sample period, 1962.07.02-2003.12.31. Over this sample period, the meanlog return on stocks was 10.46% per annum, the mean log return on bonds was 7.94%per annum, and the mean short-term interest rate was about 5.65% per annum. Thestandard deviation of stock log returns was 14.51% p.a., and the standard deviationof bond log returns is 5.01%. These numbers imply that, ex-post, the Sharpe ratioof bonds was actually slightly larger than the Sharpe ratio of stocks over this sampleperiod: 0.483 versus 0.404, based on annualized moments.

Table 1 also reports the full-sample correlation of daily stock and bond returnsand the CAPM beta of bonds. Both measures reflect a positive, but weak, covarianceof daily bond returns with daily stock returns: Correlation is estimated to be 2.95%,and beta is 0.03. However, Figure 1 and Table 2 show that these full-sample estimateshide considerable variation over time in the comovement of stock and bond returns.Figure 1 plots 3-month rolling estimates of bond CAPM betas for the period 1963through 2003 (β(t, 3)) and Table 2 reports descriptive statistics of the realized monthlystock-bond return covariance, the CAPM beta of stocks, and the volatility of stockand bond returns. The table shows the standard deviation of each measure of realizedsecond moments is at least 1.5 times its mean. Realized covariance is the momentwith the highest volatility relative to its mean, at 4.39 times. By contrast, realizedbond CAPM beta, realized stock return volatility and realized bond return volatilityhave standard deviations relative to their means an order of magnitude lower: 1.67,1.57 and 1.88 times, respectively.

Table 2 also reports the correlation of realized second moments. It shows thatrealized covariance and realized stock return volatility are negatively correlated, at-32%. This helps explain why realized CAPM beta is less volatile relative to itsmean than realized covariance. Interestingly, realized bond CAPM beta itself is alsonegatively correlated with realized stock return volatility, at -17%. Thus times of high

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volatility in the stock market tend to coincide with times of low covariation betweenstock and bond returns.

By contrast, the stock-bond return covariance, its CAPM normalization, and stockmarket volatility are all positively correlated with realized bond return volatility, withcoefficients in the 20%-30% range. Thus times of high volatility in the bond markettend to coincide with times of increased comovement between bond and stock returns,and high stock market volatility.

Finally, Table 2 also reports the autocorrelation function of realized covariance,realized bond betas, and the log of realized stock return volatility and bond returnvolatility. The table reports the autocorrelation statistics for log volatility for con-sistency with the subsequent multivariate analysis, which relies on log volatilities.5

The autocorrelation function of realized second moments shows that they are highlypersistent processes. This is a well known property of stock return volatility. Table 2shows that it also extends to bond return volatility, and to measures of the covariancebetween stock returns and bond returns. For all realized second moment series, thefirst order autocorrelation coefficient is at least 50%, and the autocorrelation functiondeclines slowly. At a lag of 12 months, all autocorrelations are above 25%.

A slowly decaying autocorrelation function might indicate the presence of long-range dependence. To test for that type of dependence, Table 2 also reports tests oflong-memory in second moments. Specifically, the table reports Lo’s (1991) modifiedrange-over-standard-deviation (R/S) statistic, which corrects the standard R/S sta-tistic for the presence of heteroskedasticity and short-range dependence–a “Newey-West corrected” R/S statistic. The last two rows of Table 2 report two values of themodified R/S statistic for each second moment: R/S (0) is the standard R/S statistic,which does not take into account short-range dependence in the series; R/S (12) usesautocovariances of the second moment up to order 12. The 5% and 1% critical valuesfor this statistic under the null of no long-memory are 1.862 and 2.098 respectively,which suggest that one cannot unambiguously reject the null in all cases.

5The use of log volatility in regression analysis ensures that the variable of interest has a contin-uous support in the real line. The log transformation also makes sense in light of the evidence aboutheteroskedasticity and approximate log normality of stock returns and exchange rates (Andersen,Bollerslev, Diebold and Labys 2000, 2001).

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3 Time variation in bond risk, bond return volatil-ity, and the term structure of interest rates.

3.1 Main results

This section explores whether the time variation in the second moments of bond re-turns is systematically related to movements in the term structure of interest rates. Inparticular, this section presents regressions of realized bond-stock return covariances,bond CAPM betas, and bond return volatility on the short-term nominal interestrate, and the spread between the yield on long-term nominal bonds and the short-term nominal interest rate. For completeness, I also present forecasting regressionsfor stock return volatility.

This exercise is motivated by two well established empirical regularities. First,there is robust empirical evidence that the yield spread forecasts positively bondexcess returns. Second, there is also ample empirical evidence that movements in thelevel of interest rates are positively related to changes in stock return volatility andexchange rate volatility. Thus it is natural to ask whether the short rate also explainstime variation in the second moments of bond returns, and whether this is related tochanges in expected bond excess returns.

Table 3 presents regressions of bond log excess returns, measured at horizonsranging from one month to three years, onto the lagged short-term nominal interestrate and the lagged yield spread. Bond excess returns are measured as the excess logtotal return on a constant 5-year maturity Treasury coupon bond over the short-termnominal interest rate, taken as the log yield on a 30-day Treasury Bill. Both series areobtained from the CRSP US Treasuries and Inflation Indices File. The yield spreadis the log yield on a 5-year artificial zero-coupon bond from the CRSP Fama-BlissDiscount Bond File minus the short-term rate. The table reports t-statistics basedon Newey-West standard errors, the R2 of the regressions, and a χ2 statistic of thesignificance of the slopes. This statistic also uses the Newey-West correction.6

The regressions in Table 3 confirm that the result that the yield spread forecasts

6The number of lags included in the Newey-West correction is the horizon in months minus one.This choice of lag order is based on the fact that under the null of no predictability, the residuals ofthis long-horizon regression are autocorrelated up to an order equal to the number of periods in thehorizon minus one.

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bond excess returns positively at all horizons holds in this sample period. The Newey-West corrected t-statistic on the yield spread is always above 2.8 at all horizons, andboth the R2 and the χ2 statistic of the regression increase with the horizon. Onepossible interpretation of these regression results is that a widening of the yield spreadis associated with an increase in bond risk. Fama and French (1989) show empiricalevidence that the yield spread is low around measured business cycle peaks, and highnear throughs. Thus the yield spread might be a proxy for counter-cyclical timevariation in bond risk.

The variation on bond risk associated with movements in the yield spread could bethe result of variation in the quantity of bond risk, variation in the aggregate marketprice of risk, or both. This paper focuses on the first possibility, i.e., on whether theyield spread is positively correlated with measures of the quantity of bond systematicrisk such as the covariance of bond returns with stock returns–as the CAPM wouldsuggest–or the covariance of bond returns with aggregate consumption growth–asthe Consumption CAPM would suggest.7

Table 3 shows that there is only weak evidence that the short-rate forecasts bondexcess returns. The slope on the short rate is positive at all horizons, but its t-statisticis always well below 2 except at the three-year horizon. However, even at this horizonit is questionable that the short rate forecasts bond excess returns. The short rate isboth highly persistent and its innovations are negatively correlated with unexpectedbond excess returns, which implies that the estimated coefficient on this variable isupward biased (Stambaugh 1999).8

We can now turn to examine whether the time variation in second moments ofbond and stock returns is related to time variation in the term structure of interestrates. Table 4 and Table 5 report regressions of realized second moments measuredat horizons up to 36 months on a constant, the lagged value of the second moment,

7Wachter (2006) considers an extension of the Campbell-Cochrane’s (1999) habit formation modelin which counter-cyclical variation in aggregate risk aversion leads to countercyclical time variationin real interest rates and real term premia.

8Stambaugh (1999) shows that estimate of the slope in return forecasting regressions suffers fromsmall-sample bias whose sign is opposite to the sign of the correlation between innovations in returnsand innovations in the predictive variable. In the bond return forecasting regression, the short rateis negatively correlated with bond returns, leading to a positive small-sample bias which helps toexplain some apparent predictability. On the other hand, the yield spread is positively correlatedwith bond returns, leading to a negative small-sample bias which suggests that the small sampleevidence of bond return predictability from the yield spread probably underestimates the true degreeof predictability.

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the lagged short-term interest rate, and the lagged yield spread. Panel A in Table 4presents regression results for σS,B (t, n), the n-month realized covariance of stock andbond returns defined in (1) and Panel B presents results for βS,B (t, n), the n-monthrealized CAPM beta of bond returns defined in (3). Table 5 reports regression resultsfor log bond return volatility (Panel A) and for log stock return volatility (Panel B).Similar to Table 3, these two tables report Newey-West corrected t-statistics, the R2

of the regressions, and the Newey-West corrected χ2 statistic of the significance ofthe slopes. The Newey-West correction takes into consideration up to (n − 1) auto-correlations. The 5% critical value of the χ2 distribution with 3 degrees of freedomis 7.82, and the 1% critical value is 11.34.

Table 4 shows that the yield spread forecasts positively both the realized covari-ance of stock and bond returns, and its normalization given by the bond CAPM beta.The coefficient on the yield spread is statistically significant up to a 6-month hori-zon in the covariance regression, and significant at all horizons in the bond CAPMbeta regression. These results suggest that there is counter-cyclical variation in thecomovement of bond returns with stock returns. This evidence suggests that at leastpart of the countercyclical variation in expected bond excess returns is driven bycountercyclical variation in bond risk as measured by the comovement of bond re-turns with stock returns.

By contrast, Table 5 shows there is only weak evidence that the yield spread isrelated to movements in bond return volatility and stock return volatility. The slopeof the yield spread in the regressions for log volatility is not statistically significant,except at a 1-month horizon for bond return volatility, and at a 12-month horizon forstock return volatility. Moreover, the point estimates of the slope change sign as weconsider one return volatility or the other. The slope of the yield spread is positive inthe bond return volatility regression, while it is negative in the stock return volatilityregression. Though not significant statistically, the negative slope of the yield spreadin the stock return volatility regression helps nonetheless explain why the coefficienton the yield spread is larger in the bond CAPM beta regression than in the covarianceregression: The yield spread forecasts positively the numerator of the bond CAPMbeta–the stock-bond return covariance–, and negatively its denominator–stockreturn volatility.

The regression results in Table 4 and Table 5 also show strong evidence thatshort rate forecasts positively both the covariance of bond and stock returns, andthe volatility of bond returns. The estimates of the coefficient on the short rate in

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the forecasting regressions for second moments are all statistically significant up toa 12-month horizon. These results thus show that the ability of the short rate toforecast positively the volatility of stock returns and exchange rates extend to thecase of bond return volatility and the covariance of bond returns with stock returns.Since the short rate tends to move procyclically, these results suggest that the shortrate captures a procyclical component in the time variation of the second momentsof bond returns. Interestingly, the short rate also has a positive slope in the bondreturn forecasting equation, though this slope is not statistically significant. Section5 explores this result in more detail.

Table 4 and 5 do not report the intercepts of the second moment forecastingregressions to save space. However, it is interesting to remark that the interceptsin the regressions for realized covariance and bond CAPM beta are all negative andhighly significant.9 Thus periods of low short-term nominal interest rates and a flatyield curve tend to be times of low or negative bond betas. To the extent that thelevel of interest rates reflects inflation expectations, these estimates of the interceptsuggest that bond risk is low when expected inflation is low.

Table 4 and Table 5 also show that lagged values of secondmoments also enter theirown second moment forecasting regression significantly, but only at short horizons.Figures 2 and 3 help understand the role of lagged values and the role of the termstructure variables in fitting the time variation in the covariance of stock and bondreturns, and the bond CAPM beta. Figure 2 plots the time series of the realized3-month covariance of bond and stock returns, σS,B (t, 3), and its fitted value. PanelA plots the fitted values from the regression shown in Table 4, which includes laggedcovariance, while Panel B plots the fitted values from a regression that excludeslagged covariance. Figure 3 repeats the exercise for the realized 3-month CAPM betaof bonds, βS,B (t, 3) and its fitted value.

Figure 2 and Figure 3 suggest that lagged second moment values capture highfrequency movements in the quantity of bond risk, while the level and slope of the yieldcurve capture movements at lower frequencies. There is considerable low frequencyvariation in bond risk; excluding lagged values from the regressions shown in Table4 still result in large R-square coefficients. For example, the R2 of a regression ofσS,B (t, 3) on a constant, the short-term interest rate, and the yield spread is 30%;

9The intercepts are also negative and statistically different from zero in regressions that omit thelagged value of the second moment.

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the R2 of a similar regression for βS,B (t, 3) is 47%.10 Thus movements in the intercept

and the slope of the term structure of interest rates capture a large fraction of the totalvariability in realized bond risk. It is important to note though that term structurevariables have a more difficult time fitting the last part of the sample period. Whilethey are able to capture the general direction of realized covariance and bond beta–positive between 1994 and 1998, and negative afterwards–, they do not fit welltheir average level. Instead, they undershoot in the 1994-1998 period, and overshootafterwards.

3.2 Stability

The period 1962-1993 was one in which the US economy experienced a long periodof generally increasing short-term nominal interest rates and inflation until mid-1981,followed by another equally long period of generally declining nominal interest ratesand inflation which brought them back to the levels prevailing in the 1950’s and early1960’s. A recent paper (Fama, 2006) examines whether this long period of decliningshort rates and inflation has weaken the evidence about bond return predictabilityshown in Fama and Bliss (1987), and concludes that this evidence is robust to thepresence of this slow mean reversion of the short rate.

Table 6 conducts a similar exercise for bond betas. It examines whether thevariation in bond CAPM beta, bond return volatility and stock return volatilityexplained by the short rate and the yield spread between two subperiods, 1962.01—1981.07 and 1981.07—2003.12. The sample is split around the date in which the shortrate reached its maximum in-sample value. The table reports regressions similarto those shown in Table 4 and Table 5, except that they add an extra term thatinteracts each variable with an indicator variable IND(t) which is equal to zerobetween 1962.01 and 1981.07, and equal to one between 1981.08 and 2003.12. PanelA reports results for realized bond CAPM betas, Panel B for the log of realized bondreturn volatility, and Panel C for the log of realized stock return volatility. I onlyconsider horizons up to 12 months, because each subperiod is not long enough toreliably test for changes at longer horizons. (In any case, the results do not change).

Table 6 shows that the coefficient on the short rate in all realized second moment10These regression results are not reported here to save space. However, they are available from

the author upon request.

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regressions is significantly smaller in the post-1981 period and in the earlier period.The coefficient on the interaction term is negative and large in all cases. In the caseof realized volatility of bond returns and stock returns, this coefficient is of roughlythe same magnitude as the coefficient on the short rate, suggesting that the abilityof the short rate to forecast changes in return volatility is concentrated in the periodof increasing inflation and interest rates of the 1960’s and 1970’s.

By contrast, the effect of the spread is remarkably stable across both subsamples:The coefficient on the interaction term, while negative, is not statistically significantin any of the regressions. Thus the spread forecasts positively realized bond betasand does not forecast realized volatilities in both subperiods.

3.3 Cochrane-Piazzesi tent-shaped bond return forecastingvariable

In a recent paper, Cochrane and Piazzesi (2005) show that a tent-shaped linear com-bination of forward rates–or, equivalently, yields–from the Fama-Bliss CRSP datafile forecasts bond excess returns and stock excess returns at a 12-month horizon.The R2 in the bond excess returns is as high 35%, and the χ2 statistic is highly signif-icant. Thus it is interesting to ask if this variable also forecasts realized bond betasand realized volatility of stock returns and bond returns. Table 7 reports the resultsof this exercise.

Panel A in Table 7 reports long horizon regressions of bond log excess returnsonto the short rate and the yield spread on one hand, and onto the short rate andthe Cochrane-Piazzesi “tent” variable on the other. Note that the sample period inthese regressions is slightly shorter than the sample period in the previous tables, asthis variable is available only from 1964.01 through 2003.12.11 The 12-month horizonrow essentially reproduces Cochrane-Piazzesi results: The R2 in the tent variableregression is 34% and the χ2 statistic is 72. These statistics are 71% and 287%larger, respectively, than the corresponding statistics in the yield spread regression.Interestingly, however, the ability of the tent variable to forecast bond excess returnsat horizons shorter and longer than 12 months is remarkably similar to the ability ofthe yield spread.

11I am grateful to John Cochrane and Monika Piazzesi for making the data on this variableavailable to me.

14

Panel B in Table 7 reports forecasting regressions for realized bond CAPM betas.Table 7 shows that the yield spread and the tent variable have similar ability toforecast realized bond betas. Both regression models exhibit very similar R2

0s and χ2

statistics, particularly at horizons of 12 months or shorter.

The regression results shown in Table 7 thus suggest that the short rate and theyield spread, on the one hand, and the short rate and the tent variable on the otherhand, capture similar components of the time series variation in expected bond excessreturns and bond risk. Accordingly, the remaining of the paper focuses on the shortrate and the yield spread as predictors of bond risk.

3.4 The short rate and inflation uncertainty

In a recent paper, David and Veronesi (2004) explore whether survey-based measuresof inflation and earnings uncertainty have forecasting power for the realized covarianceand volatility of stock and bond returns. They find that a survey-based measure ofinflation uncertainty forecasts positively and significantly the realized covariance ofbond and stock returns.

Table 8 examines whether this measure of inflation uncertainty captures informa-tion about the time series variability of realized covariances and bond CAPM betaswhich the short rate and the yield spread do not. The table reports regressions of therealized covariance of stock and bond returns and realized bond CAPM betas on theirlagged value, a measure of inflation uncertainty similar to that of David and Veronesi(2004), the short rate, and the yield spread. Following David and Veronesi (2004),I have constructed a survey-based measure of inflation uncertainty using the Surveyof Professional Forecasters. This measure is a cross-sectional standard deviation ofindividual inflation forecasts at different horizons. This results in a time series ofcross-sectional standard deviations at a quarterly frequency, the frequency at whichthe survey data is available, that goes from the last quarter of 1968 through the lastquarter of 2003.

Table 8 shows that, consistent with the results in David and Veronesi (2004),inflation uncertainty forecasts positively and significantly the covariance of stock andbond return covariances and the CAPM beta of bonds when the short rate and theyield spread are not included in the regression. However, when the short rate isincluded in the regression as an additional forecasting variable, the slope coefficient

15

on inflation uncertainty ceases to be statistically significantly different from zero inboth the realized covariance regression and the CAPM beta regression. Furthermore,the point estimate of this slope reverses its sign. By contrast, the coefficient onthe short rate has a large t-statistic. Interestingly, the inclusion of the short-rate inthe forecasting regressions does not increase the R2 of these regressions significantly,suggesting that the short rate absorbs the effect on bond risk of the survey-basedmeasure of inflation uncertainty.

These results lends support to the hypothesis in Glosten et al. (1993) that theshort rate might reflect aggregate economic uncertainty in addition to expectationsof future inflation and the real rate. Interestingly, the stability analysis of Section 3.2also lends support to this hypothesis. Table 6 shows that the ability of the short rateto forecast return volatility is greatly diminished in the post-1981 period. This is aperiod where macroeconomic volatility has decreased significantly, both in real andnominal terms (Kim and Nelson 1999, McConnell and Perez-Quiros 2000, Stock andWatson 2002).

4 Where does time variation in bond betas comefrom?

Section 3 has shown evidence that time variation in bond risk–as measured by thecovariance of bond returns with stock returns–is positively related to movements inthe level of short-term interest rates and in spread between the yield on long-termbonds and the short-term interest rate. This section further explores this relation byperforming a decomposition of bond returns into three components, and exploringthe ability of the short rate and the yield spread to explain the time variation in thecovariance of each component with realized stock returns.

This section also explores time variation in bond risk measured as the covarianceof bond returns with consumption growth. Standard representative-agent asset pric-ing models suggest that risk is also related to the covariance of asset returns withthe intertemporal marginal rate of substitution of consumption of the representativeinvestor, which is typically a function of aggregate consumption growth. Thus onemight also want to explore whether the covariance of bond returns with consumptiongrowth is time varying, and whether the yield spread is correlated with this other

16

measure of bond risk. Unfortunately we do not observe consumption at high frequen-cies. In a recent paper, Duffee (2005) examines time variation in the covariance ofstock returns and consumption growth using low frequency data. This gives us somehope that it is possible to explore, with some degree of accuracy, time variation inthe consumption risk of bonds.

4.1 Bond return decomposition

The bond return decomposition is based on the asset return decomposition of Camp-bell and Ammer (1993). Using the log linear asset pricing framework of Campbelland Shiller (1988) and Campbell (1991), Campbell and Ammer (1993) derive an ex-pression for the excess return on any asset with respect to the short-term interest rateas a function of news about future real cash flows, news about future real interestrates, and news about expected future excess returns.

A straightforward adaptation of the Campbell-Ammer approach (see Appendix)shows that we can write the unexpected log nominal return on a bond in excess ofthe log nominal short rate as the sum of three components:

xrbt+1 − Et [xrbt+1] = − (Et+1−Et)"NPj=1

ρjbπt+1+j

#

− (Et+1−Et)"NPj=1

ρjbrf,t+1+j

#

− (Et+1−Et)"NPj=1

ρjbxrbt+1+j

#, (4)

where xrbt+1 = rb$t+1 − y1,t, rb$t denotes the log of the nominal gross return on anominal bond with maturity N , y1,tdenotes the short-term nominal log interest rate,πt+1 denotes the log inflation rate, and rf,t denotes the ex-ante real interest rate(rf,t+1 = y1,t−Et [πt+1]). ρb is a loglinearization constant described in the Appendix.This constant is equal to one when the bond is a zero-coupon bond. Otherwise, theconstant is related to the average coupon yield on the bond.

The first element on the right hand side of equation (4) captures the real cash flownews component of nominal bond excess returns. Since nominal bond cash flows–coupons and principal–are fixed in nominal terms, the real cash flow on a nominal

17

bond is given by the inverse of the price level. Therefore the real cash flow newscomponent of nominal bond log excess returns is given by the negative of changes inexpected future inflation: positive inflation surprises are negative news for bond realcash flows, and cause bond prices to fall.

The second element and the third element on the right hand side of equation (4)capture, respectively, the effect of news about future real interest rates and excessreturns–or risk premia–on realized nominal bond excess returns. Unexpected in-creases in real interest rates or in the premium that investors demand for holdingbonds cause bond prices to fall.

The components of bond returns are unobservable, and must be estimated us-ing statistical methods. Campbell and Ammer (1993) suggest estimating a vectorautoregressive (or VAR) model that captures the dynamics of excess returns, andusing these estimates to compute the news components of unexpected returns. Ofcourse, this return decomposition is sensitive to the specification of the VAR system(Campbell 1991, Campbell and Ammer 1993, Campbell, Lo, and MacKinlay 1997,Campbell and Vuolteenaho 2004, Chen and Zhao 2006). I adopt their methodologyand, to minimize specification issues, I include variables in the VAR system whichone could reasonably expect to capture predictable variation in bond returns, stockreturns, growth in real consumption per capita, inflation, and the ex-ante real interestrate. I also include variables that help identify bond nominal and real cash flows.

Unfortunately, the frequency of observation of some of these variables is not higherthan a month or a quarter. Thus the empirical results shown in this section are allbased on monthly and quarterly returns. I estimate two VAR systems. The firstVAR model is based on monthly observations of excess log returns on bonds, excesslog returns on stocks, log inflation, the nominal short-term log interest rate, the yieldspread, and the log dividend-price ratio. The second VAR system augments thefirst system by including log consumption growth, and it is estimated for this reasonusing quarterly data. I estimate the systems using the same sample period as in themonthly regressions shown in Section 3. A longer sample period that includes thedecade of the 1950’s–the earliest data for which observations of the yield spread areavailable–gives very similar results, as shown in the Appendix.

The residuals and the coefficient estimates from the VAR system allow the extrac-tion of unexpected stock excess returns, unexpected bond excess returns, unexpectedconsumption growth, inflation news, and news about expected bond excess returns.Real interest rate news then obtain as a residual using equation (4). The Appendix

18

provides details of the derivation of these components under the VAR specification,and reports the estimates of the VAR models.

Given estimates of the components of bond returns, unexpected stock returns,unexpected bond returns and innovations to consumption growth, it is straightforwardto compute the time series of the realized covariance, the realized CAPM beta and therealized consumption beta of each component of bond returns, and to examine howthese moments are related to time variation in the short rate and the yield spread.

4.2 Full sample estimates of the CAPM beta and the con-sumption beta of bonds

Before exploring the time variation in the second moments of bond excess returns andtheir components, it is useful to look at the full sample estimates of these moments.Table 9 reports full-sample estimates of the CAPM beta, the consumption beta ofbond returns and the betas that result from the covariance of each component ofbond excess returns with stock returns and consumption growth. That is, Table 9reports the realized covariance of unexpected stock excess returns (Panel A) or con-sumption growth (Panel B) with real cash flow news (−(Et+1−Et)[

PNj=1 ρ

jbπt+1+j ]),

real interest rate news (−(Et+1−Et)[PN

j=1 ρjbrf,t+1+j ]), expected excess return news

(−(Et+1−Et)[PN

j=1 ρjbxrbt+1+j]), and unexpected bond excess returns (xrbt+1−Et [xrbt+1])

divided by the variance of unexpected stock excess returns (consumption growth).Note that the first three betas add up to the fourth.

Panel A shows that the full-sample estimates of the CAPM beta of bonds esti-mated from monthly returns and from quarterly returns are both positive but smallat about 0.06. Table 2 shows a similar estimate using daily returns. But Panel A alsoshows that the small total CAPM beta hides substantial offsetting covariation of thecomponents of bond returns with stock returns.

First, the estimate of the CAPM beta of the real cash flow component of bondreturns is positive and it has a size at least twice the size of the total beta in bothVAR models. A positive real cash flow beta implies that unexpected stock returnsand inflation news are unconditionally negatively correlated in this sample period.Thus inflation contributes positively to bond risk as measured by its covariance withstock returns: Unexpected negative returns in the stock market tend to coincide with

19

positive inflation surprises which in turn drive bond returns down, making bondsrisky in a CAPM sense.

Second, the estimate of the CAPM beta of the discount rate component of bondreturns–i.e., the sum of the CAPM beta of the real rate news component plus theCAPM beta of the expected excess return news component–is negative and largein both VAR models. Its magnitude offsets in large part, but not completely, theCAPM beta of the component related to bond real cash flow news. As a result, thetotal CAPM beta of bonds is positive but small.

Interestingly, Panel A shows that the contribution of each component of bonddiscount rate news to total CAPM beta depends on whether we use the monthlyVAR or the quarterly VAR to extract bond return news. At a 1-month horizon, thecontribution of real interest rate news to bond beta is large and negative, while thecontribution of expected excess return news is small and positive. By contrast, at a 1-quarter horizon, the real interest rate bond beta is essentially zero while the expectedexcess return beta is large and negative. It is important to note that this differencein results between the monthly and the quarterly VAR systems does not result fromthe fact that the quarterly VAR model also includes log consumption growth as anadditional variable. A quarterly VAR model that excludes log consumption growthand thus has an identical specification as the monthly VARmodel gives similar results.

But the net effect of the discount rate components of bond returns is the same atboth frequencies: Together they tend to attenuate the contribution of the real cashflow component of bond returns to bond CAPM risk. Unexpected negative returns inthe stock market tend to coincide with negative shocks to bond discount rates, whichin turn drive bond returns up and make bonds less risky in a CAPM sense.

Panel B in Table 9 reports the full sample estimates of the consumption beta ofbonds and its components. The unconditional full sample estimate of bond consump-tion beta is negative, with a magnitude about twice that of the bond CAPM beta,at -0.122. Thus nominal bonds help investors hedge aggregate consumption risk. Adecomposition of bond returns into real cash flow news and discount rate news helpsus understand the source of the negative correlation of realized bond returns withshocks to consumption growth.

Panel B in Table 9 shows that the real cash flow consumption beta of bonds islarge and negative, reflecting positive covariation between inflation news and shocks toconsumption growth. Thus negative shocks to consumption growth tend to coincide

20

with negative shocks to inflation which in turn drive the price of nominal bonds up.This makes nominal bonds an asset that can help investors hedge consumption risk.

The consumption beta of bond discount rate news–which results from addingup the real interest rate news beta and the expected excess return beta–is positive,implying that negative shocks to consumption tend to coincide with increases in bonddiscount rates. This negative correlation between shocks to consumption growth andnews to bond discount rates is consistent with the predictions of habit consump-tion models in which negative surprises in consumption growth drive aggregate riskaversion up and lead to unexpected declines in asset prices (Campbell and Cochrane1998, Wachter 2006). However, the negative effect of consumption growth on bondrisk premia is not large enough to offset entirely the consumption risk reducing effectsof the real cash flow component of bond returns.

The positive sign of the consumption beta of bonds is driven by a positive con-sumption beta of expected excess return news. The beta associated with real interestrate news is negative.

4.3 Bond CAPM beta predictive regressions with monthlyand quarterly returns

Prior to examining the time series variation in the CAPM betas of the components ofbond returns, it is useful to verify whether the predictive regression results shown inTable 4 for realized covariances and bond CAPM betas obtained from daily returnsalso hold when we measure returns at lower frequencies. Table 10 reports predictiveregressions of realized covariances and bond CAPM betas onto a constant, their ownlagged value, the short-term nominal interest rate and the yield spread. Realizedsecondmoments are based on realizations of unexpected stock and bond excess returnsextracted from the monthly VAR model.

Table 10 shows that the basic results shown in Table 4 hold reasonably well whenusing returns measured at a monthly frequency: Both the short rate and the yieldspread positively forecast bond CAPM betas. As one might expect, Newey-Westt-statistics, χ2 statistics and R2’s are smaller, especially at short horizons, but theoverall direction and significance of the results remain. Table 10 thus suggests thatestimates of realized covariance and beta based on monthly observations of returns,while noisier estimates than those based on daily returns, can still uncover systematic

21

patterns in the time series of second moments.

Table 11 reports predictive regressions for the realized CAPM betas of bond cashflow news (Panel A), real rate news (Panel B), and expected excess return news (PanelC) measured at horizons between three months and five years. The coefficient on theshort rate is statistically significant only in the predictive regression for the CAPMbeta of bond expected excess return news. The sign of the coefficient is positive. Thisresult suggests that the positive relation between the level of the short interest rateand the total bond CAPM beta operates through the discount rate channel: Bondsbecome riskier at times of high interest rates because at those times stock returnsand bond discount rates tend to move in opposite directions.

The coefficient on the yield spread is statistically significant in the three predictiveregressions. Interestingly, the sign of this coefficient changes across the regressions:The yield spread forecasts negatively the real cash flow CAPM beta of bonds, andpositively the discount rate CAPM beta of bonds.

A widening of the yield spread is associated with a decrease in the covarianceof bond real cash flow news with unexpected stock returns or, equivalently, with anincrease in the covariance of inflation news with unexpected stock returns. That is,while Table 9 shows that unexpected stock returns and inflation news are uncondi-tionally negatively correlated, Table 11 suggests that they tend to be less negativelycorrelated when the yield spread widens. Thus bonds are less risky from a real cashflow (or inflation) perspective at times when the yield spread widens. Since the yieldspread is a countercyclical variable, this suggests that the covariance of inflation andstock returns is procyclical: It is smallest at the through of the business cycle, andlargest at the peak. Boyd, Hu and Jagannathan (2005) find that on average thestock market responds positively to news of rising unemployment in expansions, andnegatively in contractions. The results shown here indicate that the stock marketresponds negatively to news of rising inflation in expansions, and less negatively oreven positively in contractions.

However, a widening of the yield spread is also associated with an increase in thediscount rate components of the CAPM beta of bonds. Times when the yield spreadincreases tend to be times when there is increased negative covariation between stockreturns and bond discount rates, which makes bonds riskier. To the extent that theyield spread proxies for business cycle variation in aggregate relative risk aversion, asin Wachter (2006), this result suggests that the time variation in the discount ratecomponent of bond CAPM risk is driven by countercyclical variation in aggregate

22

risk aversion, which makes investors discount more heavily all long-term assets in badtimes.

Overall, these effects suggest that in recessions–i.e., when the yield spread widens–the real cash flow (or inflation) risk of bonds decreases. But these are also times inwhich aggregate risk aversion increases and makes investors dislike all risky assets.The risk premium (or risk aversion) effect more than offsets the cash flow effect, andtherefore bond risk moves countercyclically.

Table 12 reproduces the predictive regressions shown in Table 10 and Table 11 forbond excess returns and their components extracted from the quarterly VAR model.Betas are measured over four, twelve and twenty quarters. Table 12 shows that thebasic results shown for bond returns measured at daily and monthly frequencies stillhold when returns are measured at a quarterly frequency. However, the statisticalsignificance of the coefficient on the yield spread weakens considerably in all predictiveregressions.

4.4 Bond consumption beta predictive regressions

Table 13 examines the ability of the short nominal interest rate and the yield spreadto explain time variation in the realized consumption beta of bonds and its compo-nents. The predictive results shown in this table are based on bond unexpected excessreturns, the news components of bond returns, and shocks to consumption growthextracted from the quarterly VAR model. Table 13 has a structure identical to thatof Table 12.

Table 13 shows that the statistical evidence that the short rate and the yieldspread explain time variation in bond consumption beta and its components is ingeneral much weaker than the corresponding evidence for bond CAPM betas. Inmost instances, the coefficients on either the short rate or the yield spread are notstatistically significant. However, we have already seen that the statistical evidence ofsignificance of coefficients in bond CAPM regressions weakens as the return measure-ment frequency decreases. Therefore the weak statistical inference in consumptionbeta regressions could well be the result of the fact that realized second moments arebased on returns and consumption measured at a relatively low (quarterly) frequency.Nevertheless, regression results for a longer sample period that starts in 1952–notreported here to save space–are stronger.

23

Panel A in Table 13 shows that, when statistically significant, both the short rateand the yield spread forecast positively the consumption beta of bonds. Thus increasesin short-term nominal interest rates and in the yield spread forecast are correlatedwith increases in bond risk, both in a CAPM sense and in a Consumption-CAPMsense.

Panel B, panel C, and panel D in the table report predictive regression results forthe consumption betas of the components of bond returns. These results suggest thathe short rate forecasts positively the consumption beta of bonds associated with realcash flow news, and the consumption beta associated with real rate news. The yieldspread forecasts positively the real cash flow component of bond consumption betaat all horizons, suggesting that a widening of the yield spread is associated with anincrease in the covariance of shocks to consumption growth with bond real cash flownews or, equivalently, a decrease in the covariance of shocks to consumption growthwith inflation news.

5 Conclusions

This paper documents time variation in the realized CAPM beta of bonds, the con-sumption beta of bonds, and bond return volatility, and shows that it is systematicrelated to movements in the term structure of nominal interest rates, particularlythe intercept and the slope of the yield curve, which are themselves variables thatproxy for business conditions. Thus this paper offers empirical evidence that the co-movement of bond returns with stock returns (or consumption growth) can changesignificantly at business cycle frequencies.

A robust stylized fact in empirical asset pricing is that the expected excess returnon long-term bonds is time varying, and that this time variation is positively relatedto movements in the yield spread. This paper shows evidence that time variation inbond risk, measured by both the CAPM beta of bonds and the consumption betaof bonds, is also positively related to the yield spread. This relation is robust acrossperiods of generally increasing inflation and short-term interest rates, and periodsof generally decreasing inflation and short-term interest rates. It is also robust tothe inclusion in beta forecasting regressions of variables that proxy for inflation andeconomic uncertainty.

24

A second stylized fact in empirical asset pricing is that short-term nominal in-terest rates forecast positively stock return volatility and exchange rate volatility.This paper shows that the nominal short-term interest rate also forecasts positivelyrealized bond CAPM betas, bond consumption betas, and bond return volatility.Recent research has shown that measures of inflation uncertainty forecast positivelythe covariance of bond returns with stock returns (David and Veronesi 2004). Thispaper shows evidence that the short rate absorbs the effect of these measures on thecovariance of bond and stock returns. This evidence lends support to the hypothesisin Glosten et al. (1993) that the short rate might be a market-driven empirical proxyfor inflation uncertainty and, more generally, for aggregate economic uncertainty.Consistent with this hypothesis, the paper also shows that the ability of the shortrate to explain time variation in bond betas, bond return volatility and stock returnvolatility is much weaker in the period after 1981, which has been characterized bylow macroeconomic volatility.

This paper also investigates the sources of systematic time variation in bond riskby conducting a decomposition of unexpected bond returns into a real cash flow newscomponent and a discount rate news component, and examining how the betas ofeach component are related to changes in the term structure of interest rates.

I find that the short-term interest rate forecasts positively the CAPM beta of allthe components of bond returns. By contrast, the yield spread forecasts negativelythe CAPM beta of the real cash flow news component of bond returns, which reflectsthe negative of the conditional covariance of stock returns with news about expectedfuture inflation, and positively the CAPM beta of the discount rate component ofbond returns. That is, a widening of the yield spread is associated with an increase inthe covariance of unexpected stock returns with inflation news, which makes nominalbonds less risky from a cash-flow perspective. But it is also associated with an increasein the covariance of unexpected stock returns with bond discount rate news, whichmakes bonds riskier. Overall, the discount rate effect dominates the cash flow effect.

There are several possible directions of future research. First, the empirical analy-sis in this paper builds in estimates of bond risk and volatility using standard realizedsecond moments, which weight each observation equally. It would be interesting torepeat this analysis with estimates of second moments obtained using the mixed datasampling (or MIDAS) technique of Ghysels, Santa-Clara and Valkanov (2005, 2006),where observations are weighted according to an exponentially decreasing weightingscheme which is estimated jointly with the forecasting regression equation. Alter-

25

natively, one could avoid using realized second moments by using instead a latentvariable model along the lines of the multivariate GARCH model of Bollerslev, Engleand Wooldridge (1998), the Dynamic Conditional Correlation model of Engle (2002),and other models summarized in Andersen, Bollerslev, Chrisoffersen, and Diebold(2005).

This paper offers empirical evidence that suggests that bond risk changes overtime, and that these changes are correlated with time variation in the term structure ofnominal interest rates. A second direction of future research is to develop theoreticalmodels, particularly in the area of term structure models, which can generate theempirical patterns uncovered in this paper, and at the same time link the relationbetween second moments and interest rates to underlying fundamental factors suchas real interest rates and expected inflation.

A third direction of future research would be to examine the implications for assetallocation of changing correlations between bond returns and stock returns along thelines of Buraschi, Portia, and Trujani (2006). This is particularly important in lightof the fact uncovered in this paper that this correlation appears to move counter-cyclically. Campbell and Viceira (2002) and Chacko and Viceira (2005) have shownthat persistent changes in expected returns and risk can lead long-term investors tooptimally adopt investment policies which are

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7 Appendix: Bond return decomposition.

A direct extension of the results Campbell and Ammer (1993) shows that we canexpress the unexpected log nominal return on any asset in excess of the log nominal

31

short rate as

xrt+1 − Et [xrt+1] ≈ πt+1 − Et [πt+1] + (Et+1−Et)"∞Pj=0

ρj∆cft+1+j

#

− (Et+1−Et)"∞Pj=1

ρjrf,t+1+j

#− (Et+1−Et)

"∞Pj=1

ρjxrt+1+j

#,

where xrt+1 = r$t+1 − y1,t, r$t is the log of the nominal gross return on the asset, y1,tis nominal short interest rate, πt+1 is the log inflation rate, dt is the real cash flowon the asset, and rf,t is the ex-ante real interest rate (rf,t+1 = y1,t − Et [πt+1]). Theconstant ρ is a loglinearization constant, equal to ...

[TO BE COMPLETED]

32

Figure 1 CAPM beta of bonds

(1962.07-2003.12)

Realized beta of bonds based on 3-months of daily returns on stocks and bonds.

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

62.07

64.07

66.07

68.07

70.07

72.07

74.07

76.07

78.07

80.07

82.07

84.07

86.07

88.07

90.07

92.07

94.07

96.07

98.07

00.07

02.07

Figure 2 CAPM beta of bonds and the short interest rate

(1962.07-2003.12)

Realized beta of bonds based on 3-months of daily returns on stocks and bonds (right axis), and annualized log yield on the 30-day Treasury bill (left axis).

0

2

4

6

8

10

12

14

16

18

62.07

64.07

66.07

68.07

70.07

72.07

74.07

76.07

78.07

80.07

82.07

84.07

86.07

88.07

90.07

92.07

94.07

96.07

98.07

00.07

02.07

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

CAPM Bond Beta

T-Bill Yield

Figure 3 CAPM beta of bonds and the yield spread

(1962.07-2003.12)

Realized beta of bonds based on 3-months of daily returns on stocks and bonds (right axis), and annualized log yield spread (right axis)

-4

-2

0

2

4

6

8

62.07

64.07

66.07

68.07

70.07

72.07

74.07

76.07

78.07

80.07

82.07

84.07

86.07

88.07

90.07

92.07

94.07

96.07

98.07

00.07

02.07

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

CAPM Bond Beta

Yield Spread

-

Table 1

Full sample moments of daily stock and bond returns (1962.07.02-2003.12.31)

This table reports sample statistics of daily log stock and bond total returns. Stock returns are log returns on the stock total returns on the value weighted portfolio of all stocks traded in the NYSE, the AMEX, and NASDAQ from CRSP. Bond returns are log returns on the 5-year constant maturity bond from the CRSP Fixed Term Indices File.

Stocks Bonds Stocks and Bonds

Mean return (% p.a.) 10.47 7.94

Standard Deviation (% p.a.) 14.51 5.01

Correlation (%) 2.95

CAPM beta of bonds 0.03

Table 2

Full sample moments of monthly realized covariance, bond beta, and the volatility of stock and bond returns

(1962.07-2003.12)

This table reports full sample statistics of monthly realized covariance, bond beta, and stock and bond return volatility. These estimates are based on the daily stock and bond returns described in Table 1. The estimators are given in equations (1) through (4) in text. ACF(#) is the autocorrelation coefficient of order #. R/S (#) is Lo’s (1991) modified long-memory statistic and # is the number of autocovariances included in the computation of the statistic. The 5% and 1% critical values for this statistic under the null of no long-memory are 1.862 and 2.098 respectively. Please note that we report autocorrelation coefficients for log volatilities.

γB,S(t,1) βB(t,1) ( )2 ,1B tσ ( )2 ,1S tσ

Mean 2.9 × 1e-6 0.08 (12.38% p.a.)2 (3.91% p.a.)2

S.D./Mean 4.39 1.67 1.88 1.57

Correlation with ( )2 ,1B tσ (%) 23.6 26.2 1.0 21.6

Correlation with ( )2 ,1S tσ (%) −32.7 −16.9 21.6 1.0

ACF (1) 0.479 0.513 0.706 0.800

ACF (2) 0.471 0.481 0.629 0.748

ACF (3) 0.414 0.456 0.564 0.711

ACF (12) 0.270 0.417 0.397 0.523

R/S (0) 3.165 4.412 2.880 4.248

R/S (12) 1.810 2.327 1.395 1.866

Table 3

Predictive regressions for bond excess returns (1962.07-2003.12)

Table 3 reports monthly overlapping regressions of long horizon log bond excess returns onto a constant, the log short rate y(t) and the yield spread spr(t), as described in equation (5) in text. The short rate is the log yield on the 30-day Treasury Bill from CRSP, and the spread is the difference between the log yield on a 5-year artificial zero-coupon bond from the CRSP Fama-Bliss Discount Bond File, and the log yield on the Treasury Bill. We compute bond excess returns as the difference between the log total return a 5-year constant maturity coupon bond from CRSP, and the log yield on the Treasury Bill. The table reports coefficient estimates, the R2 of the regression and, in brackets, Newey-West corrected t-statistics, and a Newey-West corrected χ2 statistic of the significance of the slopes in the regression. The lag order of the Newey-West correction is equal to the horizon (in months) minus one. The χ2 statistic has an asymptotic χ2 statistic distribution with 2 degress of freedom. The 5% critical value of this distribution is 5.99, and the 1% critical value is 9.21.

Horizon (months) y(t) [t-stat]

spr(t) [t-stat]

R2 (%) [χ2-stat]]

1 0.38

[0.74] 2.82

[2.88] 2.7

[9.03]

3 1.33 [1.09]

8.05 [3.24]

6.7 [11.09]

6 2.45 [1.26]

15.52 [4.09]

13.0 [16.85]

12 5.29 [1.38]

27.83 [4.24]

20.0 [18.95]

36 23.443 [3.38]

37.627 [3.03]

23.2 [18.95]

60 39.495 [6.87]

51.581 [4.13]

37.6 [47.68]

Table 4

Covariance and bond beta predictive regression

(1962.07-2003.12)

Table 4 reports monthly overlapping regressions of realized covariances (Panel A) and bond betas (Panel B) onto a constant, their own lagged value, the log short rate y(t) and the yield spread spr(t), as described in equation (6) in text. Please refer to Table 3 and Table 4 for a description of the data. The table reports coefficient estimates, the R2 of the regression and, in brackets, Newey-West corrected t-statistics, and a Newey-West corrected χ2 statistic of the significance of the slopes in the regression. The lag order of the Newey-West correction is equal to the horizon (in months) minus one. The χ2 statistic has an asymptotic χ2 statistic distribution with 3 degress of freedom. The 5% critical value of this distribution is 7.82, and the 1% critical value is 11.34.

(A) Covariance regression

Horizon (months)

Lagged [t-stat]

y(t) [t-stat]

spr(t) [t-stat]

R2 (%) [χ2-stat]

1 0.338 [4.690]

0.002 [5.798]

0.002 [2.959]

30.2 [91.18]

3 0.472 [4.674]

0.001 [4.657]

0.001 [1.977]

45.2 [84.98]

6 0.393 [3.304]

0.002 [5.224]

0.001 [1.391]

43.9 [66.77]

12 0.692 [3.295]

0.001 [2.749]

-0.000 [-0.232]

55.5 [72.53]

36 0.131 [0.218]

0.002 [1.736]

0.002 [1.390]

29.0 [13.75]

60 -0.834 [-1.824]

0.002 [2.866]

0.004 [2.282]

43.9 [19.71]

(B) Capm beta regression

1 0.380 [7.175]

18.394 [5.324]

29.056 [4.542]

33.4 [209.35]

3 0.502 [7.095]

14.504 [4.248]

20.053 [3.686]

47.4 [213.04]

6 0.569 [6.251]

11.412 [2.856]

17.466 [2.805]

53.0 [181.35]

12 0.588 [4.530]

11.571 [1.978]

15.558 [2.000]

52.6 [89.49]

36 -0.076 [-0.394]

18.186 [2.254]

31.053 [2.712]

20.5 [14.32]

60 -0.889 [-6.104]

26.435 [5.220]

37.702 [4.357]

58.6 [47.02]

Table 5

Volatility predictive regressions

(1962.07-2003.12)

Table 5 reports monthly overlapping regressions of realized bond return log volatility (Panel A) and stock return log volatility (Panel B) onto a constant, their own lagged value, the log short rate y(t) and the yield spread spr(t), as described in equation (6) in text. Please refer to Table 3 and Table 4 for a description of the data. The table reports coefficient estimates, the R2 of the regression and, in brackets, Newey-West corrected t-statistics, and a Newey-West corrected χ2 statistic of the significance of the slopes in the regression. The lag order of the Newey-West correction is equal to the horizon (in months) minus one. The χ2 statistic has an asymptotic χ2 statistic distribution with 3 degress of freedom. The 5% critical value of this distribution is 7.82, and the 1% critical value is 11.34.

(A) Bond volatility regression

Horizon (months)

Lagged log vol [t-stat]

y(t) [t-stat]

spr(t) [t-stat]

R2 (%) [χ2-stat]

1 0.734 [24.151]

85.236 [4.654]

92.520 [2.290]

65.7 [899.26]

3 0.756 [15.830]

78.340 [3.113]

34.248 [0.660]

69.2 [400.15]

6 0.719 [8.801]

88.434 [2.629]

-12.919 [-0.214]

66.8 [143.22]

12 0.589 [4.148]

108.413 [2.053]

-70.508 [-0.919]

59.8 [44.53]

36 0.016 [0.092]

181.148 [2.330]

49.672 [0.717]

37.6 [50.62]

60 -0.304 [-0.988]

214.969 [2.647]

226.239 [1.564]

28.2 [21.23]

(B) Stock volatility regression

1 0.699 [21.714]

23.441 [1.711]

-2.908 [-0.088]

50.2 [509.10]

3 0.661 [16.104]

32.985 [1.735]

-25.543 [-0.566]

45.8 [286.48]

6 0.626 [9.837]

36.135 [1.497]

-50.052 [-0.874]

41.9 [121.15]

12 0.599 [5.379]

14.282 [0.464]

-147.070 [-2.392]

37.8 [47.10]

36 0.176 [0.734]

-2.480 [-0.047]

-135.710 [-1.197]

7.3 [2.49]

60 -0.528 [-1.929]

28.708 [0.649]

64.118 [0.829]

21.5 [5.33]

Table 6 Stability of capm beta and return volatility predictive regressions

(1962.07-2003.12) Table 6 reports monthly overlapping regressions of realized bond beta (Panel A), bond return log volatility (Panel B) and stock return log volatility (Panel C) onto a constant, their own lagged value, the log short rate y(t) and the yield spread spr(t), as well as the interaction of each variable with an indicator variable IND(t) which is equal to zero between 1962.01 and 1981.07, and equal to one between 1981.08 and 2003.12. Please refer to Table 3 and Table 4 for a description of the data. The table reports coefficient estimates, the R2 of the regression and, in brackets, Newey-West corrected t-statistics, and a Newey-West corrected χ2 statistic of the significance of the slopes in the regression. The lag order of the Newey-West correction is equal to the horizon (in months) minus one. The χ2 statistic has an asymptotic χ2 statistic distribution with 6 degress of freedom. The 5% critical value of this distribution is 12.59, and the 1% critical value is 16.81.

(A) CAPM bond beta

Horizon (months)

lagged [t-stat]

IND*lagged [t-stat]

y(t) [t-stat]

IND*y(t) [t-stat]

spr(t) [t-stat]

IND*spr(t) [t-stat]

R2 (%) [χ2-stat]

3 0.023 [0.256]

0.604 [5.589]

22.301 [5.884]

-8.902 [-2.525]

22.014 [2.786]

-1.567 [-0.163]

51.8 [232.81]

6 0.160 [1.249]

0.494 [3.439]

19.831 [4.283]

-10.573 [-2.660]

20.365 [2.282]

3.871 [0.333]

55.7 [212.342]

12 0.360 [1.844]

0.302 [1.419]

17.249 [2.468]

-12.640 [-2.764]

13.786 [1.254]

15.638 [0.948]

55.7 [127.75]

(B) Bond return volatility

3 0.615 [12.874]

-0.126 [-5.102]

244.747 [6.012]

-225.241 [-4.927]

214.139 [2.633]

-163.543 [-1.681]

72.9 [471.36]

6 0.548 [6.872]

-0.130 [-4.017]

260.213 [4.534]

-244.638 [-3.711]

122.854 [1.320]

-79.733 [-0.739]

71.7 [149.40]

12 0.399 [2.946]

-0.129 [-4.215]

267.365 [4.207]

-273.838 [-3.979]

-10.805 [-0.111]

71.637 [0.644]

67.7 [56.03]

(C) Stock return volatility

3 0.623 [15.913]

-0.085 [-3.198]

83.917 [2.838]

-93.164 [-2.471]

28.956 [0.470]

-146.421 [-1.626]

48.8 [330.85]

6 0.578 [9.733]

-0.083 [-2.465]

79.203 [2.436]

-86.741 [-1.738]

-31.311 [-0.454]

-105.443 [-0.954]

45.7 [141.60]

12 0.542 [5.455]

-0.114 [-2.600]

73.320 [1.899]

-133.017 [-1.874]

-113.505 [-1.254]

-130.338 [-1.122]

45.0 [77.22]

Table 7 Short rate, spread, and tent variable

(64.01-03.12) Table 7 reports monthly overlapping regressions of bond excess returns (Panel A) and realized bond beta (Panel B) onto a constant, their own lagged value (in the case of bond betas only, and not reported here), the log short rate y(t), and the yield spread spr(t) or Cochrane-Piazzesi tent-shaped linear combination of forward rates. Please refer to Tables 2 through 4 for a description of the data. The table reports coefficient estimates, the R2 of the regression and, in brackets, Newey-West corrected t-statistics, and a Newey-West corrected χ2 statistic of the significance of the slopes in the regression. The lag order of the Newey-West correction is equal to the horizon (in months) minus one. The χ2 statistic in Panel A has an asymptotic χ2 statistic distribution with 2 degress of freedom. The 5% critical value of this distribution is 5.99, and the 1% critical value is 9.21. The χ2 statistic in Panel B has an asymptotic χ2 statistic distribution with 3 degress of freedom. The 5% critical value of this distribution is 7.82, and the 1% critical value is 11.34.

(A) Expected return regression

Horizon (months)

y(t) [t-stat]

spr(t) [t-stat]

tent(t) [t-stat]

R2 (%) [χ2-stat]

3 1.395

[1.092] 8.134

[3.212] 6.6

[10.71] -0.179

[-0.132] 0.003

[3.470] 6.6

[13.407]

12 5.457 [1.347]

28.082 [4.144]

19.7 [18.539]

0.196 [0.078]

0.014 [8.448]

33.6 [71.714]

36 23.344

[3.222] 37.422 [2.891]

21.5 [16.06]

15.946 [2.205]

0.015 [3.088]

21.8 [20.26]

(B) Capm beta regression

3 14.463

[4.072] 19.995 [3.573]

46.3 [194.20]

9.996 [2.930]

0.004 [1.973]

44.6 [178.62]

12 11.764

[1.918] 15.894 [1.997]

51.1 [79.22]

7.604 [1.367]

0.001 [0.270]

49.5 [80.25]

36 18.257

[2.189] 31.225 [2.642]

19.5 [12.0]

10.477 [1.172]

0.004 [0.865]

9.5 [4.13]

Table 8

Bond Risk and Inflation Uncertainty (1968.Q4 -2003.Q4)

Table 11 reports quarterly overlapping regressions of realized covariances (Panel A) and bond betas (Panel B) onto a constant, their own lagged value, inflation uncertainty, the log short rate y(t) and the yield spread spr(t). Please refer to Table 2 and Table 3 for a description of the data. Inflation uncertainty is the cross-sectional standard deviation of inflation forecasts from the Survey of Professional Forecasters. The table reports coefficient estimates, the R2 of the regression and, in brackets, Newey-West corrected t-statistics, and a Newey-West corrected χ2 statistic of the significance of the slopes in the regression. The lag order of the Newey-West correction is equal to the horizon (in months) minus one. The χ2 statistic has an asymptotic χ2 statistic distribution with 2. 3 or 4 degress of freedom, depending on the regression.

(A) Covariance regressions

Horizon (months)

Lagged [t-stat]

Infl Unc(t) [t-stat]

y(t) [t-stat]

spr(t) [t-stat]

R2 (%) [χ2-stat]

3 0.550 [4.685]

0.000 [3.315]

42.8 [44.21]

0.484 [4.205]

0.000 [0.868]

0.001 [3.601]

45.8 [47.85]

0.421 [3.131]

0.000 [0.950]

0.002 [4.244]

0.002 [1.901]

47.6 [58.01]

12 0.550 [5.216]

0.000 [2.988]

42.8 [37.93]

0.484 [4.576]

0.000 [1.038]

0.001 [3.850]

45.8 [46.92]

0.421 [3.354]

0.000 [1.116]

0.002 [3.789]

0.002 [2.173]

47.6 [46.78]

(B) Capm beta regression

3 0.636 [8.939]

1.155 [1.885]

42.7 [96.67]

0.573 [6.815]

-0.383 [-0.579]

11.813 [2.527]

45.1 [123.07]

0.518 [5.864]

-0.471 [-0.787]

16.644 [3.577]

18.076 [2.666]

46.9 [133.31]

12 0.636 [7.882]

1.155 [1.429]

42.7 [71.13]

0.573 [5.652]

-0.383 [-0.517]

11.813 [2.320]

45.1 [88.69]

0.518 [4.535]

-0.471 [-0.753]

16.644 [3.001]

18.076 [3.065]

46.9 [85.46]

Table 9

Full sample estimates of components of bond beta

Table 9 reports full sample estimates of the components of bond beta described in equations (8)-(10) in text. This decomposition is based on monthly observations of unexpected stock and bond returns extracted from the VAR(1) described in text.

(A) CAPM Bond Beta

Monthly VAR(1) (1962.07-2003.12)

Quarterly VAR(1) (1962.Q3-2003.Q4)

Bond cash flow news 0.231 0.132

Bond real rate news -0.175 0.000

Bond excess return news 0.006 -0.069

Bond CAPM beta 0.062 0.063

(B) Consumption Bond Beta

Bond cash flow news -0.632

Bond real rate news -0.231

Bond excess return news 0.740

Bond consumption beta -0.122

Table 10

Covariance and bond beta predictive regression using monthly returns

(1962.07-2003.12)

Table 8 reports monthly overlapping regressions of realized covariances (Panel A) and bond betas (Panel B) onto a constant, their own lagged value, the log short rate y(t) and the yield spread spr(t), as described in equation (6) in text. Realized covariances are based on monthly observations of unexpected stock and bond returns extracted from the VAR(1) described in text. The table reports coefficient estimates, the R2 of the regression and, in brackets, Newey-West corrected t-statistics, and a Newey-West corrected χ2 statistic of the significance of the slopes in the regression. The lag order of the Newey-West correction is equal to the horizon (in months) minus one. The χ2 statistic has an asymptotic χ2 statistic distribution with 3 degress of freedom. The 5% critical value of this distribution is 7.82, and the 1% critical value is 11.34.

(A) Covariance regression

Horizon (months)

Lagged [t-stat]

y(t) [t-stat]

spr(t) [t-stat]

R2 (%) [χ2-stat]

3 -0.008 [-0.084]

0.076 [3.886]

0.011 [0.333]

12.8 [17.01]

6 0.115 [1.381]

0.065 [4.045]

-0.003 [-0.104]

22.8 [21.90]

12 -0.005 [-0.040]

0.072 [4.539]

0.019 [0.633]

28.7 [22.67]

36 -0.154 [-0.826]

0.050 [3.960]

0.031 [1.155]

22.5 [16.32]

60 -0.843 [-4.069]

0.062 [4.483]

0.063 [2.852]

52.5 [37.73]

(B) Capm beta regression

3 0.019 [0.332]

50.677 [4.878]

21.775 [1.084]

12.2 [28.96]

6 0.125 [1.544]

39.312 [4.014]

33.516 [2.002]

22.9 [26.61]

12 -0.034 [-0.231]

39.423 [3.703]

40.289 [2.632]

25.5 [24.32]

36 -0.175 [-1.146]

26.241 [2.579]

30.781 [1.961]

18.5 [14.24]

60 -0.871 [-5.924]

28.258 [4.291]

26.555 [2.513]

63.6 [38.79]

Table 11

Components of Bond CAPM beta (monthly return data) (1962.07-2003.12)

Please refer to text for a description of this table.

(A) Bond Cash Flow News beta

Horizon (months) y(t) spr(t) R2 (%)

[χ2-stat] 3 -1.716

[-0.128] -49.106 [-2.448]

2.3 [7.10]

6 5.356 [0.655]

-39.654 [-2.699]

9.8 [14.93]

12 4.792 [0.575]

-29.469 [-2.254]

12.6 [19.61]

36 -0.647 [-0.121]

-28.730 [-2.903]

21.9 [13.93]

60 -7.477 [-1.476]

-30.608 [-3.025]

25.3 [21.07]

(B) Bond Real Rate News beta

3 6.155 [1.086]

25.340 [2.451]

1.8 [6.74]

6 3.778 [0.960]

20.925 [2.780]

5.7 [8.51]

12 1.893 [0.550]

13.067 [2.172]

5.8 [6.43]

36 0.296 [0.146]

7.350 [2.135]

11.2 [12.43]

60 2.334 [1.087]

8.667 [2.177]

20.3 [10.94]

(B) Bond Excess Return News beta

3 43.150 [3.589]

44.385 [2.767]

13.2 [30.94]

6 28.765 [3.528]

51.756 [4.146]

25.4 [42.19]

12 29.018 [3.720]

51.420 [4.286]

30.4 [44.25]

36 22.724 [2.846]

42.107 [3.719]

30.1 [23.98]

60 27.012 [4.603]

37.427 [3.641]

44.9 [22.5]

Table 12

Components of Bond CAPM beta (quarterly return data) (1962.Q3-2003.Q4)

Please refer to text for a description of this table.

(A) Total Bond Beta

Horizon (quarters) y(t) spr(t) R2 (%)

[χ2-stat] 4 16.828

[4.218] 10.168 [0.968]

12.0 [17.899]

12 11.770 [2.265]

9.297 [0.712]

13.5 [5.582]

20 20.884 [4.040]

16.838 [1.872]

48.4 [23.310]

(B) Bond Cash Flow News beta

4 -0.351 [-0.121]

-14.761 [-2.971]

12.6 [15.633]

12 2.239 [1.600]

-12.194 [-4.325]

36.4 [28.291]

20 -2.383 [-1.542]

-11.997 [-3.799]

23.6 [16.172]

(C) Bond Real Rate News beta

4 4.668 [3.114]

1.324 [0.394]

14.1 [12.667]

12 1.682 [0.953]

0.195 [0.053]

6.8 [2.387]

20 4.780 [2.259]

2.225 [0.697]

25.1 [10.180]

(D) Bond Excess Return News beta

4 11.779 [2.852]

23.232 [2.620]

10.3 [11.507]

12 7.747 [2.171]

22.664 [2.476]

17.4 [15.832]

20 8.115 [3.773]

13.223 [2.434]

48.9 [37.466]

Table 13

Components of Bond Consumption beta (quarterly return data)

(1962.Q3-2003.Q4)

Please refer to text for a description of this table.

(A) Total Bond Beta

Horizon (quarters) y(t) spr(t) R2 (%)

[χ2-stat] 4 -0.170

[-0.219] -1.837

[-1.021] 3.7

[2.914] 12 0.168

[1.155] 0.477

[1.687] 15.6

[11.350] 20 0.125

[1.880] -0.013

[-0.097] 44.7

[123.220]

(B) Bond Cash Flow News beta

4 0.063 [0.343]

0.456 [1.341]

2.8 [4.365]

12 0.031 [0.332]

0.080 [0.486]

2.7 [5.723]

20 0.118 [1.544]

0.152 [1.054]

19.1 [6.104]

(C) Bond Real Rate News beta

4 0.319 [2.237]

-0.472 [-0.962]

6.1 [5.840]

12 0.052 [1.739]

-0.078 [-0.796]

5.1 [8.98]

20 0.020 [0.629]

-0.085 [-1.624]

6.3 [12.755]

(D) Bond Excess Return News beta

4 -0.557 [-0.690]

-1.729 [-1.085]

1.8 [1.816]

12 0.021 [0.225]

0.451 [1.824]

12.7 [7.500]

20 -0.063 [-0.881]

-0.140 [-0.945]

16.1 [3.014]

Intercept R2

(1) (2) (3) (4) (5) (6)(1) log stock excess returns 0.113 -0.005 0.396 0.026 -1.376 -3.114 0.279 0.066

(3.407) (-0.096) (2.780) (3.221) (-1.700) (-2.428) (0.127)(2) log bond excess returns -0.014 -0.068 0.133 -0.003 -0.367 0.922 2.910 0.073

(-1.452) (-3.185) (2.252) (-1.196) (-1.129) (1.570) (3.108)(3) log dividend yield -0.088 0.032 -0.445 0.979 1.485 2.164 -0.996 0.984

(-2.519) (0.555) (-3.114) (115.570) (1.694) (1.618) (-.442)(4) log inflation 0.007 0.002 -0.020 0.002 0.431 0.194 -0.452 0.491

(3.858) (0.974) (-2.478) (3.584) (7.301) (2.733) (-3.561)(5) log nominal yield on T-bills 0.000 0.002 -0.010 0.000 0.012 0.963 0.087 0.942

(0.665) (2.492) (-4.630) (0.740) (1.030) (42.625) (2.345)(6) log yield spread 0.000 -0.001 0.007 0.000 -0.003 0.014 0.871 0.774

(0.283) (-1.153) (3.757) (0.018) (-.319) (0.701) (26.712)

B. Annualized percent standard deviations of residuals (diagonal) and cross-correlations of residuals (off-diagonal)

(1) (2) (3) (4) (5) (6)(1) log stock excess returns 14.486 0.169 -0.967 -0.168 0.010 -0.131(2) log bond excess returns 0.169 5.300 -0.160 -0.075 -0.481 -0.124(3) log dividend yield -0.967 -0.160 15.015 0.158 -0.008 0.123(4) log inflation -0.168 -0.075 0.158 0.751 0.132 -0.092(5) log nominal yield on T-bills 0.010 -0.481 -0.008 0.132 0.182 -0.803(6) log yield spread -0.131 -0.124 0.123 -0.092 -0.803 0.163

A. Slopes (t-statistics in parenthesis)Coefficients on lagged variables

APPENDIX: TABLES

Table A1 Estimates of Monthly VAR(1) model

(1967.07 – 2003.12)

Intercept R2

(1) (2) (3) (4) (5) (6) (7)(1) log stock excess returns 0.223 0.002 0.442 0.051 -0.552 -1.800 -0.050 -0.380 0.068

(2.203) (0.029) (1.804) (2.090) (-.518) (-1.252) (-.018) (-.212)(2) log bond excess returns -0.062 -0.052 -0.077 -0.012 0.174 0.797 3.727 0.169 0.110

(-2.433) (-2.470) (-.670) (-2.081) (0.422) (1.528) (3.370) (0.381)(3) log dividend yield -0.123 0.034 -0.459 0.971 0.882 0.335 -0.153 0.668 0.957

(-1.279) (0.423) (-1.945) (42.235) (0.877) (0.253) (-.057) (0.391)(4) log inflation 0.025 0.004 -0.045 0.005 0.301 0.135 -0.705 -0.137 0.512

(3.817) (0.665) (-2.698) (3.172) (4.131) (1.153) (-3.556) (-1.362)(5) log nominal yield on T-bills 0.005 0.005 0.005 0.001 -0.002 0.908 0.042 0.005 0.850

(2.185) (2.402) (0.367) (2.474) (-.045) (16.163) (0.395) (0.081)(6) log yield spread -0.001 -0.001 0.000 0.000 -0.008 0.034 0.771 -0.017 0.558

(-.531) (-.718) (0.044) (-.900) (-.271) (0.836) (10.911) (-.351)(7) log consumption growth 0.011 0.008 -0.037 0.002 -0.182 -0.027 0.044 0.391 0.331

(2.987) (2.701) (-3.461) (1.936) (-4.048) (-.445) (0.383) (5.942)

B. Annualized percent standard deviations of residuals (diagonal) and cross-correlations of residuals (off-diagonal)

(1) (2) (3) (4) (5) (6) (7)(1) log stock excess returns 17.126 0.188 -0.979 -0.270 -0.166 0.063 -0.067(2) log bond excess returns 0.188 5.759 -0.176 -0.339 -0.760 0.172 -0.015(3) log dividend yield -0.979 -0.176 16.223 0.256 0.151 -0.052 0.071(4) log inflation -0.270 -0.339 0.256 1.182 0.348 -0.165 0.120(5) log nominal yield on T-bills -0.166 -0.760 0.151 0.348 0.538 -0.765 0.101(6) log yield spread 0.063 0.172 -0.052 -0.165 -0.765 0.376 -0.128(7) log consumption growth -0.067 -0.015 0.071 0.120 0.101 -0.128 0.716

A. Slopes (t-statistics in parenthesis)Coefficients on lagged variables

Table A2 Estimates of Quarterly VAR(1) model

(1962.Q3-2003.Q4)

Intercept R2

(1) (2) (3) (4) (5) (6) (7)(1) log stock excess returns 0.181 0.013 0.465 0.043 -0.470 -1.328 0.934 -0.363 0.081

(3.029) (0.165) (2.018) (2.497) (-.482) (-1.252) (0.374) (-.265)(2) log bond excess returns -0.020 -0.059 -0.078 -0.003 -0.039 0.395 2.973 0.039 0.092

(-1.350) (-2.836) (-.735) (-.812) (-.110) (1.017) (2.754) (0.106)(3) log dividend yield -0.115 0.024 -0.457 0.972 0.853 0.281 -0.731 0.735 0.960

(-2.092) (0.328) (-2.049) (62.292) (0.938) (0.288) (-.311) (0.560)(4) log inflation 0.006 0.000 -0.027 0.001 0.341 0.375 -0.481 -0.035 0.514

(1.489) (0.057) (-1.590) (0.836) (5.073) (4.269) (-2.660) (-.396)(5) log nominal yield on T-bills 0.001 0.005 0.003 0.000 0.020 0.944 0.118 0.017 0.881

(1.078) (2.707) (0.300) (1.044) (0.515) (22.140) (1.136) (0.371)(6) log yield spread 0.000 -0.001 0.002 0.000 -0.016 0.027 0.742 -0.016 0.549

(0.071) (-.892) (0.293) (-.518) (-.603) (0.835) (10.811) (-.456)(7) log consumption growth 0.003 0.009 -0.042 0.000 -0.162 0.066 0.159 0.331 0.260

(1.078) (2.865) (-4.100) (-.169) (-3.451) (1.211) (1.476) (5.137)

B. Annualized percent standard deviations of residuals (diagonal) and cross-correlations of residuals (off-diagonal)

(1) (2) (3) (4) (5) (6) (7)(1) log stock excess returns 16.196 0.133 -0.977 -0.236 -0.135 0.061 -0.072(2) log bond excess returns 0.133 5.429 -0.126 -0.330 -0.763 0.172 -0.048(3) log dividend yield -0.977 -0.126 15.316 0.232 0.123 -0.052 0.084(4) log inflation -0.236 -0.330 0.232 1.174 0.325 -0.136 0.093(5) log nominal yield on T-bills -0.135 -0.763 0.123 0.325 0.502 -0.762 0.130(6) log yield spread 0.061 0.172 0.052 -0.136 -762.000 0.348 -0.143(7) log consumption growth -0.072 -0.048 0.084 0.093 0.13 -0.143 0.799

A. Slopes (t-statistics in parenthesis)Coefficients on lagged variables

Table A3 Estimates of Quarterly VAR(1) model

(1952.Q3-2003.Q4)

Table A4

Full sample estimates of components of bond beta (Quarterly results based on extended sample 1952.Q3-2003.Q4)

Table 9 reports full sample estimates of the components of bond beta described in equations (8)-(10) in text. This decomposition is based on monthly observations of unexpected stock and bond returns extracted from the VAR(1) described in text.

(A) CAPM Bond Beta

Monthly VAR(1) (1962.07-2003.12)

Quarterly VAR(1) (1952.Q3-2003.Q4)

Bond cash flow news 0.231 0.054

Bond real rate news -0.175 0.008

Bond excess return news 0.006 -0.017

Bond CAPM beta 0.062 0.045

(B) Consumption Bond Beta

Bond cash flow news -0.797

Bond real rate news -0.032

Bond excess return news 0.506

Bond consumption beta -0.323

Table A5

Components of Bond CAPM beta (quarterly return data) (1952.Q2-2003.Q4)

Please refer to text for a description of this table.

(A) Total Bond Beta

Horizon (quarters) y(t) spr(t) R2 (%)

[χ2-stat] 4 17.896

[5.578] 11.505 [1.088]

16.6 [35.29]

12 15.120 [3.121]

13.396 [0.958]

27.1 [14.782]

20 20.669 [4.337]

13.761 [1.329]

37.5 [28.092]

(B) Bond Cash Flow News beta

4 4.449 [1.590]

-13.250 [-2.162]

12.0 [10.924]

12 6.708 [2.727]

-9.409 [-1.715]

33.4 [20.568]

20 5.601 [2.017]

-5.657 [-1.081]

20.6 [11.660]

(C) Bond Real Rate News beta

4 3.488 [3.278]

2.469 [0.978]

27.9 [32.242]

12 2.428 [1.900]

2.311 [0.780]

17.0 [7.227]

20 2.223 [1.608]

-0.209 [-0.091]

10.9 [8.506]

(D) Bond Excess Return News beta

4 8.309 [3.144]

20.513 [2.960]

13.6 [15.306]

12 6.656 [4.291]

21.770 [3.037]

29.9 [27.518]

20 5.885 [4.263]

10.040 [2.096]

29.6 [24.217]

Table A6

Components of Bond Consumption beta (quarterly return data)

(1952.Q2-2003.Q4)

Please refer to text for a description of this table.

(A) Total Bond Beta

Horizon (quarters) y(t) spr(t) R2 (%)

[χ2-stat] 4 0.372

[0.951] -0.836

[-0.606] 3.1

[3.610] 12 0.223

[1.754] 0.591

[2.033] 16.7

[9.932] 20 0.104

[2.125] -0.018

[-0.143] 40.5

[74.556]

(B) Bond Cash Flow News beta

4 0.259 [1.271]

0.155 [0.295]

1.7 [1.976]

12 0.156 [1.916]

0.356 [2.127]

13.7 [7.422]

20 0.060 [1.173]

0.072 [0.661]

20.2 [9.270]

(C) Bond Real Rate News beta

4 0.182 [2.898]

-0.180 [-0.646]

7.8 [10.202]

12 0.161 [3.875]

0.093 [1.562]

28.0 [18.786]

20 0.063 [2.706]

-0.011 [-0.317]

22.4 [92.908]

(D) Bond Excess Return News beta

4 -0.079 [-0.262]

-0.799 [-1.165]

1.1 [1.441]

12 -0.90 [-1.748]

0.166 [0.714]

9.8 [14.094]

20 -0.052 [-1.193]

-0.132 [-1.436]

10.7 [7.360]