Equity Volatility as a Determinant of Future Term-structure Volatility

7/25/2019 Equity Volatility as a Determinant of Future Term-structure Volatility

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Equity volatility as a determinant of future

term-structure volatility$

Naresh Bansal a,n, Robert A. Connolly b,1, Chris Stivers c,2

aJohn Cook School of Business, Saint Louis University, St. Louis, MO, United Statesb UNC Kenan-Flagler Business School, University of North Carolina Chapel Hill, Chapel Hill, NC, United Statesc College of Business, University of Louisville, Louisville, KY, United States

a r t i c l e i n f o

Article history:

Received 28 September 2013

Received in revised form

8 May 2015

Accepted 13 May 2015

Available online 22 May 2015

JEL classication:

G12

G14

Keywords:

Equity risk

Term structure

Bond volatility

a b s t r a c t

We show that equity volatility serves as a determinant of future

Treasury term-structure volatility over the recent October 1997 to

June 2013 period. We nd that equity volatility contains incre-

mentally reliable information for the subsequent volatility of: (1)

10-year and 30-year bond futures returns, (2) the term-structure's

level, and (3) the term-structure's slope. We present additional

evidence that suggests a ight-to-quality/ight-from-quality pri-

cing avenue is a likely contributor to the volatility linkages, where

time-varying economic uncertainty can generate both a large

positive serial correlation in stock volatility and a time-variation in

the precautionary savings motive and diversication benets of

holding bonds.

& 2015 Elsevier B.V. All rights reserved.

1. Introduction

Understanding term-structure volatility is a fundamental issue in nancial economics with both

theoretical and practical importance. In this paper, we show that realized equity volatility can serve as

an important determinant of future term-structure volatility. By term-structure volatility, we refer to

Contents lists available at ScienceDirect

journal homepage: www.elsevier.com/locate/finmar

Journal of Financial Markets

http://dx.doi.org/10.1016/j.nmar.2015.05.002

1386-4181/& 2015 Elsevier B.V. All rights reserved.

We thank Andrew Lim and seminar participants at the 2013 Financial Management Association meetings, the 2013

Midwest Finance Association meetings, and the 2013 Missouri Economics Conference for helpful comments. We also gratefully

acknowledge an anonymous referee who provided thoughtful comments that improved the paper.n Corresponding author. Tel.: 1 314 977 7204.

E-mail addresses: [email protected](N. Bansal),[email protected](R.A. Connolly),

[email protected](C. Stivers).1

Tel.: 1 919 962 0053.2 Tel.: 1 502 852 4829.

Journal of Financial Markets 25 (2015) 3351
http://www.elsevier.com/locate/finmarhttp://www.elsevier.com/locate/finmarhttp://dx.doi.org/10.1016/j.finmar.2015.05.002http://dx.doi.org/10.1016/j.finmar.2015.05.002http://dx.doi.org/10.1016/j.finmar.2015.05.002mailto:[email protected]:[email protected]:[email protected]://dx.doi.org/10.1016/j.finmar.2015.05.002http://dx.doi.org/10.1016/j.finmar.2015.05.002http://dx.doi.org/10.1016/j.finmar.2015.05.002http://dx.doi.org/10.1016/j.finmar.2015.05.002mailto:[email protected]:[email protected]:[email protected]://crossmark.crossref.org/dialog/?doi=10.1016/j.finmar.2015.05.002&domain=pdfhttp://dx.doi.org/10.1016/j.finmar.2015.05.002http://dx.doi.org/10.1016/j.finmar.2015.05.002http://dx.doi.org/10.1016/j.finmar.2015.05.002http://www.elsevier.com/locate/finmarhttp://www.elsevier.com/locate/finmar


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the volatility of Treasury bond futures returns, and the volatility of both the level and slope of the

Treasury term-structure. By determinant, we refer to an intertemporal relation between the lagged

realized equity volatility and the subsequent bond-market volatility that holds in a multivariate

framework, when also controlling for the past term-structure volatility and other term-structure state

variables.

Researchers have offered both theory and empirical evidence that suggest important linkages

between equity risk and the Treasury bond market. For example,Bekaert, Engstrom, and Xing (2009)

nd that higher economic uncertainty can lead to both higher equity volatility and an increased

motive for precautionary savings that can depress interest rates. Fleming, Kirby, and Ostdiek (1998)

and Kodres and Pritsker (2002) suggest cross-asset-class effects tied to hedging and portfolio

rebalancing. Pricing effects linking the stock and bond markets have been attributed to ight-to-

quality/ight-from-quality (FTQ/FFQ), where some investors (presumably) switch between riskier

stocks and safer Treasuries as risk perceptions change (Connolly, Stivers, and Sun, 2005, 2007;

Underwood, 2009; Baele, Bekaert, and Inghelbrecht, 2010; BenRaphael, Kandel, and Wohl, 2012;

Jubinski and Lipton, 2012;Bansal, Connolly, and Stivers, 2014).3 Chordia, Sarkar, and Subrahmanyam

(2005) nd that innovations to stock volatility forecast an increase in bond bidask spreads.

Why might the realized equity volatility contain important incremental information for the

subsequentbond volatility? First, consider a FTQ/FFQ avenue, as motivated by the literature cited

above. With linkages between the economic state and stock volatility, a higher stock volatility this

month is likely to be associated both with more extreme stock price movements over the next month

(volatility clustering), and with higher economic uncertainty and volatility in that uncertainty (stock

volatility tending to be higher in stressful economic times with greater economic-state uncertainty). If

a higher stock-return volatility and a higher time series variability in economic uncertainty are likely

following months with a high realized stock volatility, then the likelihood of FTQ/FFQ pricing

inuences over the subsequent month is presumably much greater.4 Second, the return volatility of

both equities and bonds may be responding to some omitted factor or news that bears on the

volatility of each asset class, in the sense ofFama and French (1993). If there is volatility clustering in

that common factor, then equity volatility may be providing an additional signal about the underlyingvolatility environment for subsequent bond returns.5

Our empirical investigation is also motivated byAndersen and Benzoni's (2010) ndings. Under

standard afne term structure models, they note that the instantaneous yield volatility should be

spanned by the cross-section of yields. They nd evidence inconsistent with this prediction and

conclude that a broad class of afne diffusive, quadratic Gaussian, and afne jump-diffusive models

cannot accommodate the observed yield volatility dynamics, (p. 603). Their ndings suggest that

factors outside the bond market are likely to be important for understanding yield volatility. In this

paper, we examine the role of equity volatility as one potential factor.

We focus on the October 1997 to June 2013 period since the literature indicates a clear change in

the joint distribution of stock and bond returns around October 1997. Fig. 1 in Baele, Bekaert, and

Inghelbrecht (2010, p. 2376)depicts the shift in the stock-bond correlation from sizably positive topredominantly negative in the latter part of 1997.Bansal, Connolly, and Stivers (2014)argue that the

equity-risk dynamics and ight-to-quality pricing inuences may be particularly important for

understanding bond market dynamics over the post-1997 period since this period features a

predominantly negative stock-bond-return correlation, a low ination-risk environment, and several

episodes of high and volatile equity risk. We also briey examine an earlier period in the mid-1990's

3 Some authors use the phrase ight-to-safety, rather than ight-to-quality. For the purposes of our study, we consider

these terms as interchangeable.4 We focus on volatility measures over the monthly horizon, but also evaluate the quarterly horizon.5

SeeFleming, Kirby, and Ostdiek (1998),for an alternate discussion on the intuition behind these two avenues for a stock-bond volatility linkage. In their model, two distinct sources of linkages arise. One is common information, such as news about

ination, which simultaneously affects investor expectations in multiple markets. The second source is due to cross-market

hedging. When information alters expectations in one market, traders adjust their holdings across markets, producing an

information spillover,(p. 135). In our view, their cross-market hedging and our FTQ/FFQ capture a similar perspective and their

common information is similar to our omitted common factor perspective.

N. Bansal et al. / Journal of Financial Markets 25 (2015) 335134


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to evaluate an alternate period with low equity risk and a sizably positive stock-bond return

correlation.6

We analyze the volatility of returns on long-term (30-year) and medium-term (10-year) Treasury

futures contracts, the volatility in the change in term structure's level, and the volatility in the

change in term structure's slope.The volatility of long-term and medium-term bond returns may be

attributed to changes in the level of yields or the slope of the yield curve. Because principal

component representations of the yield curve are orthogonal by construction, we can identify the

separate effects of equity volatility on the principal components. Following standard practice in the

term structure literature, we measure the change in the term structure'slevelas the change in the rst

principal component (PC1) of the term structure, and the change in the term structure's slope as the

change in the second principal component (PC2) of the term structure.7 Our term-structure volatility

measures are based on either daily returns (for the 30-year and 10-year futures contracts) or daily

changes (for the principal components) over rolling one-month and one-quarter periods. For the

lagged equity volatility, we use the lagged realized stock volatility over a one-month period, as

calculated from either past daily S&P 500 futures returns or past 5-minute returns on the SPY (S&P

500) Exchange Traded Fund.

To summarize our primary empirical results, we nd that lagged equity volatility is a substantial,

reliable determinant of the subsequent T-bond and T-note futures return volatility and of the

subsequent volatility of the level and slope of the term-structure. The information content of equity

volatility is incremental in nature, in the sense that we control for the volatility information contained

in the lagged realized term-structure volatility and other term-structure state variables (e.g.,Andersen

and Benzoni, 2010;Cochrane and Piazzesi, 2005). The intertemporal aspect of our ndings supports

the notion that equity risk can help us to understand movements in the term-structure, beyond an

approach that only looks at the bond market in isolation.

We also provide additional evidence to probe the underlying mechanisms behind the

intertemporal stock-to-bond volatility (ISBV) relation. We nd evidence consistent with FTQ/FFQ

dynamics being a key contributor to the ISBV relation. For example, we nd that the ISBV relation is

linked to the economic state, with a much stronger ISBV relation in stressful uncertain economic times(such as around recessions). Further, we nd that the partial ISBV relation remains strong when

controlling for the lagged volatility of economic variables such as ination and the default yield

spread, variables that seem likely to be more linked to bond volatility but might also be embedded in

equity volatility.

The paper proceeds as follows. Section 2 describes our data and sample selection. Section 3

presents our main empirical results. Sections 4 and 5 present additional evidence that bears on

understanding the underlying mechanisms behind our ISBVndings.Section 6discusses our ndings

in relation to earlier empirical studies on stock-bond volatility linkages, and Section 7concludes.

2. Data description and sample selection

2.1. Data description

We investigate the relation between lagged equity volatility and three dimensions of the realized

term-structure volatility over the next month. First, we investigate the volatility of Treasury bond

returns, measured by the daily returns on Treasury futures contracts for two different maturities, the

10-year T-note futures contract (medium-term bond) and the 30-year T-bond futures contract (long-

term bond). By futures returns,we refer to the daily price change (close-to-close, based on the daily

6

Fleming, Kirby, and Ostdiek (1998)and Chordia, Sarkar, and Subrahmanyam (2005) consider stock and bond volatilitylinkages, but with data only through 1995 and 1998, respectively, and with substantially different empirical approaches. In

Section 6, we discuss our ndings in relation to these earlier studies.7 Researchers have shown that the term-structure's rst three principal components are closely related to its level, slope,

and curvature, respectively, and capture almost all of the variation in the yields. Diebold, Piazzesi, and Rudebusch (2005) nd

that the rst two principal components alone account for almost all (99%) of the variation in the yields.

N. Bansal et al. / Journal of Financial Markets 25 (2015) 3351 35


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mark-to-market contract price) divided by the preceding day's closing price, as obtained from

DataStream's continuous futures series. Second and third, we investigate the volatility of the term-

structure's level and the slope, based on the daily changes in the term-structure's rst principal

component (PC1) and second principal component (PC2), respectively. In our principal component

analysis, we compute the rst three principal components for every trading day from the 10 zero-

coupon-bond yields from year-one to year-ten, using the instantaneous continuously-compounded

forward-rate yields as described in Gurkaynak, Sack, and Wright (GSW) (2007).

Following fromAndersen and Benzoni (2010), we also use the lagged values of the three principal

components (from day t1) as term-structure state variables when modeling the subsequent term-

structure volatility over trading days t to tj. In robustness testing, we also use the lagged GSW

instantaneous continuously-compounded forward-rate yields at 1 year, 2 years, 3 years, 5 years, 7 years,

and 10 years out as alternate term-structure state variables, following from Cochrane and Piazzesi (2005).

For the stock volatility, we examine two alternate measures. First, to treat the stock volatility and

term-structure volatility symmetrically, we calculate the realized stock volatility from daily returns

from S&P 500 futures contracts. Second, as an alternate measure of realized monthly stock volatility,

we use the standard deviation of 5-minute returns of the SPY (S&P 500) Exchange Traded Fund (ETF).

By calculating the realized volatility from 5-minute returns, our volatility measure captures more

information so we should end up with a higher quality volatility estimate over the respective time.8

Note, in both our measures of lagged stock volatility, the lagged realized stock volatility is calculated

from thepaststock returns over trading dayst1 tot22, relative to the subsequent term-structure

volatility that is measured over trading days tto t21. In an Online Appendix, we provide details on

constructing the realized volatilityfrom the 5-minute SPY returns.

Aspects of our empirical investigation also use: (1) the Chicago Board Option Exchange's Volatility

Index (VIX), dened as the implied volatility from S&P 500 equity-index options standardized for a

one-month expiration; and (2) ination compensationdata, based on yield differences between 10-

year TIPS and 10-year nominal Treasuries per GSW (2010).

2.2. Sample selection

In our introduction, we explained why our primary sample period is over October 1997 to June 2013,

with the start date coinciding with the shift from a sizably positive to a predominantly negative stock-bond

correlation. Further, in contrast to the relatively high ination risk of the 1970s and early 1980s,Campbell,

Sunderam, and Viceira (2013)andDavid and Veronesi (2013)present evidence that, over the 19972013

period, bond investors faced relatively lower ination risk and Treasury bonds likely became more of a

hedge instrument. We note that the 19972013 period also largely postdates the earlier work on stock-

bond volatility linkage in Fleming, Kirby, and Ostdiek (1998) andChordia, Sarkar, and Subrahmanyam

(2005), whose samples end in 1995 and 1998, respectively. To expand our analysis and assist in

interpretation, we also estimate our volatility models over the 1993-1996 period to provide a lower-stress

stock-market period to contrast with the higher-stress periods from our primary sample period.

2.3. Summary statistics

Table 1 reports the mean, median, standard deviation, skewness, and excess kurtosis for the four

different term-structure volatility measures (rows 14), for the volatility of daily stock futures returns

(row 5), and for the stock volatility from 5-minute SPY returns (row 6). The four term-structure volatility

measures and the daily-stock-return volatility are monthly measures, as computed from daily

observations over the rolling 22-trading-day period. The 5-minute-stock-return volatility is also computed

in similar fashion over the same 22-trading-day periods, but with returns at ve-minute intervals.

For the different volatility measures inTable 1, we present the statistics both for the raw variable

(row a) and for the logarithmic transformation of the raw variable (row b). In our empirical regression

8 Andersen and Bollerslev (1998) and Andersen, Bollerslev, Diebold, and Ebens (2001) argue that such a high-frequency

volatility estimate should be less noisy than a comparable estimate from daily returns.



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models, we use the log transformation to transform the raw volatility variable into a series that is

closer to normally distributed.

Fig. 1exhibits the time series of the volatility of the 30-year T-bond-futures return (Panel A) and

the volatility of the S&P 500 futures return (Panel B). The graphs show a sizable relation between the

equity volatility and the bond volatility series, which is the subject of our empirical investigation.

3. Main empirical results for the ISBV relation

To investigate how equity volatility is related to the future term-structure volatility, we regress the

realized monthly volatility of our term-structure variables on our measures of lagged equity volatility,

while controlling for other relevant variables from the literature. We estimate variations of the

following regression for each of our four term-structure volatility measures:

TmStt;t21 01TmStt1;t22 2

STt1;t22

X3

j 1

jPrCompj;t1t; 1

where the dependent variable, TmStt;t21, is the logarithmic transformation of one of the four term-

structure volatility measures over trading days tto t21, calculated as the log of the square-root of the

sum of 22 squared daily values (daily returns for the T-bond and T-note futures, and daily changes for

the principal components) over the rolling 22-trading-day period. The explanatory variables are: (1)

TmStt22;t1, the rst lag of the dependent variable (to address volatility clustering); (2)STt1;t22, the log of

volatility for the S&P 500 futures returns over trading days t22 to t1; and (3) PrCompj;t1 are the

three principal components at the end of day t1. We use the three principal component at time t1

as term-structure state variables, since the principal components are well known to represent the level,

slope, and curvature in the term structure.9 The s and s are coefcients to be estimated. We also

report results for an alternate estimation where the 2stock-volatility term is based on high-frequency

Table 1

Summary data statistics.

This table reports the summary statistics for the key volatility variables. We report the mean, median, standard deviation,

skewness, and excess kurtosis for each volatility measure. The annualized volatility measures for the variables in rows 1 to 5 are

computed from the square-root of the sum of 22 squared daily returns for the futures or daily changes for the principal

components over the rolling 22-trading-day period, from t to t21. Rows 1 and 2 report the 30-year T-bond futures returnvolatility and 10-year T-note future return volatility, respectively. Rows 3 and 4 report the volatility of the rst and second

principal component, respectively. Row 5 reports the volatility of the S&P 500 futures returns. Finally, row 6 reports the

volatility calculated from 5-minute returns of the SPY ETF over the same 22-trading-day period. In all rows, the rst row

(labeled a) presents the statistics for the raw variable, and the second row (labeled b) presents the statistics for logarithmic

transformation of the variable. The sample period is from October 1997 to June 2013.

Row Variable Mean Median Std. Dev. Skewness Kurtosis

1a. TB 9.56 8.96 3.26 1.01 0.87

1b. ln(TB) 2.20 2.19 0.33 0.17 0.24

2a. TN 5.91 5.47 2.24 1.30 2.77

2b. ln(TN) 1.71 1.70 0.36 0.12 0.02

3a. PC1 1.80 1.69 0.66 1.07 1.91

3b. ln(PC1

) 0.52 0.53 0.36 0.03 0.114a. PC2 0.58 0.53 0.26 1.29 1.68

4b. ln(PC2) 0.64 0.64 0.42 0.13 0.08

5a. StFt 18.32 15.87 10.85 2.79 12.36

5b. ln(StFt) 2.78 2.76 0.48 0.46 0.38

6a. SPY5 minute 19.22 16.93 10.18 2.05 6.68

6b. lnSPY5 minute 2.84 2.83 0.46 0.45 0.13

9 Our principal componentsusage here follows fromAndersen and Benzoni (2010). In robustness checks, we replace the

lagged principal components with multiple lagged forward rates [followingCochrane and Piazzesi, 2005], and nd that the

coefcients of interest remain qualitatively similar.



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(5-minute) SPY returns, to evaluate robustness of the ISBV relation and to investigate whether the ISBV

relation is stronger with a presumably higher-quality stock volatility measure. Test statistics are

calculated based on heteroscedastic- and autocorrelation-consistent standard errors, with the number

of lags for the autocorrelation structure set to 22 since we use 22-trading-day overlapping variables.

We focus on rolling monthly periods since the monthly horizon is common in nance studies and

this approach should mitigate some of the noise that would be present in a daily-volatility analysis.

We evaluate rolling periods because this approach is likely to better use the available data to capture

the return dynamics (Richardson and Smith, 1991).

The next four subsections report estimation results for each of our four term-structure volatility

measures, respectively. In this analysis, we report results for separate estimations over our complete

primary sample period (1997:102013:06), an approximate rst-half subperiod (1997:102005.06),

and an approximate second-half subperiod (2005.072013:06).

3.1. Long-term T-bond return volatility

Table 2reports on the volatility of 30-year T-bond-futures return as the term-structure volatility

measure (i.e.,TmSti;j TBi;j) in Eq.(1). Our primary coefcient of interest here is 2, the coefcient on the

lagged stock return volatility. We report the results from estimating three variations of Eq. (1).

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

Oct-1997

Oct-1998

Oct-1999

Oct-2000

Oct-2001

Oct-2002

Oct-2003

Oct-2004

Oct-2005

Oct-2006

Oct-2007

Oct-2008

Oct-2009

Oct-2010

Oct-2011

Oct-2012

Ln(Volatili

ty)

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

4.5

5.0

Oct-19

97

Oct-19

98

Oct-19

99

Oct-20

00

Oct-20

01

Oct-20

02

Oct-20

03

Oct-20

04

Oct-20

05

Oct-20

06

Oct-20

07

Oct-20

08

Oct-20

09

Oct-20

10

Oct-20

11

Oct-20

12

Ln(Volatility)

Panel B: Ln(Volatility of Stock Futures Returns)

Panel A: Ln(Volatility of 30-year T-bond Futures Returns)

Fig. 1. Time series of rolling volatility measures. This gure displays the time series of the rolling monthly volatilities of the

long-term T-bond returns and the stock-market returns, where the rolling volatilities are constructed from the 22 daily

observations within the 22-trading-day period. Panel A reports the log of the volatility of 30-year T-bond futures returns, and

Panel B reports the log of the volatility of S&P 500 futures returns. The units are the natural log of the annualized sample

standard deviation, with the returns in percentage terms. For the x-axis, the value for day tis for the volatility over trading days

tto t21. The sample period is October 1997 to June 2013.



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The rst model has equity-risk as the only explanatory term. In the second model, we add the lagged

value of the dependent variable as an additional explanatory term. The third model is the full

specication given by Eq.(1).

In Panel A ofTable 2with the daily-return-based stock volatility, we nd that the estimates of2in

all three models are positive and statistically signicant over all three estimation periods. We also nd

that both the rst lag of the historical volatility (the 1 term) and the three lagged principal

components are reliably related to the subsequent volatility.

In Panel B ofTable 2, where the lagged 5-minute-return-based stock volatility replaces the daily-

return-based stock volatility, we nd that the ISBV relation remains reliably evident. For Panel B, the

statistical signicance of the estimated2and theR2 value is higher than the comparable model (c) in

Panel A for all three periods. This supports both the robustness of our ISBVndings and the notion

that the ISBV relation is more reliably evident when using a higher quality stock-volatility measure.

3.2. Medium-term T-bond return volatility

Next, inTable 3, we repeat a comparable analysis for the volatility for the 10-year T-note-futures

returns (i.e.,TmSti;j TNi;j ). For all three estimation periods, we againnd that the2estimates in all theregressions are positive and statistically signicant. In Panel B, we again nd that the statistical

signicance of the estimated 2and theR2 value is higher than the comparable model (c) in Panel A for

all three periods. For the other explanatory terms, the results are comparable to those for the longer-

term bond.

Table 2

Volatility in 30-year T-bond futures returns.

This table reports results for how the equity risk is related to the subsequent 30-year T-bond return volatility. Panel A reports

three variations of the following regression, denoted as models (a) to (c) in the table:

TBt;t21 0 1TBt1;t22 2STt1;t22 X3

j 1

jPrCompj;t1 t;

where TBi;j STi;j is the logarithmic transformation of volatility for the 30-year T-bond (S&P 500) futures contract over trading

daysi to j, calculated as the log of the square-root of the sum of 22 squared daily futures returns over the rolling 22-trading-day

period;PrCompj;t1are the three principal components from day t1; and thes ands are coefcients to be estimated. Panel B

reports on a similar model, except the log of the standard deviation of realized 5-minute S&P 500 ETF returns overt1 to t22

replaces the realized stock volatility from daily returns for the 2 term. We report on the October 1997 to June 2013 sample

period, along with two subperiods (1997:102005:06 and 2005:072013:06). T-statistics are in parentheses, calculated with

heteroscedastic and autocorrelation consistent standard errors. An F-test (3) test statistic is in brackets, which jointly tests the

coefcients on the three lagged principal components. nnn, nn, and n indicate 1%, 5%, and 10% p-values.

Period Model 1 2 [Pr Comp] R2

Panel A: Realized stock volatility from daily returns over t1 to t22 as the 2 TermFull Period a. 0.347 (9.96)nnn 26.4%

1997:102013:06 b. 0.594 (11.82)nnn 0.132 (3.93)nnn 51.5%

c. 0.318 (5.51)nnn 0.189 (5.92)nnn [17.62]nnn 58.8%

First Subperiod a. 0.155 (2.55)nn 5.1%

1997:102005.06 b. 0.485 (6.52)nnn 0.118 (2.22)nn 28.3%

c. 0.201 (2.77)nnn 0.199 (3.90)nnn [17.07]nnn 43.1%

Second Subperiod a. 0.466 (12.44)nnn 46.6%

2005.072013.06 b. 0.661 (8.60)nnn 0.118 (2.25)nn 64.3%

c. 0.456 (5.39)nnn 0.099 (1.92)n [9.20]nn 69.1%

Panel B: Realized stock volatility from 5-minute returns over t1 to t22 as the 2 Term

Full Period 0.286 (4.89)nnn 0.230 (7.01)nnn [19.92]nnn 60.3%

1997:102013:06

First Subperiod 0.173 (2.35)nn

0.233 (4.33)nnn

[18.58]nnn

44.4%1997:102005.06

Second Subperiod 0.397 (4.88)nnn 0.174 (3.10)nnn [7.62]nnn 70.1%

2005.072013.06



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The evidence inTables 2and3indicates that the equity volatility contains substantial and reliable

forward-looking information about the subsequent bond return volatility. The information is

incremental, in that the partial ISBV relation remains strong when controlling for the information

contained in the lagged bond volatility and the three lagged principal components.

3.3. Volatility of the term-structure's level

We also investigate the term-structure's level as proxied by the rst principal component of the

term structure. Note the term-structure's rst principal component is by construction orthogonal to

the second and third principal component (which are representative of the slope and curvature,

respectively).

Table 4reports on an estimation of Eq. (1) with the realized volatility of the change in the term-

structure's rst principal component as the term-structure volatility measure (i.e., TmSti;j PC1i;j ). In

Panel A, we nd that the estimates of 2 are positive in all cases, and are strongly statistically

signicant for all cases except regressions (b) and (c) for the second-half subperiod. The positive 2indicates that the volatility of the rst principal component tends to be larger following higher lagged

equity volatility. We note that the coefcient on the equity risk term is largely similar for both models(b) and (c) (with or without the three lagged principal components), which indicates that the

information from the equity risk term is largely distinct from the information in the lagged cross-

section of yields. In Panel B ofTable 4, we nd that the 2estimates are larger in both magnitude and

statistical signicance for all three estimation periods, as compared to regression (c) in Panel A.

Table 3

Volatility in 10-year T-note futures returns.

This table reports results for how the equity risk is related to the subsequent 10-year T-note return volatility. Panel A reports

three variations of the following regression, denoted as models (a) to (c) in the table:

TNt;t21 0 1TNt1;t22 2STt1;t22 X3

j 1

jPrCompj;t1 t;

where TNi;j STi;j is the logarithmic transformation of volatility for the 10-year T-note (S&P 500) futures contract over trading

daysi toj, calculated as the log of the square-root of the sum of 22 squared daily futures returns over the rolling 22-trading-day

period;PrCompj;t1 are the three principal components from day t1; and thes ands are coefcients to be estimated. Panel B

reports on a similar model, except the log of the standard deviation of realized 5-minute S&P 500 ETF returns overt1 to t22

replaces the realized stock volatility from daily returns for the 2 term. We report on the October 1997 to June 2013 sample

period, along with two subperiods (1997:102005:06 and 2005:072013:06). T-statistics are in parentheses, calculated with

heteroscedastic and autocorrelation consistent standard errors. An F-test (3) test statistic is in brackets, which jointly tests the

coefcients on the three lagged principal components. nnn, nn, and n indicate 1%, 5%, and 10% p-values.


Panel A: Realized stock volatility from daily returns over t1 to t22 as the 2 TermFull Period a. 0.361 (9.54)nnn 23.6%

1997:102013:06 b. 0.575 (10.36)nnn 0.237 (3.22)nnn 46.9%

c. 0.382 (6.14)nnn 0.154 (4.05)nnn [10.67]nnn 51.5%

First Subperiod a. 0.191 (2.63)nn 5.7%

1997:102005.06 b. 0.495 (5.85)nnn 0.116 (1.85)n 29.3%

c. 0.256 (2.97)nnn 0.205 (3.25)nnn [10.63]nnn 39.4%


2005.072013.06 b. 0.655 (9.85)nnn 0.099 (2.10)nn 58.0%

c. 0.444 (5.59)nnn 0.093 (1.73)n [8.21]nn 63.2%


Full Period 0.365 (5.99)nnn 0.184 (4.62)nnn [11.38]nnn 52.3%

1997:102013:06

First Subperiod 0.236 (2.80)nnn

0.236 (3.49)nnn

[11.50]nnn

40.2%1997:102005.06

Second Subperiod 0.390 (4.94)nnn 0.169 (2.92)nnn [7.12]nnn 64.1%

2005.072013.06



9/19

3.4. Volatility of the term-structure's slope

InTable 5, we report on an examination of the volatility of the term-structure's slope, as proxied

for by the term-structure's second principal component (i.e., TmSti;j PC2i;j in Eq.(1)). In Panel A, the

estimates of2 are positive in all cases, and are statistically signicant for all cases except regression

(c) for the second subperiod. The positive 2 indicates that the volatility of the second principalcomponent tends to be larger following higher lagged equity volatility. For model (a) with only the

equity risk as an explanatory term, the R2 values are again appreciable, at 24.8% for our entire sample

period and 38.4% for the second-half subperiod. Again, we note that the coefcient on the equity risk

term is largely similar for both models (b) and (c) (with or without the three lagged principal

components), which indicates that the information from the equity risk term is largely distinct from

the information in the lagged cross-section of yields. Once again, the2estimates in Panel B ofTable 5

are larger in both magnitude and statistical signicance for all three estimation periods, as compared

to regression (c) in Panel A.

3.5. Summary of main results and robustness checks

The evidence in Tables 25 indicates that the lagged equity volatility contain substantial and

reliable forward-looking information about the subsequent term-structure volatility, beyond the

information contained in the lagged own volatility and the lagged principal components (as term-

structure state variables). In all four tables, the evidence is consistent for both the subperiods, both in

Table 4

Volatility of change in the term-structure's level (rst principal component). This table reports results for how the equity risk is

related to the subsequent volatility of the term-structure's rst principal component. Panel A reports three variations of the

following regression, denoted as models (a) to (c) in the table:

PC1t;t21 0 1PC1t1;t22 2STt1;t22 X3

j 1

jPrCompj;t1 t;

wherePC1i;j (STi;j) is the logarithmic transformation of volatility of the change in the rst principal component(S&P 500 futures

contract) over trading days i to j, calculated as the log of the square-root of the sum of 22 squared daily changes in the rst

principal component (daily futures returns) over the rolling 22-trading-day period; PrCompj;t1 are the three principal

components from dayt1; and the s ands are coefcients to be estimated. Panel B reports on a similar model, except the log

of the standard deviation of realized 5-minute S&P 500 ETF returns over t1 to t22 replaces the realized stock volatility from

daily returns for the 2term. We report on the October 1997 to June 2013 sample period, along with two subperiods (1997:10

2005:06 and 2005:072013:06). T-statistics are in parenthesis, calculated with heteroscedastic and autocorrelation consistent

standard errors. An F-test (3) test statistic is in brackets, which jointly tests the coefcients on the three lagged principal

components. nnn, nn, and n indicate 1%, 5%, and 10% p-values.


Panel A: Realized stock volatility from daily returns over t1 to t22 as the 2 Term

Full Period a. 0.342 (9.41)nnn 21.0%

1997:102013:06 b. 0.191 (1.66)n 0.264 (4.99)nnn 28.0%

c. 0.097 (1.26) 0.222 (5.99)nnn [30.05]nnn 42.0%

First Subperiod a. 0.204 (3.01)nnn 7.3%

1997:102005.06 b. 0.058 (0.90) 0.194 (3.01)nnn 8.8%

c. 0.013 (0.50) 0.271 (4.76)nnn [17.16]nnn 35.4%


2005.072013.06 b. 0.693 (11.23)nnn 0.051 (1.06) 54.8%

c. 0.441 (5.52)nnn 0.065 (1.13) [10.35]nn 61.4%


Full Period 0.086 (1.12) 0.256 (6.37)nnn [31.49]nnn 43.1%

1997:10

2013:06First Subperiod 0.002 (0.08) 0.326 (5.20)nnn [20.38]nnn 38.4%

1997:102005.06

Second Subperiod 0.395 (4.96)nnn 0.131 (2.11)nn [9.58]nnn 61.9%

2005.072013.06



10/19

terms of the statistical signicance and the magnitude of the coefcients on the equity-risk terms. In

all four tables, the estimated coefcients on the lagged equity volatility changes only modestly when

adding the three lagged principal components (comparing models (b) to (c) in each table). Finally, in

all four tables, the ISBV relation is stronger in Panel B with the 5-minute-return-based stock volatility,

which supports the robustness of our ISBV ndings and indicates that the ISBV relation is more

reliably evident when using a higher-quality stock volatility measure.We next probe the robustness of these ndings. First, we re-estimate the primary regressions from

Tables 25, but with six lagged forward rates (the 1-year, 2-year, 3-year, 5-year, 7-year, and 10-year

GSW instantaneous forward rates) replacing the lagged three principal components as term-structure

state variables. This approach follows from Cochrane and Piazzesi (2005), who nd that the term-

structure of forward rates has strong explanatory power for one-year bond excess returns. We nd

qualitatively very similar results to those depicted in Tables 25. For all four term-structure

volatilities, the estimated 2on the lagged stock volatility remains reliably positive, with a p-value of

1% or better.

Second, a natural question is whether our results are evident when analyzing longer time horizons

(rather than the rolling monthly horizon analyzed in the tables) and for alternate specications. In an

Online Appendix, we evaluate the subsequent quarterly term-structure volatility (from 66 trading-dayobservations) with an alternate specication that allows for additional information from the prior

realized term-structure volatility by including multiple lags as explanatory terms. We nd that the

lagged stock volatility remains a reliable incrementally informative determinant for the subsequent

Table 5

Volatility of change in the term-structure's slope (second principal component). This table reports results for how the equity

risk is related to the subsequent volatility of the term-structure's second principal component. Panel A reports three variations

of the following regression, denoted as models (a) to (c) in the table:

PC2t;t21 0 1PC2t1;t22 2STt1;t22 X3

j 1

jPrCompj;t1 t;

where PC2i;j (STi;j) is the logarithmic transformation of volatility of the change in the second principal component (S&P 500

futures contract) over trading days i to j, calculated as the log of the square-root of the sum of 22 squared daily changes in the

second principal component (daily futures returns) over the rolling 22-trading-day period; PrCompj;t1 are the three principal

components from dayt1; and the s ands are coefcients to be estimated. Panel B reports on a similar model, except the log

of the standard deviation of realized 5-minute S&P 500 ETF returns over t1 to t22 replaces the realized stock volatility from

daily returns for the 2term. We report on the October 1997 to June 2013 period, along with approximate one-half subperiods

(1997:102005:06 and 2005:072013:06). T-statistics are in parenthesis, calculated with heteroskedastic and autocorrelation

consistent standard errors. An F-test (3) test statistic is in brackets which jointly tests the coefcients on the three lagged

principal components. nnn, nn, and n indicate 1%, 5%, and 10% p-values.


Panel A: Realized stock volatility from daily returns over t1 to t22 as the 2 Term

Full Period a. 0.534 (8.82)nnn 24.8%

1997:102013:06 b. 0.188 (1.39) 0.351 (5.14)nnn 32.1%

c. 0.093 (1.29) 0.352 (8.36)nnn [51.52]nnn 55.5%

First Subperiod a. 0.247 (3.87)nnn 10.8%

1997:102005.06 b. 0.002 (0.05) 0.247 (3.88)nnn 10.8%

c. 0.004 (0.12) 0.298 (5.36)nnn [5.51]nnn 21.7%


2005.072013.06 b. 0.777 (11.32)nnn 0.092 (1.65)n 71.7%

c. 0.459 (5.26)nnn 0.089 (1.48) [17.46]nn 77.5%


Full Period 0.063 (1.00) 0.433 (10.31)nnn [67.07]nnn 59.4%

1997:10

2013:06First Subperiod 0.010 (0.34) 0.361 (6.32)nnn [7.56]nnn 25.7%

1997:102005.06

Second Subperiod 0.422 (5.05)nnn 0.156 (2.33)nn [16.87]nnn 78.0%

2005.072013.06



11/19

term-structure volatility. Taken together, these additional ndings support the robustness of our

primary ndings.

4. A

ight-to-quality/

ight-from-quality avenue?

We next discuss and present evidence regarding FTQ (and FFQ) as a potential underlying economic

mechanism that might be an important contributor behind the documented ISBV relation.

Theoretically, we rst appeal to the framework in Bekaert, Engstrom, and Xing (BEX) (2009). BEX

consider the joint pricing of stocks and bonds in a market where both economic uncertainty and risk

aversion may change over time. One result from BEX is that the volatility of economic fundamentals

(or economic uncertainty) is very highly correlated with expected stock-market volatility, where

fundamentals refers to dividend growth. BEX also nd that stock volatility is systematically higher in

bad economic times such as recessions, which indicates positive serial correlation in stock volatility or

volatility clustering.10 Their model also features a classic FTQ avenue where bond prices are likely to

appreciate with heightened economic uncertainty due to a precautionary savings effect.

The theoretical framework ofVeronesi (1999)also provides a rationale for volatility clustering. In

his model, time-varying volatility is tied to uncertainty about the economic state, and the price impact

of news is higher when uncertainty about the underlying economic state is higher.11 Bollerslev, Chou,

and Kroner (1992) provide a survey of the empirical evidence on volatility clustering, along with a

discussion on the theoretical underpinnings of volatility clustering.

Thus, if a relatively high stock-return volatility and high time series variability in economic

uncertainty are likely following months with a relatively high realized stock volatility, then the

likelihood of FTQ pricing inuences over the subsequent month is presumably much greater. With this

economic intuition in mind, we next report ve additional evaluations that bear on the plausibility of

a FTQ avenue.

4.1. Variation in the ISBV relation with the market state

The basic premise of FTQ is that the phenomenon would be largely episodic around times with

higher stock-market stress or economic uncertainty. Accordingly, we expect that the ISBV relation

would tend to be stronger around recessions. Further, under a FTQ avenue, the ISBV relation would

presumably be largely non-existent over low stock-market stress periods (periods with a low and

stable stock volatility and with no prominent economic or international crises). In this subsection, we

explore these predictions.

For our investigation, we choose two high-stress and two low-stress stock-market subperiods and

estimate the ISBV relation over each of these four subperiods separately. While such market-state

classications are admittedly somewhat subjective, we rely on the National Bureau of Economic Research(NBER) recession classication and the VIX behavior to provide objectivity in our classications.

We categorize two subperiods as having relatively higher stock market stress: March 2001 to

November 2002 and December 2007 to June 2010. The beginning month of these higher-stress states

is the rst month of a formal NBER recession. The nal month of the higher-stress states is 12 months

following the last month of each NBER recession. Our choice of a one-year after recessionterminal

month recognizes that uncertainty and market stress typically remain past the formal end of a

recession as the market learns of the recovery. We note that the formal NBER announcement of the

10 The BEX measure of economic uncertainty is the volatility of fundamentals. In their Table 5 (p. 71), they report on

simulated moments from their model using estimated parameters from actual data for the 1927 to 2004 period. In this exercise,

they estimate that the volatility of fundamentals is highly correlated to the expected stock market volatility, with a correlationcoefcient of 0.88. Further, they estimate an autocorrelation coefcient of 0.98 for the stock-market's conditional variance, or

strong volatility clustering.11 The notion of economic uncertainty is different inVeronesi (1999)versusBEX (2009). In Veronesi, the uncertainty refers

to the notion that the true underlying economic state is unobservable and unknown by investors. In certain market states, the

uncertainty about the market state is relatively higher.



12/19

end of the respective recession occurred after the November 2002 and June 2010 end-months for the

higher-stress states, with the NBER announcements occurring in July 2003 and September 2010 for

the earlier and later recession, respectively. We also feel that this one-year postchoice is a good t

with the VIX time series behavior, as discussed further below.

We also categorize two subperiods as having relatively lower stock market stress: January 1993 to

November 1996 and April 2004 to June 2007.12 These choices rely heavily on the VIX behavior, as

follows. The beginning month has the following two criteria: (1) occurs after the end of the preceding

recession has been formally announced by the NBER, and (2) has the closing daily VIXo20% for that

entire month and for the next 11 calendar months (for one consecutive year with the closing daily

VIXo20%). The nal month for the lower-stress states is selected as the rst month that: (1) occurs

after the beginning-month criteria is met, and (2) precedes two consecutive months that have

episodes where the VIX exceeds 20%.

Our choice of the two high and low market-stress subperiods is supported by Table 6, Panel A,

which reports VIX summary statistics for each of the four subperiods. For our two high-stress

subperiods, the daily closing VIX value is above the full-sample median of 19.03% for 95.2% and 89.7%

of the days for our rst and second high-stress subperiods, respectively. For our two low-stress

subperiods, the daily closing VIX value is above the full-sample median for less than 1.8% of the days

for both low-stress subperiods.

We estimate variations of the following regression separately for these four subperiods:

TBt;t21 01TBt1;t222

STt1;t22t; 2

where the terms are as dened for Eq.(1)andTable 2, Panel A.

In Panel B of Table 6, we report the results for the T-bond futures return volatility. The row-1

specication includes only the lagged stock-futures volatility as the explanatory term. Note that the

estimated 2 coefcients on the stock volatility term are about twice as large for the high-stress

subperiods as compared to the same coefcients for the low-stress subperiods. Further, the R2 values

are strikingly different, at an average of 28.9% for the high-stress periods versus 3.6% for the low-stress

periods.

In the row-2 specication, we evaluate a variation that includes only the own lagged T-bond

volatility as the explanatory term. When comparing the row-1 model (with stock-volatility as the sole

explanatory term) to the row-2 model (with the T-bond volatility as the sole explanatory term), we

nd that the stock volatility contains more forward-looking information about the subsequent T-bond

volatility than does the own-lagged T-bond volatility for the high-stress state (in terms of the R2

values).

Finally, the row-3 specication includes both the lagged T-bond volatility and lagged stock

volatility as explanatory terms. The results again indicate that the lagged stock volatility is the more

important explanatory term for the high-stress periods, whereas the own-lagged T-bond return

volatility is the more important explanatory term for the low-stress subperiods. In untabulated

results, wend similar results when estimating a comparable model where the T-bond futures returnvolatility is replaced by the 10-year T-note futures return volatility.

Overall, the results inTable 6support the premise that the ISBV relation is stronger with higher

stress/volatility in the stock market. These ndings indicate an episodic nature of the ISBV relation in a

manner that one would expect through a FTQ avenue.

4.2. Evidence of stock market volatility clustering

With the intertemporal nature of our primary ndings, a FTQ avenue would require that a

relatively high realized stock volatility last month must be reliably associated with a higher

subsequent stock volatility over the next month. As previously discussed, Veronesi (1999) andBEX

(2009) provide theory and evidence about stock volatility clustering. Consistent with their ndings,

12 For this exercise, we extend our sample earlier back to 1993 in order to capture a second low-stress market state for

evaluation.



13/19

clustering in stock volatility is also evident over our sample. The simple correlation between the 22-

trading-day stock-futures volatility over days t to t21 and the same volatility over trading-days

t22 tot1 is 0.69 for the simple standard deviation and 0.70 for the log of the standard deviation.

To illustrate further the tendency of stock volatility to cluster, we perform the following sorting

exercise to analyze only extreme volatility episodes. With the realized stock volatility over tradingdays t22 to t1 as the sorting variable, we sort ve variables that are constructed from daily

observations over trading days t to t21: (1) the S&P 500 futures volatility, (2) the T-bond futures

volatility, (3) the 10-year T-note futures volatility, (4) the correlation between the T-bond and S&P 500

futures returns, and (5) the correlation between the T-note and S&P 500 futures returns. For the

lagged volatility in this sorting exercise, we use the more precise volatility measure from 5-minute

returns on the SPY ETF.

InTable 7, Panel A, we report the summary statistics for the volatilities and correlations that follow

the largest 10% of the realized stock volatilities. The results indicate that the subsequent stock

volatility is appreciably larger than average for months when the prior realized stock volatility was

high. For this high volatility state, the row-1 results show that the mean/median of this conditional

subsequent stock volatility was 33.3%/28.1% (versus 18.3/15.9% over our primary 1997:10 to 2013:06sample), with 96.4% of these conditional stock-volatilities being above the full-sample median.

Conversely, inTable 7, Panel B, we present comparable statistics for observations that follow the

smallest 10% of the prior realized stock volatilities. The results indicate that the subsequent stock

volatility is appreciably smaller than average for months when the prior realized stock volatility was

Table 6

The intertemporal stock-to-bond volatility relation for four key subperiods. This table reports how equity volatility is related to

the subsequent volatility of the T-bond-futures returns for four different key subperiods. Panel A reports statistics for the

CBOE's implied volatility index (VIX) from S&P 500 index options for the overall January 1993 to June 2013 period and

separately for the four subperiods to highlight subperiods differences in stock market stress. Panel B reports estimation

variations of the volatility model fromTable 2, with separate regression results for the two separate

lower market stress anduncertaintysubperiods (rows 13 and 79) and the two separate higher market stress and uncertaintysubperiods (rows 46

and 1012). The volatilities here are calculated from daily return observations over a 22-trading-day period. For the estimated

coefcients,T-statistics are in parentheses, calculated with heteroscedastic and autocorrelation consistent standard errors. nnn,nn, and nn indicate 1%, 5%, and 10% p-values.

Panel A: VIX Subperiod Statistics

Subperiod Dates Mean Median Low Max Std. Dev. %419:03

Full Sample 1993:012013:06 20.52 19.03 9.31 80.86 8.46 50%

I. Low Stress 1993:011996:11 13.75 13.23 9.31 23.87 2.17 1.8%

II. High Stress 20 01:032002:11 26.66 24.46 17.40 45.08 6.29 95.2%

III. Low Stress 20 04:0 42007:06 13.39 13.04 9.89 23.81 2.13 1.1%

IV. High Stress 2007:122010:06 30.09 25.41 15.58 80.86 12.61 89.7%

Panel B: Subperiod Regression Results for the 30-year T-bond Futures Volatility

Subperiod Dep. Var. 1 (TBt1;t22) 2 (

STt1;t22) R

2

I. Low Stress 1. TBt;t21 n/a 0.167 (2.33)nn 4.8%

1993:01 to 2. TBt;t21 0.460 (4.54)nnn n/a 21.6%

1996:11 3. TBt;t21 0.447 (4.11)nnn 0.024 (0.33) 21.7%

II. High Stress 4. TBt;t21 n/a 0.326 (4.17)nnn 21.6%

2001:03 to 5. TBt;t21 0.236 (1.63) n/a 5.5%

2002:11 6. TBt;t21 0.141 (0.82) 0.304 (3.69)nnn 23.5%

III. Low Stress 7. TBt;t21 n/a 0.134 (0.95) 2.4%

2004:04 to 8. TBt;t21 0.574 (6.72)nnn n/a 36.4%

2007:06 9. TBt;t21 0.579 (5.79)nnn 0.016 (0.14) 36.4%

IV. High Stress 10.TBt;t21 n/a 0.307 (6.31)nnn 36.2%

2007:12 to 11.TBt;t21 0.596 (5.29)nnn n/a 35.7%

2010:06 12.TBt;t21 0.325 (1.52) 0.176 (1.94)n 40.3%



14/19

low. For this low volatility state, the row-1 results show that the mean/median of this conditional

subsequent stock volatility was 9.9%/9.4%, with only 5.1% of these conditional stock volatilities being

above the full-sample median.

Finally, rows 2 and 3 of Table 7 show that the conditional volatility of the T-bond and T-note

futures returns are also strikingly different, depending upon whether the prior month's stock

volatility was extremely high or low. The T-bond and T-note return volatilities that follow a high stock

volatility (Panel A) have a mean and median that are about twice the comparable values for theobservations that follow a low stock volatility (Panel B).

4.3. VIX characteristics following an extreme stock market volatility

Under a FTQ avenue that is linked to periods of stock market stress and time-varying economic

uncertainty, we would expect that periods following a high realized stock volatility would also be

periods with both a relatively high level and high variability in the option-derived implied stock-

market volatility. We examine this proposition using VIX data. In addition to VIX's basic interpretation

as a measure of expected stock volatility, VIX has also been interpreted as a fear index that re ects

economic uncertainty and, perhaps, risk aversion (e.g., Bollerslev, Tauchen, and Zhou, 2009).

Using the same sorting exercise as inSection 4.2, we nd that the VIX is strikingly higher and morevariable for observations that follow a high realized stock volatility. InTable 7, Panels A and B, rows

4 and 5 report these conditional VIX statistics, with the VIX variability dened as the square root of

the sum of squared daily VIX-changes over tto t21. For the days that follow a high (low) realized

stock volatility in the top (bottom) decile, the conditional mean/median of VIX is 38.89/36.22 (12.51/

Table 7

Return volatilities and correlations that follow an extreme realized stock volatility. This table reports subset statistics for

volatilities and correlations that follow an extreme realized stock volatility over the prior month. For every rolling 22-trading-

day period, the monthly volatility and correlation are calculated from the 22 daily observations. Panel A (B) reports statistics for

the subset of volatilities and correlations over trading days tto t21 that follow the largest (smallest) 10% of the realized 5-

minute S&P 500 ETF return

volatilities over trading days t22 to t1. Rows 1

3 report on the S&P 500 futures return volatility,the T-bond futures return volatility, and the 10-year T-note futures return volatility, respectively, in annualized standard-

deviation percentage units. Row 4 reports on the VIX level on day t, in percentage units. Row 5 reports on the realized volatility

of the daily VIX changes over t to t21 in daily-change VIX units. Rows 6 and 7 report the correlations between the stock

futures returns and the T-bond and T-note futures returns, respectively. For each subset, we report the mean, median, 25th

percentile, and 75th percentile. The sample period is October 1997 to June 2013.

Panel A: Observations following the Largest 10% of Realized Stock Volatility

Variable Mean Median 25th Pctl 75th Pctl

1. StFt 33.29 28.13 21.94 37.11

2. TB 13.43 13.30 10.54 16.62

3. TN 8.41 8.00 6.50 9.83

4.VIX 38.89 36.22 31.19 43.67

5. VIX 2.66 2.11 1.58 2.916. StFt;TB 0.33 0.32 0.51 0.12

7. StFt;TN 0.34 0.35 0.51 0.14

Panel B: Observations following the Smallest 10% of Realized Stock Volatility

Variable Mean Median 25th Pctl 75th Pctl

1. StFt 9.92 9.35 7.81 11.35

2. TB 6.80 6.49 5.94 7.47

3. TN 4.17 4.10 3.54 4.57

4.VIX 12.51 12.08 11.19 13.34

5. VIX 0.77 0.60 0.48 0.91

6. StFt;TB

0.04 0.07 0.33 0.277. StFt;TN 0.03 0.04 0.33 0.29



15/19

12.08). For the periods that follows a high (low) realized stock volatility in the top (bottom) decile, the

conditional mean/median of the VIX variability is 2.66/2.11 (0.77/0.60).

4.4. The stock-bond return correlation and high stock market volatility

Under a FTQ avenue for understanding the ISBV relation, we would expect that a high realized

stock volatility this month would be associated with a relatively more negative stock-bond return

correlation over the next month. Again using the same sorting approach as in Section 4.2, we nd that

the stock-bond return correlations are appreciably more negative for observations that follow a high

realized stock volatility. For example, inTable 7, we nd that the median of the 22-trading-day stock-

bond return correlations (between daily S&P 500 futures returns and 10-year T-notefutures returns)

is 0.35 (0.04) for observations that follow a high (low) realized stock volatility.13

4.5. T-bond diversication benets with large stock market declines

Finally, under a FTQ avenue, one would expect that: (1) T-bonds would have actually served as agood diversication instrument against bad stock market outcomes over our sample, and (2) that such

diversication benets would have been relatively stronger following periods with a higher realized

stock volatility.14 We compute conditional average returns for 30-year T-bond futures and 10-year

T-note futures for those observations when there was a concurrent extreme stock return, with

separate evaluations for those observations that follow a relatively low stock volatility and for those

observations that follow a relatively high stock volatility (based on the lagged volatility being below or

above its median value).

Table 8 reports these conditional average returns both at the daily horizon (Panel A) and at the

weekly horizon (Panel B), where a week is dened as ve consecutive trading days with overlapping

weekly observations. Column (2) reports on the stock-return threshold for both extremely negative

returns (o5th percentile) and extremely positive returns (495th percentile), based on the realizedstock returns over the October 1997 to June 2013 sample period. Column (3) reports if the stock

volatility over dayt1 to t22 is below or above its median value. Column (4) reports the number of

observations for each stock-return threshold. Columns (5)(7) report the average returns for the 30-

year T-bond futures, the 10-year T-note futures, and the S&P 500 futures, respectively, for the

observations when the realized stock return falls in the threshold listed in column (2).

As expected, the extreme stock returns are much more likely when the lagged stock volatility is

above its median value. In all the sub-panels of the table, the number of observations (column 4) is

dramatically lower for rows when the lagged stock volatility was below its median (rows 1 and 3) as

compared to that for rows when the lagged stock volatility was above its median (rows 2 and 4).

When the stock return is extreme, we also nd a sizably opposite average T-bond return. For each row,

we nd that the conditional average of the T-bond-futures returns (columns 5 and 6) are appreciableand of opposite sign, as compared to the average of the extreme S&P 500 futures returns (column 7).

Together, these observations provide support for the notion that the diversication benet of T-bonds

looks to be appreciably greater following periods of higher realized stock volatility.

4.6. Summary discussion

This section's evidence supports the plausibility of a FTQ mechanism being an important

contributor to the ISBV relation. Perhaps most importantly, we found that the ISBV relation is stronger

during periods of stock-market stress and weaker during periods of relative calm. Further, we found

that a relatively higher stock-market volatility this month is associated with: (1) a much higher stock-

13 These ndings are consistent with related ndings in Connolly, Stivers, and Sun (2005) and Baele, Bekaert, and

Ingehelbrecht (2010) that a higher VIX is associated with a subsequently more negative stock-bond correlation.14 This intuition also ts with ndings inCampbell, Sunderam, and Viceira (2013), which indicate more of a hedge role for

T-bonds since around 2000.



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market volatility next month; (2) both a higher VIX level and higher VIX variability next month;

indicating higher risk perceptions and greater time series variation in risk perceptions; (3) a more

negative stock-bond return correlation next month; and (4) an increased diversication benet for

holding T-bonds in a stock-bond portfolio.

5. An omitted factor perspective?

In this section, we present evidence regarding a potential

omitted common-factor

avenue thatmay bear on understanding our ISBV ndings. Under this channel, presumably there might be

common factors that are important determinants of both stock and bond volatility, with the common-

factor's volatility exhibiting substantial positive serial correlation. If so, then the lagged stock volatility

might provide information about the underlying common-factor-news volatility, which could

contribute to an empirical ISBV link.

We consider two potential common-factor candidates. First, Fama and French (1993) found that

the monthly returns of both stock portfolios and government bond portfolios responded similarly to

their default yield spread (DYS) variable.15 We investigate whether or not the volatility of a DYS

variable, dened as the yield difference between Moody's Baa- and Aaa-rated bonds, is an important

omitted variable that may be relevant for understanding our primary results. We add the lagged

volatility of DYS, based on daily changes, as an additional explanatory variable in our primary

regressions inTables 2and3. In an Online Appendix, we report that: (1) the lagged stock volatility

Table 8

Average T-bond returns when stock returns are extreme. This table examines T-bond and 10-year T-note futures returns over

days and weeks that experienced extreme stock returns. We report the conditional averages of futures returns for four separate

subsets of observations using the following double-sort criteria: Returns coincident with either extremely negative/positive

stock returns (rst sort), but then separate subsets depending upon whether the prior month had a relatively low/high realized

stock volatility (second sort). An extreme stock return (column 2) is if the observation is either in the top or bottom vigintile. Aprior low/high realized stock volatility (column 3) is whether the stock volatility over the prior month was above or below its

median value. Panels A and B report on the daily and weekly horizon, respectively, where a week is a rolling 5-trading-day

period. The stock returns are S&P 500 futures returns, and the realized stock volatility is computed from the daily S&P 500

futures returns over the rolling 22-trading-day period. The sample period is October 1997 to June 2013.

Panel A: Daily Returns

Row If coincident IfStFtt1;t22 of Avg Ret Avg Ret Avg Ret

stock return is: was: Obs. 30-yr T-bond 10-yr T-note S&P500

(1) (2) (3) (4) (5) (6) (7)

1. o5th pctl Below Median 31 0.59 0.37 2.75

2. o5th pctl Above Median 167 0.61 0.41 3.22

3. 495th pctl Below Median 18 0.49 0.20 2.15

4. 495th pctl Above Median 180 0.46 0.27 3.20

Panel B: Weekly Returns

Row If coincident IfStFtt1;t22 of Avg Ret Avg Ret Avg Ret

stock return is: was: Obs. 30-yr T-bond 10-yr T-note S&P500

(1) (2) (3) (4) (5) (6) (7)

1. o5th pctl Below Median 20 0.99 0.63 5.07

2. o5th pctl Above Median 178 1.27 0.90 6.54

3. 495th pctl Below Median 10 0.32 0.12 4.56

4. 495th pctl Above Median 188 0.70 0.31 5.91

15 Chen, Roll, and Ross (1986), Fama and French (1993), and Jagannathan and Wang (1996), among others, relate yield

spreads to expected stock returns.



17/19

remains a reliable, incremental explanatory term, and (2) the lagged DYS volatility is not an important

incremental explanatory term in our setting.

Second, while ination may be more directly relevant to the valuation of nominal bonds,

researchers have proposed that ination news may drive both stocks and bond returns, sometimes in

opposite directions (e.g., Campbell and Ammer, 1993; David and Veronesi, 2013). Next, using the

method fromGurkaynak, Sack, and Wright (2010), we form a daily ination compensationvariable

based on the difference between the 10-year TIPS yield and the yield of the 10-year nominal T-note.

We add the lagged ination-compensation volatility (based on daily changes) as an additional

explanatory variable in our primary regressions in Tables 2and3. In the Online Appendix, we show

that the lagged stock volatility remains a reliable, incremental explanatory term, and the lagged

ination-compensation volatility is not an important incremental explanatory term in our setting. To

conclude, while our limited evidence in this section is inconclusive (in that other factors or

approaches could be evaluated), our ndings lend no support to the notion that this omitted

common-factor avenue is ofrst-order importance for understanding our ISBVndings.

6. Other empirical studies on stock-bond volatility linkages

In this section, we briey discuss our ndings in the context of two earlier empirical studies that

also evaluated volatility linkages between the stock and Treasury bond markets.

6.1. Relation toFleming, Kirby, and Ostdiek (1998)

Fleming, Kirby, and Ostdiek (FKO) (1998)evaluate daily return data from S&P 500, T-bond, and T-

bill futures contracts for the January 1983 to August 1995 period. They modeled information ows and

evaluated how information inuences all three markets through both a direct effect and an

information-spillover effect tied to cross-market hedging. Their results show a greater volatilitylinkage across the markets than is indicated by the modest correlations in daily returns and daily

absolute returns. Their nding of strong cross-market volatility linkages is consistent with the

premise of our main ndings.

However, FKO's empirical work is much different than ours. First, they analyze the volatility of

daily returns using a stochastic volatility model with an AR(1) process. Our focus is on monthly and

quarterly realized volatilities, estimated from daily or high frequency intraday returns. Second, their

notion of volatility linkages is based on the correlation of conditional daily variances of S&P 500 and T-

bond futures returns. Our investigation is broader in the sense that we examine the intertemporal

linkages between lagged stock volatility and four different measures of the subsequent term-structure

volatility in a multivariate setting that controls for the lagged term-structure volatility and other

term-structure state variables. Finally, their sample (which predates our sample) has a stock-bondreturn correlation of 0.35; the comparable correlation for our sample is 0.30. Such striking

correlation differences suggest differences in the relative importance of a FTQ/FFQ avenue between

our two samples.

6.2. Relation to Chordia, Sarkar, and Subrahmanyam (2005)

Chordia, Sarkar, and Subrahmanyam (CSS) (2005)evaluate the liquidity, returns, and volatility of

stock and bond markets for the June 1991 to December 1998 period. Their focus is on liquidity, but

they also evaluate cross-market volatility linkages. Their measure of daily volatility is the absolute

residual of a regression of daily returns against various calendar-related dummy variables.

In contrast to our overall ndings, they do not nd that stock volatility Granger-causes bondvolatility in a two-lag VAR model. While their empirical approach is substantially different than ours,

we believe that sample-period differences are central to understanding the differences between their

volatility ndings and ours. Only the last 15 months of their sample (October 1997 to December 1998)

overlap with our sample. Over June 1991 to September 1997 (the rst 6 years, 4 months of their



18/19

sample), the mean (median) VIX was 15.2% (14.7%), with a maximum of 25.99%, and the correlation

between daily S&P 500 and 10-year T-note futures returns was 0.42. Over our October 1997 to June

2013 sample, the mean (median) VIX was 22.2% (20.9%), with 947 different days having VIX values

greater than 25.99%. Also, in our Table 6, recall that we reported the results of our volatility model

over the low stock stress period from 1993 to 1996 (which is a subset of their sample) and did not nd

a reliable ISBV relation.

During times of predominantly low stock-market stress, the collective ndings inCSS (2005)and

our study indicate that the ISBV relation is likely to be weak. However, during times of high stock-

market stress, our ndings indicate that the ISBV relation is likely to be sizable.

7. Conclusions

Over the October 1997 to June 2013 period, we nd an economically substantial and statistically

reliable intertemporal relation between realized stock volatility and the subsequent realized volatility

of important Treasury market variables. Further, we nd that the intertemporal stock-to-bond

volatility (ISBV) relations remain substantial and reliable, even while controlling for: (1) the own

term-structure's lagged realized volatility, (2) various term-structure state variables proposed in the

literature, and (3) other potential omitted factors that might plausibly subsume the estimated ISBV

relations. Specically, we study the volatility of returns from both T-bond and 10-year T-note futures

contracts and the volatility of changes in the term-structure's rst and second principal component.

Our investigation focuses on rolling estimates of monthly realized volatilities, constructed from daily

observations over rolling 22-trading-day periods.

The intertemporal aspect of our ndings supports the notion that equity volatility can help

understand volatility behavior in bond markets, beyond an approach that only looks at the bond

market in isolation. This notion builds from the results in Andersen and Benzoni (2010),who nd that

the cross-section of yields does not span yield volatility and who suggest that linking term structure

dynamics to the general economic environment might prove productive.While the ISBV relations are substantial and reliable over our full sample, we nd that the ISBV

relations are substantially stronger during times with notable stock-market stress (such as around

recessions) and appreciably weaker in calm times (periods with a sustained low VIX and no

prominent economic or international crisis). This suggests that a ight-to-quality/ight-from-quality

(FTQ/FFQ) avenue may be important for understanding our ndings. Consistent with this premise, we

nd that a high realized stock volatility this month is associated with the following over the next

month: (1) a much higher subsequent realized stock volatility, (2) much higher day-to-day variability

in the stock-market's option-derived implied volatility (VIX), (3) a more negative stock-bond return

correlation, and (4) an appreciably greater diversication benet to holding T-bonds in a stock-bond

portfolio.

We argued that the combined intuition from Bekaert, Engstrom, and Xing (2009) andVeronesi(1999) is also consistent with our evidence favoring a FTQ/FFQ interpretation of our ISBVndings.

Higher economic uncertainty is associated with a persistently higher stock volatility, more variability

in the perceived economic uncertainty, and a greater precautionary savings motive. So, a higher stock

volatility this month is likely to be followed by more extreme stock returns next month and with more

variability in market uncertainty measures (or fear measures), such as VIX. If so, then the likelihood of

FTQ/FFQ pricing inuences over the subsequent month should be greater, and this, we believe,

contributes to our ISBVndings.

At rst glance, our ndings seem at odds with evidence in Chordia, Sarkar, and Subrahmanyam

(CSS) (2005), who nd that stock volatility does not Granger-cause bond volatility. However, their

sample is the largely low-risk period from June 1991 to December 1998. So, their ndings seem to t

with our evidence that the ISBV relation is weaker in calmer stock-market states (see our Table 6).Thus, an additional contribution of our study is to show that the intertemporal stock-to-bond

volatility ndings inCSS (2005) lack some generality.

Finally, in the sense that equity risk is related to macroeconomic uncertainty and investor

sentiment, our intertemporal ndings seem consistent with recent evidence on bond-return



19/19

predictability. For example, Cooper and Priestley (2009) and Ludvigson and Ng (2009) nd that

macroeconomic variables play a role in understanding bond risk premia. Laborda and Olmo (2014)

show that investor sentiment variables have predictive power for bond risk premia.

Appendix A. Supplementary data

Supplementary data associated with this article can be found in the online version athttp://dx.doi.

org/10.1016/j.nmar.2015.05.002.

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Equity Volatility as a Determinant of Future Term-structure Volatility