And Liquidity-Related Variation in the Correlations and Mean Returns Across Stocks and T-Bonds

Flight-to-Quality- and Liquidity-Related Variation in the

Correlations and Mean Returns across Stocks and T-Bonds1

Naresh Bansal,a Robert Connolly,b and Chris Stiversc

a John Cook School of Business Saint Louis University

b Kenan-Flagler Business School University of North Carolina at Chapel Hill

c Terry College of Business University of Georgia

This version: December 19, 2007

1We thank Tyler Henry, Lubos Pastor, Robert Savickas, Cheick Samake, John Scruggs, Jahangir Sultan, and seminar participants at the University of Georgia, the 2007 Financial Management Association meeting, the 2007 Washington Area Finance Conference, and the 2007 Southern Finance Association meeting for helpful comments. Please address comments to Naresh Bansal (e-mail: [email protected]; phone: (314) 977-7204; Robert Connolly (email: Robert [email protected]; phone: (919) 962-0053); or to Chris Stivers (e-mail: [email protected]; phone: (706) 542-3648). Flight-to-Quality- and Liquidity-Related Variation in the

Correlations and Mean Returns across Stocks and T-Bonds

Abstract

Over the crisis-rich 1997 to 2005 period, we document new time-series and cross-sectional evidence which suggests a sizable ﬂight-to-quality- and liquidity-related variation in the correlations and mean returns across stocks and T-Bonds. Our collective results support the premise of a “searching” in the relative valuation of stocks and bonds during times of market stress. First, higher levels of stock implied volatility (IV) and stock illiquidity and higher time-series variability in stock IV are associated with both: (1) a much lower correlation in the subsequent returns of stock and T-bond returns, and (2) much greater time-series variability in the subsequent stock

IV and illiquidity values. Second, daily stock returns are negatively and appreciably related to the contemporaneous stock IV change, and more-liquid stocks exhibit both: (1) greater responsiveness to the IV change, and (2) a more negative stock-bond correlation in stressful times.

Third, stock IV changes are positively related to the T-bond returns. Finally, when stock IV is relatively high, the subsequent mean returns and turnover are relatively greater for portfolios of more-liquid stocks, as compared to portfolios of less-liquid stocks.

JEL Classiﬁcation: G12, G14

Keywords: Stock and Bond Correlations, Flight-to-Quality, Liquidity 1. Introduction

For decades, researchers have recognized the central role of the joint distribution of equity and bond returns in asset pricing, portfolio allocation, and risk management problems. Recent work has examined time-variation in the comovement of stock and bond returns (see, e.g., Flem- ing, Kirby, and Ostdiek (1998), (2001), and (2003), Scruggs and Glabadanis (2003), Hartmann,

Straetmans, and Devries (2001), Connolly, Stivers, and Sun (2005), (2007), Gulko (2002), Li

(2002), and Baele, Bekaert, and Inghelbrecht (2007)). These papers document substantial time- variation in stock-bond return comovements with sustained periods of a negative correlation that seem unable to be explained by traditional long-term fundamentals in the sense of Campbell and

Ammer (1993) or Fama and French (1989).

Economic and political crises can temporarily shock financial markets in the sense of Kodres and Pritsker (2002). Recent examples of such crisis include the 1997 East Asian financial crisis, the 1998 Russian foreign debt default, the 1999 Brazilian currency crisis, the 2001 terrorism crisis, and the 2003 Iraqi war. During such crises, flight-to-quality and liquidity may be particularly important in understanding the joint stock-bond return distribution.

In this paper, we present new time-series and cross-sectional evidence on this issue by examining the crisis-rich 1997 to 2005 period. We begin by examining whether the correlation between daily 10-year T-bond futures and equity index futures can be linked to variation in stock implied volatility, stock liquidity, and futures volume. Next, to probe deeper, we evaluate conditional stock-bond correlations, mean returns, and turnover for disaggregate stock portfolios that are comprised of individual stocks with diﬀerent levels of volatility, liquidity, and market capitalization. By conditional, we mean that we evaluate these portfolio-level parameters under various market conditions.

We appeal to the following premise to motivate our empirical investigation. When expected stock volatility and illiquidity increase substantially, then riskier stocks are likely to be revalued lower relative to safer T-bonds with ﬂight-to-quality and/or liquidity pricing inﬂuences. Times of market stress are likely to be associated not only with higher levels of stock volatility and illiquidity, but also with higher time-series variability in these ‘market conditions’ variables. The high variability may induce “searching” in the relative valuation of stocks and bonds during times

1 of market stress, with a resulting negative (or, at least, lower) correlation in returns. Such market dynamics may also be associated with prominent diﬀerences in the conditional mean returns and turnover across stocks with diﬀerent levels of return volatility and liquidity.

We examine the 1997 to 2005 period, with separate evaluations for the 1997 to 2000 and 2001 to 2005 subperiods, for several reasons. First, Chordia, Sarkar, and Subrahmanyam (2005) and

Connolly, Stivers, and Sun (2005) ﬁnd that the 1997 to 2000 period is a particularly rich period, with multiple ﬁnancial crises such as those listed in our second paragraph. Second, the 2001 to

2005 subperiod is entirely out-of-sample, relative to these prior papers.

Our work features the implied volatility from equity index options, stock illiquidity derived from joint return-volume behavior, and the trading volume from stock and T-bond futures markets. These variables do not directly relate to stock-bond correlations or cross-sectional differences in realized stock means. Rather, we use these variables to describe market conditions, where relatively higher levels and/or higher time-series variability of these ‘market conditions’ variables are one way to qualify times of market stress. Presumably, flight-to-quality and liquidity pricing influences should be relatively more important during such times of market stress.

We use the CBOE’s VIX as a measure of the stock market’s implied volatility (IV). For stock illiquidity, we evaluate both a return-reversal measure (RRV) that follows from Pastor and

Stambaugh (2003), and a price-impact measure (PIM) that follows from Amihud (2002). To measure the relative degree of futures trading, we construct a standardized measure of volume for the S&P 500 index and 10-year T-bond futures. Our standardization procedure controls for both the long-term growth in futures volume and the quarterly cycle in futures volume.

Our empirical investigation has four major components. Each is intended to address different aspects of the general hypothesis that flight-to-quality (FTQ hereafter) and liquidity differences may be important in understanding variations in conditional correlations and mean returns across stocks and T-bonds. The first two major components of our work investigate the time-series behavior of market-level returns and other market-level variables. Our last two major components focus on cross-sectional differences across stocks, so they feature spot return data.

Motivated by the episodic nature of stock market crises, the ﬁrst major component of our empirical investigation features a regime-switching model. We extend prior work by estimating a bivariate regime-switching model on daily stock and T-bond futures returns in a speciﬁcation

2 that allows the stock and bond volatilities, means, and their return correlations to vary across regimes. We use the estimated regime episodes to examine how stock liquidity, implied volatility, and futures volume diﬀer across the two regimes. The intent is to use the regime-switching estimation to provide a broad description of market diﬀerences in stressful times, which should provide intuition and perspective for our subsequent analysis.

We are interested in whether the regime-switching estimation will depict a plausible ‘bad regime,’ where FTQ and liquidity pricing influences seem likely to be important in understanding joint stock-bond price formation. If so, we hypothesize that the bad-regime days will be associated with appreciably higher stock volatility, a lower stock-bond correlation, higher mean bond returns, and greater time-series variability in stock IV and stock liquidity. We also hypothesize that bad-regime day t will be preceded by higher levels of stock IV, stock illiquidity, and futures trading volume as measured through day t − 1; which would, presumably, reflect day t expectations.1 Our findings are in line with all our hypothesized empirical characteristics, with a stock-bond correlation of +0.14 in the good regime and -0.43 in the bad regime.

The second major component of our empirical investigation has a forward-looking focus. We hypothesize that high values of stock IV, stock illiquidity, and futures volume will be associated with a lower subsequent correlation between stock and T-bond returns. This aspect of our work builds on Connolly, Stivers, and Sun (2005), (2007), who ﬁnd that the stock IV level is negatively related to the correlation of subsequent stock and T-bond returns. In addition to adding the stock illiquidity and futures volume to the analysis, we also extend prior work by examining whether the recent time-series variability of stock IV is informative about the subsequent stock- bond correlation. With persistence in the volatility of stock IV, we hypothesize that a high IV variability should be associated with a lower subsequent correlation, which would support the notion of a “searching” in the relative valuation of stocks and bonds as uncertainty ﬂuctuates.

In a univariate setting, we ﬁnd a sizable and reliable negative relation between the stock-bond correlation and lagged measures of either stock IV, stock illiquidity, the stock futures volume, and the recent variability in the stock IV. For example, when the IV variability over days t − 1 to t − 5 is in its top quintile, the subsequent stock-bond correlation is -0.37 versus an average

1Higher futures volume in times of market stress may be attributed to greater hedging demand, more dispersion in beliefs, or greater information ﬂows.

3 correlation of about zero for the bottom three quintiles. In a multivariate setting, stock IV is the most informative about the subsequent stock-bond return correlation. However, measures of stock illiquidity and futures hedging do contain incremental information about the subsequent stock-bond correlation. For example, when the lagged IV is in its top quintile and stock illiquidity is relatively high (low), then the stock-bond correlation is -0.57 (-0.38).

The third major component of our investigation features a cross-sectional comparison of correlation differences. During times of market stress, we evaluate how the stock-bond correlations vary across stock portfolios comprised of individual stocks with different levels of volatility, PIM illiquidity, and market capitalization. An observation day t is considered to be a time of market stress if the stock IV at the end of day t − 1 is in its top quintile over our sample period. For our purposes here, we consider both firm size and a firm’s recent PIM illiquidity as measures to evaluate cross-sectional variation in liquidity.

Consider the following two cross-sectional propositions. First, our earlier results indicate that times of market stress are associated with sizably negative stock-bond correlations and with both higher levels and higher time-series variability in stock IV and stock illiquidity. Given this, less liquid stocks may exhibit a more negative stock-bond correlation if their prices are particularly sensitive to variations in volatility and liquidity (as compared to more liquid stocks).

The alternative proposition is that larger, more liquid stocks should primarily be the focus of

FTQ influences from a stock-versus-bond asset class perspective, because they make up the vast majority of the stock market’s capitalization and they should be less costly to trade in stressful times (as compared to smaller, less liquid stocks). Thus, it is also plausible that larger, more liquid stocks may be more affected by the pricing influences that generate a sizable negative stock-bond correlation during stressful times.

During the stressful times, we ﬁnd that the stock-bond correlations are more negative for the portfolios that contain either more liquid stocks or more volatile stocks (while controlling for the other characteristic). These ﬁndings favor our second cross-sectional proposition above.

In the ﬁnal component of our work, we evaluate how the conditional mean returns and turnover of our disaggregate stock portfolios vary with two diﬀerent dimensions of market stress.

First, while controlling for a stock’s recent volatility, we document that the daily returns of stock portfolios that contain more liquid stocks are much more negatively related to contemporaneous

4 changes in stock IV than the stock portfolios that contain less liquid stocks. For example, for the 10% of the days with the largest stock IV increases (decreases), large high-volatility stocks have a daily mean return of -2.52% (+2.35%) per day, but small high-volatility stocks are less responsive with a comparable daily mean return of -1.39% (+1.06%) per day. Subperiod results are consistent. In contrast, T-bond returns are positively related to changes in stock IV. These cross-sectional diﬀerences in the ‘contemporaneous price responsiveness to IV changes’ ﬁt with the cross-sectional correlation patterns.

From an intertemporal perspective, we ﬁnd that the mean returns and turnover of more liquid stocks are relatively higher in times of market stress as compared to less liquid stocks, while controlling for a stock’s recent volatility. For example, for the most volatile quintile of stocks, the mean return diﬀerential between the more-liquid, low-PIM stocks and the less-liquid, high-

PIM stocks is +0.241% per day in times of market stress, versus -0.085% per day for the other times. These mean-return diﬀerentials are highly statistically reliable and subperiod results are consistent. While multiple factors may contribute to this ﬁnding, the results suggest an ongoing shift in preferences to more liquid stocks during stressful times.

Our collective ﬁndings suggest a sizable ﬂight-to-quality- and liquidity-related variation in the correlations and mean returns across stocks and T-Bonds. Our results support the premise of a

“searching” in the relative valuation of stocks and bonds during stressful times, where: (1) higher

IV levels, higher illiquidity, and higher time-series IV variability are associated with both much lower correlations in the subsequent returns of stock and T-bond returns and much greater time- series variability in the subsequent IV and illiquidity; (2) stock returns are negatively related to the daily stock IV change, and the more-liquid stocks exhibit both greater responsiveness to the

IV change and a more negative stock-bond correlation in stressful times; (3) more-liquid stocks exhibit a relatively higher turnover in stressful times; and (4) stock IV changes are positively related to T-bond returns.

This article proceeds as follows. First, Section 2 provides additional discussion of the literature, and Section 3 discusses our data. Section 4 presents our regime-switching model and estimation results. Section 5 reports our forward-looking time-series empirical results. Section 6 reports on the cross-sectional variation in correlations and Section 7 reports on the cross-sectional variation in conditional mean returns and turnover. Finally, Section 8 concludes.

5 2. Additional Related Literature

Studies such as Campbell and Ammer (1993) consider the traditional fundamentals approach in understanding the stock-bond correlation. In the long-term fundamentals setting, both variation in real interest rates and common movements in long-term expected returns act to generate a positive correlation (also, see Fama and French (1989), whereas variation in expected inﬂation tends to generate a negative correlation. However, in practice, periods with relatively stable inﬂation have also been associated with sizably negative stock-bond correlations.

More recently, Bekaert, Engstrom, and Grenadier (2005) show that stochastic risk aversion may be important in understanding joint stock-bond pricing, but their model generates a correlation that is somewhat larger than the data. Baele, Bekaert, and Inghelbrecht (2007) examine the determinants of stock and bond return comovements. They focus on fundamentals over longer horizons and find that “even the best fitting economic factor model fits the dynamics of stock-bond return correlations poorly”. They note the VIX has information about stock-bond correlations, beyond their economic fundamentals model.

Thus, one contribution of our paper is to investigate other shorter-term inﬂuences that may be important in understanding time-variation in stock-bond correlations. In this sense, we build from related work in Connolly, Stivers, and Sun (2005), (2007), which focuses on the relation between VIX and stock-bond correlations in the time series.

The intent of our work is to examine times of heightened financial market uncertainty, where the uncertainty of stocks (as a more risky asset class) is presumably more variable and is a first- order effect, as compared to the uncertainty of T-bonds (as a less risky asset class). Accordingly, we examine only stock implied volatility and stock illiquidity, and not bond equivalents. We find that the variability in T-bond return volatility is much less than the variability in stock return volatility, which supports this notion.2

Fleming, Kirby, and Ostdiek (1998) propose that cross-market hedging may be important in understanding the linkages between the financial markets of different asset classes. In their analysis, demand for bonds is affected by information events that alter expected stock returns.

2In contrast, Chordia, Sarkar, and Subrahmanyam (2005) jointly study the dynamics of stock and bond market liquidity with microstructure measures of liquidity. Their ﬁndings indicate some commonality in the time-series of stock and bond market liquidity, a relation which our study does not address.

6 Expected interest rates and expected inflation may be unchanged, but bond markets can be importantly affected. They take this influence into account when estimating the volatility linkage between stocks, bonds, and bills and find stronger linkages than previously thought.3 Through the cross-market rebalancing avenue of Kodres and Pritsker (2002), investors respond to shocks in one market by optimally readjusting their positions in other markets. This action transmits the shocks, so that a shock in one asset market, which may appear to be largely asset specific, may have a material influence on other financial assets.

Some papers have tried to distinguish between pricing inﬂuences attributed to FTQ versus

flight-to-liquidity (FTL); see, e.g., Vayanos (2004) and Beber and Brandt (2006). The distinction in Vayanos (2004) considers FTQ as a flight from more volatile assets and FTL as a flight to more liquid assets. In our study, distinguishing between the two effects is not a fundamental goal.

Rather, from a stock-to-bond asset class perspective, both FTQ and FTL pricing influences are likely to occur during periods of substantial stock market stress. In the first two components of our study (Section 4 and 5), we evaluate time-series behavior of index returns only and treat FTQ and FTL as related phenomenon, without trying to distinguish between them. In the last two components of our study (Sections 6 and 7), we also consider cross-sectional differences across stocks, and we are able to make some contrasts between FTQ and liquidity pricing influences.

3. Data and Variable Construction

Our work uses the following times series over the 1997 to 2005 sample period: (1) daily futures returns on the S&P 500 futures contracts and 10-year Treasury bond futures contracts; (2) daily trading volume on each of these two futures contracts and a standardized measure of trading volume that is constructed to control for growth over time and the quarterly seasonality of futures contract trading; (3) equity-index implied volatility from option contracts; (4) two diﬀerent measures of market-wide and stock-speciﬁc illiquidity, based on the Return-Reversal measure (RRV) from Pastor and Stambaugh (2003) and the Price-Impact measure (PIM) from

Amihud (2002). For brevity in the main text, we report details and summary statistics on each

3In a similar vein, Underwood (2006) examines order ﬂow in a high frequency analysis of the stock and bond spot market. He ﬁnds evidence that cross-market hedging is an important source of linkages across the two markets during periods of elevated equity volatility.

7 of these series in Appendix A. It is important to note that we construct both the PIM and RRV illiquidity measures so that a higher value indicates a less liquid (or more illiquid) market.

In Sections 4 and 5 for the two index-level components of our study, our work features short- term futures returns, rather than spot returns, for several reasons.4 First, futures contracts on the S&P 500 and Treasury notes are very widely traded and the corresponding returns are derived from prices on a single contract, rather than an aggregation of the price quotes from many different securities. Thus, the futures returns avoid potential microstructure-related measurement concerns. Ahn, Boudoukh, Richardson, and Whitelaw (2002) elaborate on this point and find that daily futures returns do not display the positive autocorrelation that is evident in daily spot portfolio returns. Second, the futures contracts are derivatives in zero net supply with a maturity of a few months. Thus, the trading volume in the contracts can better be characterized as having a direct relation to hedging demand over a specific and modest horizon.

Our study also use daily returns and daily turnover for diﬀerent disaggregate stock portfolios, where the portfolios are constructed from sorting individual stocks based on a stock’s past volatility, liquidity, and/or market capitalization. Again, for brevity, details are in the appendix.

4. The Regime-Switching Model and Estimation Results

Given the episodic nature of ﬁnancial crises, a regime-switching perspective seems intuitive and is likely to be useful in characterizing times of market stress. The crisis-rich 1997 to 2005 period seems especially suitable for a regime-switching analysis. In regard to our overall study, the regime-switching estimation serves as an initial attempt to characterize periods of market stress and to provide intuition and perspective for our later analysis. We are interested in whether the regime-switching estimation depicts a bad regime where market characteristics imply an association between lower stock-bond correlations and FTQ and/or liquidity pricing inﬂuences.5

As discussed in Guidolin and Timmermann (2006) (GT), the vast majority of existing work

4In contrast, the aforementioned papers of Hartmann, Straetmans, and Devries (2001), Gulko (2002), and

Connolly, Stivers, and Sun (2005), (2007) investigate the correlations of spot daily returns. 5We do not evaluate whether a three-state, or greater, model is a better ﬁt, since we desire parsimony here. In this sense, we acknowledge that our regime-switching model is incorrect, in an absolute sense, but we feel it meets the hurdle of being useful.

8 with regime switching has only considered univariate models. GT model the joint distribution of monthly stock and bond returns in a bivariate regime-switching model that allows for multiple states. Our two-state, bivariate framework is a simpler approach than GT. While their focus is on asset allocation (using monthly data), our focus is on how the correlation and other market characteristics diﬀer in times of market stress (using daily data).6

Our regime-switching investigation is intended to provide evidence on the following issues.

First, we are interested in characterizing how the correlations, mean returns, and volatilities vary across the two regimes.7 Second, if FTQ and/or liquidity have a material role in understanding stock bond correlations in times of stock market stress, then we expect that the bad regime will also be associated with high VIX, lower stock liquidity, higher futures trading volume, and greater time-series variability in both VIX and stock liquidity.

4.1. The Regime-Switching Model

Our model simultaneously estimates regime-speciﬁc correlations, means, and variance for daily stock and bond futures returns. The model is given by:

j j rs,t = µs + σs ηs,t (1)

j j rb,t = µb + σb ηb,t (2) where the s and b subscripts are markers for stock and bond futures returns; the superscript j indicates the regime for the regime-specific parameters, with j = 1 for the good regime or j = 2 for the bad regime; rs,t and rb,t are the daily close-to-close returns of the stock futures and j j j j bond futures; µs and µb are the regime-specific mean returns; σs and σb are the regime-specific standard deviations; and ηs,t and ηb,t are the return shocks. The return shocks are modeled as bivariate, standard, normally-distributed, random variables.

6For other recent related work that features regime switching, see Ang and Bekaert (2002a), (2002b), (2002c), and (2004) and Baele, Bekaert, and Inghelbrecht (2007). 7Using daily spot stock and Treasury bond returns, Connolly, Stivers, and Sun (2005), (2007) present evidence that stock-bond comovements vary with the lagged VIX. However, they do not estimate the joint variation in correlations, mean returns, and volatilities in a bivariate model that allows each return parameter to vary across regimes. They evaluate either rolling 22-day correlations or a univariate regime-switching model that only allows the stock-bond comovement to vary across regimes.

9 The j state variable is modeled with constant transition probabilities, pjj, where pjj equals the probability that the regime in period t is j, given that the state in period t − 1 is j. We use the S&P 500 futures to represent the stock futures and the 10-year Treasury bonds futures to represent the longer-term Treasury-bond futures.

The means, standard deviations, correlations between the residuals, and the transition probabilities are regime-speciﬁc parameters to be simultaneously estimated. We estimate the model by maximizing the log-likelihood function for the bivariate normal density while allowing for regime-switching between the two states.

4.2. Empirical Results from Regime-Switching Estimation

4.2.1. Return Parameters

We begin by presenting the basic estimation results for our regime-switching model in Table 1.

We report separate estimations for the full 1997 to 2005 sample period and for the 1997-2000 and 2001-2005 subperiods. For our purposes, we consider the fundamental distinction for the bad regime to be higher stock return volatility.

The following picture emerges from the estimates in Table 1. Over our 1997 to 2005 sample, the good regime is the predominant regime with an expected duration of 80 days, while the bad regime has an expected duration of 44 days. For the bad regime, stock volatility is much higher, the regime-speciﬁc mean of the stock returns is lower, the stock-bond correlation is much lower, and the regime-speciﬁc mean of the bond returns is higher.8 Thus, the parameters indicate persistent regimes with sizable contrasts in correlations, volatility, and means. Subperiod results are roughly similar, but the expected regime durations are over twice as long for the 2001-2005 subperiod than for the 1997-2000 subperiod.

Figure 1, Panel A, displays the ﬁltered probability of being in the bad regime over time. In

Figure 1, 38.4% of the observations have a ≥50% chance of being in the bad regime. Thus, the results support the notion of a predominant good regime, corresponding to less stressful times, and an abnormal bad regime, corresponding to more stressful times. Consistent with the notion

8For comparison, we also estimate a comparable regime-switching model using daily spot returns of the stock market and 10-yr T-bonds. The return patterns are qualitatively comparable to those in Table 1 for daily futures returns.

10 that the bad regime corresponds to crisis periods, bad-regime episodes in Figure 1 corresponds to the Asian ﬁnancial crisis in the fall of 1997, the Russian default crisis in the fall of 1998, the Brazilian currency crisis in early 1999, the terrorism crisis in September 2001, and the Iraqi war in 2003. For comparison, Figure 1, Panel B, plots rolling stock-bond correlations calculated using days t to day t + 21, relative to regime-day t. Note the substantial overlap between the bad regime and negative stock-bond correlations.

We now turn to describing the regime-related variation in the return parameters. First, the stock-bond correlations in the bad regime are always appreciably lower than the correlations in the good regime. For the 1997-2005 period, the estimated correlation is 0.143 in the good regime versus -0.429 in the bad regime, a diﬀerence of over 0.56. For the 1997-2000 and 2001-2005 subperiods, the regime correlation diﬀerences are also sizable at 0.69 and 0.52, respectively.

Next, note that the bad regime exhibits a substantially higher stock volatility but little diﬀerence in the bond volatility. Over our full sample, the stock volatility (bond volatility) in the bad regime is 1.99 times (1.07 times) that of the stock volatility (bond volatility) in the good regime. Subperiod results are very similar.

In regard to variation in regime-specific means, we find that the stock mean is reliably positive in the good regime but negative and insignificant in the bad regime. For the bond means, the reverse is evident, with the bond mean being near zero and insignificant in the good regime but appreciably positive and statistically significant in the bad regime. Overall, in our view, the regime variations in the return parameters fit with FTQ intuition.

4.2.2. Stock Implied Volatility

We are also interested whether our estimated regimes identify systematic diﬀerences in other dimensions of market stress. Table 2 reports how important ‘market conditions’ variables vary across the two regimes. As discussed in our introduction, these ‘market condition’ variables are likely to exhibit high values and/or high variability during times of market stress.

We begin by discussing the implied volatility from option prices. Over our full sample period, the average value of the lagged VIX is 30.5% for the bad regime versus 19.8% for the good regime, or about 54% higher for the bad regime. For all three periods in Table 2, the regime diﬀerence in the lagged VIX is sizable and statistically reliable at a 0.1% p-value.

11 We also compare the VIX variability during good- vs. bad-regime days. Time-series variability in VIX is of great interest in our study, because it may signal changes in the relative attractiveness of stocks and bonds. Table 2, Panel B, reports the time-series variability of the

VIX within each regime. Over our full sample, the average absolute daily VIX change in the bad regime is 2.2 times that in the good regime. Subperiod results are similar.

4.2.3. Stock Illiquidity

Table 2 also reports on how our two stock market-wide illiquidity measures, PIM and RRV, vary across the two regimes. We estimate both illiquidity measures over the month preceding each trading day (using individual stock return and volume data over days t − 1 to t − 22) and then evaluate for regime-specific differences in the illiquidity measures. For both illiquidity measures, we find that the lagged stock illiquidity is higher for the bad- vs. the good-regime days, with the difference being statistically significant at a p-value of 0.1% or better. Subperiod results are consistent.

Similar to our discussion about VIX variability, the time-series variability in stock illiquidity is important here because it may help to explain variation in the relative attractiveness of stocks and bonds. Panel B of Table 2 reports the time-series variability of our PIM and RRV illiquidity measures within each regime. For both measures in all three periods, the time-series variability in the illiquidity measures is greater in the bad regime. The differences are statistically significant in all cases except for the PIM measure in the second sub-period. The regime-specific variability differences in RRV are particularly strong, where the variability is around 50% greater in the bad regime than in the good regime for all three periods.

4.2.4. Futures Trading Volume

In our view, futures hedging is also likely to be higher during times of market stress when

FTQ and/or liquidity are likely to have important pricing inﬂuences. Accordingly, we are also interested in how futures volume varies across regimes. As detailed in Appendix A, our work here features a standardized, detrended volume measure that controls for long-term growth in the volume over time and the quarterly seasonality in futures trading volume. Our standardized volume measure indicates the percentage increase in that day’s volume, relative to the average

12 volume over a lagged moving average that is chosen to take into account the quarterly futures cycle. We evaluate the standardized volume on day t − 1 relative to regime-day t.

Note that lagged standardized trading volume for both stock and bond futures is appreciably higher for the bad regime than for the good regime. Over the 1997 to 2005 period, the average stock trading volume preceding the bad-regime days is about 22% higher than the lagged moving average. In contrast, the average stock trading volume preceding good-regime days is only about 1.5% higher than the lagged moving average. This regime difference of nearly 21% is statistically significant at a 0.1% p-value. Subperiods results are consistent. Regime differences in the standardized bond volume are qualitatively similar and statistically significant, but somewhat weaker.

Table 2, Panel B reports the time-series variability of our standardized stock and bond futures volume measures within each regime. The variability diﬀerence across regimes is small and not statistically signiﬁcant in any case. This indicates that the volumes are consistently higher in the bad regime and lower in the good regime.

To sum up, we find that the futures trading volume is appreciably higher preceding bad- regime days, as compared to the good-regime days. The higher futures volume in the bad regime may reflect both greater information flows and greater hedging demand. Presumably, agents could be hedging the greater stock risk directly by shorting the stock futures, or collaterally with a cross-market hedge that increases their bond exposure. Episodes with appreciable FTQ pricing influences also seem likely to have higher futures hedging, so this finding is consistent with FTQ having an appreciable pricing influence during the bad regime.

5. Forward-looking Time-series Results

Our regime-switching results indicate that bad-regime days exhibit a sizably negative stock-bond correlation and that the bad-regime days are preceded by a much higher VIX, higher stock illiquidity, and higher stock and bond futures trading volume (we refer to bad-regime day t, with the ‘market condition’ variables using information through the market close on day t−1). In this section, from a forward-looking perspective, we examine whether high values of stock IV, stock illiquidity, and futures volume are informative about a lower subsequent correlation between

13 stock and T-bond returns. In addition to adding the stock illiquidity and futures volume to the mix, we also extend prior work by evaluating whether the recent time-series variability of VIX is informative about the subsequent stock-bond correlation. Given that the variability of stock IV is persistent, we expect that a high IV variability will also be associated with a lower subsequent correlation. Such a ﬁnding would support the notion of “searching” in the relative valuation of stocks and bonds as uncertainty ﬂuctuates.

We sort the day t return observations into quintile subsets, based either on: (1) the lagged level of VIX (from the end of day t − 1), (2) the recent time-series variability in VIX (calculated as the average absolute daily change in VIX over days t − 1 to t − 5), (3) the lagged level of stock illiquidity (both our RRV and PIM measures, calculated using data over days t − 22 through t−1), or (4) the lagged standardized stock and bond futures volume (our detrended, standardized measure from day t − 1). Then, we calculate the correlation for each resulting quintile subset of return observations. When calculating the subset correlation, we follow from prior literature in assuming that the daily mean return is zero for each subset.9

5.1. Empirical Findings: Univariate Perspective

Table 3 reports on correlations for alternate quintile subsets, based on separate univariate sorts on the lagged VIX, the recent variability of daily VIX changes, the RRV stock illiquidity, the

PIM stock illiquidity, our standardized stock futures volume, and our standardized T-bond futures volume. For each sort, the correlation variations are evaluated by calculating two alternate difference-in-correlations. Difference 1 is defined as the bootstrapped difference between the correlation for the highest quintile less the average correlation for the other four quintiles. Difference

2 is deﬁned as the bootstrapped diﬀerence between the average correlation for the largest two quintiles less the average correlation for the lower three quintiles.10

We begin with the VIX sorts. The stock-bond correlations vary substantially and reliably with the lagged VIX (Table 3, Sort 1). Over 1997 to 2005, the stock-bond correlation averages

9In practice, the correlations are essentially the same when using the subset mean in calculating the correlation. 10In this section and Section 6, we rely on bootstrap methods to evaluate whether the differences in the correlations across the different sorted groupings are statistically significant. For our bootstrap procedure, we make draws with replacement for 1000 cycles to generate a bootstrapped distribution. The number of draws for each cycle is equal to the total number of return observations in each grouping.

14 0.019 for the four lowest quintiles of VIX observations versus a correlation of -0.485 following the largest quintile of VIX observations, and the bootstrapped difference is -0.50. The comparable correlation differences for the two subperiods are -0.52 for the first subperiod and -0.47 for the second subperiod. These results are consistent with findings in Connolly, Stivers, and Sun (2005),

(2007).

In our introduction, we suggest that the time-series volatility of stock market uncertainty may be particularly important in understanding the stock-bond correlations. We examine this notion in Table 3, Sort 2. We calculate a 5-day VIX variability, deﬁned as the average absolute daily change in VIX over days t − 1 to t − 5. We believe that ﬁve days is a good compromise for capturing recent VIX variability and for guarding against day-of-the-week patterns. Sort 2 evaluates whether this VIX variability is informative about the stock-bond correlation in day t. If

VIX variability is substantially persistent, then we hypothesis that a high lagged VIX variability will be associated with a lower stock-bond correlation.

Accordingly, we ﬁrst investigate the persistence of this 5-day VIX variability in a simple autoregressive-one regression. The lagged VIX variability over days t − 6 to t − 10 is a highly reliably explanatory variable for the VIX variability over days t−1 to t−5, with a p-value of 0.1% on the estimated coeﬃcient of 0.56 and an R-squared of 31.6%. Subperiod results are consistent.

We conclude that this short-term VIX variability is highly persistent.

For Sort 2 on the 5-day VIX variability, the stock-bond correlations are significantly and appreciably more negative following the high VIX-variability observations. Over 1997 to 2005, the stock-bond correlation averages -0.057 for the four lowest VIX-variability quintiles versus a correlation of -0.37 following the largest VIX-variability quintile. The comparable correlation differences for the two subperiods are -0.43 for the first subperiod and -0.34 for the second subperiod. These correlation patterns are consistent with the flight-to-quality description that we offered in our introduction.

Next, we report results from our stock illiquidity sorts using our RRV measure in Table 3,

Sort 3. The stock-bond correlations also vary substantially and reliably with the lagged RRV.

Over our 1997 to 2005 sample, the stock-bond correlation averages -0.061 for the four lowest RRV quintiles versus a correlation of -0.403 following the largest RRV quintile, with a bootstrapped diﬀerence of -0.34. The comparable correlation diﬀerences for the two subperiods are similar at

15 -0.37 and -0.36.

In Table 3, Sort 4, we report on the comparable investigation using PIM illiquidity. For the PIM over our full sample, the stock-bond correlation averages -0.156 for the four lowest

PIM quintiles versus a correlation of -0.127 following the largest PIM quintile, a statistically insigniﬁcant diﬀerence. Further, the correlation variation with PIM is far from monotonic. Thus, in the time-series for our sample, PIM contains no reliable information about the subsequent stock-bond correlation.

In Table 3, Sort 5, we evaluate how the stock-bond correlation varies with our standardized stock futures volume. The stock-bond correlations also vary substantially and reliably with the lagged standardized stock futures volume. Over our full sample, the stock-bond correlation averages -0.085 for the four lowest quintiles of stock futures volume versus a correlation of -0.354 following the largest quintile of stock-future volume, with a bootstrapped diﬀerence of -0.27. The comparable correlation diﬀerences for the two subperiods are -0.32 and -0.18.

Finally, in Table 3, Sort 6, we report on the comparable investigation for our standardized measure of bond futures volume. Over our full sample, the correlation averages -0.128 for the four lowest quintiles of bond futures volume versus a correlation of -0.249 following the largest quintile of bond-future volume, with a bootstrapped difference of -0.12. The comparable correlation differences for the two subperiods are -0.31 for the 1997 to 2000 subperiod, but essentially zero at 0.013 for the 2001 to 2005 subperiod. Thus, for the T-bond futures volume, the correlation differences are less reliable and are primarily driven by the first subperiod.

5.2. Empirical Findings: Multivariate Perspective with Double-sorts

We next investigate the forward-looking information about the subsequent stock-bond correlation from a multivariate perspective. We evaluate whether there is incremental information in one of our ‘market conditions’ variables, after ﬁrst controlling for another of the ‘market conditions’ variables. We report on the Table 3 variables that performed well in the univariate sorts; speciﬁ- cally, the VIX, RRV illiquidity, and the standardized stock futures volume. Since the correlation variation is stronger for the VIX level than for the recent VIX variability, we focus on the VIX level conditioning in this subsection.

We use a double-sorting procedure that begins by ﬁrst sorting return observations into quin-

16 tiles, based on one of the selected sorting variables. Then, for each quintile of observations from the ﬁrst sort, we further subdivide the observations into groupings based on the upper half and bottom half of a diﬀerent sorting variable.

The intent of our double-sorting exercise is to capture the incremental information of the second sorting variable, while controlling for the ﬁrst sorting variable. The eﬀectiveness of this exercise relies upon observing the following qualitative variation in means across the groupings:

(1) for each quintile of the first sorting variable, the mean of the first-sort variable would be similar in value for each of the two second-sort groupings; and (2) the variation-in-means for the second sorting variable across the two second-sort groupings should be much larger than any associated variation in the first-sort variable. Accordingly, in our appendix, we report the means of each sorting variable for each respective double-sorted grouping of observations. In our view, the variations in means indicate that the sorts do well by these criteria.

For each double sort, Table 4 also reports on the bootstrapped distributions for two alternate differences in correlations across groupings. Difference 1 focuses on times of market stress and is equal to the difference between the average correlation for quintiles 4 and 5 for the observations where the second sorting variable is high and the average correlation for quintiles 4 and 5 for the observations where the second sorting variable is low. Difference 2 reports on the same comparison but across all five quintiles of the first sorting variable.

When the ﬁrst sorting variable is the VIX (Table 4, Sorts 1 and 2), we ﬁnd the following. When the lagged VIX is in its top quintile and the RRV illiquidity or stock futures are relatively high, then the subsequent stock-bond correlation averages -0.556 for these two pair-wise combinations.

On the other hand, when the lagged VIX is in its top quintile and either the RRV illiquidity or stock futures is relatively low, then the stock-bond correlation averages -0.392 for these two pair-wise combinations. Correlation differences 1 and 2, defined above, are reliably negative for all cases in Sorts 1 and 2. Thus, in regard to the subsequent stock-bond correlation, our findings suggest that the RRV illiquidity and the stock futures volume contain incremental information beyond the VIX level.

In Table 4, Sort 3, we examine the case where RRV illiquidity is the ﬁrst sort and VIX is the second sort. First, note that a high RRV combined with a high VIX yields the most negative stock-bond correlation in all of Table 4 at -0.555 for Sort 3 and -0.567 for Sort 1. Next, Sort

17 3 indicates that a high RRV must be accompanied by a relatively high VIX in order for the subsequent stock-bond correlation to be sizably negative. For Sort 3, both correlation diﬀerence

1 and 2 are reliably negative.

Finally, in Table 4, Sort 4, we examine the case where our standardized stock future volume is the ﬁrst sort and VIX is the second sort. Sort 4 indicates a monotonic decrease in the correlation with the stock futures volume for the observations where VIX is also relatively high. However, a high stock futures volume must be accompanied by a relatively high VIX in order for the subsequent stock-bond correlation to be sizably negative. For Sort 4, both correlation diﬀerences

1 and 2 are also reliably negative.

5.3. Discussion of Time-series Results

Our time-series ﬁndings in Section 4 and 5 indicate that a more negative stock-bond correlation tends to be associated with more uncertain times with higher VIX, higher stock illiquidity, higher stock futures trading volume, and higher VIX variability. Figure 2 displays the time-series of

VIX, RRV illiquidity, and our standardized stock futures trading volume. Note that each series has high levels and high volatility around prominent episodes of a sustained negative stock-bond correlation (see Figure 1, Panel B), such as the second half of 1997, the second half of 1998, and

2002-2003. In sum, the relations depicted in Tables 1 through 4 and Figure 2 seem to fit with our introduction’s premise where FTQ and/or liquidity pricing influences may materially influence joint stock-bond price formation in times of heightened market stress.

6. Cross-sectional Variation in Stock-Bond Correlations

In this section, we broaden our work beyond stock index returns and consider cross-sectional variation of the stock-bond correlations across diﬀerent disaggregate portfolios of individual stocks.

We investigate how the stock-bond correlations vary across different stock portfolios, where the different portfolios contain individual stocks sorted on volatility, PIM illiquidity, and/or market capitalization. We consider size and PIM illiquidity as two alternate measures of individual stock liquidity that may be useful in evaluating cross-sectional differences in liquidity. In our introduction, we discuss reasons why smaller, less liquid stocks might exhibit a more negative

18 stock-bond correlation in times of market stress; and other reasons why larger, more liquid stocks might exhibit a more negative stock-bond correlation in times of market stress. Thus, we are interested in whether the stock-bond correlation varies with a stock’s liquidity, after controlling for a stock’s volatility, and vice versa.

For this cross-sectional component of our analysis, we measure the total return volatility,

PIM illiquidity, and size for each individual stock over the preceding 22 trading days. Then, we perform “quintile double sorts” on two of these three lagged variables to form six sets of 25 stock portfolios (volatility-size, size-volatility, volatility-PIM, PIM-volatility, size-PIM, PIM-size). The second-stage sort is based on the sorting-variable distribution of only the stocks remaining after the ﬁrst-stage sort. Thus, each of the 25 portfolios contains approximately the same number of individual stocks.

We then measure the subsequent stock-bond correlation for each portfolio during times of market stress. By market stress, we mean times of heightened market uncertainty when market conditions are likely to include relatively high VIX, high stock illiquidity, and high futures trading.

For our ‘market stress’ categorization, we considered two alternate classiﬁcation methods, one based on a lagged VIX criteria and one based on the regime classiﬁcation from our Table

1 regime-switching estimation. We choose to examine days when the lagged VIX (the VIXt−1 value, relative to day t) is in its top quintile for several reasons. First, for this high-VIX state, the t − 1 values of RRV illiquidity and our standardized futures volumes are also quite high with means of 34.7%, 0.376, 31.0%, and 20.2% for the lagged VIX, RRV illiquidity, our standardized stock futures volume, and our standardized bond futures volume, respectively. By comparison, for the remaining daily observations in the low-VIX state, the means are 20.7%, 0.086, 3.2%, and 4.5% for the lagged VIX, RRV illiquidity, our standardized stock future volume and our standardized bond futures volume, respectively. Second, in the high-VIX state, the time-series variability in VIX is quite high with an average absolute daily change in VIX of 2.00% per day versus 0.89% per day in the low-VIX state. These contrasts between the ‘high-VIX state’ versus the ‘low-VIX state’ are greater than the comparable regime-based diﬀerences documented in

Table 2. Third, examining the top quintile seems a reasonable compromise between capturing times of market stress, yet ensuring adequate observations for parameter estimation.11 Finally,

11Alternatively, if we had used the bad regime from Section 4 as a classiﬁcation method, then 38.4% of the

19 VIX is a well-known and widely used measure of market fear or uncertainty that is observable at a point in time (rather than requiring a time-series to estimate, as for the regime estimation or for estimating the RRV illiquidity).

For our cross-sectional analysis, we present results with the PIM illiquidity. We make this choice because portfolios with stocks sorted by PIM are associated with more sizable and reliable cross-sectional variation in the stock-bond correlations than are portfolios with stocks sorted by

RRV. This is an interesting contrast because, in Section 5, the market’s RRV contained more forward-looking information about the subsequent stock-bond correlation than did PIM. This result may be because RRV is a more complex measure that is too noisy at the disaggregate level to contain reliable information.

As we discussed in Section 5.2 for the time-series double sorts, our intent for the double-sort exercise is to consider the incremental information in the second sorting variable, while controlling for the first sorting variable. Appendix A, Table A3, reports the firm-level means for each sorting variable for each double-sort portfolio. The variation in means for the sorting variables indicates that the sorts do a good job in distinguishing between volatility and size, or between volatility and PIM, but not between size and PIM. Collectively, since we interpret variation in both PIM and size as proxies for liquidity variation across firms, we feel that the double sorts largely do their jobs for our purposes. Discussion in Appendix A provides more details.

Table 5 reports the results for the double-sorting exercise for six diﬀerent two-way sorts.

Table 5, Sort 1, reports on a portfolios formed from a volatility then size sort. For all five volatility quintiles, the “Size5-minus-Size1” correlation difference is negative, with the difference being negative and significant for the two most volatile quintiles. This indicates that large-firm portfolios tend to exhibit a more negative stock-bond correlation, after controlling for a stock’s volatility.

For each sort, we also report on the diﬀerence in the average stock-bond correlation between two sets of opposing “corner portfolios” in the 5x5 matrix of correlations, where one set of “corner portfolios” are the four portfolios that have extreme values of the two sorting variables in one observations would be considered as times of market stress, which seems high in our view. In regard to our regime- switching estimation in Section 4, the top VIX quintile of days are also classiﬁed as a “bad regime” day for 93.4% of the time.

20 direction and the other set of “corner portfolios” are the four portfolios that have extreme values of the two sorting variables in the opposite direction. For example, for our volatility-size Sort 1, this correlation diﬀerence in “corner portfolios” is equal to the average correlation of the four less- volatile and smaller-ﬁrm portfolios (the Volatility1-Size1, Volatility1-Size2, Volatility2-Size1, and

Volatility2-Size2 portfolios) minus the average correlation of the four more-volatile and larger-

ﬁrm portfolios (the Volatility5-Size5, Volatility5-Size4, Volatility4-Size5, and Volatility4-Size4 portfolios). Note that the “corner portfolios” are underlined in the table for each sort. For Sort

1, the diﬀerence in “corner portfolio” correlations is 0.133 (p-value < 1%). For our two primary subperiods, this diﬀerence in “corner portfolio” correlations is similar at 0.142 and 0.155 (both with p-values < 1%).

Sort 2 reverses the sorting order and indicates that more volatile stocks have a more negative correlation, when controlling for size, especially for the largest size quintile. For this sort, the

“Vol5-Vol1” correlation difference is negative for all five volatility quintiles and negative and significant for the largest volatile quintile. For this size-then-volatility sort, the correlation difference between “corner portfolios” (the smaller-firm, less volatile portfolios and the larger-firm, more volatile portfolios) is 0.123 for the entire sample (p-value < 1%), 0.127 for the 1997-2000 subperiod (p-value < 5%), and 0.143 for the 2001-2005 subperiod (p-value < 1%). Thus, both

Sorts 1 and 2 indicate that more volatile and larger stocks have a more negative stock-bond correlation in times of market stress, as compared to less volatile and smaller stocks.

Sorts 3 and 4 in Table 5 involve volatility and PIM. Sort 3 uses volatility first, then PIM. For all five volatility quintiles in Sort 3, the “PIM5-minus-PIM1” correlation difference is positive, with the difference being positive and significant for the high-volatility quintile. This indicates that lower PIM stocks have a more negative stock-bond correlation, after controlling for volatility.

Sort 4 sorts on PIM first, then volatility. For Sort 4, the “Vol5-minus-Vol1” difference is negative for all five PIM quintiles and negative and significant for three of the five PIM quintiles. This indicates that more volatile stocks have a more negative stock-bond correlation, after controlling for PIM. For both Sort 3 and 4, the correlation differences between the “corner portfolios” of interest are statistically reliable, both for the overall periods and one-half subperiods.12

12For Sort 3, the subperiod diﬀerences in “corner portfolio” correlations are 0.129 (p-value < 5%) and 0.161

(p-value < 1%) for the 1997-2000 and 2001-2005 subperiods, respectively. For Sort 4, the subperiod diﬀerences

21 Finally, Sorts 5 and 6 in Table 5 involve size and PIM illiquidity. The results suggest that, for the larger and more-liquid stocks, PIM and size capture much of the same information about the subsequent stock-bond correlation since there is little variation down the columns for the larger and more liquid quintiles. However, for both Sort 5 and 6, the correlation diﬀerences between the “corner portfolios” of interest are statistically signiﬁcant and indicate that larger-cap and lower PIM stocks have a more negative correlation in times of market stress, as compared to smaller-cap and higher PIM stocks.13

Overall, our results indicate that the stock-bond correlations tend to be more negative for the portfolios that contain more liquid stocks (while controlling for a stock’s recent volatility) or more volatile stocks (while controlling for a stock’s recent illiquidity). The evidence suggests a practical view where larger, more-liquid stocks tend to exhibit a more negative stock-bond correlation in stressful times, because: (1) any sizable allocation changes across the stock and bond asset classes must be weighted towards larger stocks since large stocks make up the substantial majority of market cap, and (2) with the lower stock liquidity in stressful times, investors would tend to trade the larger, more-liquid stocks because they are less costly to trade. In the next section, we present additional evidence consistent with this view.

7. Cross-sectional Variation in Means and Turnover with VIX

The correlation variations in Table 5 are sizable and indicate that more liquid stocks and more volatile stocks have a more negative stock-bond correlation in the times of market stress than do less liquid stocks and less volatile stocks. Some readers may feel that the liquidity aspect of this result is counterintuitive if they believe that the prices of smaller, less liquid stocks should be more sensitive to the underlying economic forces behind the sizably negative stock-bond correlations.

In this section, we evaluate how the conditional means of disaggregate stock portfolios vary in “corner portfolio” correlations are -0.146 and -0.154 for the 1997-2000 and 2001-2005 subperiods (both with p-values < 1%), respectively. 13The diﬀerence in correlations for the “corner portfolios” in our two primary subperiods are very similar and statistically reliable. For Sort 5, the subperiod diﬀerences in “corner portfolio” correlations are 0.116 (p-value <

5%) and 0.137 (p-value < 1%) for the 1997-2000 and 2001-2005 subperiods, respectively. For Sort 6, the subperiod diﬀerences in “corner portfolio” correlations are -0.122 (p-value < 5%) and -0.154 (p-value < 1%) for the 1997-2000 and 2001-2005 subperiods, respectively.

22 with two different aspects of VIX behavior, where the stock portfolios are formed from a 5x5 double-sort as described in Section 6. In our tables, we focus on a set of 25 portfolios that is formed from a firm-level volatility sort first and a size sort second. We also discuss comparable results for a second set of 25 portfolios that is formed from a firm-level volatility sort first and a PIM-illiquidity sort second. For our purposes here, as before, we consider size and PIM as alternate measures of cross-sectional variation in liquidity. We also investigate the cross-sectional differences in turnover for the stock portfolios.

We perform two exercises to help understand our prior correlation results and to explore further pricing influences related to FTQ and liquidity differences. The first exercise explores for cross-sectional variation in how stock returns move contemporaneously with the daily changes in VIX.14 Given our cross-sectional correlation results in Table 5, we would expect that changes in VIX would be more negatively related to contemporaneous price changes in larger stocks, lower PIM stocks, and higher volatility stocks (after controlling for the other individual stock characteristics). Affirmative evidence would support the premise offered in our introduction, that relates a negative stock-bond correlation to FTQ with “searching” in the relative valuation of stocks and bonds as uncertainty fluctuates in times of market stress.

The second exercise in this section is intertemporal in nature. We explore how mean returns vary across the disaggregate stock portfolios for both the high-VIX state and the low-VIX state, where (as in Section 6) the high-VIX state refers to day t observations when the VIXt−1 is above its 80th percentile and the remaining observations are considered the low-VIX state. In Section

7.2, we discuss several reasons why the VIX-state may be related to variation in the realized means across the stock portfolios.

Before proceeding, we acknowledge that any analysis of mean returns over modest periods must be considered skeptically. With conditional mean returns, one must be especially careful whether the means represent risk premia or realized means associated with a speciﬁc outcome or

14VIX is calculated under the assumptions of the traditional Black-Scholes framework; but, in reality, its value will also incorporate stochastic volatility risk. In our view, this is not an important distinction for our use because both higher expected volatility and higher volatility uncertainty should lead to a higher VIX value and both relate to more stock market uncertainty. See related discussion in Ang, Hodrick, Xing, and Zhang (2006) when explaining their use of a daily ∆VIX series.

23 microstructure distortion. For example, realized means are lower during recessions although the interpretation is that the risk premia increases in bad economic times (Fama and French (1989).

7.1. Cross-sectional Variation in Mean Returns and a Day’s VIX Change

Table 6 reports how the mean returns vary across the stock portfolios, depending upon whether there was a large increase in VIX that day (Table 6, Panel A) or a large decrease (Table 6, Panel

B). We note that over 46% of the most extreme VIX changes (deﬁned as either the top 10% of

VIX changes (large increases) or the bottom 10% of VIX changes (large decreases)) occur in the high-VIX state. Since large VIX changes are much more likely when the lagged VIX level is high, this mean-return analysis relates to our earlier results that linked the correlation to the lagged

VIX level.

Table 6 reports on 25 stock portfolios formed by a double sort with individual stock volatility

first and size second. There are three prominent findings that seem to fit with our cross-sectional correlation results in Section 6. First, for every volatility quintile, the larger-cap stocks responds much more strongly to the changes in VIX. For example, for the most volatile quintile, the larger- cap stocks have a daily mean of -2.52% for the 10% of days with the largest VIX increases and a daily mean of 2.35% for the 10% of days with the largest VIX decreases. In contrast, for the most volatile quintile, the smaller-cap stocks have a daily mean of -1.39% for the 10% of days with the largest VIX increases and a daily mean of 1.06% for the 10% of days with the largest

VIX decreases. This sizable contrast between the larger-cap and smaller-cap stocks is highly statistically reliable for all ﬁve volatility quintiles (see rows 4, 5, 12, and 13 in Table 6).

Second, we note that the more volatile stocks respond appreciably more than the less volatile stocks to the changes in VIX. For example, for the 10% of days with the largest VIX increases, the larger-cap stocks in the most volatile quintile have a daily mean of -2.52% versus a daily mean of -1.08% for the larger-cap stocks in the least volatile quintile.

Third, we note that the T-bond return is reliably positive for the decile of observations with the largest VIX increase and reliably negative for the decile of observations with the largest VIX decreases. The diﬀerences between the T-bond return and the stock returns are highly reliably positive for the large VIX-increase days and reliably negative for the large VIX-decrease days

(see rows 7, 8, 15, and 16 in the table).

24 We also repeat the same exercise as in Table 6, but where we sort stocks ﬁrst based on their volatility and then on their PIM illiquidity (recall a lower PIM is more liquid). The results are quite similar to those in Table 6, where the lower PIM stocks respond much more to the VIX change. For example, for the most volatile quintile, the lower PIM stocks have a daily mean of

-2.53% for the 10% of days with the largest VIX increases and a daily mean of 2.36% for the

10% of days with the largest VIX decreases. In contrast, for the most volatile quintile, the higher

PIM stocks have a daily mean of -1.47% for the 10% of days with the largest VIX increases and a daily mean of 1.25% for the 10% of days with the largest VIX decreases.

Overall, the results in Table 6 reinforce our conjecture about the premise behind the sizably negative stock-bond correlations in times of market stress. The large changes in VIX: (1) tend to occur when the level is already high, (2) are sizably and negatively related to stock valuations with a greater relation to larger and more liquid stocks, and (3) are positively related to the

T-bond valuations. In our appendix, we document that subperiod results are quite similar.

7.2. Cross-sectional Variation in Mean Returns and the Lagged VIX Level

From a risk-return tradeoﬀ perspective, some readers might feel that the less-liquid stocks should have relatively higher means in the high-VIX state (as compared to the more-liquid stocks), if one assumes that the less-liquid stocks face greater risk in this state of higher volatility and lower liquidity. This view assumes that the prices of the stocks adjust contemporaneously to the higher risk as VIX ascends, and then the subsequent average returns could be interpreted as conditional ex ante risk premia.

However, given our results in Section 7.1, there would seem to be several possibilities why the more-liquid stocks might have higher conditional means in the high-VIX state than the less- liquid stocks (and vice versa for the low-VIX state). First, if the ascent to the high-VIX state is associated with an ongoing, gradual preference shift towards more-liquid stocks over less-liquid stocks, then this could translate to a positive return differential between the more-liquid stocks and less-liquid stocks. Such an effect could be termed a flight-to-liquidity preference shift across stocks. Second, we know that the prices of the more-liquid stocks are more responsive to the large VIX changes associated with the high-VIX state (see Table 6). Then, from a conditional risk-return tradeoff perspective, investors may demand a relatively higher risk premium for the

25 more-liquid stocks in recognition of this higher VIX-volatility risk. Third, given the results in

Table 6, it may be possible that the more-liquid stocks take an excessive price hit during the ascent to the high-VIX state. If the prices move more than justiﬁed by fundamentals, possibly due to temporary microstructure imbalances, then this might translate to a higher realized mean after the VIX has ascended to the high VIX state, as prices recover. Finally, the less-liquid stocks may be slower to react to the bad news leading up to the high-VIX state (because of their lower liquidity), which might contribute to a lower realized means in the high-VIX state, as compared to the more-liquid stocks.

Table 7 reports how the means vary across the stock portfolios, depending upon the lagged level of VIX. We report on 25 stock portfolios formed by a double sort with individual stock volatility ﬁrst and size second, as explained in Section 6.15

We note several prominent and reliable regularities in Table 7. First, for every volatility quintile, the mean of the larger-cap stocks is reliably greater than the mean of the smaller-cap stocks in the high-VIX state (see rows 4 and 5 in the table).16 In contrast, for every volatility quintile, the mean of the smaller-cap stocks is reliably greater than the mean of the larger-cap stocks in the low-VIX state (see rows 9 and 10 in the table). The contrasts in mean differentials are quite sizable. For example, for the most volatile quintile, the differential in the daily mean between the larger-cap stocks and smaller-cap stocks is 0.246% per day for the high-VIX state, which would annualize to over a 60% return difference. The same differential for the low-VIX state reverses sign and is a sizable -0.118% per day, which would annualize to around a -30% return difference.

Next, we contrast the ‘large-minus-small mean return’ diﬀerential for the high-VIX state (in

Panel A) versus the low-VIX state (in Panel B). For all five volatility quintiles, we find that the ‘large-minus-small mean return’ differential for the high-VIX state is sizably and reliably

15We note that the daily mean of the 10-year T-bond return is 0.020% per day in the low-VIX state versus

0.044% in the high-VIX state, a difference that is not statistically significant. 16After finishing this analysis, we discovered that Copeland and Copeland (1999) and Leistikow and Yu (2007) had explored VIX-based trading rules for small-cap versus large-cap allocation decisions. They find that the return differential between larger-cap stocks and smaller-cap stocks tends to be higher following relatively high values of

VIX, using various large-cap and small-cap indices or futures on indices. The goals and method of our exploration here are appreciably diﬀerent, but our mean-return ﬁndings in Table 7 are consistent with their results.

26 greater than the same diﬀerential for the low-VIX state (see rows 11 and 12 in the table). In our appendix, we document that subperiod results are quite similar.

In results not documented in a table, we note that similar patterns are also evident when we sort stocks by volatility ﬁrst, then by PIM. For this case, for the most volatile quintile, the diﬀerential in the daily mean between the low-PIM stocks and the high-PIM stocks is 0.241% per day for the high-VIX state and -0.085% per day for the low-VIX state.

To sum up, the mean-return differentials in Table 7 seem too large to be attributed solely to ex ante risk premia. Further, the negative means for the small, illiquid stocks in the more stressful times cast doubt on an ex ante risk premia interpretation for the mean returns. Rather, the patterns in Table 7 seem likely to reflect some combination of the pricing influences discussed in paragraph two of this subsection. Future research to distinguish among these potential expla- nations may prove interesting. Collectively, our results in Tables 5 and 7 provide an interesting contrast between the correlation patterns (where the larger, more-liquid stocks have the most negative correlation in the high-VIX state) and the mean-return differentials (where the smaller, less-liquid stocks have lower realized means in the high-VIX state).

7.3. Cross-sectional Variation in Stock Turnover and the Lagged VIX Level

As compared to the less-liquid stocks, our prior ﬁndings indicate that the daily returns of the more-liquid stocks: (1) exhibit a more negative stock-bond correlation in the high-VIX state,

(2) are substantially more responsive to the daily changes in VIX, and (3) have relatively higher means in the high-VIX state. In regard to FTQ pricing inﬂuences between the stock and bond asset classes, these results suggest that the larger, more liquid stocks are relatively more inﬂuenced.

Presumably, liquidity is fundamental in understanding these three diﬀerences.

In this subsection, to foster a better understanding of our main return-based results, we brieﬂy contrast the turnover behavior across the diﬀerent stock portfolios featured in Section

6. We follow the same sorting procedure for individual stocks as we did in Section 6; however, instead of the portfolio return, we calculate a daily value-weighted portfolio turnover.

We are interested in two empirical questions. First, in the high-VIX state, do the stock portfolios that exhibit a more negative stock-bond correlation and a greater sensitivity to VIX changes also exhibit a relatively higher turnover, as compared to the portfolios that exhibit a less

27 negative stock-bond correlation and a lower sensitivity to VIX changes? This question addresses cross-sectional variation in turnover levels in the high-VIX state.

Our second question evaluates how turnover varies between the high-VIX state and low-VIX state, for a given stock portfolio. Our earlier ﬁndings suggest that the larger, more-liquid stocks should experience more of an increase (or less of a decrease) in turnover during the high-VIX state, as compared to the smaller, less-liquid stocks. This question addresses cross-sectional variation in turnover diﬀerences between the high-VIX state and low-VIX state.

To examine the first empirical question for this section, we construct the portfolio turnover for the same sets of 25 stock portfolio that are examined in Table 5, Panels A through D. For brevity, we only report representative turnover results that focus on the cases in Table 5 where the portfolio correlation differences are statistically significant. In all these cases, the more liquid and more volatile stock portfolios exhibit appreciably and reliably higher turnover in the high-

VIX state. For Sort 1 with the volatility then size sort, the turnover for the highest-volatility, largest-size stock portfolios is 0.80% per day versus 0.46% for the highest-volatility, smallest-size portfolio. Or, for Sort 2 with the size then volatility sort, the turnover for the largest-size, highest- volatility portfolio is 0.70% per day versus 0.34% for the largest-size, lowest-volatility portfolio.

Or, for Sort 3 with the volatility then PIM sort, the turnover for the highest-volatility, lowest-

PIM portfolio is 0.84% per day versus 0.21% for the highest-volatility, highest PIM portfolio.

Our findings provide a clear affirmative answer to the first empirical question in this subsection.

To examine the second empirical question for this section, we focus on two sets of quintile stock portfolios, one set is constructed with a size sort and the other with a PIM sort. For the size sort, we find that the larger-cap (smaller-cap) quintiles have higher (lower) turnover in the high-VIX state. For size quintiles four and five, the average daily turnover is 0.495% per day in the low-VIX state versus 0.536% per day in the high-VIX state, with this 8.3% increase being statistically significant at a 0.1% p-value. In contrast, for size quintiles one and two, the average daily turnover is 0.41% in the low-VIX state and 0.35% in the high-VIX state, with this 13.4% decrease being statistically significant at a 0.1% p-value. For the PIM sort, we find a similar result where the turnover for the two most liquid stock quintiles increases 8.9% in the high-VIX state, and the turnover for the two least liquid stock quintiles decreases 9.2% in the high-VIX state (with both changes statistically reliable with p-values of 0.1%). Thus, our findings provide

28 a clear aﬃrmative answer to the second empirical question in this section.

We also calculate turnover differences for days with large VIX changes. For day’s when the absolute VIX change is in its top 20th percentile, the turnover for the largest quintile and most liquid quintile exhibits a reliable increase, while the turnover for the two smallest quintiles and two least liquid quintiles exhibit a reliable decrease. Overall, the turnover results support our interpretation that the larger, more-liquid stocks are more influenced by FTQ affects, since these stocks exhibit higher levels of turnover in the high-VIX state, relative to both: (1) the turnover of smaller, less-liquid stocks in the high-VIX state, and (2) their own turnover in the low-VIX state.

8. Conclusions

In this article, we examine the crisis-rich 1997 to 2005 period with the goal to evaluate ﬂight- to-quality- and liquidity-related variation in the correlations and mean returns across stocks and

T-Bonds. Our investigation has four components. Our first two components evaluate the time- series of index-level returns and ‘market conditions’ variables that proxy for stock market stress; specifically, the option-derived implied volatility, stock illiquidity, and standardized futures volume. Our second two components examine cross-sectional comparisons. We evaluate differences in stock-bond correlations, conditional mean returns, and turnover across disaggregate stock portfolios, where the portfolios contain stocks with differing volatility and illiquidity.

The ﬁrst major component of our work proposes and estimates a bivariate, two-state, regime- switching model on daily stock and T-bond futures returns. The estimation identiﬁes a bad regime with much higher stock volatility, a much lower stock-bond correlation, and higher mean bond returns. The bad-regime days are preceded by much higher VIX, higher stock illiquidity, and higher stock and bond futures trading volume. The time-series variability of VIX and stock illiquidity are also appreciably higher during the bad regime.

The paper’s second major component has a forward-looking focus. Individually, we document that the levels of VIX, stock illiquidity, and our standardized stock futures volume, each has a sizable negative relation with the subsequent stock-bond correlation. Additionally, the recent time-series variability in VIX is appreciably negatively related to the subsequent stock-bond

29 correlation. Jointly, while VIX appears the most informative about the subsequent stock-bond correlation, measures of stock illiquidity and futures volume do contain incremental information about the subsequent stock-bond correlation. The collective results of these first two components of our study indicate that the diversification benefits of holding both stocks and bonds increases during times of heightened market stress.

The third major component of our work evaluates how the stock-bond correlations vary across disaggregate stock portfolios comprised of individual stocks with diﬀerent levels of liquidity and volatility. In times of market stress, we ﬁnd that the stock-bond correlations are more negative for disaggregate portfolios that contain more liquid stocks (while controlling for the stock’s volatility) and more volatile stocks (while controlling for the stock’s liquidity). The fourth major component of our work investigates patterns in conditional mean returns and stock turnover, as related to various VIX conditions.

Collectively, our results indicate that FTQ and liquidity pricing inﬂuences have an important role in understanding variations in conditional correlations and mean returns across stock and

T-bonds. The following four aspects of our results support the premise of a “searching” in the relative valuation of stocks and bonds during times of market stress. First, higher VIX levels, higher stock illiquidity, and higher time-series variability in VIX are associated with both: (1) a much lower correlation in the subsequent returns of stock and T-bond returns, and (2) much greater time-series variability in the subsequent VIX and illiquidity values. Second, daily stock returns are negatively and appreciably related to the contemporaneous VIX change, and more- liquid stocks exhibit both: (1) greater responsiveness to the VIX change, and (2) a more negative stock-bond correlation in stressful times. Third, VIX changes are positively related to the T- bond returns. Fourth, when VIX is relatively high, the subsequent turnover is relatively greater for portfolios of more-liquid stocks, as compared to portfolios of less-liquid stocks.

Our findings suggest that stock-bond FTQ influences the prices of larger and more liquid stocks relatively more, presumably because: (1) larger stocks comprise the substantial majority of the market’s capitalization and, thus, take the brunt of stock-bond asset class FTQ effects, and (2) larger, more liquid stocks are less expensive to trade in stressful times.

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32 Table 1: A Regime-Switching Model for Daily Stock and 10-yr T-Bond Futures Returns

This table reports results from estimating the following two-state, bivariate regime-switching model on the daily close-to-close returns of stock and 10-yr T-bond futures:

j j rs,t = µs + σs ηs,t

j j rb,t = µb + σb ηb,t

Where the s and b subscripts indicate stock futures and bond futures terms, respectively; the superscript j indicates regime-specific parameters, with j = 1 for the good regime or j = 2 for the bad regime; rs,t j j j j and rb,t are the stock and bond futures returns; µs and µb are the regime-specific mean returns; σs and σb are the regime-specific standard deviations; and ηs,t and ηb,t are the return shocks, which are modeled as bivariate, standard, normally-distributed, random variables. The j state variable is modeled with constant transition probabilities, pjj , which equals the probability that the regime in period t is j, given that the regime in period t − 1 is j. The means, standard deviations, correlations, and transition probabilities are regime-specific parameters to be estimated. We estimate the model by maximizing the log-likelihood function for the bivariate normal density with regime switching. The sample period is 1997 to 2005 for Panel A, 1997 to 2000 for Panel B, and 2001 to 2005 for Panel C. Standard errors are in parentheses. ∗∗∗, ∗∗, and ∗ indicate 1%, 5%, and 10% p-values for whether the estimated regime-specific means and correlations are different than zero. The final row reports the expected duration of the respective regime.

Panel A: 1997-2005 Panel B: 1997-2000 Panel C: 2001-2005 Good reg. Bad reg. Good reg. Bad reg. Good reg. Bad reg. Coeﬀ. (j = 1) (j = 2) (j = 1) (j = 2) (j = 1) (j = 2) ρj 0.143 -0.429 0.404 -0.289 -0.018 -0.547 (0.032)∗∗∗ (0.038)∗∗∗ (0.047)∗∗∗ (0.056)∗∗∗ (0.037) (0.035)∗∗∗ j σs 0.858 1.706 1.000 1.734 0.738 1.651 (0.028) (0.056) (0.038) (0.076) (0.0208) (0.0615) j σb 0.370 0.397 0.295 0.373 0.416 0.419 (0.0079) (0.013) (0.0100) (0.017) (0.011) (0.015) j µs 0.050 -0.077 0.079 -0.068 0.030 -0.083 (0.024)∗∗ (0.062) (0.043)∗ (0.087) (0.027) (0.081) j µb -0.0007 0.048 -0.0030 0.035 -0.0021 0.061 (0.0102) (0.015)∗∗∗ (0.013) (0.021)∗ (0.0138) (0.021)∗∗∗

p21 0.0125 0.0194 0.0061 (0.0045) (0.0094) (0.0030)

p12 0.0229 0.0308 0.0134 (0.0079) (0.015) (0.0061) Duration 80 days 44 days 52 days 32 days 164 days 75 days

33 Table 2: Implied Volatility, Futures Volume, and Liquidity across Regimes

This table provides good-regime (Gd Rg) and bad-regime (Bd Rg) statistics for implied volatility, futures volume, and stock illiquidity. An observation-day is considered to be in a given regime when the filtered probability is greater than 50%, based on the Table 1 model. The table reports separately for the 1997-2005, 1997-2000, and 2001-2005 periods. The regime-specific mean and its standard deviation (in parentheses) are reported for: (1) the implied volatility from equity index options, VIX; (2) our stock market-wide illiquidity measures, PIM and RRV , where higher values are more illiquid; and (3) our standardized volume measure for stocks and 10-year T-bond futures, StV ol and BdV ol. Panel A report statistics on the lagged (t − 1) observation of these variables that corresponds to a day t regime classification. Panel B reports on the variability of these variable; where abs(∆VIXt), abs(∆St V olt), and abs(∆Bd V olt) are the absolute value of the daily change in the variable in parentheses. The abs(∆PIMt) and abs(∆RRVt) are the absolute change in the illiquidity from day t back to day t − 22, since it takes 22 day to obtain an illiquidity value. The Dif Mns is the difference-in-means between the bad-regime mean and good-regime mean with statistical significance indicated by 1, 2, 3 for p-values at the 0.1%, 1%, and 5% level, respectively.

Panel A: Regime Variation in the Levels of the ‘Market Condition’ Variables 1997-2005 Period 1997-2000 Sub-period 2001-2005 Sub-period Gd Rg Bd Rg Dif Mns Gd Rg Bd Rg Dif Mns Gd Rg Bd Rg Dif Mns

1 1 1 VIXt−1 19.83 30.45 10.62 23.19 28.97 5.78 16.89 31.39 14.49 (0.18) (0.30) (0.36) (0.14) (0.39) (0.411) (0.230) (0.436) (0.496)

1 1 2 PIMt−1 0.158 0.197 0.039 0.194 0.240 0.046 0.129 0.159 0.030 (0.006) (0.0089) (0.0098) (0.009) (0.008) (0.011) (0.004) (0.0080) (0.0082)

1 1 RRVt−1 0.068 0.285 0.218 0.078 0.261 0.182 0.063 0.289 0.226 (x 100) (0.022) (0.055) (0.058) (0.035) (0.093) (0.096) (0.030) (0.059) (0.066)

1 2 2 St V olt−1 0.015 0.223 0.208 0.0035 0.142 0.138 0.011 0.299 0.288 (0.69) (0.041) (0.044) (0.037) (0.059) (0.063) (0.023) (0.050) (0.054)

3 3 Bd V olt−1 0.044 0.138 0.094 -0.033 0.083 0.116 0.092 0.202 0.111 (0.028) (0.60) (0.046) (0.44) (0.046) (0.079) (0.032) (0.042) (0.052)

34 Table 2: (continued)

Panel B: Regime Variation in the Time-series Variability of the ‘Market Condition’ Variables 1997-2005 Period 1997-2000 Sub-period 2001-2005 Sub-period Gd Rg Bd Rg Dif Mns Gd Rg Bd Rg Dif Mns Gd Rg Bd Rg Dif Mns abs 0.781 1.72 0.9361 0.966 1.91 0.9391 0.642 1.49 0.8511

(∆VIXt) (0.020) (0.045) (0.064) (0.035) (0.104) (0.110) (0.021) (0.067) (0.070) abs 0.015 0.021 0.00542 0.017 0.024 0.00743 0.014 0.018 0.0038

(∆PIMt) (0.0011) (0.0017) (0.0020) (0.0013) (0.0028) (0.0029) (0.0017) (0.0020) (0.0026) abs 0.333 0.500 0.1671 0.373 0.560 0.1872 0.301 0.443 0.1422

(∆RRVt) (0.024) (0.035) (0.041) (0.038) (0.059) (0.066) (0.032) (0.034) (0.047) (x 100) abs 0.205 0.215 0.0030 0.183 0.208 0.025 0.224 0.218 -0.006

(∆St V olt) (0.0057) (0.0079) (0.0097) (0.0075) (0.012) (0.014) (0.0083) (0.010) (0.013) abs 0.332 0.307 -0.025 0.354 0.322 -0.032 0.315 0.294 -0.020

(∆Bd V olt) (0.010) (0.011) (0.013) (0.014) (0.017) (0.022) (0.010) (0.020) (0.016)

35 Table 3: Correlation Variation with VIX, Futures Volume, and Stock Illiquidity

This table reports how the correlation between daily stock and T-bond futures returns varies with six different lagged measures related to stock market uncertainty and illiquidity. We report on correlations for quintile subsets of daily return observations, where observations are sorted into quintiles based on the t-1 value of the respective variable. Sorts 1 through 6 report on VIX, the recent 5-day variability of VIX (defined as the average absolute daily change in VIX over days t − 1 to t − 5), our RRV and PIM measures of stock illiquidity, and our standardized stock and bond futures volumes, respectively. Results for the overall 1997- 2005 period, the 1997-2000 sub-period, and the 2001-2005 sub-period are reported separately. The final two columns report on correlation differences. Difference 1 is defined as the bootstrapped difference between the correlation for the highest quintile less the average correlation for the other four quintiles. Difference 2 is defined as the bootstrapped difference between the average correlation for the largest two quintiles less the average correlation for the lower three quintiles. For these differences, statistical significance that the higher-quintile correlation is less than the lower-quintile correlation is indicated by 1, 2, 3 for p-values at the 1%, 5%, and 10% level, respectively, based on the bootstrapped distribution.

Sample Period Quintile 1 Quintile 2 Quintile 3 Quintile 4 Quintile 5 Diﬀ 1 Diﬀ 2 Sort 1: Sorted by VIX 1997 - 2005 0.031 0.022 0.093 -0.070 -0.485 -0.5041 -0.3281 1997 - 2000 0.260 0.318 0.113 0.067 -0.330 -0.5211 -0.3611 2001 - 2005 0.032 -0.045 -0.165 -0.229 -0.577 -0.4741 -0.3421

Sort 2: Sorted by the 5-day Variability of Daily VIX Changes 1997 - 2005 0.041 0.031 -0.078 -0.224 -0.373 -0.3201 -0.2971 1997 - 2000 0.215 0.322 0.011 0.061 -0.275 -0.4321 -0.2911 2001 - 2005 0.046 -0.102 -0.194 -0.405 -0.509 -0.3421 -0.3701

Sort 3: Sorted by Stock Illiquidity, RRV 1997 - 2005 -0.147 -0.014 0.070 -0.154 -0.403 -0.3411 -0.2471 1997 - 2000 -0.080 0.183 0.214 0.037 -0.217 -0.3741 -0.2861 2001 - 2005 -0.259 -0.121 -0.085 -0.251 -0.540 -0.3611 -0.2401

Sort 4: Sorted by Stock Illiquidity, PIM 1997 - 2005 -0.101 -0.036 -0.341 -0.148 -0.127 0.028 0.021 1997 - 2000 0.236 0.018 -0.008 0.033 -0.171 -0.2401 -0.1482 2001 - 2005 -0.184 -0.111 -0.225 -0.464 -0.330 -0.086 -0.2241

Sort 5: Sorted by the Standardized Stock Futures Volume 1997 - 2005 -0.033 -0.029 -0.060 -0.216 -0.354 -0.2711 -0.2461 1997 - 2000 0.133 0.094 0.145 -0.030 -0.236 -0.3211 -0.2531 2001 - 2005 -0.184 -0.161 -0.278 -0.332 -0.414 -0.1751 -0.1641

Sort 6: Sorted by the Standardized T-Bond Futures Volume 1997 - 2005 -0.066 -0.064 -0.141 -0.240 -0.249 -0.1202 -0.1521 1997 - 2000 -0.039 0.235 0.064 0.072 -0.225 -0.3071 -0.1652 2001 - 2005 -0.318 -0.183 -0.244 -0.402 -0.277 0.013 -0.0923

36 Table 4: Correlation Variation with Double-Sorting on the ‘Market Conditions’ Variables

This table reports how the correlation between daily stock and T-bond futures returns varies with the lagged levels of VIX, stock RRV illiquidity, and our standardized stock futures volume, when using a double-sort procedure. For the double sorts, we first sort the daily return observations into quintiles based on the t − 1 value of the primary sorting variable. Then, each quintile of observations are sorted again into the upper or lower 50 percentile of the t − 1 value of the secondary sorting variable, based on the distribution of the secondary variable within the first-stage quintile subset. Sorts 1 through 4 below report on four different two variable combinations. The sample period is 1997 to 2005. The final two columns report on the difference in correlations. ‘Diff 1’ is equal to the difference between the average correlation for quintiles 4 and 5 for the observations where the second sorting variable is high and the average correlation for quintiles 4 and 5 for the observations where the second sorting variable is low.

‘Diff 2’ reports on the same comparison but across all five quintiles of the first sorting variable. We use bootstrapped distributions to evaluate whether ’Diff 1’ and ’Diff 2’ are reliably negative, where 1, 2, 3 indicate p-values at the 1%, 5%, and 10% significance level, respectively.

Quintile 1 Quintile 2 Quintile 3 Quintile 4 Quintile 5 Diﬀ 1 Diﬀ 2 Sort 1: Sorted by VIX (columns 1-5), then RRV Illiquidity (rows 1-2) Low RRV 0.109 -0.027 0.148 0.020 -0.378 -0.1971 -0.1131 High RRV -0.052 0.081 0.041 -0.186 -0.567

Sort 2: Sorted by VIX (columns 1-5), then the standardized Stock Futures Volume (rows 1-2) Low Stock Volume 0.011 0.037 0.115 0.032 -0.406 -0.1611 -0.0733 High Stock Volume 0.052 0.002 0.073 -0.155 -0.544

Sort 3: Sorted by RRV Illiquidity (columns 1-5), then VIX (rows 1-2) Low VIX -0.057 0.059 0.115 -0.018 -0.078 -0.3471 -0.2011 High VIX -0.195 -0.052 0.045 -0.227 -0.555

Sort 4: Sorted by the standardized Stock Futures Volume (columns 1-5), then VIX (rows 1-2) Low VIX -0.024 0.060 0.079 0.056 -0.007 -0.4251 -0.2431 High VIX -0.041 -0.090 -0.135 -0.322 -0.487

37 Table 5: Variation in Stock-Bond Correlations, based on Stocks’ Volatility and Liquidity

This table reports on cross-sectional variation in the stock-bond correlations for various disaggregate stock portfolios. We examine 25 different stock portfolios for each sort, where the individual stocks are double-sorted into portfolios based on a stock’s total volatility, market capitalization, or PIM illiquidity. For the total volatility, we use the stock’s volatility estimated from daily returns over the preceding 22 trading days. For the market capitalization, we use the value as of day t−22 to match the beginning of the volatility and illiquidity estimation period (Size 1 is the smallest grouping). Finally, for the illiquidity, we use our Price Impact Measure (PIM), calculated for each stock over the preceding 22 trading days (PIM 1 is the most liquid grouping). For the double sorts, individual stocks are first sorted into quintiles based on the column variable. Then, each quintile of stocks are sorted again into subset quintiles, based on the row variable. We report the stock-bond correlation for each of the 25 stock portfolios, for observations where the lagged VIX is greater than its 80th percentile. Six different double-sorts, denoted as Sort 1 through Sort 6, are examined. The final value in columns one through five reports on the bootstrapped difference between the correlation for the high grouping of the second sorting variable and the low grouping of the second sorting variable. The final value in column 6 in brackets, [*], indicates the bootstrapped difference between the average of the four underlined correlations in the denoted left-hand corner of the panel and the four underlined correlations in the opposing right-hand corner of the panel. For these differences, p-values that the correlation for the portfolio with stocks that have higher volatility, larger size, or higher liquidity is lower are indicated by 1, 2, 3 for the 1%, 5%, and 10% significance level, respectively, based on the bootstrapped distributions. The sample period is 1997 to 2005.

Sort 1: Sorted by a Stock’s Total Volatility (columns 1-5), then Size (rows 1-5) 1. Low Vol. 2. Vol. 2 3. Vol. 3 4. Vol. 4 5. High Vol. 6. Row Aver. 1. Size 1 -0.225 -0.357 -0.360 -0.378 -0.338 -0.332 2. Size 2 -0.321 -0.385 -0.385 -0.375 -0.364 -0.366 3. Size 3 -0.383 -0.407 -0.432 -0.420 -0.389 -0.406 4. Size 4 -0.378 -0.433 -0.448 -0.461 -0.436 -0.431 5. Size 5 -0.290 -0.384 -0.444 -0.479 -0.444 -0.408 6. Column Aver. -0.319 -0.393 -0.414 -0.423 -0.394 7. Diﬀ. 5-1/ [*] -0.065 -0.025 -0.082 -0.0973 -0.1063 [0.133]1

Sort 2: Sorted by a Stock’s Size (columns 1-5), then Total Volatility (rows 1-5) 1. Size 1 2. Size 2 3. Size 3 4. Size 4 5. Size 5 6. Row Aver. 1. Low Vol. -0.263 -0.368 -0.374 -0.382 -0.294 -0.336 2. Vol. 2 -0.314 -0.381 -0.411 -0.433 -0.376 -0.383 3. Vol. 3 -0.354 -0.384 -0.409 -0.446 -0.439 -0.406 4. Vol. 4 -0.379 -0.383 -0.418 -0.451 -0.463 -0.419 5. High Vol. -0.327 -0.377 -0.424 -0.446 -0.459 -0.407 6. Column Aver. -0.328 -0.379 -0.407 -0.432 -0.406 7. Diﬀ. 5-1/ [*] -0.065 -0.005 -0.045 -0.065 -0.1661 [0.123]1

38 Table 5: (continued)

Sort 3: Sorted by a Stock’s Total Volatility (columns 1-5), then PIM Illiquidity (rows 1-5) 1. Low Vol. 2. Vol. 2 3. Vol. 3 4. Vol. 4 5. High Vol. 6. Row Aver. 1. PIM 1 -0.287 -0.382 -0.443 -0.478 -0.441 -0.406 2. PIM 2 -0.377 -0.431 -0.462 -0.464 -0.445 -0.436 3. PIM 3 -0.368 -0.434 -0.420 -0.443 -0.401 -0.413 4. PIM 4 -0.334 -0.385 -0.416 -0.383 -0.376 -0.379 5. PIM 5 -0.206 -0.349 -0.368 -0.396 -0.319 -0.328 6. Column Aver. -0.314 -0.396 -0.422 -0.433 -0.396 7. Diﬀ. 5-1/ [*] 0.081 0.030 0.073 0.082 0.1212 [0.136]1

Sort 4: Sorted by a Stock’s PIM Illiquidity (columns 1-5), then Total Volatility (rows 1-5) 1. PIM 1 2. PIM 2 3. PIM 3 4. PIM 4 5. PIM 5 6. Row Aver. 1. Low Vol. -0.290 -0.370 -0.403 -0.344 -0.222 -0.325 2. Vol. 2 -0.369 -0.439 -0.427 -0.388 -0.323 -0.389 3. Vol. 3 -0.435 -0.457 -0.413 -0.385 -0.347 -0.407 4. Vol. 4 -0.454 -0.472 -0.434 -0.388 -0.392 -0.428 5. High Vol. -0.454 -0.479 -0.404 -0.386 -0.320 -0.408 6. Column Aver. -0.400 -0.443 -0.416 -0.378 -0.321 7. Diﬀ. 5-1 /[*] -0.1631 -0.1113 -0.004 -0.045 -0.0993 [-0.146]1

Sort 5: Sorted by a Stock’s Size (columns 1-5), then PIM Illiquidity (rows 1-5) 1. Size 1 2. Size 2 3. Size 3 4. Size 4 5. Size 5 6. Row Aver. 1. PIM 1 -0.366 -0.391 -0.417 -0.426 -0.439 -0.408 2. PIM 2 -0.358 -0.375 -0.425 -0.461 -0.451 -0.414 3. PIM 3 -0.329 -0.403 -0.413 -0.466 -0.453 -0.413 4. PIM 4 -0.326 -0.355 -0.417 -0.452 -0.475 -0.405 5. PIM 5 -0.238 -0.358 -0.400 -0.429 -0.493 -0.383 6. Column Aver. -0.323 -0.376 -0.414 -0.447 -0.462 7. Diﬀ. 5-1 [*] 0.1253 0.033 0.023 -0.006 -0.053 [0.125]1

Sort 6: Sorted by a Stock’s PIM Illiquidity (columns 1-5), then Size (rows 1-5) 1. PIM 1 2. PIM 2 3. PIM 3 4. PIM 4 5. PIM 5 6. Row Aver. 1. Size 1 -0.434 -0.409 -0.400 -0.360 -0.210 -0.363 2. Size 2 -0.445 -0.442 -0.384 -0.373 -0.242 -0.377 3. Size 3 -0.436 -0.454 -0.408 -0.391 -0.345 -0.407 4. Size 4 -0.459 -0.446 -0.437 -0.387 -0.370 -0.420 5. Size 5 -0.445 -0.502 -0.435 -0.399 -0.372 -0.431 6. Column Aver. -0.444 -0.451 -0.413 -0.382 -0.308 7. Diﬀ. 5-1 [*] -0.008 -0.0933 -0.038 -0.037 -0.1622 [-0.167]1

39 Table 6: Cross-sectional Variation in Mean Returns on Days with Extreme VIX Movements

This table reports how the realized mean returns of the 10-year T-bond and disaggregate stock portfolios vary on days with extreme VIX movements, where the stock portfolios are formed based on the recent volatility and market-cap of the individual stocks. We examine 25 stock portfolios, where the individual stocks are double-sorted into portfolios based first on the stock’s recent total volatility and then on the stock’s market cap, as explained for Table 5. We report the mean stock returns, in daily percentage units, both for days with extreme VIX increases and for days with extreme VIX decreases. Panel A reports on days when the VIX-change is in its top decile (or the largest increases). Panel B reports on days when the VIX-change is in its lowest decile (or the largest decreases). Rows 4, 5, 7, 8, 12, 13, 15, and 16 report on return differences between rows; for example, row 4 reports on the difference-in-means between row 3 and row 1 (the difference-in-means between the large-size quintile and small-size quintile for each respective volatility column). Row 6 and row 14 report the mean daily T-bond return for the respective observations in each panel. T-statistics are in parentheses, based on heteroskedastic- and autocorrelation-consistent standard errors. Size 1 contains the smallest stocks. The sample period is 1997 to 2005.

Panel A: Grouping One - Top Decile of Daily VIX Changes (Increases) 1. Low Vol. 2. Vol. 2 3. Vol. 3 4. Vol. 4 5. High Volat. 6. Col. 5-1 1. Size 1 -0.38 (-10.9) -0.77 (-13.1) -0.92 (-13.9) -1.04 (-12.5) -1.39 (-13.2) -1.01 (-11.7) 2. Size 2 -0.65 (-14.5) -1.07 (-15.0) -1.22 (-15.7) -1.36 (-15.1) -1.65 (-15.0) -1.00 (-12.4) 3. Size 5 -1.08 (-10.9) -1.38 (-17.3) -1.74 (-19.0) -2.10 (-19.9) -2.52 (-18.6) -1.44 (-12.3) 4. Row 3-1 -0.70 (-12.4) -0.61 (-10.5) -0.83 (-12.1) -1.06 (-14.6) -1.12 (-11.6) 5. Row 3-2 -0.43 (-7.7) -0.32 (-5.02) -0.52 (-7.3) -0.74 (-10.7) -0.87 (-8.8) 6. T-Bd Rt. 0.166 (4.5) 0.166 (4.5) 0.166 (4.5) 0.166 (4.5) 0.166 (4.5) 7. Row 3-6 -1.25 (-14.4) -1.55 (-15.2) -1.91 (-16.7) -2.26 (-17.5) -2.68 (-17.3) 8. Row 1-6 -0.55 (-9.2) -0.94 (-11.5) -1.08 (-12.2) -1.20 (-11.4) -1.56 (-12.4)

Panel B: Grouping Two - Bottom Decile of Daily VIX Changes (Decreases) 1. Low Vol. 2. Vol. 2 3. Vol. 3 4. Vol. 4 5. High Volat. 6. Col. 5-1 9. Size 1 0.18 (5.6) 0.58 (10.3) 0.75 (11.7) 0.83 (12.6) 1.06 (11.1) 0.88 (9.5) 10. Size 2 0.48 (12.1) 0.88 (13.1) 1.08 (13.9) 1.19 (13.5) 1.39 (13.1) 0.91 (10.8) 11. Size 5 0.91 (13.6) 1.33 (17.0) 1.66 (18.4) 1.96 (18.7) 2.35 (14.7) 1.44 (9.3) 12. Row 11-9 0.73 (10.6) 0.75 (9.5) 0.91 (11.1) 1.14 (12.8) 1.29 (9.5) 13. Row 11-10 0.43 (6.6) 0.45 (5.9) 0.58 (7.8) 0.77 (9.4) 0.97 (8.1) 14. T-Bd Rt. -0.095 (-2.8) -0.095 (-2.8) -0.095 (-2.8) -0.095 (-2.8) -0.095 (-2.8) 15. Row 11-14 1.01 (13.4) 1.43 (15.9) 1.75 (16.7) 2.06 (17.1) 2.44 (14.2) 16. Row 9-14 0.27 (5.9) 0.68 (9.6) 0.84 (10.6) 0.92 (11.4) 1.15 (10.6)

40 Table 7: Cross-sectional Variation in Mean Returns, Conditional on the Lagged VIX Level

This table reports how the realized mean returns of disaggregate stock portfolios differ, depending upon the lagged VIX level (VIXt−1), where the stock portfolios are formed based on the recent volatility and market-cap of the individual stocks. We examine 25 different stock portfolios, where the individual stocks are double-sorted into portfolios based on a stock’s recent total volatility first and market cap second, as explained for Table 5. Panel A reports the mean stock returns for observations that fall in the high-VIX state, when VIXt−1 is greater than its 80th percentile. Panel B reports the mean stock returns for observations that fall in the low-VIX state, when VIXt−1 is less than its 80th percentile. Panel C compares the differences- in-means between the two states, or an across-group comparison. T-statistics are in parentheses, based on heteroskedastic- and autocorrelation-consistent standard errors. Size 1 contains the smallest stocks. The sample period is 1997 to 2005. The mean stock returns are reported in daily percentage units. Rows 4, 5, 9, 10, 11, and 12 report on differences-in-means between rows. For example, row 4 reports on the difference- in-means between row 3 and row 1 (the difference-in-means between the large-size quintile and small-size quintile for each respective volatility column).

Low Volat. Vol. 2 Vol. 3 Vol. 4 High Volat. Col. 5-1 Panel A: Grouping One - High VIX State 1. Size 1 -0.083 (-3.64) -0.049 (-1.01) -0.011 (-0.18) -0.078 (-1.15) -0.050 (-0.59) 0.032 (0.44) 2. Size 2 -0.045 (-1.24) -0.021 (-0.34) -0.011 (-0.16) -0.004 (-0.05) -0.037 (-0.38) 0.008 (0.11) 3. Size 5 0.058 (1.04) 0.105 (1.52) 0.126 (1.48) 0.129 (1.23) 0.196 (1.44) 0.137 (1.18) 4. Row 3-1 0.141 (2.72) 0.154 (2.97) 0.137 (2.15) 0.206 (2.71) 0.246 (2.57) 5. Row 3-2 0.103 (2.29) 0.126 (2.57) 0.138 (2.44) 0.133 (1.95) 0.233 (2.74)

Panel B: Grouping Two - Low VIX State 6. Size 1 0.059 (5.98) 0.078 (5.31) 0.090 (5.29) 0.110 (5.92) 0.101 (4.53) 0.042 (2.18) 7. Size 2 0.061 (4.83) 0.061 (3.38) 0.081 (3.93) 0.093 (4.01) 0.085 (3.11) 0.024 (1.23) 8. Size 5 0.014 (0.75) 0.035 (1.69) 0.029 (1.24) 0.014 (0.53) -0.017 (-0.46) -0.030 (-0.94) 9. Row 8-6 -0.045 (-2.73) -0.043 (-2.31) -0.060 (-2.95) -0.096 (-4.26) -0.118 (-3.91) 10. Row 8-7 -0.048 (-3.05) -0.025 (-1.47) -0.051 (-2.70) -0.078 (-3.78) -0.102 (-3.64)

Panel C: Across Group Comparison of Return Diﬀerences 11. Row 4-9 0.186 (3.38) 0.197 (3.59) 0.197 (3.04) 0.302 (4.03) 0.364 (3.78) 12. Row 5-10 0.151 (3.14) 0.151 (2.91) 0.189 (3.26) 0.211 (3.12) 0.335 (3.85)

41 Figure 1: Time-Series of Regime Movements and Correlations for Stock and T-Bond Futures

Panel A displays the time-series of the regime movements from our Table 1 estimation on the daily returns of stock and 10-yr T-bond futures contracts. The ﬁltered probability of being in the bad regime is plotted over time. Panel B displays the time-series of 22-trading day correlations between the stock and 10-yr T-bond futures, where the day t value is the correlation over days t to t + 21 with the restriction that the daily mean is zero. The sample period is 1997 to 2005.

Panel A: Filtered Probability of Being in the Bad Regime, 1997 - 2005 Period

0.9

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

0 1/2/1997 7/2/1997 1/2/1998 7/2/1998 1/2/1999 7/2/1999 1/2/2000 7/2/2000 1/2/2001 7/2/2001 1/2/2002 7/2/2002 1/2/2003 7/2/2003 1/2/2004 7/2/2004 1/2/2005 7/2/2005

Panel B: 22-Trading-Day Correlations for Stock and T-bond Futures (days t to t+21)

0.8

0.6

0.4

0.2

-0.2

-0.4

-0.6

-0.8

-1 1/2/97 7/2/97 1/2/98 7/2/98 1/2/99 7/2/99 1/2/00 7/2/00 1/2/01 7/2/01 1/2/02 7/2/02 1/2/03 7/2/03 1/2/04 7/2/04 1/2/05 7/2/05

42 Figure 2: Time-Series of VIX, RRV Illiquidity, and our Standardized Stock Futures Volume

Panel A displays the time-series of the stock implied volatility index, VIXt−1. Panel B displays the time-series of the RRV stock illiquidity measure, RRVt−1, formed from returns and volume over days t − 22 through t − 1. Panel C displays the time-series of our standardized stock futures volume for day t − 1, constructed to control for the positive trend in volume and the quarterly volume cycle in futures. Note that the t − 1 value for all three series is plotted, relative to regime-day t or correlation t from Figure 1. The sample period is 1997 to 2005.

Panel A: VIX(t-1)

0 1/2/1997 7/2/1997 1/2/1998 7/2/1998 1/2/1999 7/2/1999 1/2/2000 7/2/2000 1/2/2001 7/2/2001 1/2/2002 7/2/2002 1/2/2003 7/2/2003 1/2/2004 7/2/2004 1/2/2005 7/2/2005

Panel B: Return Reversal Measure (t-1)

2 1.5 1 0.5 0 -0.5 -1 -1.5 -2 1/2/1997 7/2/1997 1/2/1998 7/2/1998 1/2/1999 7/2/1999 1/2/2000 7/2/2000 1/2/2001 7/2/2001 1/2/2002 7/2/2002 1/2/2003 7/2/2003 1/2/2004 7/2/2004 1/2/2005 7/2/2005

Panel C: Standardized Stock Futures Volume (t-1)

1.5

0.5

-0.5

-1 1/2/1997 7/2/1997 1/2/1998 7/2/1998 1/2/1999 7/2/1999 1/2/2000 7/2/2000 1/2/2001 7/2/2001 1/2/2002 7/2/2002 1/2/2003 7/2/2003 1/2/2004 7/2/2004 1/2/2005 7/2/2005 43 ` Appendix A: Data Description and Supplementary Resutls

Futures Data. As explained in our introduction, the first two major components of our empirical investigation, corresponding to Sections 4 and 5, rely on futures market data rather than spot market data. We collect daily data from Datastream on six specific futures contracts, covering the sample period of January 1, 1997 through December 31, 2005. For computing returns, we use the continuous futures series computed by Datastream for the S&P 500 futures contract and the Treasury Bond and Note contracts. The continuous series uses the price of the nearest to maturity contract until the month in which the contract expires. Then, the series switches at that point to the next nearest to maturity contract.17 The principal S&P500 contract is traded on the Chicago Mercantile Exchange (CME) both in an open outcry and electronic market. Pit trading takes place between 8:30 a.m. and 3:15 p.m. The E-mini S&P500 contract, introduced in September 1997, trades on the CME’s Globex electronic trading system, with the E-mini contract being one-fifth the size of the full contract. The T-bond futures contracts trades on the Chicago Board of Trade (CBOT), both in an open outcry and electronic market. Open outcry trading begins at 7:20 a.m. and closes at 2:00 p.m. Our trading volume data for the electronic trading of the Treasury Bond futures commences in August 2000, which is the earliest available from the exchange. Electronic trading began in 1994, but trading volume was very small, relative to the open outcry trading, until about 2000. Thus, there are some difference between the stock and bond futures trading times.18 Our empirical work focuses on the 10-year T-bond futures contract for several reasons. First, for our principal analysis, we desire a futures contract where the underlying asset is a longer-term bond, whose maturity should roughly correspond to the bond holdings in a portfolio that is allocated across stock, bonds, and the money market. Second, we desire a very widely-traded contract where prices should rapidly respond to changing conditions. In our sample, the 10-yr T-bond futures contract has the largest trading volume. Over 1997-2005, the average daily trading volumes of the four Treasury debt contracts (including both open-auction and electronic volume) are 367,958 for the 10-yr contract, 287,036 for the 30-yr contract, 192,860 for the 5-yr contract, and 19,251 for the 2-yr contract. Table A1, Panel A, reports on the means, correlations, and volatility for the stock and bond futures returns for our full sample and both subperiods. We report on daily returns from close-to-close prices.19

17The switch of the series as one rolls into the maturity month will result in an artiﬁcial return on that day. Accordingly, when computing returns, we discard those four days a year. 18Making comparisons across the markets using close-to-close returns entails some timing mismatch because the S&P 500 market closes later than the T-bond market. Fleming, Kirby, and Ostdiek (1998) assess the impact of this mismatch by using stock futures prices from 2 PM (CST). Their results when using the stock futures returns from 2 PM were not qualitatively diﬀerent from those obtained using close-to-close returns. 19The average futures returns may seem low, but remember that these average returns do not include the risk-free

44 As expected, note that the stock volatility is much higher than the bond volatility. For the futures trading volume, we wish to capture the economically relevant trading so we include both open outcry pit trading and electronic trading. For the stock futures, we gather the total futures trading volume for the regular S&P 500 futures contract and for the E-mini S&P 500 futures contract. We divide the E-mini volume by five to reflect that the contract is one-fifth the size of the full contract. For the bond futures, we gather the total futures trading volume for both the 10-year Treasury Note contract and the 30-year Treasury Bond contract. We feel that this summed volume will fairly reflect the trading in longer-term Treasury debt futures. As one would expect, there is a significant increase in the futures volume over time. Further, there are substantial quarterly trends associated with the maturity cycle of the contracts. In order to compare trading volume over time, we need a method to detrend and standardize the trading volume. We propose a simple metric that measures the percentage increase over a lagged moving average. Our standardized, detrended, futures volume measure is calculated as follows. First, we calculate a lagged moving average using observations from the prior four months that fall at the same point in the quarterly cycle as the observation in question. For example, consider the trading day of May 18, 2005. May is the second month in the quarterly cycle, where the quarterly contracts mature in March, June, September, and December. Thus, we calculate the average trading volume for all trading days that fall in February 2005, November 2004, August 2004, and May 2004 (the prior four months that occur at the same point in the quarterly cycle). We then take the trading volume from May 18, 2005 and subtract this lagged moving average. Finally, we take the resulting difference and divide by the same lagged moving average. In our view, this measure is intuitively appealing in several dimensions. First, it is in the spirit of a “volume return” in that it is a difference in levels divided by the lagged level. The resulting number can be interpreted as the percentage increase in that day’s volume, relative to the average volume over the lagged moving average. Second, the method serves to standardize the volume by detrending the series without relying on a linear trend that may not be a good fit to the data. Third, the measure does not require the estimation of any coefficients that rely upon observing the entire sample, so the measure can be calculated in real time as soon as the day’s volume is observed. Finally, our moving average uses observations going back one year but only for months that are in the same point in the quarterly cycle. Thus, hopefully, our procedure should act to standardize and detrend the volume series by controlling both for the long-term growth and quarterly seasonalities that are evident in the futures volume series. Stock Implied Volatility and Stock Liquidity Measures. To measure the implied volatility of the U.S. stock market, we rely on the original VIX measure produced by the Chicago Board Options interest rate because of the nature of futures returns.

45 Exchange (CBOE), now denoted as VXO by the CBOE. This daily series measures the implied volatility of a hypothetical at-the-money option on the S&P 100 stock index with 22 trading days until expiration. The CBOE constructs this VIX as a weighted average of the implied volatilities extracted from eight different options. Specifically, these are call and put options written at the two strike prices closest to the money plus the two options (both puts and calls) nearest to expiration, excluding options that are within one week of expiration. The implied volatilities account for dividend payments and the possibility of early exercise. For the stock liquidity measures, we consider two well-known measures that are constructed using only the daily return and volume data from CRSP. We construct measures based on Amihud’s (2002) price impact measure, denoted as PIM, and Pastor and Stambaugh’s (2003) return reversal measure, denoted as RRV. While PIM is closely correlated with price-impact measures based on high-frequency data (Hasbrouck (2005)), RRV adequately captures many of the known historical properties of the stock market liquidity (Pastor and Stambaugh (2003)). We construct our two liquidity measures such that they measure market illiquidity; in other words, higher values indicates markets that are more illiquid. Note that we measure illiquidity over a rolling 22-consecutive-trading-day period. This allows us to construct a daily time-series for aggregate illiquidity. In our time series, the aggregate illiquidity on day t refers to illiquidity measured over a backward-looking 22-trading-day period ending on day t. Therefore, PIMt−1 or RRVt−1 refers to the illiquidity estimated over the period t − 22 to t − 1. From a computational perspective, the two illiquidity measures are similar in some ways. In both, we first estimate the illiquidity measures for individual stocks, and then take a cross-sectional average to get the market-wide illiquidity measure, and finally scale-up the series to make it relatively stationary. For the cross-sectional average, we include only those stocks that meet the following conditions: (a) there should be more than 15 observations to estimate illiquidity measure of individual stocks, (b) it should be a ordinary share (CSRP share code 10 or 11), (c) it should be listed on NYSE/AMEX (CRSP exchange code 1 or 2), (d) share price should be between $5 and $1000, (e) the first (or the last day) that stock appears (or disappears) on CRSP should not fall between the 22-trading-day period. The values for share code, exchange code and share price for purpose of sample stock selection is the values as of the beginning

mt of the 22-trading-day period. To scale up the resulting series, we multiply by , where mt is the total m1 dollar value of the stocks (included in the cross-sectional average) as of the beginning of that period and m1 is the corresponding value for the first 22-trading-day period in January 1997. Our two liquidity measures differ in the first step, i.e. the estimation of illiquidity measure of individual stocks. The return reversal measure of Pastor and Stambaugh is based on the idea that the price changes accompanying large volumes tend to be reversed when market-wide liquidity is low. Specifically, the

46 liquidity value for stock i in a 22-consecutive-trading-day period ending on day t is given by

e e ri,d+1,t = θi,t + φi,t · ri,d,t + γi,t · sign(ri,d,t).voli,d,t + εi,d+1,t (3)

RRVi,t = −γi,t (4) where ri,d,t and voli,d,t are the return and the dollar volume (measured in millions), respectively, of stock

e i on day d in the 22-trading-day period, and ri,d+1,t is the excess return given by ri,d,t −rm,d,t where rm,d,t e is the CRSP value-weighted market return on day d. If we regard sign(ri,d,t) as a proxy for order-flow, then γi,t represents an order-flow return reversal. Note that we flip the sign of return reversal measure in order to make it a measure of illiquidity. The price impact measure of Amihud (2002) measure is based on the idea that there is a positive relationship between the price change and the net order flow which results from the information asymmetry between market makers and traders. Following Amihud (2002), we use the illiquidity ratio as a price impact proxy. We remove the stock-days with zero volume and measure the illiquidity value for stock i in a 22-consecutive-trading-day period ending on day t as:

DXi,t 1 |ri,d,t| PIMi,t = (5) Di,t voli,d,t d=1 where ri,d,t and voli,d,t are the return and the dollar volume (measured in millions), respectively, of stock i on day d in the 22-trading-day period, and Di,t is the number of days the stock i traded (non-zero volume) in the 22-trading-day period. Thus, both our measures are illiquidity measures, where a higher value indicates a less liquid market (or more illiquid market). Table A1 reports on summary statistics for the ‘market conditions’ variables featured in this study; specifically the stock implied volatility, our standardized measures of stock and bond futures trading, and the RRV and PIM measures of stock illiquidity. Table A1, Panel B, reports on the levels of the variables, and Table A1, Panel C, reports on their time-series variability. Variation in Means of Sorting Variables for the Double-sort Time-series Groupings in Table 4. In regard to the subsequent stock-bond correlation, the intent of the double-sorting is to capture the incremental information of the second sorting variable, while controlling for the first sorting variable. With this goal, we hope that variation in means for the sorting variables will exhibit: (1) for each quintile based on the first sorting variable, the mean of the first-sort variable would be similar in value for each of the two second-sort groupings; and (2) the variation-in-means for the second sorting variable across the two second-sort grouping should be much larger than any associated variation in the first-sort variable. Appendix A, Table A2, reports the subset means for each sorting variable for each respective double-sort of observations. In our view, the variations in means indicate that the sorts do well by these criteria. Data Description for the Cross-sectional Analysis in Sections 6 and 7. For our cross- sectional analysis, we form disaggregate stock portfolios comprised of individual stocks with different

47 levels of volatility, illiquidity, and/or size. We begin by measuring these three variables for each individual stock. For a stock’s recent volatility, we use the realized volatility in daily returns in the previous 22- trading-days. For a stock’s illiquidity, we use the Price Impact Measure (PIM) for the stock, also estimated over the same 22-trading-days. For size, we use the market capitalization as of day t-22 to match the beginning of the volatility and illiquidity estimation period. Then, we perform “quintile double sorts” on two of these three lagged variables to form six sets of 25 stock portfolios (volatility-size, size-volatility, volatility-PIM, PIM-volatility, size-PIM, PIM-size). For instance, for the volatility-PIM set of portfolios, we first assign stocks into five quintiles based on their realized volatility; next, within each volatility quintile, we further assign stocks into five quintiles based on their illiquidity. Note that in our portfolio formation, the portfolios are being rebalanced on a daily basis. Finally, to calculate the returns on the disaggregate stock portfolios, we compute the value-weighted average return of all the stocks that comprise that portfolio. We also calculate a value-weighted turnover for each stock portfolio by value-weighting the turnover of each stock that comprises the respective portfolio. Turnover is defined as a day’s trading volume divided by shares outstanding. Variation in Means of Sorting Variables for the Double-sort Cross-sectional Groupings, as Featured in Tables 5 through 7. As previously explained, the intent of our double-sorting exercise is to capture the incremental information of the second sorting variable, while controlling for the first sorting variable. Appendix A, Table A3, reports the subset means for each sorting variable for each respective double-sort of individual stocks. Here, the mean patterns indicate that the sorts do their job for the volatility-size and volatility-PIM sorts. For these four cases, Sort 1 through Sort 4 in Table A3, the variation in means for the first sorting variable is relatively modest down the column, as compared to the substantial variation in means for the second sorting variable down the column. However, for the size-PIM sorts, Sort 5 and 6 in Table A3, the variation in means down the column and across the rows indicate that size and PIM are capturing much of the same information with smaller firms also having a high PIM value. Thus, the double sorts are of limited effectiveness in isolating the incremental effect of the second sorting variable for the size-PIM double sorts. Subperiod Results for Cross-sectional Variation in Mean Investigation in Tables 6 through 7. We also repeat the exercise for mean returns in Tables 6 and 7 separately for our 1997 to 2000 subperiod and our 2001 to 2005 subperiod. The results are in Table A4 (corresponding to Table 6) and Table A5 (corresponding to Table 7) . We find that the results are quite similar in both subperiods and indicate that our results in Tables 6 and 7 are also reliably evident in subperiods.

48 Appendix A: Table A1: Summary Statistics for the Market-level Data Series

Panel A reports the correlations, means, and standard deviations for the stock and 10-yr Treasury bond futures returns. The means and standard deviations are in daily percentage units. Panel B reports statistics on the levels of the following ‘market conditions’ variables: (1) the implied volatility from equity index options (VIX), (2) our standardized, detrended daily trading volume on both the stock and T-bond futures contracts, (3) two different measures of market-wide stock illiquidity, based on the Price-Impact measure (PIM) from Amihud (2002) and the Return-Reversal (RRV) measure from Pastor and Stambaugh (2003). Each variable is explained in detail in Appendix A. Panel C reports statistics on the time-series variability of the Panel B variables, defined as follows: For the stock IV, the table reports on the absolute value of the daily change from close-to-close values. For the futures volumes, we report on the daily change of our standardized, detrended futures measures, as equal to the absolute value of the day t value minus the day t − 1 value. For the liquidity measures, we report on the absolute value of the difference between the day t value (which is constructed from data over days t back to t − 21) and the day t − 22 value (which is constructed from data over days t − 22 back to t − 43). Statistics are reported in separate columns for the overall sample, 1997 to 2005, and for the two subperiods, 1997 to 2000 and 2001 to 2005.

Panel A: Statistics for the Futures Returns 1997-2005 1997-2000 2001-2005 Stock-bond Corr. -0.160 0.011 -0.289 Mean Std. Dev. Mean Std. Dev. Mean Std. Dev. Stock Futures Return 0.0052 1.229 0.024 1.323 -0.010 1.148 T-bond Futures Returns 0.0161 0.380 0.011 0.327 0.020 0.418

Panel B: Levels of the ‘Market Conditions’ Variables 1997-2005 1997-2000 2001-2005 Mean Std. Dev. Mean Std. Dev. Mean Std. Dev. Stock Implied Volatility (VIX) 23.5 7.49 25.3 4.77 22.1 8.87 Stock Futures Volume 0.088 0.387 0.053 0.388 0.115 0.384 Bond Futures Volume 0.077 0.441 0.0083 0.471 0.132 0.407 PIM Illiquidity 0.171 0.058 0.211 0.056 0.140 0.037 RRV Illiquidity (x100) 0.144 0.373 0.144 0.404 0.145 0.346

Panel C: Time-series Variability in the ‘Market Conditions’ Variables 1997-2005 1997-2000 2001-2005 Mean Std. Dev. Mean Std. Dev. Mean Std. Dev. Stock Implied Volatility (VIX) 1.11 1.18 1.30 1.34 0.95 1.01 Stock Futures Volume 0.209 0.189 0.192 0.176 0.222 0.193 Bond Futures Volume 0.323 0.272 0.343 0.299 0.308 0.251 PIM Illiquidity 0.017 0.015 0.020 0.016 0.016 0.015 RRV Illiquidity (x100) 0.392 0.357 0.440 0.386 0.353 0.327

49 Appendix A:

Table A2: Group Means of Sorting Variables for the Time-series Double Sorts in Table 4

This table reports how the means of the ‘market conditions’ sorting variables vary across the groupings for the time-series double sorts in Table 4. First, the daily return observations are sorted into quintiles based on the lagged t − 1 value of the primary sorting variable. Then, each quintile of observations are sorted again into the upper or lower 50 percentile of the lagged t−1 value of the secondary sorting variable, based on the distribution of the secondary variable within the ﬁrst-stage quintile subset. Sorts 1 through

4 below report on four diﬀerent two variable combinations. Each cell in the table reports two means as

X/Y, where ‘X’ is the mean of the ﬁrst sorting variable for that grouping and ‘Y’ is the mean of the second sorting variable for that grouping. The sample period is 1997 to 2005.

Quintile 1 Quintile 2 Quintile 3 Quintile 4 Quintile 5 Sort 1: Sorted by VIX (columns 1-5), then RRV Illiquidity (rows 1-2) Low RRV 13.3/-0.10 20.3/-0.11 23.0/-0.16 26.4/-0.17 34.2/-0.02 High RRV 13.5/0.22 19.7/0.29 23.1/0.28 26.5/0.44 35.3/0.77

Sort 2: Sorted by VIX (columns 1-5), then standardized Stock Futures Volume (rows 1-2) Low Stock Volume 13.4/-0.25 19.9/-0.22 23.1/-0.28 26.3/-0.17 32.9/-0.02 High Stock Volume 13.5/0.31 20.1/0.24 23.1/0.30 26.6/0.33 36.6/0.64

Sort 3: Sorted by RRV Illiquidity (columns 1-5), then VIX (rows 1-2) Low VIX -0.05/18.5 -0.01/15.8 0.02/16.4 0.05/17.9 0.11/22.7 High VIX -0.07/28.4 -0.01/26.0 0.02/25.8 0.06/28.7 0.15/35.0

Sort 4: Sorted by the standardized Stock Futures Volume (columns 1-5), then VIX (rows 1-2) Low VIX -0.38/16.8 -0.12/17.1 0.04/18.3 0.23/18.0 0.62/19.7 High VIX -0.39/25.6 -0.12/26.8 0.05/28.4 0.24/27.5 0.71/35.3

50 Appendix A: Table A3: Firm-level Means of Sorting Variables for Cross-sectional Sorts

This table reports how the means of the firm-level sorting variables vary across the portfolios formed from the cross-sectional double sorts featured in Tables 5 through 7. We form 25 different stock portfolios for each sort, where the individual stocks are double-sorted into portfolios based on sorting on a stock’s total volatility, market capitalization, or illiquidity, as explained in Table 5. For each stock portfolio, we calculate the mean of each sorting variable across the stocks in the portfolio for each observation day. The reported mean in the table is the time-series average of the portfolio’s mean for each portfolio-day, for days when the lagged VIX is greater than its 80th percentile. Each cell in the table reports two means as X/Y, where ‘X’ is the mean of the first sorting variable for the stocks in that portfolio and ‘Y’ is the mean of the second sorting variable for the stocks in that portfolio. For brevity, we only report on the high quintile (quintile-5) and the low quintile (quintile-1) for the second sorting variable.

Quintile 1 Quintile 2 Quintile 3 Quintile 4 Quintile 5 Sort 1: Sorted by Volatility (columns 1-5), then Size (rows 1-2) Small Size 0.178/0.073 0.445/0.152 0.703/0.163 1.125/0.139 3.434/0.104 Large Size 0.238/19.81 0.450/29.64 0.697/28.35 1.105/23.07 2.693/15.17

Sort 2: Sorted by Size (columns 1-5), then Volatility (rows 1-2) Low Volatility 0.099/0.136 0.381/0.237 0.952/0.244 2.377/0.229 21.87/0.260 High Volatility 0.121/3.769 0.386/3.462 0.923/3.029 2.427/2.658 20.86/2.301

Sort 3: Sorted by Volatility (columns 1-5), then PIM Illiquidity (rows 1-2) Low PIM 0.226/0.049 0.449/0.034 0.698/0.035 1.105/0.050 2.712/0.075 High PIM 0.181/74.05 0.446/52.65 0.701/47.17 1.121/52.25 3.239/63.62

Sort 4: Sorted by PIM Illiquidity (columns 1-5), then Volatility (rows 1-2) Low Volatility 0.044/0.234 0.222/0.244 0.853/0.249 3.78/0.232 53.75/0.141 High Volatility 0.046/2.236 0.224/2.783 0.864/3.159 3.68/3.631 57.29/3.426

Sort 5: Sorted by Size (columns 1-5), then PIM Illiquidity (rows 1-2) Low PIM 0.167/4.67 0.460/0.943 1.085/0.286 3.059/0.085 72.36/0.0098 High PIM 0.052/185.21 0.326/18.98 0.81/4.37 1.916/2.00 6.22/0.246

Sort 6: Sorted by PIM Illiquidity (columns 1-5), then Size (rows 1-2) Small Size 0.071/3.60 0.291/1.031 1.069/0.430 4.808/0.173 144.99/0.034 Large Size 0.116/75.80 0.154/5.264 0.659/1.973 2.929/0.939 20.84/0.415

51 Appendix A: Table A4: Cross-sectional Variation in Mean Returns and a Day’s VIX Change: Subperiods

This table reports on the cross-sectional variation in mean returns for days with extreme VIX movements for our two subperiods. The analysis is the same as that in Table 6, but for our two subperiods. Panel A reports on the 1997 to 2000 subperiod and Panel B reports on the 2001 to 2005 subperiod. Grouping One reports on days when the VIX-change is in its top decile (or largest increases). Grouping Two reports on days when the VIX-change is in its lowest decile (or largest decreases).

Panel A: 1997 - 2000 Subperiod (Sorted by a Stock’s Total Volatility (columns 1-5), then Size (rows 1-2)) Low Volat. Vol. 2 Vol. 3 Vol. 4 High Volat. Col. 5-1 Grouping One - Top Decile of Daily VIX Changes (Increases) 1. Size 1 -0.39 (-6.2) -0.65 (-6.5) -0.76 (-7.0) -0.94 (-7.1) -1.36 (-7.8) -0.97 (-7.2) 2. Size 5 -1.07 (-9.0) -1.49 (-11.6) -1.78 (-11.8) -2.10 (-13.5) -2.50 (-11.1) -1.43 (-7.1) 3. Row 2-1 -0.68 (-7.1) -0.84 (-8.6) -1.02 (-9.7) -1.16 (-10.8) -1.14 (-7.6) 4. T-Bd Rt. 0.11 (1.86) 0.11 (1.86) 0.11 (1.86) 0.11 (1.86) 0.11 (1.86)

Grouping Two - Bottom Decile of Daily VIX Changes (Decreases) 1. Size 1 0.11 (2.4) 0.32 (4.8) 0.46 (6.2) 0.57 (6.7) 0.75 (5.7) 0.64 (5.0) 2. Size 5 0.91 (9.3) 1.41 (11.6) 1.69 (13.7) 1.96 (14.3) 2.39 (11.3) 1.48 (6.9) 3. Row 2-1 0.80 (8.1) 1.09 (8.5) 1.23 (9.5) 1.39 (10.1) 1.64 (8.4) 4. T-Bd Rt. 0.026 (0.61) 0.026 (0.61) 0.026 (0.61) 0.026 (0.61) 0.026 (0.61)

Panel B: 2001 - 2005 Subperiod (Sorted by a Stock’s Total Volatility (columns 1-5), then Size (rows 1-2)) Low Volat. Vol. 2 Vol. 3 Vol. 4 High Volat. Col. 5-1 Grouping One - Top Decile of Daily VIX Changes (Increases) 1. Size 1 -0.38 (-10.1) -0.93 (-13.4) -1.14 (-14.5) -1.24 (-12.5) -1.51 (-12.8) -1.13 (-11.0) 2. Size 5 -1.09 (-15.3) -1.37 (-15.4) -1.75 (-15.9) -2.18 (-17.1) -2.67 (-16.8) -1.58 (-10.9) 3. Row 3-1 -0.71 (-10.4) -0.44 (-7.0) -0.61 (-7.4) -0.94 (-10.4) -1.16 (-9.0) 4. T-Bd Rt. 0.32 (7.6) 0.32 (7.6) 0.32 (7.6) 0.32 (7.6) 0.32 (7.6)

Grouping Two - Bottom Decile of Daily VIX Changes (Decreases) 1. Size 1 0.26 (6.6) 0.85 (11.2) 1.02 (11.0) 1.07 (10.6) 1.29 (9.6) 1.02 (7.9) 2. Size 5 0.93 (12.8) 1.28 (14.1) 1.65 (12.6) 2.01 (12.5) 2.35 (10.1) 1.42 (6.7) 3. Row 3-1 0.67 (8.5) 0.43 (5.4) 0.63 (7.1) 0.94 (8.1) 1.06 (6.3) 4. T-Bd Rt. -0.25 (-5.04) -0.25 (-5.04) -0.25 (-5.04) -0.25 (-5.04) -0.25 (-5.04)

52 Appendix A: Table A5: Cross-sectional Variation in Mean Returns and the Lagged VIX Level: Subperiods

This table reports on the cross-sectional variation in mean stock returns, conditional on the VIXt−1 value. The analysis is the same as that in Table 7, but for our two subperiods. Panels A and B report on the 1997-2000 and 2001-2005 subperiods, respectively.

Panel A: 1997 - 2000 Subperiod Low Volat. Vol. 2 Vol. 3 Vol. 4 High Volat. Col. 5-1

Grouping One - Times of Market Stress (Top 20 Percentile of VIXt−1 observations) 1. Size 1 -0.109 (-3.04) -0.088 (-1.63) -0.044 (-0.65) -0.094 (-1.14) -0.126 (-1.07) -0.017 (-0.17) 2. Size 2 -0.066 (-1.56) 0.015 (0.23) 0.026 (0.30) 0.071 (0.74) -0.060 (-0.48) 0.006 (0.06) 3. Size 5 0.263 (3.91) 0.294 (3.27) 0.296 (2.71) 0.299 (2.24) 0.293 (1.53) 0.029 (0.17) 4. Row 3-1 0.372 (5.62) 0.383 (4.71) 0.340 (3.44) 0.394 (3.51) 0.419 (2.72) 5. Row 3-2 0.330 (4.84) 0.279 (3.33) 0.270 (2.91) 0.228 (2.21) 0.353 (2.55) Grouping Two - Remainder of Observations 6. Size 1 0.029 (2.19) 0.053 (2.66) 0.085 (3.81) 0.094 (3.71) 0.060 (1.78) 0.031 (1.02) 7. Size 2 0.038 (2.40) 0.034 (1.43) 0.056 (2.05) 0.070 (2.30) 0.046 (1.24) 0.010 (0.34) 8. Size 5 0.004 (0.14) 0.043 (1.18) 0.031 (0.76) 0.017 (0.37) -0.031 (-0.56) -0.032 (-0.64) 9. Row 8-6 -0.025 (-0.84) -0.009 (-0.28) -0.054 (-1.46) -0.077 (-1.96) -0.091 (-1.87) 10. Row 8-7 -0.034 (-1.18) 0.009 (0.28) -0.026 (-0.76) -0.053 (-1.49) -0.077 (-1.70)

Across Group Comparison of Return Diﬀerences 11. Row 4-9 0.397 (5.58) 0.392 (4.32) 0.394 (3.77) 0.471 (4.09) 0.510 (3.34) 12. Row 5-10 0.363 (5.08) 0.270 (2.90) 0.296 (2.95) 0.281 (2.55) 0.431 (3.02)

Panel B: 2001 - 2005 Subperiod Low Volat. Vol. 2 Vol. 3 Vol. 4 High Volat. Col. 5-1

Grouping One - Times of Market Stress (Top 20 Percentile of VIXt−1 observations) 1. Size 1 -0.072 (-2.41) -0.032 (-0.45) -0.019 (-0.22) -0.100 (-1.07) -0.004 (-0.04) 0.067 (0.66) 2. Size 2 -0.045 (-0.85) -0.040 (-0.44) -0.014 (-0.14) -0.074 (-0.63) -0.053 (-0.40) -0.008 (-0.09) 3. Size 5 -0.055 (-0.70) 0.013 (0.13) 0.067 (0.56) 0.042 (0.28) 0.176 (1.01) 0.232 (1.62) 4. Row 3-1 0.016 (0.21) 0.044 (0.71) 0.086 (1.06) 0.142 (1.43) 0.181 (1.56) 5. Row 3-2 -0.010 (-0.18) 0.053 (0.90) 0.081 (1.17) 0.116 (1.33) 0.229 (2.34) Grouping Two - Remainder of Observations 6. Size 1 0.084 (5.98) 0.102 (4.75) 0.103 (4.17) 0.133 (4.98) 0.137 (4.63) 0.053 (2.12) 7. Size 2 0.084 (4.45) 0.079 (3.02) 0.094 (3.12) 0.114 (3.28) 0.125 (3.16) 0.042 (1.56) 8. Size 5 0.008 (0.38) 0.015 (0.57) 0.009 (0.31) 0.001 (0.01) -0.020 (-0.39) -0.028 (-0.64) 9. Row 8-6 -0.076 (-4.30) -0.088 (-4.54) -0.094 (-4.09) -0.133 (-5.01) -0.157 (-4.00) 10. Row 8-7 -0.076 (-4.73) -0.065 (-3.66) -0.085 (-4.02) -0.114 (-4.57) -0.145 (-4.01)

Across Group Comparison of Return Diﬀerences 11. Row 4-9 0.092 (1.21) 0.132 (2.08) 0.180 (2.27) 0.275 (2.93) 0.338 (2.86) 12. Row 5-10 0.065 (1.06) 0.117 (2.02) 0.166 (2.46) 0.230 (2.86) 0.374 (3.62)