Bond Risk Premia and Realized Jump Risk

8/12/2019 Bond Risk Premia and Realized Jump Risk

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Bond Risk Premia and Realized Jump Risk

Jonathan Wrightand Hao Zhou

First Draft: November 2006This Draft: December 2008

Abstract

We find that augmenting a regression of excess bond returns on the term structure

of forward rates with a rolling estimate of the mean realized jump sizeidentifiedfrom high-frequency bond returns using the bi-power variation techniqueincreasestheR2 of the regression from around 30 percent to 60 percent. This result is consistentwith the setting of an unspanned risk factor in which the conditional distribution ofexcess bond returns is affected by a state variable that does not lie in the span of theterm structure of yields or forward rates. The return predictability from augmentingthe regression of excess bond returns on forward rates with the jump mean easilydominates the return predictability offered by instead augmenting the regression withoptions-implied volatility or realized volatility from high frequency data. In out-of-sample forecasting exercises, inclusion of the jump mean can reduce the root meansquare prediction error by up to 40 percent. The unspanned risk factoras proxied byrealized jump mean in this paperhelps to account for the countercyclical movements

in bond risk premia.

JEL Classification Numbers: G12, G14, E43, C22.Keywords:Unspanned Stochastic Volatility, Expected Excess Bond Returns, ExpectationsHypothesis, Countercyclical Risk Premia, Realized Jump Risk, Bi-Power Variation.

We thank Torben Andersen, Ravi Bansal, Darrell Duffie, Cam Harvey, Jay Huang, Chris Jones, SydneyLudvigson, Nour Meddahi, Bruce Mizrach, Monika Piazzesi, George Tauchen, Fabio Trojani, and seminarparticipants at Federal Reserve Board, Stanford SITE Workshop, Econometric Society Annual Meeting (NewOrleans), Rutgers University, Banque de France Workshop on Financial Market and Real Activity for helpfulcomments and suggestions. The views presented here are solely those of the authors and do not necessarilyrepresent those of the Federal Reserve Board or its staff.

Department of Economics, Johns Hopkins University, 3400 N. Charles St., Baltimore MD 21218 USA,E-mail [email protected], Phone 410-516-5728.

Division of Research and Statistics, Federal Reserve Board, Mail Stop 91, Washington DC 20551 USA,E-mail [email protected], Phone 202-452-3360.


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Abstract

We find that augmenting a regression of excess bond returns on the term structure of forward

rates with a rolling estimate of the mean realized jump sizeidentified from high-frequency

bond returns using the bi-power variation techniqueincreases the R2 of the regression from

around 30 percent to 60 percent. This result is consistent with the setting of an unspanned

risk factor in which the conditional distribution of excess bond returns is affected by a state

variable that does not lie in the span of the term structure of yields or forward rates. The

return predictability from augmenting the regression of excess bond returns on forward rates

with the jump mean easily dominates the return predictability offered by instead augmenting

the regression with options-implied volatility or realized volatility from high frequency data.

In out-of-sample forecasting exercises, inclusion of the jump mean can reduce the root mean

square prediction error by up to 40 percent. The unspanned risk factoras proxied by

realized jump mean in this paperhelps to account for the countercyclical movements in

bond risk premia.

JEL Classification Numbers: G12, G14, E43, C22.Keywords:Unspanned Stochastic Volatility, Expected Excess Bond Returns, Expectations

Hypothesis, Countercyclical Risk Premia, Realized Jump Risk, Bi-Power Variation.


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1 Introduction

The Expectations Hypothesis (EH) is well known to be a miserable failure, with bond risk

premia being large and time-varying. A regression of yield changes on yield spreads pro-

duces a negative slope coefficient instead of unity, as would be implied by the Expectations

Hypothesis (Campbell and Shiller, 1991), and forward rates can predict future excess bond

returns (Fama and Bliss, 1987). Indeed Cochrane and Piazzesi (2005) recently showed that

using multiple forward rates to predict excess bond returns generates a very high degree of

predictability, withR2 values of around 30-40 percent.

As pointed out by Collin-Dufresne and Goldstein (2002), if the bond market is complete,

then bond yields can be written as an invertible function of the state variables and so the

state variables lie in the span of the term structure of yields. On the other hand, under the

unspanned stochastic volatility (USV) hypothesis, some state variables do not lie in the span

of the term structure of yields. Since expected excess bond returns are a function of all the

state variables (see, Singleton, 2006, pages 322-325), we argue that this gives a direct test

of the USV hypothesis. Similar reasoning is used by Almeida, Graveline, and Joslin (2006)

and Joslin (2007).

If the bond market is complete, then expected excess bond returns should be spanned bythe term structure of yields and so in a regression of excess bond returns on term structure

variables and anyother predictors, the inclusion of enough term structure control variables

should always cause the other predictors to become insignificant. On the other hand, if the

USV hypothesis is correct, then the other predictors may be significant as long as they are

correlated with the unspanned state variable that does not drive innovations in bond yields

but affects the conditional mean of bond yields.

In the last few years, much progress has been made in using high-frequency data to obtain

realized volatility estimates. Further, if we observe high-frequency data on the price of an

asset, and assume that jumps in the price of this asset are both rare and large, then Barndorff-

Nielsen and Shephard (2004), Andersen, Bollerslev, and Diebold (2007), and Huang and

Tauchen (2005) show how to detect the days on which jumps occur and how to estimate

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the magnitude of these jumps. These estimates all have the advantage of being model-

freeit has to be assumed that asset prices follow a jump diffusion process, but no specific

parametric model needs to be estimated. It then seems natural to try to relate realized

volatility and the distribution of jumps to bond risk premia and indeed to financial riskpremia in general (Tauchen and Zhou, 2008). With this motivation, this paper augments

some standard regressions for excess bond returns with measures of realized volatility, jump

intensity, jump mean and jump volatility constructed from five-minute returns on Treasury

bond futures.

Our first finding is that augmenting regressions of excess bond returns on forward rates

with the realized bond jump mean greatly increases the predictability of excess bond returns,

withR2 values nearly doubling from 34-38 percent to 60-62 percent. In contrast, inclusion of

other high-frequency jump measuresjump intensity and jump volatilityin the equation

for predicting excess bond returns raises the R2 by at most a couple of percentage points.

And, if we instead augment the regression of excess bond returns on forward rates with

options-implied and realized volatilities, the coefficients on these volatility variables are not

significantly different from zero and the R2s are little changed.

The fact that the term structure of yields cannot well explain implied volatility (Collin-

Dufresne and Goldstein, 2002) or realized volatility (Andersen and Benzoni, 2008) indicates

the existence of an unspanned risk factor. However, since we find that these volatility

measures offer only a modest improvement in the predictability of excess bond returns,

while our jump mean variable gives a large improvement, we are led to conclude that the

realized jump mean is more highly correlated with the unspanned risk factor.

The unspanned risk factor interpretation is also consistent with the earlier ARCH-M

evidence (Engle, Lilien, and Robins, 1987) that controlling for the term structure does not

eliminate the return-risk tradeoff in the government bond market. Recent work by Ludvigson

and Ng (2008) finds that some extracted macroeconomic factors have additional forecasting

power for expected bond returns in addition to the information in forward rates and by the

same token this is also evidence for the USV hypothesis (see also, Duffee, 2008; Joslin, Prieb-

sch, and Singleton, 2008). The bond jump mean however produces a larger improvement

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in predictive power than these macroeconomic factors and has the extra benefit of being

available in real time.

We perform a number of robustness checks, including shortening the holding period,

changing the size of the rolling window used to construct the jump measures, using onlynon-overlapping data, and continue to find an important role for the bond jump mean in

forecasting excess bond returns. The information content of the bond jump mean seems to

complement that of forward rates, such that theR2 of the regression on both the jump mean

and forward rates is a good bit larger than the sum of the R2s from the regression on either

variable alone.

As pointed by Duffee (2002), certain affine term structure models may perform miserably

in out-of-sample forecasting, even though the models in-sample fit is quite good. Accord-

ingly, we predicted excess bond returns in a standard recursive out-of-sample forecasting

exercise starting half way through the sample using (i) forward rates alone and (ii) forward

rates in conjunction with other variables. We find that inclusion of the jump mean can

reduce the out-of-sample root mean square prediction error by up to 40 percent, while no

other variable offers any out-of-sample improvement over the forward rates alone.

Standard affine models can explain the violation of the EH only with quite unusual model

specifications that may be inconsistent with the second moments of interest rates (Roberds

and Whiteman, 1999; Dai and Singleton, 2000; Bansal, Tauchen, and Zhou, 2004). However,

much progress has been made recently in constructing models that may explain some of the

predictability patterns in excess bond returns. These include models with richer specifi-

cations of the market prices of risk or preferences (Duffee, 2002; Dai and Singleton, 2002;

Duarte, 2004; Wachter, 2006) and models with regime shifts. For example, Bansal and Zhou

(2002), Ang and Bekaert (2002), Evans (2003), and Dai, Singleton, and Yang (2007) use

regime-switching models of the term structure to identify the effect of economic expansions

and recessions on bond risk premia. Such nonlinear regime-shifts models and the unspanned

stochastic volatility hypothesis may be conceptually connected and observationally equiva-

lent.

The rest of the paper is organized as follows: the next section provides the economic mo-

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tivation of our approach from the perspective of incomplete markets and discusses the jump

identification mechanism based on high-frequency intradaily data, then Section 3 contains

the empirical work on using realized jump risk measures to forecast excess bond returns

together with various robustness checks, and Section 4 concludes.

2 Predicting Bond Returns under Incomplete Markets

In this section, we provide motivation for regressions of excess bond returns on term struc-

ture control variables and other predictors under the incomplete market setting with an un-

spanned risk factor. Under incomplete markets with affine factor dynamics (Collin-Dufresne

and Goldstein, 2002), bond yields alone cannot hedge the unspanned volatility risk. How-

ever, an important feature of the USV model is that the unspanned risk factor affects the

conditional mean and/or conditional volatility of other spanned risk factors (see, Singleton,

2006, pages 322-325). Therefore expected excess bond returns depend not only on the bond

yields through the spanned risk factors, but also on a proxy variable that is related to the

unspanned risk factor.

2.1 Unspanned Risk Factor

Recent empirical tests find strong evidence for the existence of unspanned stochastic volatil-

ity (see, Collin-Dufresne and Goldstein, 2002; Heidari and Wu, 2003; Li and Zhao, 2006;

Casassus, Collin-Dufresne, and Goldstein, 2005; Collin-Dufresne, Goldstein, and Jones, 2008;

Andersen and Benzoni, 2008, among others).1 The regression of options-implied or realized

volatilities on bond yields is a valid test for the existence of USV, subject to the proper con-

trols for specification error and measurement error in extracting these volatility measures.

However, an alternative way is to test the USV implication that the unspanned volatilitymust predict the excess bond returns above and beyond what can be predicted by the current

yields or forward rates.

We illustrate the idea with a particular A1(3) specification (Dai and Singleton, 2000)

1There is also empirical evidence against specificunspanned volatility models (see, e.g., Fan, Gupta, andRitchken, 2003; Bikbov and Chernov, 2008; Thompson, 2008).

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which is a three factor affine model with one square-root volatility factor, examined by Collin-

Dufresne and Goldstein (2002). The risk-neutral factor dynamics for the state vectorthe

short rate, its central tendency, and its stochastic volatilityis given by

drt = r(t rt)dt + r+vtdWQt +rdBQt (1)dt = ( 2rt+ 1

rvt)dt+dB

Qt (2)

dvt = (v Qvvt)dt +vvtdZ

Qt (3)

where [dWQt , dBQt , dZ

Qt ]

is a three-dimensional vector of independent Brownian motions un-

der the risk-neutral measure, andv >0 andr 0 guarantee that the model is admissible.Then under the general setting of complete markets, all trivariate affine models have

bond prices of the following generic form

P(t, T; rt, t, Vt) =eA()B()rtC()tD()Vt , (4)

where T t, and A(), B(), C(), and D() are solutions to a system of ordinarydifferential equations (Duffie and Kan, 1996). However, the particular model specification

in eqs. (1)-(3) satisfies the incomplete market condition, such that D() = 0 for any

(Collin-Dufresne and Goldstein, 2002). Therefore the bond prices can be reduced to

P(t, T; rt, t, Vt) P(t, T; rt, t) =eA()B()rtC()t , (5)

where2

A() =

0

r+

2r

2 B(s)2 +

22C(s)2 +rB(s)C(s) C(s)

ds (6)

B() = 1

r(1 er) (7)

C() = 12r

(1 er)2 (8)

In other words, bond prices do not depend on the unspanned volatility, but only depend on

the short rate and the instantaneous mean.

2Although A() can also be solved in closed form, it is not particularly important for the conceptualexercise considered here, although it may be useful for future numerical analysis.

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2.2 Predicting Excess Bond Returns

However, the unspanned volatility is still a relevant risk factor, in the sense that it must

affect the conditional distribution of other spanned factors. In particular, the drift function

of the instantaneous mean ( 2rt+ 1/rvt) allows the conditional expectationEt[T] tolinearly depend on vt; and the drift function of short rate r(t rt) allows the conditionalexpectation Et[rT] to linearly depend on vt recursively through Et[T]. The fact that un-

spanned stochastic volatility affects the conditional mean of the state vector has important

implications for predicting excess bond returns.

Following Collin-Dufresne and Goldstein (2002), if we let the market price of risk process

be such that dWQt = dWt, dBQt = dBt, and dZ

Qt = dZt+

vtdt, then we can transform

to the objective dynamics, dvt = (v vvt)dt+ vvtdZt which governs the time-seriesevolution of the state vector.3 The instantaneous drift under the physical measure has the

same form as under the risk-neutral measure, with risk-adjustment given byv Qv + v.Therefore, the conditional expectation under the physical measure of the state vector can be

easily verified as

Et[rT] = rter(Tt) +t

er(Tt) e2r(Tt)

+

+

vrv

1

2r 1 er(Tt)

2

+vt v

v

1

(2r v)

1

v rer(Tt) ev(Tt) 1

r

ev(Tt) e2r(Tt)

(9)

Et[T] = te2r(Tt) +

+

vrv

1

2r

1 e2r(Tt)

+

vt v

v

1

r(2r v)ev(Tt) e2r(Tt) (10)

Et[vT] = vtev(Tt) +

vv1

ev(Tt) (11)where we can see the dependence of the spanned risk factors rtand ton the unspanned risk

factorvt.4

3This assumption is consistent with the affine specification as examined by Dai and Singleton (2000).More flexible assumptions on the market price of risk process result in the similar affine transformationsbetween risk-neutral and objective dynamics (Duffee, 2002; Dai and Singleton, 2002; Duarte, 2004).

4This feature mirrors an important USV model in the equity option literature (Heston, 1993), where theunspanned risk factor affects the conditional volatility of the state vector.

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Under complete markets, the bond pricing solution given in eq. (4) implies that the

expected excess bond return for holding an n-period bond over the returns on holding a m-

period bond for a holding period ofmperiods can be written as

Et[exm,nt+m] = A(nm)A(n) + A(m)m

+

B(m)

m B(n)

rt+

C(m)

m C(n)

t+

D(m)

m D(n)

vt

+B(nm)Et[rt+m] +C(nm)Et[t+m] +D(nm)Et[vt+m] (12)

and all the terms in eq. (12) are in the span of yields.

In contrast, under incomplete markets, with the bond pricing solution given in eq. (5),

the expected excess returns can be written as

Et[exm,nt+m] = A(nm)A(n) +

A(m)

m

+

B(m)

m B(n)

rt+

C(m)

m C(n)

t

+B(nm)Et[rt+m] +C(nm)Et[t+m] (13)

where the conditional meansEt[rt+m] andEt[t+m] are linear functions ofvt, which is not in

the span of yields. In fact the exact impact coefficient of the unspanned stochastic volatility

on the expected excess return can be shown to be equal to

Et[exm,nt+m]

vt=

1

r

1 er(nm) 1

(2r v)

1

v rer(m) ev(m) 1

r

ev(m) e2r(m)

+ 1

2r

1 er(nm)2 1

r(2r v)ev(m) e2r(m) (14)

which is non-zero and cannot be signed in general without further restrictions on the param-

eters of physical and risk-neutral dynamics.5

Thus, if the bond market is complete, then there should exist no macroeconomic or

financial variable that can improve the population forecastability of excess bond returns

5However, our empirical finding that realized jump mean has a significant negative impact on expectedexcess bond returns, has an intuitive economic interpretationif investors have sudden demand for long-maturity bonds such that they will accept a lower excess return in the future, then this will cause the bondprice to jump upwards today.

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once we control for the term structure of bond yields. However, if markets are incomplete,

then any variable that is correlated with this unspanned volatility factor may add to the

predictability of excess bond returns. Of course, the challenge is how to find such a proxy for

unspanned stochastic volatility. The current methods include implied volatility from fixed-income derivatives markets (e.g., Collin-Dufresne and Goldstein, 2002) and realized volatility

from intraday bond prices (Andersen and Benzoni, 2008). However, there is evidence that

such a regression based test on unspanned volatility hypothesis can be misleading, if the

measurement error of realized volatility or specification error of implied volatility is not well

under control (Bikbov and Chernov, 2008).

We argue that the necessary and sufficient condition for the unspanned stochastic volatil-

ity is that not only the volatility measures are uncorrelated with the current bond yields but

also the unspanned risk factor should predict the bond excess returns. Indeed, Almeida,

Graveline, and Joslin (2006) and Joslin (2007) find that an unspanned stochastic volatility

factor constructed from fixed income options markets is important for predicting excess bond

returns. We instead adopt a realized measure of jump risk from the bond futures market,

which has an orthogonal innovation relative to the current term structure, and find that this

variable has more substantial forecasting power for excess bond returns than any other of

the other proxies for an unspanned risk factor that have been considered in the literature.

2.3 Econometric Estimation of Realized Jump Risk

We discuss our econometric method for constructing market jump risk measures, which may

potentially constitute unspanned risk factors for predicting excess returns.

Assuming that the price of an asset (a bond in this paper) follows a jump-diffusion

process (Merton, 1976), this paper takes a direct approach to identify realized jumps based

on the seminal work by Barndorff-Nielsen and Shephard (2004, 2006). This approach uses

high-frequency data to disentangle realized volatility into separate continuous and jump

components (see, Huang and Tauchen, 2005; Andersen, Bollerslev, and Diebold, 2007, as

well) and hence to detect days on which jumps occur and to estimate the magnitude of

these jumps. The methodology for filtering jumps from bi-power variation is by now fairly

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standard, but we review it briefly, to keep the paper self-contained.

Letst= log(St) denote the time t logarithmic price of an asset, which evolves in contin-

uous time as a jump diffusion process:

dst= tdt+tdWt+Jtdqt

where tand tare the instantaneous drift and diffusion functions that are completely general

and may be stochastic (subject to the regularity conditions), Wt is a standard Brownian

motion,dqt is a Poisson jump process with intensity J It, andJt refers to the corresponding

(log) jump size distributed as Normal(JMt, JV2t ). Note that the jump intensity, mean and

volatility are all allowed to be time-varying in a completely unrestricted way. Time is

measured in daily units and the intradaily returns are defined as follows:

rst,j st,j st,(j1)

where rst,j refers to the jth within-day return on day t, and is the sampling frequency

within each day.

Barndorff-Nielsen and Shephard (2004) propose two general measures for the quadratic

variation processrealized variance and realized bi-power variationwhich converge uni-

formly (as

0 orm= 1/

) to different functionals of the underlying jump-diffusion

process,

RVt m

j=1

|rst,j|2 tt1

2udu+

tt1

J2udqu (15)

BVt 2

m

m 1m

j=2

|rst,j||rst,j1| tt1

2udu. (16)

Therefore the difference between the realized variance and bi-power variation is zero when

there is no jump and strictly positive when there is a jump (asymptotically). This is the

basis of the method for identifying jumps.

A variety of specific jump detection techniques are proposed and studied by Barndorff-

Nielsen and Shephard (2004). Here we adopted the ratio statistic favored by the findings in

Huang and Tauchen (2005) and Andersen, Bollerslev, and Diebold (2007),

RJtRVt BVtRVt

(17)

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which converges to a standard normal distribution with an appropriate scaling

ZJt RJt[(

2)2 + 5] 1

mmax(1, TPt

BV2t

)

d N(0, 1) (18)

This test has excellent size and power properties and is quite accurate in detecting jumps as

documented in Monte Carlo work (Huang and Tauchen, 2005).6

Following Tauchen and Zhou (2008), we further assume that there is at most one jump

per day and that the jump size dominates the return when a jump occurs, i.e. that the

jumps are rare and large as in Merton (1976). These assumptions allow us to filter out the

daily realized jumps as

Jt= sign(rst )(RVt BVt) I(ZJt1 )where is the cumulative distribution function of a standard normal random variable, is

the significance level of the z-test, and I(ZJt1 ) is the resulting indicator function that is

one if and only if there is a jump during that day (asymptotically).

Once the jumps have been identified, we can then estimate the jump intensity, mean and

volatility as,

JIt = Number of Realized Jump DaysNumber of Total Trading Days

JMt = Mean of Realized Jumps

JVt = Standard Deviation of Realized Jumps

with appropriate formulas for the standard error estimates.

These realized jump risk measures can greatly facilitate our identification of jumps and

hence our estimation of risk premia. The reason is that jump parameters are generally very

hard to pin down even with both underlying and derivative assets prices, due to the fact

6Note thatT Pt is the so-calledTri-Power Quarticity robust to jumps, and as shown by Barndorff-Nielsenand Shephard (2004),

T Pt m34/3m

m 2mj=3

|rst,j2|4/3|rst,j1|4/3|rst,j |4/3 tt1

4sds (19)

with k 2k/2((k+ 1)/2)/(1/2) fork >0.

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that jumps are latent in daily return data and are rare events in financial markets.7 Direct

identification of realized jumps and the characterization of time-varying jump risk measures

have important implications for interpreting financial market risk premia.

3 Predicting Excess Bond Returns

The methodology that we described in the previous section allows us to identify and estimate

jumps and to construct backward-looking rolling estimates of jump mean, jump intensity and

jump volatility. These have the important advantage of being completely model-freeno

model has to be specified or estimated to construct them. These rolling jump risk measures

may be correlated with the unspanned risk factor(s), and may accordingly be useful for

predicting excess bond returns. Investigating this possibility empirically is the focus of the

remainder of this section.

3.1 Variable Definitions, Empirical Strategy and Data Summary

Our measures of bond market realized volatility, jump mean, jump intensity and jump volatil-

ity are based on data on 30-year Treasury bond futures at the five-minute frequency from

July 1982 to September 2006, obtained from RC Research. The data cover the period from

8:20am to 3:00pm New York time each day for a total of 80 observations per day. We calcu-

lated continuously compounded returns as the log difference in futures quotes and, using the

methods described in the previous section, we then constructed the realized volatility at the

daily frequency, tested for jumps on each day, and estimated the magnitude of the jumps on

those days when jumps were detected. LetDDAILYt denote the dummy that is 1 if and only

if a jump is detected on day t and recall that Jt denotes the estimated magnitude of the

jump on dayt. For our empirical work, let theh-month rolling average realized volatility,7Existing studies have relied heavily on complex numerical procedures or simulation methods like EMM

or MCMC (see, e.g., Bates, 2000; Andersen, Benzoni, and Lund, 2002; Pan, 2002; Chernov, Gallant, Ghysels,and Tauchen, 2003; Eraker, Johannes, and Polson, 2003; At-Sahalia, 2004).

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jump intensity, jump mean, and jump volatility be defined as, respectively,

RVht = 1

h 22h221j=0 RVtj ,

JIht

= 1

h 22h221

j=0

DDAILYtj

,

JMht =h221j=0 JtjD

DAILYtj

h221j=0 DDAILYtj

,

JVht =

h221j=0 (Jtj JMht)2DDAILYtjh221j=0 D

DAILYtj

where the jump means and volatilities are calculated only over days where jumps are detected.

Realized volatility can be estimated arbitrarily accurately with a fixed span of sufficiently

high-frequency data (abstracting from issues of market microstructure noise), whereas this

is not true for jump intensity, mean or volatility, given the presumption that jumps are rare

and large. For this reason, while we use a relatively short rolling window for estimating

realized volatility (hequal to one month), we use much longer rolling windows for measuring

jump intensity, jump mean and jump volatility, setting the parameterh to 24 months or 12

months. The tradeoff in selectingh is, of course, that a shorter window gives a more noisy,

but more timely, measure of agents perceptions of jump risk.

Figure 1 plots the monthly options-implied volatility (top panel),8 monthly realized

volatility from 5-minute returns (middle panel), and daily realized jumps identified using

the method discussed earlier (bottom panel). Treasury bond options-implied and realized

volatility seems highest in the late 1980s and around 2002-2004. Bond market jumps oc-

cur on about 8 percent of days and are large (standard deviation of about 0.4 percent per

jump). There is significant time-variation in the identified jump dynamics. Figure 2 shows

plots of the 24-month rolling jump intensity, jump mean and jump volatility for Treasury

bond futures. The mean jump size fell noticeably during the economic booms in the late

1980s and 1990s, but rose sharply during the 1990-91 and 2001 recessions. These cyclical

patterns in realized jump dynamics will have important implications for predicting the bond

8These are the Black-Scholes implied volatilities from options on the front at-the-money bond futurescontract, expressed at an annualized rate. Where the front futures contract will expire within one month,we use the next futures contract instead.

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market risk premia as evidenced below.

In this paper, we use realized volatility, our realized jump risk measures and options-

implied volatility to forecast excess bond returns. The excess return on holding an n-month

bond over the return on holding anm-month bond for a holding period ofm months is givenbyexm,nt+m =p

nmt+m pnt (m/12)ymt where yjt denotes the annual continuously compounded

yield on a j -month zero coupon bond and pjt = (j/12)yt is the log price of this bond. Weused end-of-month data on zero-coupon yields and the three-month risk-free rate from the

CRSP Fama-Bliss data, and hence constructed these excess returns.

All the regressions for excess bond returns that we consider in this paper are nested

within the specification

exm,nt+m= 0 +1f12t +2f

36t +3f

60t +4IV

1t +5RV

1t +6JM

ht +7JI

ht +8JV

ht +t+m (20)

wherefjt =pj12t pjt denotes thej/12-year forward rate with a 12-months period and RV1t ,

JMht, JIht and JV

ht denote the realized volatility and jump measures in rolling windows

ending on the last day of month t, constructed from high-frequency bond data as defined

earlier, andIV1t is options-implied volatility observed at the end of the month. Using just

the forward rates gives the regression of Cochrane and Piazzesi (2005), except that, following

Bansal, Tauchen, and Zhou (2004), we use three forward rates instead of five, to minimize

the near-perfect collinearity problem. But we also assess the incremental predictive power

of implied volatility and rolling realized volatility and jump risk variables.

Some summary statistics for the key time series are given in Table 1. On average,

realized bond volatility is about 8.8 percentage points (expressed in annualized terms) with a

standard deviation of about 2 percentage points, while the options-implied volatility averages

about 10.4 percent with a similar standard deviation. The averages of our jump intensity,

jump mean and jump volatility measures are 8 percent, 0.03 percentage points and 0.41

percentage points, respectively. Turning to the correlation structure, the excess returns and

forward rates are highly collinear. Jump means and jump volatility both have a fairly strong

negative correlation with excess returns (about -0.4 and -0.3, respectively), while implied and

realized volatilities have a smaller positive correlation with excess returns (between zero and

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0.10).

Table 2 reports the results of regressions of options-implied volatility, realized volatility

and jump risk measures on the term structure of forward rates. With complete markets, the

R2

values from these regressions should be large. In fact, they are small, ranging from 4 to20 percent. This has previously been shown for realized volatility (Andersen and Benzoni,

2008) and for implied volatility (Collin-Dufresne and Goldstein, 2002). But it is true for

our jump risk measures too. All this gives tentative evidence for the unspanned stochastic

volatility (USV) hypothesis. Now we turn to testing if any of these apparently unspanned

risk measures help to forecast excess bond returns.

3.2 Predicting Excess Bond Returns

Table 3 shows coefficient estimates, associated t-statistics and R2 values for several specifi-

cations of the form of equation (23), setting m= 12 (one-year holding period) and h= 24

(two-year rolling windows in constructing jump risk measures) where the maturity of the

longer-term bond n is set to 24, 36, 48 and 60 months. Forward rates are omitted in all

specifications in this table. The jump mean is, on its own, a significantly negative predictor

of future excess returns for all values ofn and the R2 is about 15 percent. This implies

that downward jumps in bond prices are followed by large positive excess returns. The

t-statistics is around 4.6 to 5.2 for the jump mean, contrasting with near zero predictability

using implied volatility, realized volatility, and jump intensity. Such a degree of predictability

is at least comparable with that from using a single forward rate with a matching maturity

(Fama and Bliss, 1987). The coefficient on jump volatility is significantly negative (perhaps

the opposite of the sign one might expect from a standard argument of return-risk trade-off)

when the maturity of the longer-term bond, n, is 24 or 36 months, but is not significantly

different from zero when n is 48 or 60 months.

Table 4 shows results from other specifications of equation (23), in which forward rates

are now included. The forward rates show the familiar tent-shaped pattern, are often in-

dividually significant, and always jointly overwhelmingly significant. Running the regression

of excess bond returns on the forward rates alone gives an R2 in the range 34-38 percent,

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which is considerable and similar to the range reported by Cochrane and Piazzesi (2005).

But if we add the jump mean to the regression of excess returns on forward rates, the coeffi-

cient on the jump mean is negative and statistically significant for each n and theR2 rises to

60-62 percent. And, the information content of the jump mean seems to complement that offorward rates in that theR2 of the regression on both jump mean and forward rates is larger

than the sum ofR2s on each of the variables separately. Controlling for jump mean does not

change the coefficients on the forward rates greatly, suggesting that the term structure of

forward rates and jump mean are measuring different components of bond risk premia. This

suggests that the bond jump mean may act as an unspanned stochastic mean factor that

cannot be hedged with the current yields but can forecast excess bond returns. Meanwhile

implied volatility, realized volatility, jump volatility and jump intensity have no significant

predictive power for excess bond returns. This is true for all choices of the maturity of the

longer-term bond, n.

The ex-post excess returns on holding long term bonds (Figure 3, top panel) averaged

around zero during the expansion of the mid and late 1990s, with positive excess returns at

some times being offset by negative excess returns as the Federal Open Market Committee

(FOMC) was tightening monetary policy during 1994 and around the time of the 1998 Long-

Term Capital Management (LTCM) crisis. On the other hand, the excess returns were large

and positive during and immediately after the 1990-91 and 2001 recessions. The predicted

excess returns using only the forward rate term structure (middle panel) shows some of

this countercyclical variation, but overpredicted excess bond returns in 1994 and around

the time of the LTCM crisis, while underpredicting these excess bond returns during the

most recent recession. Adding the bond jump mean risk measure (bottom panel), the model

does better at predicting the high excess returns in the 2001 recession and also fits better

in 1994 and 1998, and it is also much more successful in predicting high excess returns

during and immediately after the most recent recession. The jump mean increased after

the 2001 recession, and this may indeed help explain some of the recent decline in longer

maturity yields that was referred to by former Federal Reserve Chairman Greenspan as a

conundrum (discussed further in Kim and Wright, 2005; Backus and Wright, 2007).

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Table 5 shows results from some more return prediction equations, in which excess bond

returns are predicted using forward rates and all possible combinations of jump mean, real-

ized volatility and implied volatility. The motivation is to run a horse-race comparing the

jump mean with more conventional volatility risk measures. The jump mean is consistentlysignificantly negative, while neither implied nor realized volatility is significant in any case.

We can clearly see from Figure 4 that the relationship between excess bond returns and

implied and realized volatilities (top and middle panels) is very noisysometime converging

and diverging at other times. But the realized jump mean seems to be consistently nega-

tively correlated with excess bond returns (bottom panel); and both the jump mean and the

excess return are fairly persistent, with the peaks and troughs usually several years apart.

3.3 Robustness Checks

This subsection addresses the robustness of the finding that the jump mean variable helps

forecast excess bond returns. It is well known that severe small-sample size distortions

may arise in return prediction regressions with highly persistent regressors and overlapping

returns.9 To mitigate this problem, we first re-run our regressions using non-overlapping

data. Table 6 shows the results from the same regressions as in Table 4 (i.e. settingm = 12

and h = 24 and various values ofn), but using only the forward rates of each December, so

that the holding periods do not overlap. The results are even stronger than those in Table

4, and the jump mean is highly significant. The overall predictability increases from 44-56

percent when using only forward rates to 69-74 percent when the jump mean is included.

Tables 3-6 follow Cochrane and Piazzesi (2005) in considering a one-year holding period

(m= 12). But, since we have only 22 non-overlapping periods in our sample, giving a quite

small effective sample size, it also seems appropriate to consider results with a one-quarter

holding period (m= 3), for which there are four times as many non-overlapping periods. 10

9Inference issues related to the use of highly persistent predictor variables have been studied extensivelyin the literature, see, e.g., Stambaugh (1999), Ferson, Sarkissian, and Simin (2003), and Campbell and Yogo(2006) and the references therein. For recent discussions of some of the difficulties associated with the useof overlapping data see, e.g., Valkanov (2003) and Boudoukh, Richardson, and Whitelaw (2008).

10The Fama-Bliss zero-coupon data consist of 1-, 2-, 3-, 4- and 5-year zero-coupon bonds only. To constructthree-month excess returns (m = 3), we need some yields that are not available, such as 1.75 year yields.We use the approximation yn3t ynt to obtain these yields. The problem does not arise for one-year excess

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These results are shown in Table 7. TheR2 values are lower at this shorter horizon for all

specifications, perhaps not surprisingly; but the jump mean is again consistently negatively

and significantly related to future excess bond returns, once we control for the forward rates.

Implied and realized volatility and the other realized jump measures have no significantassociation with future excess returns.

For the parameterh(rolling window size for jump risk measures), we want to pick a value

that is large enough not to give noisy estimates, but small enough to give timely measures of

agents perceptions of jump risk. A range of reasonable choices might be from 6 to 24 months,

and our results are not very sensitive to varying h over this range. For example, Table 8

shows the results with overlapping data at the one-year horizon (m = 12) and choosing a

shorter rolling window for estimating jump statistics (h = 12). These results are similar to

those in Table 4, in that the total predictability of bond risk premia is substantially increased

by the inclusion of the jump mean, though the improvement is not quite as large. The slope

coefficient for the jump mean remains negative and significant.

3.4 Out-of-Sample Forecasting

This subsection investigates the usefulness of our jump risk measures in forecasting excess

bond returns in a standard recursive out-of-sample forecasting experiment. Starting half way

through the sample, we ran the regression (23) using only data available at that time and then

made a forecast for 12-month-ahead excess bond returnsthe out-of-sample counterpart to

the forecasting exercise considered in Table 4. We then repeated this for each subsequent

month through the end of the sample. We compared the root mean square prediction errors

from the regression using just the term structure of forward rates (the baseline regression)

with those from the regression using the term structure of forward rates and one other

risk measure (the augmented regression). Table 9 reports the root mean square prediction

error from the augmented regression relative to that from the baseline regression. Entries

below one mean that the augmented regression gives lower out-of-sample root mean square

prediction error. Table 9 also shows the same results for the out-of-sample analogs of the

returns (m= 12).

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forecasting exercises conducted as robustness checks as in Tables 6, 7 and 8.

As can be seen in Table 9, the inclusion of options-implied-volatility, realized volatility,

jump intensity and jump volatility generally actually increase mean square prediction error

out of sample. On the other hand, the inclusion of the jump mean consistently lowersroot mean square prediction errors. Forecasting 12-month-ahead excess bond returns using

overlapping data with 24-month windows to form the jump risk measures, the inclusion of the

jump mean reduces out-of-sample root mean square prediction errors by about 40 percent.

In the other out-of-sample forecasting exercises, the inclusion of the jump mean leads to a

reduction of 9 to 26 percent in root mean square prediction error, which are still sizeable

improvements.

Testing the significance of the difference in root mean square prediction error in the

baseline and augmented regression is a nonstandard econometric problem because the models

are nested and so the well known test of Diebold and Mariano (1995) does not have its usual

normal asymptotic distribution. Clark and McCracken (2005) however derive the limiting

distribution of this test (MSE-t in their notation) and we can use their critical values. Cases

in which the recursive out-of-sample mean square prediction error in the augmented model is

significantly smaller than in the baseline model are marked in Table 9. For the jump mean,

all entries are significant at the 1 percent level.

Figure 5 shows the out-of-sample predictions for overlapping forecasts of 12-month-ahead

excess bond returns using the baseline regression and the regression including the 24-month

rolling jump mean as well. The model including the jump mean did a considerably better

job of predicting the large positive bond excess returns around 2001-2002 than the model

using forward rates alone, even using only data that were available at the time that the

forecast was being made. Including the jump mean also gave better forecasts around the

time of the 1998 international financial crisis, suggesting that the unspanned risk factor as

proxied by the realized jump mean may be particularly important in capturing the rise in

risk premia during economic slowdowns and/or financial crises.

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4 Conclusion

There is considerable evidence of predictability in excess returns on a range of assets, but

it is especially strong for longer maturity bonds, perhaps because their pricing is not com-

plicated by uncertain cash flows. Part of the predictability may owe to time-variation in

the distribution of jump risk, but empirical work on this has been hampered until recently

by econometricians difficulties in identifying jumps. Recently studies (Barndorff-Nielsen

and Shephard, 2004; Andersen, Bollerslev, and Diebold, 2007; Huang and Tauchen, 2005;

Tauchen and Zhou, 2008) have shown how high-frequency data can be used reliably to de-

tect jumps and to estimate their size, under the assumption that jumps are rare and large

(Merton, 1976), and these methods can then easily be used to construct rolling estimates of

jump intensity, jump mean and jump volatility.

In this paper, we have found evidence that jump risk measures can help predict future ex-

cess bond returns. Augmenting a standard regression of excess bond returns on forward rates

with the rolling realized jump mean constructed from high-frequency bond price data, we

find that the coefficient on the jump mean is both economically and statistically significant.

The predictability of bond risk premia can be increased substantially by its inclusion.

The bond jump mean appears to be procyclical, peaking around the 1990 and especiallythe 2001 recessions, and predicts the countercyclical bond excess returns well. In a number

of important episodes, predicted excess returns constructed using both the bond jump mean

and the term structure of forward rates track the actual ex-postexcess returns a good bit

more closely than predicted excess returns constructed using forward rates alone, including

during the period in 1994 when the FOMC was tightening monetary policy, the 1998 LTCM

crisis, and in the 2001 recession. A rise in the bond jump mean may also account for part of

the decline in long-term yields during the FOMC tightening cycle that began in the middle of

2004. Our result is robust to shortening the holding period, changing the size of the rolling

window used to construct the jump measures and using only non-overlapping data. The

tent-shape pattern of regression coefficients on the forward term structure however exists

regardless of whether or not we control for the jump mean. In out-of-sample forecasting

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exercises, inclusion of the jump mean can reduce the root mean square prediction error by

up to 40 percent.

Finally, the existing literature has used implied volatility from fixed-income derivatives

(e.g., Collin-Dufresne and Goldstein, 2002) and realized volatility from intraday bond prices(Andersen and Benzoni, 2008) to test the unspanned stochastic volatility hypothesis. Our

finding that the bond jump mean has significant forecasting power for excess bond returns

above and beyond that obtained from forward rates alone, is consistent with this hypothesis.

More importantly, the time-variation in bond risk premia accounted for by the realized jump

mean seems to have been particularly large during some recent cyclical downturns and times

of financial market turmoil. A good part of the countercyclical movements in excess bond

returns may be due to the unspanned risk factor, as opposed to the spanned risk factor(s).

We leave further examination of the economic role of the unspanned risk factor to future

research.

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Table 1: Summary statistics and correlation matrix for excess returns, forward rates, and jump measures

ex12,24t+12 ex12,36t+12 ex

12,48t+12 ex

12,60t+12 f

12t f

36t f

60t IV

1t RV

1t JI

24t JM

24t JV

24t

Summary statisticsMean 0.94 1.70 2.37 2.68 5.33 6.50 6.81 10.38 8.80 0.08 0.03 0.41

Std. Dev 1.54 2.93 4.16 5.12 2.24 2.05 1.88 2.03 2.15 0.02 0.09 0.07Skewness 0.19 0.14 0.20 0.17 0.00 0.41 0.62 1.32 1.49 0.37 -0.44 0.32Kurtosis 2.22 2.28 2.48 2.59 2.70 3.12 3.19 4.96 6.39 2.22 2.98 2.80

Correlation matrix

ex12,24t+12 1.00 0.98 0.96 0.93 0.39 0.52 0.51 0.00 0.09 0.07 -0.38 -0.31

ex12,36t+12 1.00 0.99 0.98 0.34 0.49 0.48 0.01 0.09 0.06 -0.41 -0.30

ex12,48t+12 1.00 0.99 0.33 0.49 0.50 0.03 0.10 0.05 -0.38 -0.27

ex12,60t+12 1.00 0.30 0.48 0.49 0.05 0.10 0.03 -0.38 -0.23f12t 1.00 0.92 0.83 0.16 0.10 -0.01 0.32 -0.06f36t 1.00 0.97 0.30 0.24 -0.09 0.29 -0.12f60t 1.00 0.36 0.29 -0.16 0.29 -0.11IV1t 1.00 0.81 -0.59 0.19 -0.01

RV1

t 1.00 -0.53 0.15 -0.12JI24t 1.00 0.05 0.34JM24t 1.00 0.33JV24t 1.00

Notes: This table summarizes the main variables used in the empirical exercise. The variable definitions are

the same as given in Section 3.1: exm,nt+n are the excess returns on holding an n-month bond over the return

on holding anm-month bond for a holding period ofm months;fjt denotes the one-year forward rate ending

j months; and RV1t , JIht, JM

ht and JV

ht denote the realized bond volatility and jump measures in rolling

windows ending on the last day of month t.

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Table 2: Regression of Risk Measures on Term Structure of Forward Rates

Dependent Variable Intercept f12t f36t f

60t R

2

IV1t 7.11 (21.94) -0.44 (-3.04) 0.03 (0.09) 0.79 (2.80) 0.20RV1t 6.27 (16.16) -0.72 (-3.97) 0.87 (1.94) 0.11 (0.33) 0.14JI24t 0.10 (18.05) -0.00 (-0.39) 0.02 (3.30) -0.02 (-4.77) 0.11

JM24t -0.06 (-2.64) 0.03 (3.26) -0.05 (-2.73) 0.04 (2.83) 0.13JV24t 0.44 (23.33) 0.02 (2.66) -0.04 (-2.43) 0.02 (1.49) 0.04

Notes: This table reports the coefficient estimates in regressions of options-implied volatility, a one-month

rolling window of realized bond volatility and 24 month rolling windows of bond jump mean, jump intensity,

and jump volatility, constructed as described in the text onto the term structure of forward rates. Observa-

tions are at the monthly frequency (end-of-month). T-statistics are shown in parentheses and are based on

standard errors that are heteroskedasticity-robust, but make no correction for serial correlation.

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Table 3: Excess returns on holding an n-month bond for a holding period of one year(Newey-West t-statistics in parentheses)

Intercept IV1t RV1t JI

24t JM

24t JV

24t R

2

n=240.93 (0.74) 0.00 (0.01) 0.000.50 (0.62) 0.05 (0.58) 0.00

0.58 (0.81) 4.56 (0.50) 0.011.14 (4.48) -6.67 (-4.69) 0.153.63 (2.74) -6.57 (-2.23) 0.10

n=361.50 (0.64) 0.02 (0.09) 0.000.84 (0.57) 0.10 (0.61) 0.011.15 (0.86) 6.98 (0.43) 0.002.10 (4.47) -13.52 (-5.23) 0.176.63 (2.55) -12.03 (-2.12) 0.09

n=481.64 (0.50) 0.07 (0.22) 0.001.01 (0.49) 0.15 (0.68) 0.011.66 (0.91) 8.99 (0.40) 0.00

2.90 (4.37) -18.14 (-4.84) 0.158.58 (2.32) -15.17 (-1.88) 0.07

n=601.47 (0.36) 0.12 (0.30) 0.000.97 (0.38) 0.19 (0.70) 0.012.19 (1.00) 6.23 (0.24) 0.003.33 (4.18) -22.22 (-4.57) 0.159.35 (2.03) -16.28 (-1.63) 0.05

Notes: This table reports the coefficient estimates in a regression of the excess returns on a n-month bond

over those on a 12 month bond, with a holding period of 12 months, on options-implied volatility, a one-month

rolling window of realized bond volatility and 24 month rolling windows of bond jump mean, jump intensity,and jump volatility, constructed as described in the text. Observations are at the monthly frequency (end-

of-month). T-statistics are shown in parentheses and are based on Newey-West standard errors with a lag

truncation parameter of 11.

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Table 4: Excess returns on holding an n-month bond for a holding period of on(Newey-West t-statistics in parentheses)

Intercept f12t f36t f

60t IV

1t RV

1t JI

24t

n=24-1.75 (-1.99) -0.58 (-2.37) 1.46 (3.11) -0.55 (-1.30)

-0.41 (-0.36) -0.66 (-2.73) 1.47 (3.09) -0.40 (-0.92) -0.19 (-1.94) -1.16 (-1.24) -0.65 (-2.55) 1.54 (3.15) -0.54 (-1.22) -0.09 (-1.50) -2.60 (-1.78) -0.57 (-2.26) 1.34 (2.54) -0.40 (-0.83) 8.34 (1.00)-2.32 (-5.06) -0.34 (-1.50) 1.02 (2.45) -0.19 (-0.59) 0.27 (0.15) -0.50 (-2.31) 1.30 (3.13) -0.47 (-1.22)

n=36-3.25 (-2.08) -1.35 (-2.88) 3.19 (3.56) -1.26 (-1.62) -0.82 (-0.39) -1.50 (-3.24) 3.20 (3.52) -0.99 (-1.20) -0.34 (-1.79) -2.05 (-1.21) -1.49 (-3.07) 3.35 (3.59) -1.24 (-1.51) -0.19 (-1.65) -4.67 (-1.80) -1.34 (-2.74) 2.97 (2.97) -1.02 (-1.14) 13.83 (0.95)-4.35 (-5.38) -0.88 (-2.23) 2.34 (3.09) -0.57 (-0.97) 0.26 (0.08) -1.22 (-3.00) 2.90 (3.64) -1.13 (-1.54)

n=48-5.11 (-2.53) -2.13 (-3.48) 4.62 (3.90) -1.65 (-1.60) -1.68 (-0.60) -2.34 (-3.88) 4.64 (3.84) -1.26 (-1.14) -0.48 (-1.76) -3.29 (-1.49) -2.34 (-3.70) 4.87 (3.93) -1.61 (-1.48) -0.29 (-1.82) -7.27 (-2.20) -2.11 (-3.28) 4.30 (3.24) -1.28 (-1.09) 21.08 (1.12)-6.59 (-5.87) -1.51 (-2.91) 3.48 (3.40) -0.73 (-0.89) -1.17 (-0.26) -1.98 (-3.68) 4.30 (3.91) -1.50 (-1.51)

n=60-6.73 (-2.81) -2.62 (-3.55) 5.20 (3.62) -1.54 (-1.24) -2.60 (-0.75) -2.87 (-3.92) 5.22 (3.56) -1.08 (-0.79) -0.58 (-1.69) -4.44 (-1.66) -2.88 (-3.76) 5.52 (3.67) -1.50 (-1.14) -0.37 (-1.86) -9.14 (-2.43) -2.60 (-3.34) 4.84 (3.03) -1.13 (-0.81) 23.52 (1.10)-8.53 (-6.08) -1.86 (-3.04) 3.82 (3.11) -0.41 (-0.42)

-2.92 (-0.53) -2.47 (-3.71) 4.89 (3.57) -1.39 (-1.15)

Notes: As for Table 3, except that the term structure of forward rates is also being controlled for.

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Table 5: Excess returns on holding an n-month bond for a holding period of one yea(Newey-West t-statistics in parentheses)


60t IV

1t RV

1t

n=24-0.43 (-0.39) -0.63 (-2.69) 1.38 (3.02) -0.34 (-0.80) -0.28 (-1.75) 0.10 (1.11) -1.40 (-1.87) -0.40 (-1.75) 1.05 (2.54) -0.11 (-0.32) -0.13 (-1.66) -9

-2.08 (-4.67) -0.37 (-1.60) 1.06 (2.56) -0.19 (-0.58) -0.04 (-0.75) -9-1.43 (-2.10) -0.36 (-1.69) 0.93 (2.52) -0.03 (-0.09) -0.24 (-1.89) -0.13 (1.61) -9

n=36-0.84 (-0.41) -1.45 (-3.21) 3.08 (3.40) -0.91 (-1.10) -0.46 (-1.51) 0.13 (0.82) -2.72 (-2.00) -1.00 (-2.49) 2.38 (3.20) -0.42 (-0.68) -0.22 (-1.59) -1-3.82 (-4.55) -0.95 (-2.36) 2.42 (3.25) -0.58 (-0.95) -0.08 (-0.97) -1-2.77 (-2.16) -0.93 (-2.50) 2.21 (3.17) -0.31 (-0.52) -0.39 (-1.67) 0.19 (1.36) -1

n=48-1.71 (-0.61) -2.30 (-3.86) 4.52 (3.68) -1.19 (-1.05) -0.60 (-1.37) 0.13 (0.58) -4.23 (-2.23) -1.67 (-3.23) 3.54 (3.54) -0.51 (-0.59) -0.32 (-1.62) -2-5.65 (-4.65) -1.62 (-3.10) 3.63 (3.63) -0.73 (-0.88) -0.14 (-1.27) -2-4.28 (-2.36) -1.60 (-3.28) 3.36 (3.47) -0.38 (-0.45) -0.50 (-1.52) 0.21 (1.09) -2

n=60-2.62 (-0.77) -2.83 (-3.88) 5.12 (3.37) -1.01 (-0.72) -0.68 (-1.27) 0.12 (0.45) -5.71 (-2.43) -2.06 (-3.39) 3.88 (3.25) -0.15 (-0.15) -0.39 (-1.58) -2-7.32 (-4.63) -2.01 (-3.26) 4.01 (3.35) -0.42 (-0.42) -0.19 (-1.39) -2-5.76 (-2.53) -1.98 (-3.41) 3.70 (3.11) -0.03 (-0.02) -0.57 (-1.42) 0.22 (0.95) -2

Notes: As for Table 4.

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Table 7: Excess returns on holding an n-month bond for a holding period of 3 m(Newey-West t-statistics in parentheses)


60t IV

1t RV

1t JI

24t

n=24-1.00 (-2.62) -0.28 (-2.00) 0.80 (2.66) -0.35 (-1.49)

-0.66 (-1.35) -0.31 (-2.15) 0.81 (2.69) -0.31 (-1.33) -0.05 (-1.21) -0.93 (-2.05) -0.29 (-1.99) 0.82 (2.64) -0.35 (-1.49) -0.01 (-0.37) -1.52 (-2.53) -0.29 (-2.09) 0.75 (2.54) -0.27 (-1.13) 4.55 (1.40)-1.26 (-3.63) -0.15 (-1.07) 0.58 (1.92) -0.18 (-0.79) -0.41 (-0.59) -0.26 (-1.88) 0.76 (2.54) -0.33 (-1.41)

n=36-1.44 (-2.60) -0.54 (-2.61) 1.43 (3.24) -0.66 (-1.91) -0.96 (-1.34) -0.58 (-2.73) 1.44 (3.27) -0.60 (-1.74) -0.07 (-1.07) -1.31 (-1.97) -0.56 (-2.60) 1.46 (3.23) -0.65 (-1.90) -0.02 (-0.42) -2.23 (-2.52) -0.56 (-2.70) 1.36 (3.14) -0.54 (-1.53) 6.92 (1.38)-1.81 (-3.56) -0.36 (-1.73) 1.11 (2.50) -0.41 (-1.23) -0.71 (-0.68) -0.52 (-2.53) 1.38 (3.15) -0.63 (-1.84)

n=48-1.89 (-2.55) -0.74 (-2.68) 1.85 (3.16) -0.81 (-1.78) -1.30 (-1.36) -0.79 (-2.79) 1.86 (3.18) -0.74 (-1.63) -0.09 (-0.98) -1.72 (-1.94) -0.77 (-2.67) 1.88 (3.15) -0.81 (-1.78) -0.03 (-0.42) -2.96 (-2.53) -0.77 (-2.77) 1.75 (3.06) -0.65 (-1.41) 9.27 (1.41)-2.37 (-3.38) -0.51 (-1.84) 1.44 (2.45) -0.49 (-1.12) -1.16 (-0.83) -0.72 (-2.63) 1.80 (3.10) -0.78 (-1.74)

n=60-2.56 (-2.85) -0.76 (-2.20) 1.74 (2.34) -0.57 (-1.00) -1.70 (-1.44) -0.83 (-2.34) 1.76 (2.37) -0.47 (-0.82) -0.13 (-1.13) -2.27 (-2.08) -0.80 (-2.24) 1.79 (2.37) -0.56 (-0.99) -0.05 (-0.55) -4.02 (-2.84) -0.80 (-2.30) 1.60 (2.21) -0.35 (-0.60) 12.80 (1.58)-3.15 (-3.72) -0.47 (-1.38) 1.23 (1.66) -0.17 (-0.31)

-1.95 (-1.13) -0.75 (-2.17) 1.70 (2.31) -0.55 (-0.96)

Notes: This table reports the coefficient estimates in a regression of the excess returns on a n-month bond over holding period of 3 months, on the term structure of forward rates, options-implied volatility, a one-month rolling wand a 24 month rolling window of bond jump mean, jump intensity, and jump volatility, constructed as describedthe monthly frequency (end-of-month). T-statistics are shown in parentheses and are based on Newey-West stanparameter of 2.

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Table 8: Excess returns on holding an n-month bond for a holding period of one year (12 month ro(Newey-West t-statistics in parentheses)


60t IV

1t RV

1t JI

12t

n=24-1.75 (-1.99) -0.58 (-2.37) 1.46 (3.11) -0.55 (-1.30)

-0.41 (-0.36) -0.66 (-2.73) 1.47 (3.09) -0.40 (-0.92) -0.19 (-1.94) -1.16 (-1.24) -0.65 (-2.55) 1.54 (3.15) -0.54 (-1.22) -0.09 (-1.50) -2.57 (-1.89) -0.62 (-2.48) 1.41 (2.88) -0.43 (-0.96) 7.11 (1.04)-1.84 (-2.62) -0.50 (-2.44) 1.30 (2.88) -0.42 (-1.04) -0.65 (-0.43) -0.58 (-2.73) 1.32 (2.96) -0.40 (-1.00)

n=36-3.25 (-2.08) -1.35 (-2.88) 3.19 (3.56) -1.26 (-1.62) -0.82 (-0.39) -1.50 (-3.24) 3.20 (3.52) -0.99 (-1.20) -0.34 (-1.79) -2.05 (-1.21) -1.49 (-3.07) 3.35 (3.59) -1.24 (-1.51) -0.19 (-1.65) -4.74 (-1.95) -1.42 (-2.97) 3.10 (3.29) -1.05 (-1.27) 12.92 (1.10)-3.43 (-2.81) -1.19 (-3.23) 2.85 (3.47) -1.00 (-1.37) -1.51 (-0.54) -1.35 (-3.25) 2.96 (3.41) -1.03 (-1.34)

n=48-5.11 (-2.53) -2.13 (-3.48) 4.62 (3.90) -1.65 (-1.60) -1.68 (-0.60) -2.34 (-3.88) 4.64 (3.84) -1.26 (-1.14) -0.48 (-1.76) -3.29 (-1.49) -2.34 (-3.70) 4.87 (3.93) -1.61 (-1.48) -0.29 (-1.82) -7.37 (-2.38) -2.24 (-3.56) 4.48 (3.59) -1.33 (-1.24) 19.62 (1.32)-5.36 (-3.33) -1.91 (-4.04) 4.16 (3.81) -1.28 (-1.33) -3.30 (-0.90) -2.14 (-3.83) 4.39 (3.69) -1.40 (-1.35)

n=60-6.73 (-2.81) -2.62 (-3.55) 5.20 (3.62) -1.54 (-1.24) -2.60 (-0.75) -2.87 (-3.92) 5.22 (3.56) -1.08 (-0.79) -0.58 (-1.69) -4.44 (-1.66) -2.88 (-3.76) 5.52 (3.67) -1.50 (-1.14) -0.37 (-1.86) -9.45 (-2.64) -2.75 (-3.59) 5.03 (3.32) -1.15 (-0.90) 23.63 (1.42)-7.04 (-3.68) -2.34 (-4.28) 4.62 (3.57) -1.08 (-0.94)

-5.26 (-1.18) -2.62 (-3.78) 5.01 (3.41) -1.34 (-1.06)

Notes: As for Table 4, except that 12 month rolling windows are used to measure the jump risk measures.

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Table 9: Out-of-sample forecasting comparison

Additional Variable n=24 n=36 n=48 n=60 Additional Variable n=24 n=36 12-month overlapping returns, 2-year window 12-month overlapping returns, 1IV1t 1.13 1.13 1.14 1.13 IV

1t 1.13 1.13

RV1t 1.06 1.07 1.08 1.07 RV1t 1.06 1.07

JI24t 1.31 1.27 1.25 1.20 JI

12t 1.24 1.20

JM24t 0.60*** 0.59*** 0.61*** 0.64*** JM12t 0.91*** 0.87***

JV24t 1.02 1.03 1.05 1.07 JV12t 1.03 1.03

12-month non-overlapping returns, 2-year window 3-month overlapping returns, 2-IV1t 1.01 1.01 0.99 0.98 IV

1t 1.02 1.02

RV1t 1.01 0.98 0.95* 0.93* RV1t 1.00 1.00

JI24t 1.18 1.12 1.08 1.04 JI24t 1.09 1.07

JM24t 0.75*** 0.74*** 0.77*** 0.80*** JM24t 0.86*** 0.89***

JV24t 0.94** 0.96** 0.98* 1.00 JV24t 1.03 1.03

Notes: This table reports the out-of-sample forecasting exercise for excess bond returns. Starting half way through thexcess bond returns using (i) three forward rates alone and (ii) forward rates in conjunction with one other variamean-square-prediction-error (RMSPE) of (ii) relative to (i). Numbers below 1 mean that the extra variable improvalso report the Diebold and Mariano (1995) test for equality of the two out-of-sample RMSPEs, using the nonstana nested forecast comparison Clark and McCracken (2005). Results are shown for forecasting 12-month excess retform the jump risk measures (as in Table 4) in the upper left quadrant of the table and for the forecasting exercisesin Tables 6, 7 and 8, respectively in the other three quadrants. Cases where the extra regressor improves mean sqand 1 percent significance levels are marked with one, two and three stars.

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1986 1988 1990 1992 1994 1996 1998 2000 2002 2004

8

10

12

14

16

18

OptionImplied Volatility

1986 1988 1990 1992 1994 1996 1998 2000 2002 2004

6

8

10

12

14

16

18

Realized Volatility

1986 1988 1990 1992 1994 1996 1998 2000 2002 2004

1.5

1

0.5

0

0.5

1

1.5

Realized Jumps

Figure 1: Volatility and Jump Risks of Treasury Bond Futures

Notes: This figure plots monthly options-implied and realized volatilities (in annualized percentage points)

and daily realized jumps (in percentage points) for Treasury bond futures, constructed as described in the

text.

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1986 1988 1990 1992 1994 1996 1998 2000 2002 2004

0.04

0.06

0.08

0.1

0.12

0.14Realized Jump Intensity

1986 1988 1990 1992 1994 1996 1998 2000 2002 20040.2

0.1

0

0.1

0.2

Realized Jump Mean

1986 1988 1990 1992 1994 1996 1998 2000 2002 2004

0.3

0.35

0.4

0.45

0.5

0.55

0.6

Realized Jump Volatility

Figure 2: Realized Jump Risk Measures

Notes: This figure plots 24-month rolling estimates of realized bond jump mean, jump intensity, and jump

volatility, constructed as described in the text.

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1986 1988 1990 1992 1994 1996 1998 2000 2002 2004

5

0

5

10

ExPost Excess Bond Return

1986 1988 1990 1992 1994 1996 1998 2000 2002 2004

5

0

5

10

Expected Excess Bond Return from Forward Rates

1986 1988 1990 1992 1994 1996 1998 2000 2002 2004

5

0

5

10

Expected Excess Bond Return from Forward Rates and Jump Mean

Figure 3: Excess Bond Returns and Predicted Risk Premia

Notes: This figure plots the excess returns onn year bonds over those on one-year bonds for a twelve-month

holding period, ending in the month shown, averaged overn from two to five. The top panel gives realized

ex-post excess returns, and the lower two panels are the projected ex-ante estimates using the coefficient

estimates from Table 4.

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1986 1988 1990 1992 1994 1996 1998 2000 2002 20043

2

1

0

1

2

3

4Excess Bond Return and OptionImplied Volatility (Standardized)

Implied Volatility

Excess Return

1986 1988 1990 1992 1994 1996 1998 2000 2002 20043

2

1

0

1

2

3

4Excess Bond Return and Realized Volatility (Standardized)

Realized Volatility

Excess Return

1986 1988 1990 1992 1994 1996 1998 2000 2002 20043

2

1

0

1

2

3

4Excess Bond Return and Realized Jump Mean (Standardized)

Negative Jump Mean

Excess Return

Figure 4: Excess Bond Return and Volatility or Jump Risks

Notes: This figure plots the monthly excess returns on n year bonds for a one-year holding period averaged

over n from two to five, with monthly options-implied bond volatility, monthly realized bond volatility, and

the realized bond jump mean over the previous 24 months (with the sign flipped).

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1996 1997 1998 1999 2000 2001 2002 2003 2004 20052

0

2

4OutofSample Forecast of 2Year Excess Bond Return

1996 1997 1998 1999 2000 2001 2002 2003 2004 20054

20

2

4

6

8OutofSample Forecast of 3Year Excess Bond Return

1996 1997 1998 1999 2000 2001 2002 2003 2004 2005

5

0

5

10

OutofSample Forecast of 4Year Excess Bond Return

1996 1997 1998 1999 2000 2001 2002 2003 2004 2005

5

0

5

10

OutofSample Forecast of 5Year Excess Bond Return

Figure 5: Out-of-Sample Forecasting of Excess Bond Returns

Notes: This figure plots the monthly out-of-sample forecasting of excess returns on 2 5 year bonds for a

Documents

Bond Risk Premia and Realized Jump Risk