Bond Risk Premia and Realized Jump Risk

Bond Risk Premia and Realized Jump Risk
∗
Jonathan Wright
†
and Hao Zhou
‡
First Draft: November 2006
This 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 size—identified
from high-frequency bond returns using the bi-power variation technique—increases
the R
2
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 factor—as proxied by
realized jump mean in this paper—helps 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.
∗
We thank Torben Andersen, Ravi Bansal, Darrell Duffie, Cam Harvey, Jay Huang, Chris Jones, Sydney
Ludvigson, Nour Meddahi, Bruce Mizrach, Monika Piazzesi, George Tauchen, Fabio Trojani, and seminar
participants at Federal Reserve Board, Stanford SITE Workshop, Econometric Society Annual Meeting (New
Orleans), Rutgers University, Banque de France Workshop on Financial Market and Real Activity for helpful
comments and suggestions. The views presented here are solely those of the authors and do not necessarily
represent 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.
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 size—identified from high-frequency
bond returns using the bi-power variation technique—increases the R
2
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 factor—as proxied by
realized jump mean in this paper—helps 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.
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, with R
2
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 by
the term structure of yields and so in a regression of excess bond returns on term structure
variables and any other 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
1
the magnitude of these jumps. These estimates all have the advantage of being model-
free—it 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 risk
premia 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,
with R
2
values nearly doubling from 34-38 percent to 60-62 percent. In contrast, inclusion of
other high-frequency jump measures—jump intensity and jump volatility—in the equation
for predicting excess bond returns raises the R
2
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 R
2
s 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
2
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 only
non-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 the R
2
of the regression on both the jump mean
and forward rates is a good bit larger than the sum of the R
2
’s 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 model’s 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-
3
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 volatility
must 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 A
1
(3) specification (Dai and Singleton, 2000)—
1
There is also empirical evidence against specific unspanned volatility models (see, e.g., Fan, Gupta, and
Ritchken, 2003; Bikbov and Chernov, 2008; Thompson, 2008).
4
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 vector—the
short rate, its central tendency, and its stochastic volatility—is given by
dr
t
= κ
r
(θ
t
−r
t
)dt +
√
α
r
+ v
t
dW
Q
t
+ σ
rθ
dB
Q
t
(1)
dθ
t
= (γ
θ
−2κ
r
θ
t
+
1
κ
r
v
t
)dt + σ
θ
dB
Q
t
(2)
dv
t
= (γ
v
−κ
Q
v
v
t
)dt + σ
v
√
v
t
dZ
Q
t
(3)
where [dW
Q
t
, dB
Q
t
, dZ
Q
t
]
′
is a three-dimensional vector of independent Brownian motions un-
der the risk-neutral measure, and γ
v
> 0 and α
r
≥ 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; r
t
, θ
t
, V
t
) = e
A(τ)−B(τ)rt−C(τ)θt−D(τ)Vt
, (4)
where τ ≡ T − t, and A(τ), B(τ), C(τ), and D(τ) are solutions to a system of ordinary
differential equations (Duffie and Kan, 1996). However, the particular model specification
in eqs. (1)-(3) satisfies the incomplete market condition, such that D(τ) = 0 for any τ
(Collin-Dufresne and Goldstein, 2002). Therefore the bond prices can be reduced to
P(t, T; r
t
, θ
t
, V
t
) ≡ P(t, T; r
t
, θ
t
) = e
A(τ)−B(τ)rt−C(τ)θt
, (5)
where
2
A(τ) =
_
τ
0
_
α
r
+ σ
2
rθ
2
B(s)
2
+
σ
2
θ
2
C(s)
2
+ σ
rθ
σ
θ
B(s)C(s) −γ
θ
C(s)
_
ds (6)
B(τ) =
1
κ
r
(1 −e
κrτ
) (7)
C(τ) =
1
2κ
r
(1 −e
κrτ
)
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.
2
Although A(τ) can also be solved in closed form, it is not particularly important for the conceptual
exercise considered here, although it may be useful for future numerical analysis.
5
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 (γ
θ
−2κ
r
θ
t
+1/κ
r
v
t
) allows the conditional expectation E
t
[θ
T
] to
linearly depend on v
t
; and the drift function of short rate κ
r
(θ
t
−r
t
) allows the conditional
expectation E
t
[r
T
] to linearly depend on v
t
recursively through E
t
[θ
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 dW
Q
t
= dW
t
, dB
Q
t
= dB
t
, and dZ
Q
t
= dZ
t
+ λ
√
v
t
dt, then we can transform
to the objective dynamics, dv
t
= (γ
v
− κ
v
v
t
)dt + σ
v
√
v
t
dZ
t
which governs the time-series
evolution 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 by κ
v
≡ κ
Q
v
+σ
v
λ.
Therefore, the conditional expectation under the physical measure of the state vector can be
easily verified as
E
t
[r
T
] = r
t
e
−κr(T−t)
+ θ
t
_
e
−κr(T−t)
−e
−2κr(T−t)
_
+
_
γ
θ
+
γ
v
κ
r
κ
v
_
1
2κ
r
_
1 −e
−κr(T−t)
_
2
+
_
v
t
−
γ
v
κ
v
_
1
(2κ
r
−κ
v
)
×
_
1
κ
v
−κ
r
_
e
−κr(T−t)
−e
−κv(T−t)
_
−
1
κ
r
_
e
−κv(T−t)
−e
−2κr(T−t)
_
_
(9)
E
t
[θ
T
] = θ
t
e
−2κr(T−t)
+
_
γ
θ
+
γ
v
κ
r
κ
v
_
1
2κ
r
_
1 −e
−2κr(T−t)
_
+
_
v
t
−
γ
v
κ
v
_
1
κ
r
(2κ
r
−κ
v
)
_
e
−κv(T−t)
−e
−2κr(T−t)
_
(10)
E
t
[v
T
] = v
t
e
−κv(T−t)
+
γ
v
κ
v
_
1 −e
−κv(T−t)
_
(11)
where we can see the dependence of the spanned risk factors r
t
and θ
t
on the unspanned risk
factor v
t
.
4
3
This 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 transformations
between risk-neutral and objective dynamics (Duffee, 2002; Dai and Singleton, 2002; Duarte, 2004).
4
This feature mirrors an important USV model in the equity option literature (Heston, 1993), where the
unspanned risk factor affects the conditional volatility of the state vector.
6
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 of m periods can be written as
E
t
[ex
m,n
t+m
] = A(n −m) −A(n) +
A(m)
m
+
_
B(m)
m
−B(n)
_
r
t
+
_
C(m)
m
−C(n)
_
θ
t
+
_
D(m)
m
−D(n)
_
v
t
+B(n −m)E
t
[r
t+m
] + C(n −m)E
t
[θ
t+m
] + D(n −m)E
t
[v
t+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
E
t
[ex
m,n
t+m
] = A(n −m) −A(n) +
A(m)
m
+
_
B(m)
m
−B(n)
_
r
t
+
_
C(m)
m
−C(n)
_
θ
t
+B(n −m)E
t
[r
t+m
] + C(n −m)E
t
[θ
t+m
] (13)
where the conditional means E
t
[r
t+m
] and E
t
[θ
t+m
] are linear functions of v
t
, 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
∂E
t
[ex
m,n
t+m
]
∂v
t
=
1
κ
r
_
1 −e
κr(n−m)
_
1
(2κ
r
−κ
v
)
×
_
1
κ
v
−κ
r
_
e
−κr(m)
−e
−κv(m)
_
−
1
κ
r
_
e
−κv(m)
−e
−2κr(m)
_
_
+
1
2κ
r
_
1 −e
κr(n−m)
_
2 1
κ
r
(2κ
r
−κ
v
)
_
e
−κv(m)
−e
−2κr(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
5
However, our empirical finding that realized jump mean has a significant negative impact on expected
excess bond returns, has an intuitive economic interpretation—if 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 bond
price to jump upwards today.
7
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
8
standard, but we review it briefly, to keep the paper self-contained.
Let s
t
= log(S
t
) denote the time t logarithmic price of an asset, which evolves in contin-
uous time as a jump diffusion process:
ds
t
= µ
t
dt + σ
t
dW
t
+ J
t
dq
t
where µ
t
and σ
t
are the instantaneous drift and diffusion functions that are completely general
and may be stochastic (subject to the regularity conditions), W
t
is a standard Brownian
motion, dq
t
is a Poisson jump process with intensity JI
t
, and J
t
refers to the corresponding
(log) jump size distributed as Normal(JM
t
, JV
2
t
). 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:
r
s
t,j
≡ s
t,j·∆
−s
t,(j−1)·∆
where r
s
t,j
refers to the j
th
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 process—realized variance and realized bi-power variation—which converge uni-
formly (as ∆ →0 or m = 1/∆ →∞) to different functionals of the underlying jump-diffusion
process,
RV
t
≡
m

j=1
|r
s
t,j
|
2
→
_
t
t−1
σ
2
u
du +
_
t
t−1
J
2
u
dq
u
(15)
BV
t
≡
π
2
m
m−1
m

j=2
|r
s
t,j
||r
s
t,j−1
| →
_
t
t−1
σ
2
u
du. (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),
RJ
t
≡
RV
t
−BV
t
RV
t
(17)
9
which converges to a standard normal distribution with an appropriate scaling
ZJ
t
≡
RJ
t
_
[(
π
2
)
2
+ π −5]
1
m
max(1,
TPt
BV
2
t
)
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
ˆ
J
t
= sign(r
s
t
) ×
_
(RV
t
−BV
t
) ×I
(ZJt≥Φ
−1
α
)
where Φ is the cumulative distribution function of a standard normal random variable, α is
the significance level of the z-test, and I
(ZJt≥Φ
−1
α
)
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,
JI
t
=
Number of Realized Jump Days
Number of Total Trading Days
JM
t
= Mean of Realized Jumps
JV
t
= 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
6
Note that TP
t
is the so-called Tri-Power Quarticity robust to jumps, and as shown by Barndorff-Nielsen
and Shephard (2004),
TP
t
≡ mµ
−3
4/3
m
m−2
m

j=3
|r
s
t,j−2
|
4/3
|r
s
t,j−1
|
4/3
|r
s
t,j
|
4/3
→
_
t
t−1
σ
4
s
ds (19)
with µ
k
≡ 2
k/2
Γ((k + 1)/2)/Γ(1/2) for k > 0.
10
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-free—no
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. Let D
DAILY
t
denote the dummy that is 1 if and only
if a jump is detected on day t and recall that
ˆ
J
t
denotes the estimated magnitude of the
jump on day t. For our empirical work, let the h-month rolling average realized volatility,
7
Existing 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; A¨ıt-Sahalia, 2004).
11
jump intensity, jump mean, and jump volatility be defined as, respectively,
RV
h
t
=
1
h ×22
Σ
h×22−1
j=0
RV
t−j
,
JI
h
t
=
1
h ×22
Σ
h×22−1
j=0
D
DAILY
t−j
,
JM
h
t
=
Σ
h×22−1
j=0
ˆ
J
t−j
D
DAILY
t−j
Σ
h×22−1
j=0
D
DAILY
t−j
,
JV
h
t
=
¸
¸
¸
_
Σ
h×22−1
j=0
(
ˆ
J
t−j
−JM
h
t
)
2
D
DAILY
t−j
Σ
h×22−1
j=0
D
DAILY
t−j
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 (h equal to one month), we use much longer rolling windows for measuring
jump intensity, jump mean and jump volatility, setting the parameter h to 24 months or 12
months. The tradeoff in selecting h 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
8
These are the Black-Scholes implied volatilities from options on the front at-the-money bond futures
contract, expressed at an annualized rate. Where the front futures contract will expire within one month,
we use the next futures contract instead.
12
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 an m-month bond for a holding period of m months is given
by ex
m,n
t+m
= p
n−m
t+m
− p
n
t
− (m/12)y
m
t
where y
j
t
denotes the annual continuously compounded
yield on a j-month zero coupon bond and p
j
t
= −(j/12)y
t
is the log price of this bond. We
used 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
ex
m,n
t+m
= β
0
+β
1
f
12
t
+β
2
f
36
t
+β
3
f
60
t
+β
4
IV
1
t
+β
5
RV
1
t
+β
6
JM
h
t
+β
7
JI
h
t
+β
8
JV
h
t
+ε
t+m
(20)
where f
j
t
= p
j−12
t
−p
j
t
denotes the j/12-year forward rate with a 12-months period and RV
1
t
,
JM
h
t
, JI
h
t
and JV
h
t
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, and IV
1
t
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
13
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
R
2
values from these regressions should be large. In fact, they are small, ranging from 4 to
20 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 R
2
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 of n and the R
2
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 R
2
in the range 34-38 percent,
14
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 the R
2
rises to
60-62 percent. And, the information content of the jump mean seems to complement that of
forward rates in that the R
2
of the regression on both jump mean and forward rates is larger
than the sum of R
2
s 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).
15
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 consistently
significantly 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 noisy—sometime 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. setting m = 12
and h = 24 and various values of n), 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
9
Inference issues related to the use of highly persistent predictor variables have been studied extensively
in 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 use
of overlapping data see, e.g., Valkanov (2003) and Boudoukh, Richardson, and Whitelaw (2008).
10
The Fama-Bliss zero-coupon data consist of 1-, 2-, 3-, 4- and 5-year zero-coupon bonds only. To construct
three-month excess returns (m = 3), we need some yields that are not available, such as 1.75 year yields.
We use the approximation y
n−3
t
≈ y
n
t
to obtain these yields. The problem does not arise for one-year excess
16
These results are shown in Table 7. The R
2
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 significant
association with future excess returns.
For the parameter h (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 returns—the 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).
17
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 lowers
root 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.
18
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 especially
the 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-post excess 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
19
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.
20
References
A¨ıt-Sahalia, Yacine (2004). “Disentangling Diffusion from Jumps.” Journal of Financial
Economics, 74, 487–528.
Almeida, Caio, Jeremy J. Graveline, and Scott Joslin (2006). “Do Interest Rate Options
Contain Information About Excess Returns.” Working Paper, Standford University GSB.
Andersen, Torben G. and Luca Benzoni (2008). “Can Bonds Hedge Volatility Risk in the
U.S. Treasury Market? A Specification Test for Affine Term Structure Models.” Working
Paper, Carlson School of Management, University of Minnesota.
Andersen, Torben G., Luca Benzoni, and Jesper Lund (2002). “An Empirical Investigation
of Continuous-Time Equity Return Models.” Journal of Finance, 57, 1239–1284.
Andersen, Torben G., Tim Bollerslev, and Francis X. Diebold (2007). “Roughing It Up:
Including Jump Components in the Measurement, Modeling and Forecasting of Return
Volatility.” Review of Economics and Statistics, 89, 701–720.
Ang, Andrew and Geert Bekaert (2002). “Regime Switches in Interest Rates.” Journal of
Business and Economic Statistics, 20, 163–182.
Backus, David and Jonanthan Wright (2007). “Cracking the Conundrum.” Brookings Papers
on Economic Activity, 1, 293–329.
Bansal, Ravi, George Tauchen, and Hao Zhou (2004). “Regime Shifts, Risk Premiums in the
Term Structure, and the Business Cycle.” Journal of Business and Economic Statistics,
22, 396–409.
Bansal, Ravi and Hao Zhou (2002). “Term Structure of Interest Rates with Regime Shifts.”
Journal of Finance, 57, 1997–2043.
Barndorff-Nielsen, Ole and Neil Shephard (2004). “Power and Bipower Variation with
Stochastic Volatility and Jumps.” Journal of Financial Econometrics, 2, 1–48.
Barndorff-Nielsen, Ole and Neil Shephard (2006). “Econometrics of Testing for Jumps in
Financial Economics Using Bipower Variation.” Journal of Financial Econometrics, 4,
1–30.
Bates, David S. (2000). “Post-’87 Crash Fears in the S&P 500 Futures Option Market.”
Journal of Econometrics, 94, 181–238.
Bikbov, Ruslan and Mikhail Chernov (2008). “Term Structure and Volatility: Lessons from
the Eurodollar Markets.” Management Sciences. forthcoming.
21
Boudoukh, Jacob, Matthew Richardson, and Robert F. Whitelaw (2008). “The Myth of
Long-Horizon Predictability.” Review of Financial Studies, 21, 1577–1605.
Campbell, John Y. and Robert J. Shiller (1991). “Yield Spreads and Interest Rate Move-
ments: A Bird’s Eye View.” Review of Economic Studies, 58, 495–514.
Campbell, John Y. and Motohiro Yogo (2006). “Efficient Tests of Stock Return Predictabil-
ity.” Journal of Financial Economics, 81, 27–60.
Casassus, Jaime, Pierre Collin-Dufresne, and Robert S. Goldstein (2005). “Unspanned
Stochastic Volatility and Fixed Income Derivatives Pricing.” Journal of Banking and
Finance, 29, 2723–2749.
Chernov, Mikhail, A. Ronald Gallant, Eric Ghysels, and George Tauchen (2003). “Alterna-
tive Models for Stock Price Dynamics.” Journal of Econometrics, 116, 225–258.
Clark, Todd and Michael McCracken (2005). “Evaluating Direct Multistep Forecasts.”
Econometric Reviews, 24, 369–404.
Cochrane, John H. and Monika Piazzesi (2005). “Bond Risk Premia.” American Economic
Review, 95, 138–160.
Collin-Dufresne, Pierre and Robert S. Goldstein (2002). “Do Bonds Span the Fixed Income
Markets? Theory and Evidence for Unspanned Stochastic Volatility.” Journal of Finance,
57, 1685–1730.
Collin-Dufresne, Pierre, Robert S. Goldstein, and Christopher S. Jones (2008). “Can Interest
Rate Volatility be Extracted from the Cross Section of Bond Yields? An Investigation of
Unspanned Stochastic Volatility.” Working Paper, Marshall School of Business, University
of Southern California.
Dai, Qiang and Kenneth J. Singleton (2000). “Specification Analysis of Affine Term Structure
Models.” Journal of Finance, 55, 1943–1978.
Dai, Qiang and Kenneth J. Singleton (2002). “Expectation Puzzles, Time-varying Risk
Premia, and Dynamic Models of the Term Structure.” Journal of Financial Economics,
63, 415–441.
Dai, Qiang, Kenneth J. Singleton, and Wei Yang (2007). “Regime Shifts in a Dynamic Term
Structure Model of the U.S. Treasury Yields.” Review of Financial Studies, 20, 1669–1706.
Diebold, Francis X. and Roberto S. Mariano (1995). “Comparing Predictive Accuracy.”
Journal of Business and Economic Statistics, 13, 253–263.
22
Duarte, Jefferson (2004). “Evaluating An Alternative Risk Preference in Affine Term Struc-
ture Models.” Review of Financial Studies, 17, 379–404.
Duffee, Gregory (2002). “Term Premia and Interest Rate Forecasts in Affine Models.” Jour-
nal of Finance, 57, 405–443.
Duffee, Gregory R. (2008). “Information in (and Not in) the Term Structure.” Working
Paper, Johns Hopkins University.
Duffie, Darrell and Rui Kan (1996). “A Yield-Factor Model of Interest Rates.” Mathematical
Finance, 6, 379–406.
Engle, Robert F., David M. Lilien, and Russell P. Robins (1987). “Estimating Time Varying
Risk Premia in the Term Structure: The ARCH-M Model.” Econometrica, 55, 391–407.
Eraker, Bjørn, Michael Johannes, and Nicholas Polson (2003). “The Impact of Jumps in
Equity Index Volatility and Returns.” Journal of Finance, 53, 1269–1300.
Evans, Martin D. D. (2003). “Real Risk, Inflation Risk, and the Term Structure.” Economic
Journal , 113, 345–389.
Fama, Eugene F. and Robert T. Bliss (1987). “The Information in Long-Maturity Forward
Rates.” American Economic Review, 77, 680–692.
Fan, Rong, Anurag Gupta, and Peter Ritchken (2003). “Hedging in the Possible Presence of
Unspanned Stochastic Volatility: Evidence from Swaption Markets.” Journal of Finance,
58, 2219–2248.
Ferson, Wayne E., Sergei Sarkissian, and Timothy T. Simin (2003). “Spurious Regressions
in Financial Economics?” Journal of Finance, 1393–1414.
Heidari, Massoud and Liuren Wu (2003). “Are Interest Rate Derivatives Spanned by the
Term Structure of Interest Rates?” Journal of Fixed Income, 13, 75–86.
Heston, Steven (1993). “A Closed-Form Solution for Options with Stochastic Volatility with
Applications to Bond and Currency Options.” Review of Financial Studies, 6, 327–343.
Huang, Xin and George Tauchen (2005). “The Relative Contribution of Jumps to Total
Price Variance.” Journal of Financial Econometrics, 3, 456–499.
Joslin, Scott (2007). “Pricing and Hedging Volatility Risk in Fixed Income Markets.” Work-
ing Paper, Standford University GSB.
23
Joslin, Scott, Marcel Priebsch, and Kenneth J. Singleton (2008). “Risk Premium Account-
ing in Macro-Dynamic Term Structure Models.” Working Paper, MIT Sloan School of
Management.
Kim, Don H. and Jonathan H. Wright (2005). “An Arbitrage-Free Three-Factor Term Struc-
ture Model and the Recent Behavior of Long-Term Yields and Distant-Horizon Forward
Rates.” Finance and Economics Discussion Series. Federal Reserve Board.
Li, Haitao and Feng Zhao (2006). “Unspanned Stochastic Volatility: Evidence from Hedging
Interest Rate Derivatives.” Journal of Finance, 61, 341–378.
Ludvigson, Sydney C. and Serena Ng (2008). “Macro Factors in Bond Risk Premia.” Review
of Financial Studies. forthcoming.
Merton, Robert C. (1976). “Option Pricing When Underlying Stock Returns Are Discontin-
uous.” Journal of Financial Economics, 3, 125–144.
Pan, Jun (2002). “The Jump-Risk Premia Implicit in Options: Evidence from an Integrated
Time-Series Study.” Journal of Financial Economics, 63, 3–50.
Roberds, William and Charlse H. Whiteman (1999). “Endogenous Term Premia and Anoma-
lies in the Term Struture of Interest Rates: Explaining the Predictability Smile.” Journal
of Monetary Economics, 44, 555–580.
Singleton, Kenneth J. (2006). Empirical Dynamic Asset Pricing: Model Specification and
Econometric Assessment. Princeton University Press.
Stambaugh, Robert F. (1999). “Predictive Regressions.” Journal of Financial Economics,
54, 375–421.
Tauchen, George and Hao Zhou (2008). “Realized Jumps on Financial Markets and Predict-
ing Credit Spreads.” Journal of Econometrics. forthcoming.
Thompson, Samuel (2008). “Identifying Term Structure Volatility from the LIBOR-Swap
Curve.” Review of Financial Studies, 21, 819–854. Harvard University.
Valkanov, Rossen (2003). “Long-Horizon Regressions: Theoretical Results and Applica-
tions.” Journal of Financial Economics, 68, 201–232.
Wachter, Jessica A. (2006). “A Consumption-Based Model of the Term Structure of Interest
Rates.” Journal of Financial Economics, 79, 365–399.
24
Table 1: Summary statistics and correlation matrix for excess returns, forward rates, and jump measures
ex
12,24
t+12
ex
12,36
t+12
ex
12,48
t+12
ex
12,60
t+12
f
12
t
f
36
t
f
60
t
IV
1
t
RV
1
t
JI
24
t
JM
24
t
JV
24
t
Summary statistics
Mean 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 0.07
Skewness 0.19 0.14 0.20 0.17 0.00 0.41 0.62 1.32 1.49 0.37 -0.44 0.32
Kurtosis 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
ex
12,24
t+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
ex
12,36
t+12
1.00 0.99 0.98 0.34 0.49 0.48 0.01 0.09 0.06 -0.41 -0.30
ex
12,48
t+12
1.00 0.99 0.33 0.49 0.50 0.03 0.10 0.05 -0.38 -0.27
ex
12,60
t+12
1.00 0.30 0.48 0.49 0.05 0.10 0.03 -0.38 -0.23
f
12
t
1.00 0.92 0.83 0.16 0.10 -0.01 0.32 -0.06
f
36
t
1.00 0.97 0.30 0.24 -0.09 0.29 -0.12
f
60
t
1.00 0.36 0.29 -0.16 0.29 -0.11
IV
1
t
1.00 0.81 -0.59 0.19 -0.01
RV
1
t
1.00 -0.53 0.15 -0.12
JI
24
t
1.00 0.05 0.34
JM
24
t
1.00 0.33
JV
24
t
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: ex
m,n
t+n
are the excess returns on holding an n-month bond over the return
on holding an m-month bond for a holding period of m months; f
j
t
denotes the one-year forward rate ending
j months; and RV
1
t
, JI
h
t
, JM
h
t
and JV
h
t
denote the realized bond volatility and jump measures in rolling
windows ending on the last day of month t.
25
Table 2: Regression of Risk Measures on Term Structure of Forward Rates
Dependent Variable Intercept f
12
t
f
36
t
f
60
t
R
2
IV
1
t
7.11 (21.94) -0.44 (-3.04) 0.03 (0.09) 0.79 (2.80) 0.20
RV
1
t
6.27 (16.16) -0.72 (-3.97) 0.87 (1.94) 0.11 (0.33) 0.14
JI
24
t
0.10 (18.05) -0.00 (-0.39) 0.02 (3.30) -0.02 (-4.77) 0.11
JM
24
t
-0.06 (-2.64) 0.03 (3.26) -0.05 (-2.73) 0.04 (2.83) 0.13
JV
24
t
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.
26
Table 3: Excess returns on holding an n-month bond for a holding period of one year
(Newey-West t-statistics in parentheses)
Intercept IV
1
t
RV
1
t
JI
24
t
JM
24
t
JV
24
t
R
2
n=24
0.93 (0.74) 0.00 (0.01) 0.00
0.50 (0.62) 0.05 (0.58) 0.00
0.58 (0.81) 4.56 (0.50) 0.01
1.14 (4.48) -6.67 (-4.69) 0.15
3.63 (2.74) -6.57 (-2.23) 0.10
n=36
1.50 (0.64) 0.02 (0.09) 0.00
0.84 (0.57) 0.10 (0.61) 0.01
1.15 (0.86) 6.98 (0.43) 0.00
2.10 (4.47) -13.52 (-5.23) 0.17
6.63 (2.55) -12.03 (-2.12) 0.09
n=48
1.64 (0.50) 0.07 (0.22) 0.00
1.01 (0.49) 0.15 (0.68) 0.01
1.66 (0.91) 8.99 (0.40) 0.00
2.90 (4.37) -18.14 (-4.84) 0.15
8.58 (2.32) -15.17 (-1.88) 0.07
n=60
1.47 (0.36) 0.12 (0.30) 0.00
0.97 (0.38) 0.19 (0.70) 0.01
2.19 (1.00) 6.23 (0.24) 0.00
3.33 (4.18) -22.22 (-4.57) 0.15
9.35 (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.
27
Table 4: Excess returns on holding an n-month bond for a holding period of one year
(Newey-West t-statistics in parentheses)
Intercept f
12
t
f
36
t
f
60
t
IV
1
t
RV
1
t
JI
24
t
JM
24
t
JV
24
t
R
2
n=24
-1.75 (-1.99) -0.58 (-2.37) 1.46 (3.11) -0.55 (-1.30) 0.34
-0.41 (-0.36) -0.66 (-2.73) 1.47 (3.09) -0.40 (-0.92) -0.19 (-1.94) 0.39
-1.16 (-1.24) -0.65 (-2.55) 1.54 (3.15) -0.54 (-1.22) -0.09 (-1.50) 0.35
-2.60 (-1.78) -0.57 (-2.26) 1.34 (2.54) -0.40 (-0.83) 8.34 (1.00) 0.35
-2.32 (-5.06) -0.34 (-1.50) 1.02 (2.45) -0.19 (-0.59) -9.63 (-5.73) 0.60
0.27 (0.15) -0.50 (-2.31) 1.30 (3.13) -0.47 (-1.22) -4.59 (-1.74) 0.38
n=36
-3.25 (-2.08) -1.35 (-2.88) 3.19 (3.56) -1.26 (-1.62) 0.35
-0.82 (-0.39) -1.50 (-3.24) 3.20 (3.52) -0.99 (-1.20) -0.34 (-1.79) 0.39
-2.05 (-1.21) -1.49 (-3.07) 3.35 (3.59) -1.24 (-1.51) -0.19 (-1.65) 0.36
-4.67 (-1.80) -1.34 (-2.74) 2.97 (2.97) -1.02 (-1.14) 13.83 (0.95) 0.36
-4.35 (-5.38) -0.88 (-2.23) 2.34 (3.09) -0.57 (-0.97) -18.54 (-6.23) 0.62
0.26 (0.08) -1.22 (-3.00) 2.90 (3.64) -1.13 (-1.54) -7.98 (-1.57) 0.39
n=48
-5.11 (-2.53) -2.13 (-3.48) 4.62 (3.90) -1.65 (-1.60) 0.38
-1.68 (-0.60) -2.34 (-3.88) 4.64 (3.84) -1.26 (-1.14) -0.48 (-1.76) 0.42
-3.29 (-1.49) -2.34 (-3.70) 4.87 (3.93) -1.61 (-1.48) -0.29 (-1.82) 0.40
-7.27 (-2.20) -2.11 (-3.28) 4.30 (3.24) -1.28 (-1.09) 21.08 (1.12) 0.39
-6.59 (-5.87) -1.51 (-2.91) 3.48 (3.40) -0.73 (-0.89) -24.89 -6.15 0.62
-1.17 (-0.26) -1.98 (-3.68) 4.30 (3.91) -1.50 (-1.51) -8.96 (-1.30) 0.40
n=60
-6.73 (-2.81) -2.62 (-3.55) 5.20 (3.62) -1.54 (-1.24) 0.37
-2.60 (-0.75) -2.87 (-3.92) 5.22 (3.56) -1.08 (-0.79) -0.58 (-1.69) 0.41
-4.44 (-1.66) -2.88 (-3.76) 5.52 (3.67) -1.50 (-1.14) -0.37 (-1.86) 0.39
-9.14 (-2.43) -2.60 (-3.34) 4.84 (3.03) -1.13 (-0.81) 23.52 (1.10) 0.38
-8.53 (-6.08) -1.86 (-3.04) 3.82 (3.11) -0.41 (-0.42) -30.33 (-6.04) 0.61
-2.92 (-0.53) -2.47 (-3.71) 4.89 (3.57) -1.39 (-1.15) -8.67 (-1.01) 0.39
Notes: As for Table 3, except that the term structure of forward rates is also being controlled for.
2
8
Table 5: Excess returns on holding an n-month bond for a holding period of one year
(Newey-West t-statistics in parentheses)
Intercept f
12
t
f
36
t
f
60
t
IV
1
t
RV
1
t
JM
24
t
R
2
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) 0.39
-1.40 (-1.87) -0.40 (-1.75) 1.05 (2.54) -0.11 (-0.32) -0.13 (-1.66) -9.22 (-5.23) 0.62
-2.08 (-4.67) -0.37 (-1.60) 1.06 (2.56) -0.19 (-0.58) -0.04 (-0.75) -9.50 (-5.47) 0.61
-1.43 (-2.10) -0.36 (-1.69) 0.93 (2.52) -0.03 (-0.09) -0.24 (-1.89) -0.13 (1.61) -9.32 (-5.53) 0.64
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) 0.40
-2.72 (-2.00) -1.00 (-2.49) 2.38 (3.20) -0.42 (-0.68) -0.22 (-1.59) -17.81 (-5.84) 0.64
-3.82 (-4.55) -0.95 (-2.36) 2.42 (3.25) -0.58 (-0.95) -0.08 (-0.97) -18.25 (-5.99) 0.62
-2.77 (-2.16) -0.93 (-2.50) 2.21 (3.17) -0.31 (-0.52) -0.39 (-1.67) 0.19 (1.36) -17.96 (-6.09) 0.64
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) 0.43
-4.23 (-2.23) -1.67 (-3.23) 3.54 (3.54) -0.51 (-0.59) -0.32 (-1.62) -23.84 (-6.00) 0.64
-5.65 (-4.65) -1.62 (-3.10) 3.63 (3.63) -0.73 (-0.88) -0.14 (-1.27) -24.38 (-5.99) 0.63
-4.28 (-2.36) -1.60 (-3.28) 3.36 (3.47) -0.38 (-0.45) -0.50 (-1.52) 0.21 (1.09) -24.00 (-6.16) 0.65
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) 0.41
-5.71 (-2.43) -2.06 (-3.39) 3.88 (3.25) -0.15 (-0.15) -0.39 (-1.58) -29.08 (-6.07) 0.63
-7.32 (-4.63) -2.01 (-3.26) 4.01 (3.35) -0.42 (-0.42) -0.19 (-1.39) -29.68 (-5.96) 0.61
-5.76 (-2.53) -1.98 (-3.41) 3.70 (3.11) -0.03 (-0.02) -0.57 (-1.42) 0.22 (0.95) -29.24 (-6.18) 0.63
Notes: As for Table 4.
2
9
Table 6: Excess returns on holding an n-month bond for a holding period of one year with non-overlapping data
(Heteroskedasticity-robust t-statistics in parentheses)
Intercept f
12
t
f
36
t
f
60
t
IV
1
t
RV
1
t
JI
24
t
JM
24
t
JV
24
t
R
2
n=24
-0.98 (-0.98) -0.83 (-2.79) 1.89 (3.85) -0.89 (-2.16) 0.44
-0.05 (-0.05) -0.89 (-3.06) 1.99 (4.00) -0.84 (-1.94) -0.15 (-3.05) 0.51
-0.52 (-0.50) -0.85 (-2.82) 1.97 (3.98) -0.91 (-2.11) -0.08 (-1.23) 0.46
-0.86 (-0.46) -0.83 (-2.82) 1.92 (3.19) -0.92 (-1.58) -1.15 (-0.10) 0.44
-1.87 (-2.62) -0.51 (-2.12) 1.40 (2.95) -0.50 (-1.29) -9.11 (-4.59) 0.69
2.20 (1.41) -0.67 (-2.52) 1.72 (3.38) -0.88 (-2.18) -7.17 (-2.91) 0.55
n=36
-2.40 (-1.41) -1.80 (-3.37) 3.86 (4.43) -1.70 (-2.26) 0.51
-0.59 (-0.35) -1.93 (-3.66) 4.04 (4.66) -1.59 (-2.01) -0.30 (-3.20) 0.57
-1.41 (-0.78) -1.85 (-3.38) 4.02 (4.64) -1.73 (-2.20) -0.17 (-1.38) 0.53
-1.98 (-0.58) -1.81 (-3.43) 3.94 (3.53) -1.79 (-1.62) -3.75 (-0.18) 0.51
-4.07 (-3.87) -1.21 (-3.07) 2.94 (3.74) -0.96 (-1.47) -17.16 (-4.78) 0.74
3.08 (1.04) -1.54 (-3.26) 3.56 (3.99) -1.69 (-2.26) -12.33 (-2.59) 0.60
n=48
-3.88 (-1.76) -2.81 (-3.99) 5.59 (4.74) -2.27 (-2.19) 0.56
-1.51 (-0.67) -2.97 (-4.27) 5.83 (5.01) -2.12 (-1.96) -0.39 (-3.06) 0.62
-2.49 (-1.05) -2.88 (-3.99) 5.81 (5.01) -2.31 (-2.14) -0.24 (-1.45) 0.59
-3.34 (-0.75) -2.82 (-4.06) 5.69 (3.76) -2.38 (-1.57) -4.82 (-0.17) 0.56
-6.02 (-4.76) -2.05 (-3.86) 4.41 (4.12) -1.31 (-1.47) -22.09 (-4.28) 0.75
2.46 (0.60) -2.50 (-3.94) 5.25 (4.29) -2.25 (-2.16) -14.28 (-2.13) 0.62
n=60
-5.28 (-2.04) -3.45 (-4.13) 6.30 (4.22) -2.20 (-1.65) 0.55
-2.45 (-0.88) -3.65 (-4.39) 6.59 (4.53) -2.03 (-1.45) -0.46 (-2.72) 0.60
-3.52 (-1.21) -3.54 (-4.12) 6.59 (4.55) -2.25 (-1.63) -0.31 (-1.44) 0.58
-4.31 (-0.81) -3.47 (-4.23) 6.49 (3.37) -2.41 (-1.25) -8.76 (-0.25) 0.55
-7.87 (-5.78) -2.54 (-4.19) 4.88 (3.70) -1.04 (-0.93) -26.65 (-4.15) 0.73
1.32 (0.26) -3.13 (-4.07) 5.95 (3.82) -2.18 (-1.62) -14.87 (-1.68) 0.59
Notes: As for Table 4, except that only end-of-year observations are used to avoid overlapping holding periods. Standard errors are heteroskedasticity-
robust, but make no correction for serial correlation.
3
0
Table 7: Excess returns on holding an n-month bond for a holding period of 3 months
(Newey-West t-statistics in parentheses)
Intercept f
12
t
f
36
t
f
60
t
IV
1
t
RV
1
t
JI
24
t
JM
24
t
JV
24
t
R
2
n=24
-1.00 (-2.62) -0.28 (-2.00) 0.80 (2.66) -0.35 (-1.49) 0.18
-0.66 (-1.35) -0.31 (-2.15) 0.81 (2.69) -0.31 (-1.33) -0.05 (-1.21) 0.19
-0.93 (-2.05) -0.29 (-1.99) 0.82 (2.64) -0.35 (-1.49) -0.01 (-0.37) 0.18
-1.52 (-2.53) -0.29 (-2.09) 0.75 (2.54) -0.27 (-1.13) 4.55 (1.40) 0.19
-1.26 (-3.63) -0.15 (-1.07) 0.58 (1.92) -0.18 (-0.79) -4.82 (-4.46) 0.30
-0.41 (-0.59) -0.26 (-1.88) 0.76 (2.54) -0.33 (-1.41) -1.39 (-1.28) 0.19
n=36
-1.44 (-2.60) -0.54 (-2.61) 1.43 (3.24) -0.66 (-1.91) 0.18
-0.96 (-1.34) -0.58 (-2.73) 1.44 (3.27) -0.60 (-1.74) -0.07 (-1.07) 0.18
-1.31 (-1.97) -0.56 (-2.60) 1.46 (3.23) -0.65 (-1.90) -0.02 (-0.42) 0.18
-2.23 (-2.52) -0.56 (-2.70) 1.36 (3.14) -0.54 (-1.53) 6.92 (1.38) 0.19
-1.81 (-3.56) -0.36 (-1.73) 1.11 (2.50) -0.41 (-1.23) -6.94 (-4.12) 0.28
-0.71 (-0.68) -0.52 (-2.53) 1.38 (3.15) -0.63 (-1.84) -1.70 (-0.98) 0.18
n=48
-1.89 (-2.55) -0.74 (-2.68) 1.85 (3.16) -0.81 (-1.78) 0.17
-1.30 (-1.36) -0.79 (-2.79) 1.86 (3.18) -0.74 (-1.63) -0.09 (-0.98) 0.17
-1.72 (-1.94) -0.77 (-2.67) 1.88 (3.15) -0.81 (-1.78) -0.03 (-0.42) 0.17
-2.96 (-2.53) -0.77 (-2.77) 1.75 (3.06) -0.65 (-1.41) 9.27 (1.41) 0.18
-2.37 (-3.38) -0.51 (-1.84) 1.44 (2.45) -0.49 (-1.12) -8.87 (-3.90) 0.26
-1.16 (-0.83) -0.72 (-2.63) 1.80 (3.10) -0.78 (-1.74) -1.72 (-0.74) 0.17
n=60
-2.56 (-2.85) -0.76 (-2.20) 1.74 (2.34) -0.57 (-1.00) 0.15
-1.70 (-1.44) -0.83 (-2.34) 1.76 (2.37) -0.47 (-0.82) -0.13 (-1.13) 0.16
-2.27 (-2.08) -0.80 (-2.24) 1.79 (2.37) -0.56 (-0.99) -0.05 (-0.55) 0.15
-4.02 (-2.84) -0.80 (-2.30) 1.60 (2.21) -0.35 (-0.60) 12.80 (1.58) 0.16
-3.15 (-3.72) -0.47 (-1.38) 1.23 (1.66) -0.17 (-0.31) -11.06 (-3.93) 0.25
-1.95 (-1.13) -0.75 (-2.17) 1.70 (2.31) -0.55 (-0.96) -1.43 (-0.49) 0.15
Notes: This table reports the coefficient estimates in a regression of the excess returns on a n-month bond over those on a 3 month bond, with a
holding period of 3 months, on the term structure of forward rates, options-implied volatility, a one-month rolling window of realized bond volatility
and a 24 month rolling window 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 2.
3
1
Table 8: Excess returns on holding an n-month bond for a holding period of one year (12 month rolling jump risk estimates)
(Newey-West t-statistics in parentheses)
Intercept f
12
t
f
36
t
f
60
t
IV
1
t
RV
1
t
JI
12
t
JM
12
t
JV
12
t
R
2
n=24
-1.75 (-1.99) -0.58 (-2.37) 1.46 (3.11) -0.55 (-1.30) 0.34
-0.41 (-0.36) -0.66 (-2.73) 1.47 (3.09) -0.40 (-0.92) -0.19 (-1.94) 0.39
-1.16 (-1.24) -0.65 (-2.55) 1.54 (3.15) -0.54 (-1.22) -0.09 (-1.50) 0.35
-2.57 (-1.89) -0.62 (-2.48) 1.41 (2.88) -0.43 (-0.96) 7.11 (1.04) 0.36
-1.84 (-2.62) -0.50 (-2.44) 1.30 (2.88) -0.42 (-1.04) -4.49 (-2.23) 0.45
-0.65 (-0.43) -0.58 (-2.73) 1.32 (2.96) -0.40 (-1.00) -2.93 (-1.35) 0.37
n=36
-3.25 (-2.08) -1.35 (-2.88) 3.19 (3.56) -1.26 (-1.62) 0.35
-0.82 (-0.39) -1.50 (-3.24) 3.20 (3.52) -0.99 (-1.20) -0.34 (-1.79) 0.39
-2.05 (-1.21) -1.49 (-3.07) 3.35 (3.59) -1.24 (-1.51) -0.19 (-1.65) 0.36
-4.74 (-1.95) -1.42 (-2.97) 3.10 (3.29) -1.05 (-1.27) 12.92 (1.10) 0.36
-3.43 (-2.81) -1.19 (-3.23) 2.85 (3.47) -1.00 (-1.37) -9.08 (-2.54) 0.47
-1.51 (-0.54) -1.35 (-3.25) 2.96 (3.41) -1.03 (-1.34) -4.67 (-1.13) 0.37
n=48
-5.11 (-2.53) -2.13 (-3.48) 4.62 (3.90) -1.65 (-1.60) 0.38
-1.68 (-0.60) -2.34 (-3.88) 4.64 (3.84) -1.26 (-1.14) -0.48 (-1.76) 0.42
-3.29 (-1.49) -2.34 (-3.70) 4.87 (3.93) -1.61 (-1.48) -0.29 (-1.82) 0.40
-7.37 (-2.38) -2.24 (-3.56) 4.48 (3.59) -1.33 (-1.24) 19.62 (1.32) 0.40
-5.36 (-3.33) -1.91 (-4.04) 4.16 (3.81) -1.28 (-1.33) -12.48 (-2.66) 0.50
-3.30 (-0.90) -2.14 (-3.83) 4.39 (3.69) -1.40 (-1.35) -4.85 (-0.87) 0.39
n=60
-6.73 (-2.81) -2.62 (-3.55) 5.20 (3.62) -1.54 (-1.24) 0.37
-2.60 (-0.75) -2.87 (-3.92) 5.22 (3.56) -1.08 (-0.79) -0.58 (-1.69) 0.41
-4.44 (-1.66) -2.88 (-3.76) 5.52 (3.67) -1.50 (-1.14) -0.37 (-1.86) 0.39
-9.45 (-2.64) -2.75 (-3.59) 5.03 (3.32) -1.15 (-0.90) 23.63 (1.42) 0.39
-7.04 (-3.68) -2.34 (-4.28) 4.62 (3.57) -1.08 (-0.94) -15.83 (-2.85) 0.49
-5.26 (-1.18) -2.62 (-3.78) 5.01 (3.41) -1.34 (-1.06) -3.94 (-0.57) 0.38
Notes: As for Table 4, except that 12 month rolling windows are used to measure the jump risk measures.
3
2
Table 9: Out-of-sample forecasting comparison
Additional Variable n=24 n=36 n=48 n=60 Additional Variable n=24 n=36 n=48 n=60
12-month overlapping returns, 2-year window 12-month overlapping returns, 1-year window
IV
1
t
1.13 1.13 1.14 1.13 IV
1
t
1.13 1.13 1.14 1.13
RV
1
t
1.06 1.07 1.08 1.07 RV
1
t
1.06 1.07 1.08 1.07
JI
24
t
1.31 1.27 1.25 1.20 JI
12
t
1.24 1.20 1.18 1.14
JM
24
t
0.60*** 0.59*** 0.61*** 0.64*** JM
12
t
0.91*** 0.87*** 0.85*** 0.84***
JV
24
t
1.02 1.03 1.05 1.07 JV
12
t
1.03 1.03 1.04 1.05
12-month non-overlapping returns, 2-year window 3-month overlapping returns, 2-year window
IV
1
t
1.01 1.01 0.99 0.98 IV
1
t
1.02 1.02 1.01 1.01
RV
1
t
1.01 0.98 0.95* 0.93* RV
1
t
1.00 1.00 1.00 1.00
JI
24
t
1.18 1.12 1.08 1.04 JI
24
t
1.09 1.07 1.05 1.04
JM
24
t
0.75*** 0.74*** 0.77*** 0.80*** JM
24
t
0.86*** 0.89*** 0.91*** 0.91***
JV
24
t
0.94** 0.96** 0.98* 1.00 JV
24
t
1.03 1.03 1.02 1.03
Notes: This table reports the out-of-sample forecasting exercise for excess bond returns. Starting half way through the sample, we recursively forecast
excess bond returns using (i) three forward rates alone and (ii) forward rates in conjunction with one other variable. Then we calculate the root-
mean-square-prediction-error (RMSPE) of (ii) relative to (i). Numbers below 1 mean that the extra variable improves forecasting out of sample. We
also report the Diebold and Mariano (1995) test for equality of the two out-of-sample RMSPE’s, using the nonstandard critical values that apply to
a nested forecast comparison Clark and McCracken (2005). Results are shown for forecasting 12-month excess returns using 24-month windows to
form the jump risk measures (as in Table 4) in the upper left quadrant of the table and for the forecasting exercises conducted as robustness checks
in Tables 6, 7 and 8, respectively in the other three quadrants. Cases where the extra regressor improves mean square prediction error at the 10, 5
and 1 percent significance levels are marked with one, two and three stars.
3
3
1986 1988 1990 1992 1994 1996 1998 2000 2002 2004
8
10
12
14
16
18
Option−Implied 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.
34
1986 1988 1990 1992 1994 1996 1998 2000 2002 2004
0.04
0.06
0.08
0.1
0.12
0.14
Realized Jump Intensity
1986 1988 1990 1992 1994 1996 1998 2000 2002 2004
−0.2
−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.
35
1986 1988 1990 1992 1994 1996 1998 2000 2002 2004
−5
0
5
10
Ex−Post 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 on n year bonds over those on one-year bonds for a twelve-month
holding period, ending in the month shown, averaged over n 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.
36
1986 1988 1990 1992 1994 1996 1998 2000 2002 2004
−3
−2
−1
0
1
2
3
4
Excess Bond Return and Option−Implied Volatility (Standardized)
Implied Volatility
Excess Return
1986 1988 1990 1992 1994 1996 1998 2000 2002 2004
−3
−2
−1
0
1
2
3
4
Excess Bond Return and Realized Volatility (Standardized)
Realized Volatility
Excess Return
1986 1988 1990 1992 1994 1996 1998 2000 2002 2004
−3
−2
−1
0
1
2
3
4
Excess 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).
37
1996 1997 1998 1999 2000 2001 2002 2003 2004 2005
−2
0
2
4
Out−of−Sample Forecast of 2−Year Excess Bond Return
1996 1997 1998 1999 2000 2001 2002 2003 2004 2005
−4
−2
0
2
4
6
8
Out−of−Sample Forecast of 3−Year Excess Bond Return
1996 1997 1998 1999 2000 2001 2002 2003 2004 2005
−5
0
5
10
Out−of−Sample Forecast of 4−Year Excess Bond Return
1996 1997 1998 1999 2000 2001 2002 2003 2004 2005
−5
0
5
10
Out−of−Sample Forecast of 5−Year 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
one-year holding period. The thick line is the realized excess bond return, the dotted line is the forecast from
using forward rates only, and the thin line is the forecast from augmenting forward rates with the realized
jump mean over the previous 24 months.
38