Short-Run Bond Risk Premia

TLDR: The authors empirically examined the predictability of the market variance risk premium, a proxy of economic uncertainty, for bond risk premia and showed the strong predictive power for the one month horizon that almost entirely disappears for horizons above one year.

Abstract: In the short-run, bond risk premia exhibit pronounced spikes around major economic and financial crises. In contrast, long-term bond risk premia feature cyclical swings. We empirically examine the predictability of the market variance risk premium – a proxy of economic uncertainty – for bond risk premia and we show the strong predictive power for the one month horizon that almost entirely disappears for horizons above one year. The variance risk premium is largely orthogonal to well-established bond return predictors – forward rates, jumps, yield curve factors, and macro variables. We rationalize our empirical findings in an equilibrium model of uncertainty about consumption and inflation which is coupled with recursive preferences. We show that the model can quantitatively explain the levels of bond and variance risk premia as well as the predictive power of the variance risk premium while jointly matching salient features of other asset prices.

Figures

Figure 4. Model-Implied and Estimated Slopes and R2s for Two to Six Month T-bills The figure shows the calibrated model-implied slope coefficients and R2s (thick lines) for regressing the two to six months Treasury bill one month excess returns on the variance risk premium, along with their estimated empirical counterparts (thin lines with circles) and 95 percent confidence bands of the slope coefficients (thin dashed lines with circles).

Figure 3. Ex Post versus Ex Ante Bond Risk Premia in Short and Long Horizons

Figure 1. Market Variance Risk Premium

Figure 2. Estimated Slope Coefficients from Univariate Regressions

Table 4 Multivariate excess bond portfolio returns - one month

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Short-Run Bond Risk Premia
∗
Philippe Mueller
Warwi ck Business School
†
Andrea Vedolin
Boston University
‡
Hao Zhou
Tsinghua University
§
Abstract
In the short-run, bon d risk premia exhibit pronoun ced spikes around major economic and financial
crises. In contrast, long-term bond risk premia feature cyclical swings. We empirically examine the
predictability of the market variance risk premium—a proxy of economic uncertainty—for bond risk
premia and we show the strong p redictive power for the one month horizon that quickly recedes for
longer horizons. The variance risk premium is largely orthogonal to well-established bond return
predictors—forward rates, jumps, and macro variables. We rationalize our empirical findings in an
equilibrium model of u ncertainty about consumption and inflation which is coupled with recursive
preferences. We show that the model can quantitatively explain the levels of bond and variance
risk pr emia as well as the predictive power of the variance risk premium while jointly matching
salient f eatures of other asset prices.
This Version: February 2019.
JEL Classification Codes: G12, G13, G17, E43, E44.
Keywords: Variance risk premium , bond risk premia, expectations hypothesis, inflation dynamics,
economic u ncertainty.
∗
We would like to thank Tim Bollerslev, Mike Chernov, Frank Diebold, Urban Jermann, George Tauchen,
and seminar participants at Lund, Duke, LSE, and HKUST for their helpful comments. Muelle r and Vedolin
gratefully acknowledge the financial support from STICERD and the Systemic Risk Centre at LSE, and the
British Academy.
†
Finance Group, Gibbet Hill Road, Coventry CV4 7AL, United Kingdom, Phone: +44 24 761 50505,
Email:
philippe.mueller@wbs.ac.uk.
‡
Questrom Scho ol of Business, 595 Commonwe alth Avenue, Bosto n, MA 02215, Phone: +1 617 35 3 4168,
Email:
avedolin@bu.edu.
§
PBC School of Finance, 43 Chengfu Road, Haidian Distr ict, Beijing, 100083, P. R. China, Phone: +86
10 62790655, Email:
zhouh@pbcsf.tsinghua.edu.cn.

Short-Run Bond Risk Premia
Abstract
In the short-run, bond risk premia exhibit pro nounced spikes around major economic and
financial crises. In contrast, long-term bond risk premia feature cyclical swings. We empiri-
cally examine the predictability o f the market variance risk premium—a proxy of economic
uncertainty—for bond risk premia and we show the strong predictive power for the one
month horizon that quickly recedes for longer horizons. The variance risk premium is largely
orthogonal to well-established bond return predictors—forward rates, jumps, and macro vari-
ables. We rationalize our empirical findings in an equilibrium model of uncertainty about
consumption and inflatio n which is coupled with recursive preferences. We show t hat the
model can quantitatively explain the levels of b ond and variance risk premia as well as the
predictive power of the variance risk premium while joint ly matching salient f eat ur es of other
asset prices.
JEL Classi fication Codes: G12 , G13, G17, E43, E44.
Keywords: Variance risk premium, bond risk premia, expectations hypothesis, inflation dy-
namics, economic uncertainty.

1 Introduct ion
The failur e of the expectations hypothesis of the term structure of interest rates, first doc-
umented in Fama and Bliss (1987) and Campbell and Shiller (1991), has received unprece-
dented attention in bot h the empirical and theoretical academic literature over the past 20
years. In this paper, we first document the large and significant predictive power of the
variance risk premium, defined as the difference between the risk-neutral and statistical ex-
pectations of realized variance, for bond risk premia at very short horizons. This short-run
forecastability is orthogonal to the well documented long horizon predictability from forward
rates (Cochrane and Piazzesi, 2005), macro variables ( Ludvigson and Ng, 2009), and jump
risk (Wright and Zhou, 2009).
1
We then posit an economy with time-varying uncertainty
risk about real and nominal quantities coupled with agents’ preferences for an early reso-
lution of uncertainty and show that these ingredients are enough to quantitatively explain
the violation of the expectation hypothesis while matching the moments of the variance risk
premium, t he equity premium, and risk-free rate.
To capture this short-r un uncertainty component of bond r isk premia, we rely on the
market variance r isk premium—or the difference between risk-neutral and objective expec-
tations of the return variation. Following the path of previous work, we proxy the risk
neutral expected variance by the popula r VIX
2
index, which is termed as “market gauge of
fear” (Whaley, 2000). With high frequency intraday data of futures o n the S&P 500, we use
heterogeneous auto r egr essive models of realized variance (HAR-RV model, see Corsi, 2009)
augmented by lags of implied variances (Dr echsler and Yaron, 2 011) for estimating the ob-
jective expectation of variance risk. Our average variance risk premium is 21.5 7 (percenta ge
squared monthly basis) and falls within the typical range of recent empirical estimates. More
importantly, our t ime-series of variance risk premium always remains positive, which makes
it a natural candidat e measure for economic uncertainty or even stochastic risk aversion.
We document t he predictive power of the bond variance risk premium for short-run bond
risk premia using various data. We show that the variance risk premium is a significant
1
Cieslak and Povala (2010) de comp ose long-term yields into a pe rsistent component and cycles and find
that the cyclical component is a strong predictor encompassing several o ther ones. Duffee (2011) estimates
a five factor Gaussian model using Treasury y ie lds and extracts a latent factor, that is “hidden” from—or
weakly spanned by—the cross-section of yields but has bearing on excess bond r e turns. Huang and Shi
(2010) construct a single macro factor using a group lasso method and show that this factor almost doubles
the R
2
compared to Ludvigson and Ng (2009).
1

predictor for one month excess returns on bond portfolios with underlying maturities ranging
between zero and ten years obtained from CRSP. The same results hold when calculating
one month excess returns on bonds with maturities ranging from two months to ten years
calculated using the G¨urkaynak, Sack, and Wright ( 2007) dataset. The same results hold
when using the variance risk premium to fo r ecast one month Treasury bill excess returns.
However, the variance risk premium only has negligible forecasting power for longer horizon
excess returns and in particular, it has zero predictive power f or one year excess returns
on two to five year Treasury bonds, which are in general used to run bond pr edictability
regressions. We show that the short-run forecasting power is robust to the inclusion of other
well established bond risk premium predictors such as forward rates, macro variables, and
jump risk. While these variables have previously been shown to predict bond risk premia
for longer maturities, they do not subsume the significance of the variance risk premium at
shorter horizons and in some cases even have zero predictive power.
The intuition fo r our empirical result becomes more evident when we look at the time
series of short term bond risk premia. Bond risk premia at short horizons exhibit pronounced
spikes around major economic and financial crises. This pattern is distinctly different f r om
the cyclical swings with a length of up to several years typically observed in long term bond
risk premia (see Fama and Bliss, 1987 and Cochrane and Piazzesi, 2005). Interestingly,
the variance risk premium exhibits a similar time-series behavior as short-term bonds: It
rises sharply before economic or financial crises and then drops again. On the o ther hand,
standard predictors like the CP factor display a strong cyclical behavior (see Koijen, Lustig,
Van Nieuwerburgh, 2010). The upshot is that short-term variation in bo nd risk premia
are related to economic uncertainty which are short- lived (see Bloom, 200 9) rather than a
business cycle component which is more apparent in bond risk pr emia of longer maturities.
We propose a potential explanation for this short-run predictability in an economy with
time-varying economic uncertainty about real and nominal quantities, extending the real
uncertainty model of Bollerslev, Tauchen, and Zhou (2009).
2
In an economy with stochas-
tic inflation volatility but with only exogenous shocks, money neutrality holds and there
is no inflation risk premium except for the standard Jensen’s inequality term (see Zhou,
2
Wu (2008), Hasseltoft (2010), and Doh (2010) study the long-run risk models for term structure with
both real and nominal uncertainty. However, they also rely on the small persistent gr owth component and
they do not exa mine predictability of the variance risk premium.
2

Frequently asked questions

Q1. What have the authors contributed in "Short-run bond risk premia" ?

The authors empirically examine the predictability of the market variance risk premium—a proxy of economic uncertainty—for bond risk premia and they show the strong predictive power for the one month horizon that quickly recedes for longer horizons. The authors show that the model can quantitatively explain the levels of bond and variance risk premia as well as the predictive power of the variance risk premium while jointly matching salient features of other asset prices. 

Q2. What is the key to a reasonable and rich bond risk premia?

Combining both a consumption growth channel and a uncertainty channel of non-neutral inflation dynamics, leads to reasonable and rich bond risk premia. 

Q3. Why is the VXO used as a variance proxy?

In order to limit the loss of available data and because VIX is only available starting in January 1990, the authors use the squared VXO as their implied variance proxy on the RHS of the regression. 

Q4. What is the upshot of the short-term variation in bond risk premia?

The upshot is that short-term variation in bond risk premia are related to economic uncertainty which are short-lived (see Bloom, 2009) rather than a business cycle component which is more apparent in bond risk premia of longer maturities. 

Q5. What is the powerful predictor for long-term bond returns?

Cochrane and Piazzesi (2005) find that a linear combination of forward rates is the most powerful predictor for long-term bond returns. 

Q6. How do the authors calculate the excess returns on Treasury bonds?

To have a consistent source of yields for calculating monthly and yearly excess returns, the authors also use the Gürkaynak, Sack, and Wright (2007, GSW dataset) dataset, which allows constructing one month excess returns for longer maturity bonds. 

Q7. What is the reason for the idiosyncratic behavior in the shortest maturity?

There is significant idiosyncratic behavior in the shortest maturity yields, which may be partly due to increased market segmentation. 

Q8. What is the variance risk premium for the Treasury bond portfolios?

For the bond portfolios, the variance risk premium has marginal predictive power for portfolios with short underlying maturities, while no predictability exists for longer maturity portfolios. 

Q9. What are the results of limiting the setup to only three risk factors?

The overshooting of the risk-free rate volatility and underfitting of long term bond risk premia are closely related outcomes of limiting the setup to only three risk factors. 

Q10. What are the main characteristics of the short-run forecastability of the bond risk premia?

This short-run forecastability is orthogonal to the well documented long horizon predictability from forward rates (Cochrane and Piazzesi, 2005), macro variables (Ludvigson and Ng, 2009), and jump risk (Wright and Zhou, 2009).1 

Q11. How does the real economy model match the observed inflation risk premium?

Their real economy model can match the observed (pre-crisis) variance risk premium reasonably well with a mean of 10.84 and a standard deviation of 10.34. 

Q12. What is the prediction power of the variance risk premium for bond risk premia?

16Their calibration result suggests that the proposed inflation uncertainty model not only has the capability to replicate the predictability pattern of the variance risk premium for bond risk premia documented in recent research, but also matches the level of bond risk premia typically hard to pin down in structural economic models. 

Q13. How can the modelimplied bond risk premia be lowered?

It is interesting to note that when money neutrality is violated as in Model III, the modelimplied bond risk premia can be dramatically lowered to around 86-113 bps, compared to the exogenous inflation Model II (around 185-385 bps). 

Q14. How do the authors calculate the excess returns on T-bills?

the authors use the Fama T-bill structures from CRSP to compute short horizon excess returns on T-bills ranging between two and six months. 

Q15. What is the variance risk premium for Treasury bill regressions?

The variance risk premium contains relevant information for short-run bond risk premia for bonds of any maturity while it has no predictive power for longer horizon excess returns. 

Q16. What is the conditional variance of the time t to t+1 return?

The conditional variance of the time t to t+1 return, σ2r,t ≡ Vart(rt+1), is given by: σ2r,t = σ2g,t + κ 2 1 ( A2σ + A 2 qϕ 2 q ) qt.