Measuring Tail Risks at High Frequency Internet Appendix Brian Weller⇤ Duke University

November 1, 2016

Additional Properties of the Tail Risk Measure I. Are Bid-Ask Spreads Enough to Recover Factor Risks? Are bid-ask spreads sufficient to recover factor information? Effective spreads impound valuable information reflecting market-making risks, and spreads have been studied extensively in the context of liquidity commonality, or comovement in liquidity as driven by aggregate factors rather than security-specific characteristics. However, as Figure I illustrates, the cross section of spreads rather than of spreads multiplied by volume fails to capture information on aggregate factor risks. Neglecting volume results in negative implied tail risks for nearly two years of the sample, and the resulting cross-sectional estimates are only 35.0% correlated. As the model implies, volume is an essential component for balancing potential gains from intermediation and losses from sources of the spread. Spreads measure only one component of the market maker’s profits and thus cannot be directly compared with stock-level risks. The simple modification to include volume and depth markedly improves the link between asset-pricing risks and microstructure liquidity measures. Repeating my analysis with volume alone in place of the liquidity composite V h/d delivers the purple path in Figure I. This path is only moderately correlated (44%) with that derived with the liquidity composite. Although volume increases roughly proportionally with betas in the run-up to the financial crisis in 2007 and 2008, the liquidity composite and volume-only time series diverge dramatically in exactly the extreme states ⇤ [email protected].

Tel: +1 919 660 1720. Website: https://sites.google.com/site/brianmweller/.

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in which tail risk is likely to be high. The sharp decline in equity shares traded in early 2009 manifests here as an inversion of the usual increasing relationship between volumes and betas, and the resulting cross-sectional slope of volumes with respect to betas becomes negative. This pattern emphasizes the necessity of including bid-ask spreads and traded volume as parts of a composite variable for studying the evolution of tail risk. Spreads and the 2010 Flash Crash The 2010 Flash Crash serves as further illustration that spreads alone fail to capture the evolution of market risks. Without accounting for the increase in volume preceding the crash, the cross section of spreads fails to identify the event as a market factor tail event rather than a level change in liquidity across stocks. In addition, bid-ask spreads alone only weakly anticipate the plunge in prices—the combination of spreads, depth, and volume together provide the most potent signal of near-term price changes. To make these points, I contrast information obtainable from spreads from information obtainable from spreads times volume divided by depth (the dependent variable implied by the market maker’s liquidity provision trade-off). Figure II depicts the evolution of raw spreads and adjusted spreads surrounding the May 6, 2010 Flash Crash. The top figure depicts median spreads by market beta quintile. Throughout the Flash Crash, level differences in spreads across market exposure quintiles 2–5 remain roughly constant. Spreads begin to increase significantly during the 2:15–2:29pm time interval, and spike between 2:45–2:59pm. The smallest

quintile sees especially sharp increases in half-spreads, but this quintile also has the lowest

volume. Levels of spreads adjusted for volume and depth also increase significantly during the same time intervals, but the beta quintiles begin to diverge as early as 2:00–2:14pm. The largest increases to double that of the smallest

quintile’s adjusted spread

quintile during the 2:45–2:59pm interval. This widening disparity

manifests in the enormous increase in implied market tail risks during and preceding the crash. Indeed, the main text demonstrates that this increase in implied market risks completely drives out the increase in implied idiosyncratic risks until the crash is under way. Importantly, adjusted spreads and beta exposures are monotone across beta quintiles throughout the entire period. Higher exposure to market risks indeed translates into larger equilibrium spreads, all else equal. Raw spreads do not share this property. Spreads alone also do not identify which factor is affected. To replicate the Flash Crash plots in the main text using spreads as the dependent variable, I compute the time series of spread slopes with respect to market beta and the date-specific intercept term. I then difference the value at the same quarter hour on May 4, 2010, and divide by the standard deviation of differences for the same quarter hour over the

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63 trading days up to and including May 4, 2010 (a calendar quarter). Figure III presents these implied “tail risks” gleaned from spreads. Slopes of spreads with respect to market exposure do not register an increase until 2:45–2:59pm, and the level effect in spreads begins to respond at 2:15–2:29pm rather than at 2:00–2:14pm. The implied change in idiosyncratic jump risk dominates the change in market factor risk throughout. In sum, spreads provide less forewarning relative to volume and depth-adjusted spreads, and they do not capture meaningful information on aggregate risks until the recovery phase of the crash. II. Robustness to Choice of Assets Selecting the right set of test assets is an important outstanding challenge in empirical asset pricing. Mathematically, the cross-sectional slope of average returns with respect to covariances of portfolio returns and factor returns depends on the set of test assets. It is natural to question whether ⇠M KT is similarly unstable given the parallels between Fama and MacBeth (1973)’s and my two-stage procedure. One approach to assessing the robustness of my measure to different sets of test assets entails trying several combinations of test assets in an attempt to break my model. To discipline my analysis and in keeping with the paper, I focus on the financial sector as a candidate non-market factor and build two “antithetical” sets of portfolios in an attempt to generate large differences in recovered slopes on financial sector risk exposure. I also attempt to break my methodology using a placebo or noise factor on which no assets should reliably load. This latter test stands in for badly chosen test assets with no factor exposures. Challenge Portfolios A typical approach for testing whether new factors are priced consists of grouping assets into characteristic-sorted portfolios and regressing average returns on portfolio betas with respect to a candidate expected return factor. I take this grouping approach here, and portfolios are formed as follows. Because the SPY and XLF have similar dynamics over the sample period, I double sort stocks based on their SPY and XLF betas. For each date t, I generate annual betas from t 252 to t 1 trading days using daily data for each stock i selected in the bootstrap sample. I first assign stocks to XLF beta deciles, and then I assign stocks to XLF-beta conditional market beta deciles to form 100 equally sized portfolios. The ordering of the sort ensures that there is sufficient dispersion in exposure to the financial sector to identify cross-sectional slopes with respect to

F IN .

The second set of portfolios is specifically chosen to minimize this dispersion and thwart my approach. Using the assignments from the first set of portfolios, I mix stocks as follows. For each market beta decile, I take a random permutation of XLF decile assignments. The conditional double sort ensures that each

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market-XLF bin has an equal number of stocks (up to rounding), and the random draws ensure that XLF exposures for the resampled portfolios should be almost identical while maintaining dispersion in market betas. Thus I am left with two sets of portfolios consisting of the same underlying set of stocks: one in which XLF dispersion is maximized and one in which XLF dispersion is low, all while maintaining variation in market factor exposure. The first set of portfolios should identify ⇠XLF well, whereas the second set offers little variation for identifying ⇠XLF at all. In forming portfolios, I take equal-weighted averages of left- and right-hand side variables across portfolio constituents. Betas and liquidity composites for the portfolio are averages of betas and liquidity composites across assets within the portfolio. Averaging is valid because betas and percent risk exposure per hour (r)

aggregate linearly. Averaging occurs after stocks are drawn and before estimation of ⇠t

for each bootstrap

resample r. In other words, the assets included in the test portfolios are themselves random, and the bootstrap standard errors account for this randomness. Figure IV presents the time series of ⇠XLF extracted from both cross sections. The top plot depicts XLF-implied tail risks from the full sample of individual stocks. The bottom-left plot depicts these risks using the 100 XLF-MKT sorted portfolios, and the bottom-right plot depicts these risks using the 100 XLFrandomized, MKT-sorted portfolios. Two features are worth noting. First, point estimates differ across the three sets of test assets. The ⇠XLF series extracted from the XLF-MKT sorted portfolios agrees well with the series extracted from the full cross-section of individual stocks. By contrast, point estimates are typically smaller in the XLF-randomized series, and occasionally these values become negative. Second, the standard error bands depend heavily on the choice of cross section. The variability of ⇠XLF across bootstrap samples is quite similar throughout the sample period for the full cross section and for the XLF-MKT sorted cross section. The XLF-randomized portfolio unsurprisingly features much larger standard errors: the challenge portfolios with randomized XLF assignments do not retain enough cross-sectional variation in financial-sector exposure to identify ⇠XLF well. This feature mirrors the sensitivity of risk premia estimates to the choice of test portfolios in the asset pricing analogue to my approach. Placebo Factors I construct a placebo factor, “PBO,” to test the limits of the two-stage recovery of tail risks. For each date, PBO draws a normally distributed value with mean zero and standard deviation one. I then estimate multivariate betas with respect to the market and PBO, and I use these beta estimates for estimating cross-sectional slopes ⇠P BO . Whereas the XLF-randomized portfolios retain enough variation to obtain sensible point estimates for

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⇠XLF , my model should fail to identify an economically meaningful PBO tail risk series. Because the firststage betas with respect to the random factor are tightly clustered around zero, the second-stage slope estimates with respect to betas are only weakly identified. Importantly, poor identification of ⇠P BO is very different from zero point estimates, and factors with very weak explanatory power for the time series of realized returns may spuriously lead to strong rejections of the null of ⇠P B0 = 0. Figure V plots spread-implied tail risks for the market and placebo factors. The market factor is wellidentified notwithstanding the noise factor, and its estimate matches the one-factor market tail risk estimate. By contrast, the spread-implied ⇠P BO is highly volatile, sometimes negative, and has much larger standard errors. There is little relation between the time series. Notwithstanding these features, the two-stage approach breaks down “too quietly.” The time series of ⇠P BO is not so irregular as to be recognizable immediately as noise. This brief analysis confirms a drawback in my approach and in any cross-sectional approach that relies on betas as generated regressors. The instruments obtained in the first stage (i.e., betas) must be strong enough to identify cross-sectional slopes in the second stage. For the placebo factor, this condition clearly is not satisfied, and the user of my methodology must exercise care in selecting factors for which assets are non-trivially exposed. This problem has parallels and solutions in the literature on cross-sectional asset pricing. For example, a beta-penalization approach as in Bryzgalova (2015) is expressly designed to squeeze cross-sectional slopes to zero when variation in betas is too small to identify them. III. Revisiting the Assumption of i.i.d Idiosyncratic Tail Risk My risk measures are contaminated when an omitted variable correlates with observed factor exposures and the liquidity composite. Market microstructure focuses on inventory risk and adverse selection as the main causes of stock illiquidity, so omitted variables should relate to these sources of risk. The main text includes a stock-date control for the probability of informed trading and discusses how the omission of nonjump adverse selection risk might bias my tail risk estimates. In addition, I control for inventory risk using the adverse selection component of the bid-ask spread as a dependent variable in place of the effective bid-ask spread. These analyses significantly narrow the scope of potential omitted variables. A large literature supports a link between idiosyncratic risk and the bid-ask spread, and several papers also find that idiosyncratic risk correlates positively with market betas in the cross section (e.g., Bali and Cakici (2008) and Bollerslev, Li and Todorov (2016)). For these reasons, omitted idiosyncratic risk might contaminate ⇠M KT . I now investigate the extent to which omitted idiosyncratic tail risk contaminates my 5

factor tail risk estimates. An idiosyncratic-volatility augmented first-stage regression is

rit = ↵i +

X

(t) ik fkt

+

idio i it

k

(1)

+ ✏it , 8i.

In light of the strong one-factor structure in idiosyncratic volatility (Herskovic, Kelly, Lustig and Nieuwerburgh (2016)), it makes sense to consider idiosyncratic volatility as a stock-specific fixed term plus a loading on a time-varying idiosyncratic volatility factor. I can then rewrite Equation (1) as

rit = (↵i +

i ⌘i0 )

+

X

(t) ik fkt

+

idio i ⌘i1 t

+ ✏it = ↵ ˆi +

k

X

(t) ik fkt

+ ˆi

idio t

+ ✏it .

(2)

k

where ⌘s are obtained from a time series regression of idiosyncratic volatility on an idiosyncratic volatility factor,

idio it

= ⌘i0 + ⌘i

idio . t

The short-form regression

ik

is contaminated if and only if idiosyncratic

volatility is correlated with returns and factor realizations, and this concern is readily addressed either by including estimated idiosyncratic volatility directly to the regression or by adding the idiosyncratic volatility loading ˆi only. Empirically this bias is virtually nonexistent: the panel correlation of without an auxiliary idiosyncratic volatility term

idio t

M KT

with and

is 99.5%, and the corresponding average cross-sectional

correlation is also 99.5%. More generally, slow-moving characteristics such as firm size or book-to-market ratios may be omitted from the first stage, as their effect is largely absorbed by the (rolling window’s) stock fixed effect ↵i . More concerning is the potential for bias in the second-stage regression. The baseline second-stage regression is

✓

Vh d

◆

= ⇠˜t + it

X

⇠tk

ik

+

it .

(3)

k

Bollerslev and Todorov (2011) cannot reject a rank-one covariance matrix for continuous and left- or rightjump variation for S&P 500 futures. I assume that a comparable relation holds for idiosyncratic continuous and jump variation and that idiosyncratic jump variation inherits the factor structure of idiosyncratic volatility. The model of Equation (3) is then misspecified because a time- and asset-dependent idiosyncratic tail risk term is missing on the right-hand side. The correct specification is ✓

Vh d

◆

= ⇠˜it + it

X

⇠tk

k

6

ik

+

it ,

(4)

= ⇠ˆt +

X

⇠tk

ik

k

⇣ + ⇠˜t ˜i + ⇠˜i +

it

⌘

(5)

,

where again it makes sense to differentiate fixed and factor-driven components of idiosyncratic tail risk. The addition of the idiosyncratic tail risk term highlights two related sources of omitted variable bias. The first is omitted variable bias originating from cross-sectional variation in the fixed component of idiosyncratic tail risk, ⇠˜i . In a market model, the bias on ⇠tk is proportional to the residual covariance of

i,M KT

and

⇠˜i after orthogonalizing both with respect to ˜i . This bias is constant over time if the relationship between betas and idiosyncratic tail risk is stable. Hence one simple approach to addressing this issue, and the one taken throughout the paper, is to interpret recovered tail risks ⇠tk relative to a baseline value—how does variation in ⇠tk relate to realized jumps, for example. The downside to the approach is that ⇠tk cannot be interpreted in absolute terms if the bias is not small. This approach also does not address the second source of omitted variable bias. Omitting regression biases ⇠tk if cov

Vh d it

, ˜i and cov (˜i ,

ik )

i

from the

are different from zero. Market makers must protect

themselves against idiosyncratic tail risks, so the first covariance is positive on economic grounds. The second covariance between idiosyncratic tail risk factor loadings and betas is also likely to be positive on the basis of a positive correlation between total idiosyncratic risk and

i,M KT .

I address this potential contamination in two ways. First, I add an idiosyncratic volatility factor in the second-stage regression as motivated by Equation (5), ✓

Vh d

◆

= ⇠ˆt + it

X

⇠tk

ik

+ ⇠˜t ˜i +

it .

k

This approach cleans my estimates of time-varying bias in ⇠M KT . Second, I add estimated stock-date idiosyncratic volatility as a control in the second stage as in Equation (4). This addition should eliminate both time-varying and additive biases to ⇠M KT . For both approaches, I construct idiosyncratic volatility as the standard deviation of the difference between realized stock returns and CAPM-implied stock returns over the previous 63 trading days. The idiosyncratic volatility factor is the simple average of idiosyncratic volatility across stocks for a given date.1 Figure VI compares the two idiosyncratic-volatility controlled estimates (bottom row) with the baseline market model estimates (top plot). Adding the idiosyncratic volatility factor loading has virtually no effect on extracted market tail risks ⇠M KT despite the comovement of the idiosyncratic volatility factor and mar1 Results are robust to using Fama-French three-factor residuals or total returns in the volatility estimate. Across models, the panel correlations of idiosyncratic volatility and time-series correlations of the idiosyncratic volatility factor exceed 99%. Herskovic et al. (2016) confirm this high correlation among idiosyncratic volatility measures in their Figure 2.B.

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ket volatility (98.7% hourly correlation with the market-only series). There are two reasons for the close alignment of the series. First, the correlation between idiosyncratic volatility and market betas for the full cross section of individual stocks is only 4.1% in the panel and 3.1% for the average cross section, far weaker than for (1) large portfolios (Bali and Cakici (2008)) and (2) S&P 500 stocks only (Bollerslev et al. (2016)). Second, although theory and empirics relate idiosyncratic volatility with bid-ask spreads, the connection between the liquidity composite and idiosyncratic volatility is tenuous because idiosyncratic volatility is also strongly inversely related to volume. This feature is also present in Table 5 of Bali and Cakici (2008) in their use of the Amihud illiquidity measure, |r| /V . Idiosyncratic volatility is effectively proportional to total volatility, so the six-fold increase in idiosyncratic volatility between their quintiles 1 and 5 should be matched by a six-fold increase in Amihud illiquidity if V is constant across quintiles. Instead, we see a 48-fold increase in Amihud illiquidity, which roughly corresponds with a coincident eight-fold decrease in volume. Comparing the top and bottom-left plot changes the picture only slightly. Adding idiosyncratic volatility directly rather than stocks’ factor loadings delivers the same time-series pattern (98.4% hourly correlation with the market-only series), but the idiosyncratic-volatility control has a first-order effect of shifting the series down by a few tenths of a percent per hour. The original series is slightly upward-biased because of the small positive correlation of idiosyncratic volatility with s and the liquidity composite. IV. Limitations Completely Predictable and Unpredictable Events The tail risk measurement technique assumes that the timing of jump events is imperfectly predictable using information available to market makers. Although the two-stage regression technique mechanically delivers estimates in other instances, the resulting ⇠ coefficients should be interpreted with care. First, market makers cannot adjust their spreads in advance of events such as natural disasters that are not anticipated by any agents in the economy and accessible to market makers in signals such as order flows. This methodology recovers the market’s perception of tail risks, which may differ sharply from true latent tail risks in such settings. However, after the initial shock, these quantities may again coincide if the source of risk persists beyond the initial event. At the opposite extreme, the tail risk extraction technique may fail if jumps occur at precise and prescheduled times. In the limit as the event’s arrival time becomes predetermined, the key equilibrium condition breaks down. Indeed, several important macroeconomic news disclosures—including FOMC announcements from 2013-onward—share this property. 8

The key failure as local jump arrival rates go to infinity is in the assumption that the jump always exceeds the half-spread. In the case of scheduled arrivals, the zero-profit condition can only be satisfied when the spread weakly exceeds the entirely of the anticipated jump distribution. It follows immediately from this equation that as the density of the waiting time distribution collapses to zero except at a single point (which can be approximated informally with

k

! 1), half-spreads h converge to a multiple of the conditional

maximum of rkd . Because uninformed volume in a tight enough neighborhood around the announcement is effectively zero and jump sizes scale with factor betas, the cross section of half-spreads without a volume adjustment reveals this conditional maximum factor jump size. In the case of FOMC announcements previously considered, there is enough residual uncertainty about the precise announcement time to remain in the Poisson arrival regime, as the main text’s FOMC discussion suggests graphically. Although spreads are elevated and quoted depth is reduced around these events, they do not achieve the extreme values that would suggest a “worst case” and a scheduled-arrival regime. In practice, even when announcement dates are known, uncertainty is not immediately resolved as information continues to be digested after the initial announcement event, and the latent asset value may continue to jump to the market maker’s detriment. For this reason, the “echo” of the initial announcement may contribute to the empirical success of this methodology even when the model assumptions are not satisfied at each instant. Long-Horizon Forecasting I exposit a model of calculated market maker liquidity provision in response to potential factor jumps in the very near term. The conclusion of the main text suggests that we can view elevated tail risks as a necessary but not sufficient forewarning of more persistent macroeconomic fluctuations or rare disasters. As additional support of this view, Figure VII plots the autocorrelation function of market tail risk estimates by hour from 2004–2013. The top subfigure plots raw autocorrelations, and the bottom subfigure illustrates autocorrelations net of intraday patterns, where I adjust for intraday variation in jump risk by dividing each value by the corresponding average market tail risk for the same hour across all dates in the sample. The raw series reveals substantial persistence in tail risks, but the persistence is masked by the pronounced intraday patterns in when the market is likely to see and respond to major news events. The pattern-adjusted series indicates that tail risk levels are highly persistent: the half-life of abnormal tail risks is six weeks. The tail risk measure thus typically informs on longer-run risks far exceeding the market maker’s holding horizon. Persistent elevations in market tail risk agree with the Andersen, Fusari and Todorov (2015) perspective of serially correlated sequences of small crashes contributing to more dramatic economic

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downturns. V. Tail Risk and Realized Jump Decay Rates Existing studies find that realized jump variation decays much more quickly than realized continuous or total variation. As seen in the previous section, latent jump risk appears to decay more slowly. There is an important theoretical distinction between the autocorrelation structure of realized jump variation and latent jump risk. Latent jump risk resides continuously in the background and only occasionally delivers a jump realization. Consequently realized jump variation and latent tail risk may have very different persistences. Andersen, Bollerslev and Diebold (2007), Andersen, Bollerslev and Huang (2011), and Bollerslev and Todorov (2011), among others, consider the persistence of realized jump and continuous variation processes. I focus on Andersen, Bollerslev and Huang (2011) as a representative paper in this literature. While continuous variation’s serial correlation has effectively zero probability of having occurred by chance (as assessed by Ljung–Box autocorrelation tests), jump variation has much weaker serial correlations. The authors further decompose the jump variation measure and find that the realized jump intensity It is somewhat more persistent than total jump variation Jt , and jump sizes St are close to independently distributed. Adding measurement error to a process may severely bias autocorrelation estimates. However, the firstorder measurement error here is qualitatively different from a pure noise process. Consider the building blocks of jump variation and latent jump risk: It (St ) is a single realization of a process governed by t ⇥ ⇤ ¯ k ), and under mild conditions the average of It (St ) across time is a consistent estimate of (E rkd |rkd > h ⇥ ⇤ ¯ k ). Notwithstanding this link, persistences of It , St , and their product Jt may the average t (E rkd |rkd > h ⇥ ⇤ ¯ k , and their product ⇠t because jumps are so diverge dramatically from the persistences of t , E rkd |rkd > h rarely realized.

I investigate this intuition with simulations based on the parameters in Table 1 of Andersen, Bollerslev and Huang (2011). For each date, the count of realized jumps is distributed Poisson with intensity the jump indicator It equals one if a day has at least one jump. I initialize of days (

0

= 0.486) and let the log of

µ = log (0.486) ⇡

t

0

t,

and

to deliver jumps on 8.6%

evolve as a discrete-time AR(1) process with a long-run mean of

0.722,2 log

t+1

= µ + ⇢ (log

t

µ) + ✏t .

(6)

Errors ✏t are serially independent. Each simulation runs for 3,000 “days” with a 500-day burn-in period to 2 More

than 8.6% of days contain jumps when

> 0. The simulation depicted features jumps on 13.6% of days.

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avoid sensitivity to initial conditions. Processes have log-intensity persistence of ⇢ = 0.99 and log-intensity volatility of

= 0.10. I select this choice of volatility to ensure that jump intensities typically do not become

large relative to

= 2 (referenced as “unlikely” in Huang and Tauchen (2005)). Notably, my simulations

simplify the dynamics of jump realizations by excluding interdependencies between

t

and It , e.g., those

responsible for clusters of jumps within short periods. The top panel of Figure VIII presents a representative sample path for

t

and It . In the sample, the

intensity and magnitudes of realized jumps are strongly related to the tail risk time series. Periods of high jump intensities are associated with more jump realizations, and periods of low

t

have streaks of zero-jump

days. By contrast, realized jump variation reveals little about the high and stable persistence of latent jump risk. The bottom panels of Figure VIII present distributions of autocorrelation functions for latent jump intensities (left) and realized jump intensities (right) across 1,000 simulations. Autocorrelations decay far more slowly for the latent jump intensity process the average autocorrelation for

t

t

than for the realized jump intensity process It . Throughout,

is roughly 5-6 times the comparable autocorrelation for It , and the distri-

butions of autocorrelation functions only overlap for the most extreme simulations at 18 or more lags. This feature arises despite a tenfold spread in

t

over typical simulation paths.

Taken together, these results confirm a disconnect between the dynamics of latent jump intensities and realized jumps. This disconnect occurs because the autocorrelation of realizations cannot disentangle lack of persistence in latent risk from strings of non-realizations of a low-probability Bernoulli random variable. Even for higher values of , the rarity of jump realizations ensures that jump risk and jump variation dynamics necessarily differ and that the latter are not very informative for the former. An important contribution of my paper is that it opens the door to improved estimation of existing jump-diffusion (or more sophisticated) models of asset price dynamics. Existing approaches struggle to evaluate the high-frequency dynamics of latent jump risk as a distinct object from realized jump variation, although they may have quite different properties in practice. For this reason, I have few benchmarks for assessing whether the persistence in my hourly tail risk measure is “too high.” I address this problem by comparing the persistence of my measure to the weekly tail risk measures of Bollerslev and Todorov (2014). At the weekly frequency, the memory of the tail risk series are roughly equal and much shorter than that of the VIX. The one- to six-week autocorrelations of the Bollerslev and Todorov (2014) left-tail risk series are 0.90, 0.82, 0.76, 0.73, 0.69, and 0.64, and the same autocorrelations for the right-tail risk series are 0.84, 0.75, 0.74, 0.77, 0.73, and 0.66. The comparable values for weekly averages

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of my series are 0.89, 0.79, 0.74, 0.69, 0.67, and 0.65. Although the left-tail risk series is contaminated by persistence in risk premia, the right-tail risk series is much less likely to be affected, and persistences are very similar across all series regardless. By contrast, the VIX is far more persistent than any of the tail risk measures, with corresponding values of 0.97, 0.94, 0.91, 0.87, 0.84, and 0.80. These high persistences reflect the trying macroeconomic environment of the 2004–2013 period: a low-frequency component likely contributes more than is typically the case to variation in tail risk, resulting in relatively high persistence of my and related measures of tail risk.

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Comparison with Alternative Tail Risk Measures VI. Comparison with SPY-Implied Tail Risk I use a single asset in developing the theory underlying my cross-sectional approach, and I only later overlay a factor structure on discontinuous returns. Focusing on the SPY rather than cross-sectional slopes is almost exactly analogous to using E [ft ] rather than cross-sectional regression estimates of

in standard

two-stage pricing regressions. Indeed the forward-looking nature of bid-ask spreads implies that even a conditional version of the analogy may be appropriate, “ t ” = Et [ft ]. The main text offers theoretical motivations for additional robustness of a cross-sectional measure relative to using a single factor-mimicking asset or portfolio such as the SPY.3 In short, for single assets, model omissions and measurement errors need not wash out—there are no guiding large N asymptotics on which to rely. Nevertheless, the SPY-implied version may suffice for some applications, and I explore below the properties of tail risk implied by variation in liquidity of the SPY. I start by contrasting the tail risk series obtained from the SPY only to the tail risk series obtained via the paper’s two-stage cross-sectional methodology. To construct the SPY’s pseudo-tail risk series, I compute hourly averages of minutely V h/d, where V is the total realized volume in shares, h is the quoted half-spread, and d is the bid and offer depth summed across exchanges. I use the quoted half-spread rather than the effective half-spread because the effective half-spread for the SPY is occasionally negative, which in turn delivers negative implied risks about 20% of the time. Reassuringly, the SPY’s liquidity composite delivers a series broadly similar to the cross sectional approach: the initial, one-asset model is not too far off the mark. At an hourly frequency, the tail risk series and the SPY series have a correlation of 71.0%. Applying rolling 10-day smoothing gives a correlation with the tail risk series of 92.3%. To put these values in context, the hourly correlation of ⇠M KT and VIX is also 71.0%, and the weekly correlation of ⇠M KT and realized volatility is 93.0%. Figure IX provides a visual comparison with raw and 10-day smoothed versions of the cross-section implied and SPY-implied tail risk series. The smoothed plot confirms that the shapes of the cross-section implied and SPY-implied series are similar up to rescaling. The close link between series breaks down in late 2008 or conversely, in all dates except for 2008—which statement better describes the series cannot be obtained from scale information alone.4 Before speculating on the cause of disagreement between the series, I evaluate which series better captures 3A

priori I rule out applicability of this simple alternative to non-tradable factors for obvious reasons of not having observable liquidity characteristics. 4 The magnitudes of the SPY-implied tail risk series are much larger than those of the cross section-implied series, so only statements relative to a normalization such as standard deviation can be made.

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market tail risk, the object of my study. To this end, I modify the baseline associative and predictive SP Y XS regressions for realized SPY jump tails to compare the performance of ⇠M KT and ⇠M KT in a horse race

setup. This test poses a difficult challenge for the the cross-sectional approach because the left-hand side jump realizations may be linked to liquidity deterioration in the SPY through channels that do not spill over into other assets. For example, a sudden break in data feeds may dramatically affect an index asset without immediate consequences for the underlying securities (as postulated by Menkveld and Yueshen (2015)). Table I presents results of this comparison. The SPY-only liquidity composite captures substantial variation in realized medium-scale SPY jumps. The leading coefficient in the ⇠-only specifications with the SPY composite varies from about 55% to 70% as large as the comparable coefficient obtained via the cross-sectional approach. This reduced explanatory power may suffice in applications in which only XS coarse assessments of market tail risk are required. However, when paired with cross-section implied ⇠M KT , SP Y ⇠M KT performs far worse. Its explanatory power falls dramatically in the basis-point jump specifications XS SP Y and disappears entirely in the spread jump specifications. In short, ⇠M KT dominates ⇠M KT for explaining

contemporaneous jumps and for predicting jump realizations. SP Y XS Columns (3) and (6) suggest that the useful residual variation in ⇠M KT net of ⇠M KT is largely spanned by SP Y XS continuous variation and VPIN, i.e., ⇠M KT picks up volatility information that ⇠M KT does not. This feature SP Y XS helps explain the large discrepancy between ⇠M KT and ⇠M KT at the height of the Great Recession (or at all

other dates). Both volatility and jump risk are greatly elevated during this period, and the elevated volatility may translate into greater inventory risk associated with liquidity provision in all assets. However, unless this heightened inventory risk is cross-sectionally correlated with market betas, my two-stage estimate assigns XS the additional cost of liquidity provision to an idiosyncratic term rather than to ⇠M KT . By contrast, the

additional costs of crisis-era market making—suggested by Nagel (2012), among others—translate directly SP Y into greater hV /d for ⇠M KT regardless of whether the cost aligns with risk factors. These other risk sources SP Y XS likely also contribute to the much larger magnitudes of ⇠M KT relative to ⇠M KT . The lack of robustness of SP Y ⇠M KT to this contamination suggests that the simple index-asset alternative breaks down precisely when a

tail risk measure is most necessary. As a final comparison, I assess how the respective tail risk measures fare during an extreme event, the 2010 Flash Crash. Like the volume-synchronized probability of informed trading (VPIN) measure of Easley, López de Prado and O’Hara (2012), V h/d can be constructed using a single time series, and hence may be faster to compute in real time. Figure X compares performance of the ⇠M KT measures by quarter hour around the Flash Crash. The

14

figure plots market tail measures from 12:45pm on May 5, 2010 through 12:45pm on May 7, 2010 for each quarter hour from 9:45am to 3:45pm. To capture innovations and place risk changes in context of normal intraday and slow-moving macroeconomic variation, I difference the value at the same quarter hour on May 4, 2010, and divide by the standard deviation of differences for the same quarter hour over the preceding 63 trading days up to and including May 4, 2010 (a calendar quarter). Two features are readily apparent. SP Y First, the ⇠M KT measure almost perfectly tracks realized volatility prior to the Flash Crash. Unlike the

cross-sectional variant, the SPY-only measure provides no additional predictive power. Second, the Flash XS Crash itself is a much more dramatic event for the SPY than for the aggregate market: ⇠M KT increases SP Y by 99 standard deviations during the event, whereas ⇠M KT increases by 737 standard deviations. Menkveld

and Yueshen (2015) explain this phenomenon as a cross-market arbitrage breakdown between the S&P 500 E-mini futures and the SPY index tracker securities. If their conjecture is true, the sharp difference in XS SP Y responses between ⇠M KT and ⇠M KT again emphasizes the robustness of my cross-sectional procedure over

single-security alternatives for the evaluation of market risks. VII. Extended Comparison with VPIN and Return Reversal Measures With the exception of options-based measures and VPIN, to the best of my knowledge, no papers claim to introduce anticipatory measures of extreme event risk. In this section, I consider a second variant of VPIN as well as Nagel (2012)’s return reversal measure. Nagel (2012) investigates daily reversal returns on a large set of stocks, and he relates this proxy for returns to liquidity provision to the VIX. The Nagel measure may capture similar phenomena to my tail risk measure because it reflects the withdrawal of liquidity supply in bad economic times. In my model, liquidity suppliers rationally withdraw liquidity in response to anticipated near-term shocks. Nagel by contrast interprets his reversal returns as a measure of intermediary constraints. To distinguish between stories and the information content of our measures, I augment my “horse race” regressions to include an hourly analogue to the daily measure used in Nagel (2012). Defining VPIN has been a central conflict in the VPIN dispute (e.g., Andersen and Bondarenko (2014b), Andersen and Bondarenko (2014a), and Easley et al. (2014)). To ensure that I give VPIN the best shot for driving out my extreme event risk measure, I reached out to both sets of authors to obtain their preferred VPIN measures. Torben Andersen and Oleg Bondarenko responded with a battery of 20 VPIN measures for February 10, 2006, through March 22, 2011 (used in Andersen and Bondarenko (2015)). In the main text I include only bulk-volume classification with volume bars advocated most recently in Easley, de Prado and 15

O’Hara (2016). Here I also include minutely time bar VPIN of the originating VPIN paper (Easley, López de Prado and O’Hara (2012)). Each measure I average by hour and day to match the frequencies of my tail risk measure and the Nagel risk measure. The Nagel reversal measure follows Lehmann (1990) and defines a long-short trading strategy with portfolio weights of N

wi,t where Rm,t

t

t

1X |Ri,t 2 i=1

=

t

Rm,t

t|

!

1

(Ri,t

t

Rm,t

t) ,

is an equal-weighted market index return over an interval starting at t

(7) t. The first paren-

thetical term represents a rescaling to obtain portfolios weights with $1 of invested capital. wit comprises a “reversal strategy” because assets with lower-than-average returns over the prior period get higher weights, whereas assets with higher-than-average returns get negative weights. Investing wit earns a “liquidity provision” return per dollar of invested capital of N

Lt =

1X |Ri,t 2 i=1

t

Rm,t

t|

!

1 N X

(Ri,t

t

Rm,t

t ) Rit .

(8)

i=1

As Nagel notes, there is little economic guidance for the frequency at which Lt should be constructed or the return horizon

t over which reversal effects should operate. Nagel hedges this uncertainty by constructing

daily illiquidity series using weights from prior returns

t = 1, . . . , 5. Empirically, as depicted in Figure

XI, the entirety of Nagel strategy returns are earned from loading on returns of the prior date ( t = 1). The high short-horizon reversal return also motivates my TAQ-based version constructed over 60-minute horizons. I expand my comparison of candidate volatility and tail risk measures in Table II. Nagel’s reversal returns, a proxy for intermediary constraints, are moderately correlated with tail risk and volatility measures at the weekly frequency. With rare exceptions, intermediary constrainedness likely does not vary much at daily or intraday horizons, so return reversal dynamics at higher frequencies are likely driven by other economic forces. Volume-bar VPIN is roughly equally correlated with all measures at daily and weekly frequencies, and it becomes slightly less correlated with my tail risk measure at the hourly frequency. I show later that this lower correlation translates into picking up distinct variation in explaining and forecasting realized jumps. By contrast, neither time-bar VPIN or return reversal alternatives share meaningful covariation with my measure at high frequency. To confirm that the negative relation with VPIN is not associated with a

16

particular choice of measure, I supplement the correlational analysis in the main text with an additional VPIN-TIC column. VPIN-TIC is the tick-rule classification VPIN with one-minute time bins used in the original VPIN paper. This measure is weakly positively correlated with the newer bulk-volume classification version, and it is negatively correlated with all other measures with the exception of the high-frequency Nagel measure. Tables III and IV report results of regressions of tail realizations on my tail risk measure and controls. ⇠t,M KT and VPIN are normalized by their standard deviations in both panels. I do not normalize VIX or the Nagel measure because these have clear economic interpretations as percent volatility and percent returns. The first set of regressions (Table III) adds lags of the dependent and independent variable to both sides of the associative regression. Contemporaneous tail risk remains a powerful explanator of tail realizations. By contrast, VPIN measures often lose their statistical significance when controlling for lagged values. VPIN hence appears to derive its explanatory power from loading on a persistent latent variable, as Andersen and Bondarenko (2014a,b, 2015) also find. The second set of regressions (Table IV) substitutes the dependent variable with its forward counterpart to form predictive regressions. As before, my spread-implied tail risk measure strongly forecasts tail risk realizations in all specifications. VPIN often loses its statistical significance and flips sign. This feature provides additional evidence of VPIN loading on a slow-moving latent variable and possessing little incremental explanatory power. The Nagel measure is also inconsistent throughout. Taken together, this regression evidence suggests that my tail risk measure and VPIN load on different latent variables. My measure explains concurrent and future tail risk realizations with or without lags and controls. Volume-bar VPIN appears to draw its associative power for explaining tail risks from a separate, slower-moving component of volume or volatility. Time-bar VPIN and high-frequency return reversals contribute little explanatory power throughout. VIII. Comparison with Other Constructions of Continuous Variation I face a trade-off in using the hourly or daily CV measure. In favor of the hourly measure, my approach captures high-frequency intraday variation in tail risk, and only hourly (or higher-frequency) CV measures the corresponding intraday variation in volatility. In favor of the daily measure, CV may be contaminated by microstructure noise in minutely returns, even on an asset as liquid as the SPY. If CV is noisily estimated, less noisy versions of continuous variation may mechanically drive it out in multivariate regressions. My approach follows recent literature in using a hybrid daily-intraday approach to construct the contin17

uous variation measure. I follow Bollerslev, Todorov and Li (2013) to define the truncation threshold for jump variation for each date-time interval as a function of the outer product of intraday volatility patterns (estimated across dates) and daily estimated realized volatility (estimated within a date). The truncation threshold relies on a daily realized variation estimate. Whereas Bollerslev et al. (2013) focus on 5-minute jump realizations to obtain JV , I focus on total variation net of jump variation, or CV . Monte Carlo evidence from Lee and Mykland (2008, 2012) and Bollerslev et al. (2013) suggest high-frequency JV detects well the noise and scale of realized jumps. By corollary, continuous variation should also be estimated without too much contamination so long as I have “good enough” estimates of hourly realized variation. It is difficult to determine whether hourly realized volatility is estimated well enough because no perfect benchmark for realized volatility exists, even ex post. By contrast, even were hourly CV to be distorted by microstructure noise, it is not evident a priori whether the conditioning frequency-noise trade-off favors daily CV . For this reason, I run a horse race including my hourly measure of continuous variation as well as a daily measure constructing using CV d = RV d

JV d , for RV estimated as the sum of squared 5-minute

midpoint returns.5 I also run a horse race with the hourly measure and a hybrid CV measure estimated as the outer product of daily CV and intraday patterns in hourly CV , estimated by year as the average hourly CV across dates for a given hour divided by the average hourly CV across all hours and dates. This hybrid measure accounts for intraday patterns in volatility (Andersen and Bollerslev (1997)) without relying on single high-frequency intraday volatility observations. Table V compares tail risk coefficients with controls for different variants of continuous variation. The CVhour specification includes CVhour and V IX as in the main text. To this specification I add CVdaily and CVhybrid as alternate ways to work around the concern of noisy estimation of continuous variation. In contemporaneous regressions, CVhour dominates other CV measures in two respects. First, CVhour typically features a positive point estimate in multivariate specifications, whereas other CV proxies do not. The residual variation in lower-frequency continuous variation is often negatively associated with realized jumps. Second, where point estimates agree in sign, the CVhour coefficients are much larger than those on alternative measures. This feature is consistent with CVhour and other CV measures having imperfectly correlated noise terms with CVhour ’s noise term having a lower standard deviation. CVhour dominates less in predictive regressions (lower panel). CVhour has much larger economic magnitudes than daily CV , but estimates are imprecise. This reduction in precision results from the inclusion 5 Although there are many more sophisticated methods for estimating volatility, Liu, Patton and Sheppard (2015) justify this approach in finding that no volatility estimation technique significantly outperforms simple five-minute realized variation for estimating daily quadratic variation.

18

of lagged CVhour but not lagged CVday (lagged CVday is the same value in hourly regressions). Indeed, omitting lagged CVhour restores the statistical significance of CVhour in specifications 2 and 4. More telling is the comparison between hourly and hybrid continuous variation estimates. Hybrid continuous variation estimates have the correct sign throughout, and they drive out CVhour as a predictor of future jump realizations. In this forecasting application, the cleaning of daily average volatility and intraday patterns appears to be more important than isolating within-day variation in volatility. Taken together, it is not clear which continuous variation estimate is preferable. Differences in tail risk point estimates among continuous variation specifications are on the order of 1-2%, and variation in R2 is on the order of 0.1% in associative regressions and 1% in predictive regressions. Choosing the “best” version of continuous variation for each specification rather than CVhour does little to affect my main result. It is universally the case that the choice of continuous variation measure has minimal impact on the headline associative or predictive relation between my tail risk measure and realized medium-scale jumps.

19

Extensions IX. Distinguishing Upside and Downside Risks The main text estimates jump tails for each factor under the assumption that left and right jump tails are symmetric. In most applications considered, little is lost in assuming symmetry because statistical jump distributions are very similar for left and right tail events. In this section, I describe how to estimate up and down jump tail risks separately under a key identifying assumption on the “true price.” The multi-factor version of the key estimation equation from Appendix B gives a pair of estimation equations for upper and lower jump tails: ✓

✓

Vh d Vh d

◆R

◆L

=

⇠˜R +

X

⇠kR

ik 1

ik >0

k

=

⇠˜L +

X

X

⇠kL

ik 1

ik <0

+ ✏i1 ,

(9)

⇠kR

ik 1

ik >0

+ ✏i2 .

(10)

k

⇠kL

ik 1

ik <0

k

X k

Signed depth is directly observed, and signed volume can be obtained using the Lee and Ready (1991) algorithm or initiator-labeled data. Decomposing the bid-ask spread into “up” and “down” components is more difficult. I use the weighted midpoint as a reference price to decompose the spread into (potentially) unequal parts, where the weighted midpoint is defined as

wmid =

bid ⇥ ask depth + ask ⇥ bid depth . ask depth + bid depth

(11)

Replacing the effective half-spread with the difference of bid and ask quotes from the weighted midpoint price, the left-hand sides of Equations (9) and (10) simplify as ✓

Vh d

◆R

= =

✓

Vh d

◆L

=

✓ ◆ buy volume 1 bid ⇥ ask depth + ask ⇥ bid depth ask ask depth wmid ask depth + bid depth ✓ ◆ (ask bid) /wmid (buy volume) , ask depth + bid depth ✓ ◆ (ask bid) /wmid (sell volume) . ask depth + bid depth

Reassuringly, the unsigned quantity

Vh d

average of up and down quantities

Vh R d

(12) (13)

used as the dependent variable in the main analysis is the simple and

Vh L d

up to the use of the midpoint price rather than a

weighted reference price. Intuitively, both upper and lower tail risks are increasing in the bid-ask spread,

20

decreasing in quoted depth, and increasing in volume (holding the other two quantities fixed). Figure XII compares left and right jump tail risk over the 2004–2013 period. With rare exceptions, the physical jump tail risks are nearly indistinguishable for positive and negative market events. The tight coevolution of negative and positive jump tail risks agrees with Bollerslev and Todorov (2011), who show in their Table III that statistical left- and right-jump tails share a single stochastic driver. Moreover, it suggests that the implied tail risk series is recovered under the physical rather than the risk-neutral measure, which does not share this property. The 2010 Flash Crash offers a striking occasion in which up and down jumps occur at distinct times. Extreme left and right jump realizations occur in quick succession as the price rapidly declines and recovers. If market makers indeed anticipate a crash event followed by a delayed, but sharp recovery, the time series of expected jump tail risks should reveal a run-up in left-tail risk before right-tail risk, and the single factor structure for jumps under P should break, albeit temporarily. Figure XIII reveals exactly this property and confirms that the measure may distinguish between extreme up and down movements in real time. Not only are implied left tail risks much larger than implied right tail risks before the crash, but large abnormal values are registered up to 30 minutes before the Flash Crash begins, long before realized volatility increases. Likewise, the recovery period of the Flash Crash is characterized by greater right tail risk than left tail risk, consistent with a sharp expected recovery.

21

References Andersen, Torben G. and Oleg Bondarenko, “Reflecting on the {VPIN} Dispute,” Journal of Financial Markets, 2014, 17, 53 – 64. and

, “VPIN and the Flash Crash,” Journal of Financial Markets, 2014, 17, 1 – 46.

and

, “Assessing Measures of Order Flow Toxicity and Early Warning Signals for Market Turbulence,”

Review of Finance, 2015, 19 (1), 1–54. and Tim Bollerslev, “Intraday Periodicity and Volatility Persistence in Financial Markets,” Journal of Empirical Finance, 1997, 4 (2–3), 115–158. , Nicola Fusari, and Viktor Todorov, “The Risk Premia Embedded in Index Options,” Journal of Financial Economics, 2015. , Tim Bollerslev, and Francis X. Diebold, “Roughing It Up: Including Jump Components in the Measurement, Modeling, and Forecasting of Return Volatility,” Review of Economics and Statistics, 2016/10/19 2007, 89 (4), 701–720. ,

, and Xin Huang, “A Reduced Form Framework for Modeling Volatility of Speculative Prices Based

on Realized Variation Measures,” Journal of Econometrics, 2011, 160 (1), 176 – 189. Realized Volatility. Bali, Turan G. and Nusret Cakici, “Idiosyncratic Volatility and the Cross Section of Expected Returns,” Journal of Financial and Quantitative Analysis, March 2008, 43 (1), 29–58. Bollerslev, Tim and Viktor Todorov, “Tails, Fears, and Risk Premia,” The Journal of Finance, 2011, 66 (6), 2165–2211. and

, “Time-Varying Jump Tails,” Journal of Econometrics, 2014, 183 (2), 168–180. Analysis of

Financial Data. , Sophia Zhengzi Li, and Viktor Todorov, “Roughing up Beta: Continuous vs. Discontinuous Betas, and the Cross-Section of Expected Stock Returns,” Journal of Financial Economics, 2016, 120 (3), 464– 490. , Viktor Todorov, and (Sophia) Zhengzi Li, “Jump Tails, Extreme Dependencies, and the Distribution of Stock Returns,” Journal of Econometrics, 2013, 172 (2), 307–324.

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Bryzgalova, Svetlana, “Spurious Factors in Linear Asset Pricing Models,” Working Paper 2015. Easley, David, Marcos Lopez de Prado, and Maureen O’Hara, “Discerning Information from Trade Data,” Journal of Financial Economics, 2016, 120 (2), 269 – 285. , Marcos M. López de Prado, and Maureen O’Hara, “Flow Toxicity and Liquidity in a Highfrequency World,” Review of Financial Studies, 2012, 25 (5), 1457–1493. , Marcos M. López de Prado, and Maureen O’Hara, “VPIN and the Flash Crash: A Rejoinder,” Journal of Financial Markets, 2014, 17, 47 – 52. Fama, Eugene F. and James D. MacBeth, “Risk, Return, and Equilibrium: Empirical Tests,” Journal of Political Economy, 1973, 81 (3), 607–636. Herskovic, Bernard, Bryan Kelly, Hanno Lustig, and Stijn Van Nieuwerburgh, “The Common Factor in Idiosyncratic Volatility: Quantitative Asset Pricing Implications,” Journal of Financial Economics, 2016, 119 (2), 249 – 283. Huang, Xin and George Tauchen, “The Relative Contribution of Jumps to Total Price Variance,” Journal of Financial Econometrics, 2005, 3 (4), 456–499. Lee, Charles M. C. and Mark J. Ready, “Inferring Trade Direction from Intraday Data,” The Journal of Finance, 1991, 46 (2), 733–746. Lee, Suzanne S. and Per A. Mykland, “Jumps in Financial Markets: A New Nonparametric Test and Jump Dynamics,” Review of Financial Studies, 2008, 21 (6), 2535–2563. and

, “Jumps in Equilibrium Prices and Market Microstructure Noise,” Journal of Econometrics, 2012,

168 (2), 396 – 406. Lehmann, Bruce N., “Fads, Martingales, and Market Efficiency,” The Quarterly Journal of Economics, 1990, 105 (1), 1–28. Liu, Lily Y., Andrew J. Patton, and Kevin Sheppard, “Does Anything Beat 5-Minute RV? A Comparison of Realized Measures Across Multiple Asset Classes,” Journal of Econometrics, 2015, 187 (1), 293 – 311. Menkveld, Albert J. and Bart Zhou Yueshen, “The Flash Crash: A Cautionary Tale about Highly Fragmented Markets,” Working Paper 2015. 23

Nagel, Stefan, “Evaporating Liquidity,” Review of Financial Studies, 2012, 25 (7), 2005–2039.

24

Figure I: Risk Measures Implied by Spreads and Volume This figure plots rolling one-month means of hourly cross-sectional slope estimates for each trading date in 2004–2013. The red series repeats the regression using spreads on the left-hand side rather than the product of spreads and volume. The gold dashed series uses spreads as the left-hand side variable without making a depth adjustment. The purple series uses share volume as the dependent variable. All series are scaled by their respective median absolute values for comparison. NBER recession dates are marked in gray.

25

Figure II: Spreads During May 6, 2010 Figures plot average percent effective half-spreads (top) and effective half-spreads scaled by realized volume and the inverse of offered depth (bottom). Median values are computed by market beta quintile for each 15-minute interval during May 6, 2010, where “12:00” denotes 12:00–12:14pm. The dashed black lines bracket the 2:30–3:00pm interval during which the Flash Crash occurs. (a) Half-Spreads by Market

Quintile

45 40 35

1, low 2 3 4 5, high

basis points

30 25 20 15 10 5 0 10:00 10:30 11:00 11:30 12:00 12:30 13:00 13:30 14:00 14:30 15:00 15:30 (b) Adjusted Half-Spreads by Market

Quintile

14

12

1, low 2 3 4 5, high

basis points

10

8

6

4

2

0 10:00 10:30 11:00 11:30 12:00 12:30 13:00 13:30 14:00 14:30 15:00 15:30

26

Figure III: Quarter-Hourly Jump Tail Deviations During May 6, 2010 Figures plot standardized deviations in market (top) and idiosyncratic (bottom) jump expectations around the May 6, 2010 Flash Crash. For each quarter hour, I normalize each value by subtracting the value during the same quarter hour on May 4, 2010, and dividing by the 15-minute specific standard deviation of this value across all dates in the 63 trading days up to and including May 4, 2010. The figure plots the normalized value for the market factor before (green), during (red), and after (orange) May 6, 2010. The dotted line plots the normalized 15-minute estimate for realized volatility. Black circles denote the 2:30–3:00pm interval during which the crash and reversion occurs. (a) Market Jump Tail Deviations 60 50

scaled deviation

40 30 20 10 0 -10 05/06

05/07

(b) Idiosyncratic Jump Tail Deviations 60 50

scaled deviation

40 30 20 10 0 -10 05/06

05/07

27

Figure IV: XLF Forecast Tail Risks with Sorted and Thwarting Portfolios Each figure plots rolling one-month means of hourly estimated financial sector tail risks for each trading date in 2004–2013. The factor model for realized returns has market and financial sector factors. The top plot uses all stocks for estimating financial sector tail risk. The bottom-left plot uses conditional double-sorted XLF-SPY beta portfolios with ten deciles for each dimension. The bottom-right plot uses identical market exposure portfolio assignments and randomizes XLF exposure portfolio assignments within each market exposure decile. Dashed blue bands depict the corresponding 95% confidence intervals estimated via pairs bootstrap. NBER recession dates are marked in gray. (a) Forecast Financial Sector Tail Risks – All Stocks

(b) Forecast Financial Sector Tail Risks – XLF-MKT Portfolios

(c) Forecast Financial Sector Tail Risks – XLFR -MKT Portfolios 40

30

jump tails x 100 realized

jump tails x 100 realized

25

30

% variation per hour

% variation per hour

20

15

10

20

10

0

5 -10

0

-5 2004

2005

2006

2007

2008

2009

2010

2011

2012

2013

28

2014

-20 2004

2005

2006

2007

2008

2009

2010

2011

2012

2013

2014

Figure V: Market and PBO Forecast Tail Risks in the Market-Placebo Model This figure plots rolling one-month means of hourly estimated market and placebo (PBO) tail risks for each trading date in 2004–2013. The factor model for realized returns has market and placebo factors. Dashed blue bands depict the corresponding 95% confidence intervals estimated via pairs bootstrap. Realized volatility (red) is estimated using minutely squared returns on the SPY and scaled to the hourly frequency. The VIX is plotted on the right axis for comparison with the SPY. (a) Forecast Market Tail Risks

(b) Forecast Placebo Tail Risks

29

Figure VI: Hourly Market Tail Risks, 2004–2013 with Idiosyncratic Volatility Controls This figure plots rolling one-month means of hourly cross-sectional slope estimates for each trading date in 2004–2013. Each subfigure plots rolling one-month means of hourly estimated financial sector tail risks (blue) for each trading date in 2004–2013. Dashed blue bands depict the corresponding 95% confidence intervals estimated via pairs bootstrap. Realized volatility (red) is estimated using minutely squared returns on the SPY and scaled to the hourly frequency. The VIX is plotted using the right axis (gold) for comparison. The top plot is the implied tail risk series from the market-factor model. The bottom-left plot adds an idiosyncratic volatility control equal to each stock’s idiosyncratic volatility factor loading. The bottom-right plot adds an idiosyncratic volatility control by stock-date. NBER recession dates are marked in gray. (a) Market Factor Only

(b) Market Factor with Idiosyncratic Volatility Factor Loading Control

30

(c) Market Factor with Idiosyncratic Volatility Control

Figure VII: Sample Autocorrelation Function for Market Tail Risks, 2004–2013 This figure plots the sample autocorrelation function for implied market tail risks. The market tail risk series consists of hourly estimates extending from 2004–2013. The top subfigure plots the unadjusted autocorrelation function of market tail risks for 0 through 180 hourly lags, or equivalently, 0 through 30 trading days. The bottom subfigure plots the same autocorrelation function for a normalized market tail risk series, where I divide each tail risk estimate by the sample average of tail risk estimates for the same hour across all dates in the sample. The blue line depicts two standard errors from zero serial correlation. (a) Autocorrelation Function of Raw Market Tail Risk Series 0.9 0.8

Sample Autocorrelation

0.7 0.6 0.5 0.4 0.3 0.2 0.1 0

0

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

21

22

23

24

25

26

27

28

29

30

26

27

28

29

30

Lag (days)

(b) Autocorrelation Function of Intraday Pattern-Adjusted Market Tail Risk Series 0.9 0.8

Sample Autocorrelation

0.7 0.6 0.5 0.4 0.3 0.2 0.1 0

0

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

Lag (days)

31

17

18

19

20

21

22

23

24

25

Figure VIII: Persistence of

t

and It

Figure plots simulations of latent jump intensity and realized jump processes. For each date, the count of realized jumps is distributed Poisson with intensity t , and the jump indicator It equals one if a day has at least one jump. The latent jump intensity process evolves according to an AR(1) of the form log

t+1

= µ + ⇢ (log

µ) + ✏t .

t

The initial intensity 0 and long-run mean µare set to deliver jumps on 8.6% of days ( 0 = 0.486) from Table 1 of Andersen, Bollerslev and Huang (2011). The process has log-intensity persistence ⇢ = 0.99 and log-intensity volatility = 0.1. Innovations ✏t are serially independent. Each simulation runs for 3,000 “days” with a 500-day burn-in period, and I repeat for 1,000 simulations. The top plot depicts a representative simulation outcome for jump intensities t and (binary) realizations It . The bottom two plots summarize the distribution of 20-lag autocorrelation structures of t (left) and It (right). Darker blue indicates more paths. t

and It 1

2

0.5

0 0

500

1000

1500

2000

It

t

(a) Representative Simulation for 4

0 3000

2500

simulation day (c) Autocorrelations of Realized Jump Intensity It

t

1

0.9

0.8

autocorrelation coefficient

autocorrelation coefficient

(b) Autocorrelations of Latent Jump Intensity 1

0.8

0.7

0.6

0.6

0.4

0.2

0

0.5

-0.2

0.4 0

2

4

6

8

10

12

14

16

18

20

0

2

4

6

8

10

lag

lag

32

12

14

16

18

20

Figure IX: Comparison of Implied Market Jump Risks Using Cross-Sectional Approach and SPY Only Figures plot raw (top) and 10-day rolling means (bottom) of hourly estimated market tail risks for each trading date in 2004–2013. The blue line is the baseline tail risk extracted from the cross-section of market betas. The red line plots V h/d for the SPY. The gold line is the VIX. All series are normalized by their respective standard deviations. (a) Cross-Section and SPY-Implied Tail Risks

30 jump tails SPY-implied risk VIX

standardized tail risk

25

20

15

10

5

0 2004

2005

2006

2007

2008

2009

2010

2011

2012

2013

2014

(b) Cross-Section and SPY-Implied Tail Risks (10-Day Smoothed)

9 jump tails SPY-implied risk VIX

8

standardized tail risk

7 6 5 4 3 2 1 0 2004

2005

2006

2007

2008

2009

33

2010

2011

2012

2013

2014

Figure X: Standardized Deviations in Jump Expectations around the 2010 Flash Crash — Cross Section Versus SPY This figure plots standardized deviations in jump expectations around the May 6, 2010 Flash Crash. Tail risks on top are assessed with a market factor model, and tail risks on top are assessed with the SPY liquidity composite only. For each quarter hour, I normalize each value by subtracting the value during the same quarter hour on May 4, 2010, and dividing by the 15-minute specific standard deviation of this value across all dates in the 63 trading days up to and including May 4, 2010. The top figure plots the normalized value for the market factor before (green), during (red), and after (orange) May 6, 2010. The dotted line plots the normalized 15-minute estimate for realized volatility. Black circles denote the 2:30–3:00pm interval during which the crash and reversion occur. The bottom plot is truncated for the 2:45–3:00pm interval (737 standard deviations). 100

scaled deviation

80

60

40

20

0

-20 05/06

05/07 XS ) (a) Anticipated Market Factor Jumps (⇠M KT

100

scaled deviation

80

60

40

20

0 05/06

05/07 SP Y ) (b) Anticipated Market Factor Jumps (⇠M KT

34

Figure XI: Nagel Reversal Returns by Lag Horizon This figure plots rolling Nagel reversal returns Lt for weights taken from t = 1 to t = 5 days. The left figure ¯ t presented in Nagel disaggregates returns by reversal horizon, and the right figure replicates the reversal return L (2012) that is earned with an equal-weighted average of weights wi,t 1 , . . . , wi,t 5 . Both sets of series are smoothed over the preceding 90 days to match Figure 1 of Nagel (2012). 3 t=1 t=2 t=3 t=4 t=5

2.5

% return

2

1.5

1

0.5

0

-0.5 1997

2000

2002 2005 2007 (a) Nagel Strategies by Date

2010

2012

2015

2012

2015

0.8

0.7

0.6

% return

0.5

0.4

0.3

0.2

0.1

0 1997

2000

2002

2005

2007

2010

(b) Nagel Strategy Average by Date ( t = 1, . . . , 5)

35

Figure XII: Anticipated Signed Jumps by Hour, 2004–2013 This figure plots rolling one-month means of hourly cross-sectional slope estimates ⇠mt in a one-factor market model for each trading date in 2004–2013. Up (blue) and down (red) jump tails are estimated using Lee and Ready (1991) signed volume. Realized volatility (magenta) is estimated using minutely squared returns on the SPY and scaled to the hourly frequency. The VIX is plotted using the right axis (gold) for comparison. NBER recession dates are marked in gray.

36

Figure XIII: Standardized Deviations in Signed Jump Expectations around the 2010 Flash Crash This figure plots standardized deviations in jump expectations around the May 6, 2010 Flash Crash. Tail risks are assessed with a market model with 15-minute increments. For each quarter hour, I normalize each value by subtracting the value during the same quarter hour on May 4, 2010, and dividing by the 15-minute specific standard deviation of this value across all dates in the 63 trading days up to and including May 4, 2010. The top figure plots the normalized value for down market factor jump tails before (green), during (red), and after (orange) May 6, 2010. The dotted purple line is the normalized 15-minute estimate for realized volatility. Black circles denote the 2:30–3:00pm interval during which the crash and reversion occur. The bottom plot provides the corresponding information for up market factor jump tails. 80 70

scaled deviation

60 50 40 30 20 10 0 -10 05/06

05/07

(a) Market Jump Tail Down Deviations 100

scaled deviation

80

60

40

20

0

-20 05/06

05/07

(b) Market Jump Tail Up Deviations

37

38

= +↵

+

,M KT

1 tail_realizationt 1

↵ + ⇠tSP Y,M KT + ⇠tXS SP Y 1 ⇠t 1

+ ,M KT

+

+ ⌘V P INt

XS 1 ⇠t 1

+ ⇣CVt

,M KT

+ V IXt 1V

IXt 1

+⇣

1 CVt 1

+⌘

1V

P INt 1

+ ✏t ,

9036 0.723

4.135 (4.647)

⇤⇤⇤

(2)

(1)

9036 0.775

1.909 (2.452) 2.768⇤⇤⇤ (7.232) 0.649⇤⇤⇤ (4.574)

⇤⇤

Jump Count

11280 0.873

1.151 (2.476) 2.434⇤⇤⇤ (5.108) 0.598⇤⇤⇤ (4.386)

⇤⇤

11280 0.848

2.841 (5.571)

⇤⇤⇤

Jump Count

(2)

(4)

(5)

(4)

11280 0.878

45.764 (7.087)

⇤⇤⇤

(5)

11280 0.909

17.350 (3.431) 40.339⇤⇤⇤ (7.813) 10.861⇤⇤⇤ (7.831)

⇤⇤⇤

5124 0.794

0.438 (0.635) 2.670⇤⇤⇤ (5.728) 0.827⇤⇤⇤ (4.762) 6.688⇤⇤ (1.992) 0.561 (0.947)

⇤⇤⇤

9036 0.739

62.063 (4.831)

⇤⇤⇤

p < 0.01,

⇤⇤

9036 0.807

⇤⇤⇤

5124 0.828

1.760 (0.152) 41.033⇤⇤⇤ (6.407) 15.245⇤⇤⇤ (6.533) 109.839⇤⇤ (2.407) 22.126⇤⇤⇤ (2.855)

(6)

6400 0.915

13.805 (2.611) 38.598⇤⇤⇤ (6.873) 10.728⇤⇤⇤ (6.781) 6.222 (0.421) 23.521⇤⇤⇤ (6.832)

(6)

p < 0.05, ⇤ p < 0.1.

23.598 (2.234) 46.281⇤⇤⇤ (7.986) 13.617⇤⇤⇤ (7.228)

⇤⇤

SPY Basis-Point Jumps Jump Sum

(3)

6400 0.878

0.859 (1.667) 2.780⇤⇤⇤ (6.178) 0.638⇤⇤⇤ (4.487) -0.920 (-0.661) 0.806⇤⇤ (2.278)

⇤

SPY Basis-Point Jumps Jump Sum

(3)

t-statistics are given in parentheses with stars indicating

Obs. R2

V P IN

CV

V IX

XS ⇠M KT

SP Y ⇠M KT

Variable

Obs. R2

V P IN

CV

V IX

XS ⇠M KT

SP Y ⇠M KT

Variable

(1)

9036 0.574

5.075 (3.549)

⇤⇤⇤

(7)

11280 0.737

3.047 (4.076)

⇤⇤⇤

(7)

9036 0.649

0.029 (0.030) 6.489⇤⇤⇤ (7.073) 1.705⇤⇤⇤ (8.691)

Jump Count

(8)

11280 0.779

-0.752 (-1.744) 5.380⇤⇤⇤ (5.312) 1.806⇤⇤⇤ (9.562)

⇤

Jump Count

(8)

(10)

(10)

11280 0.806

34.001 (5.732)

5124 0.715

-0.762 (-0.627) 5.684⇤⇤⇤ (6.607) 1.494⇤⇤⇤ (6.259) 2.018 (0.504) 5.037⇤⇤⇤ (7.702)

9036 0.642

49.641 (4.406)

⇤⇤⇤

SPY Spread Jumps

(9)

6400 0.816

-0.525 (-0.789) 5.134⇤⇤⇤ (6.000) 1.473⇤⇤⇤ (6.863) -2.728⇤ (-1.925) 4.919⇤⇤⇤ (8.521)

⇤⇤⇤

SPY Spread Jumps

(9)

9036 0.709

12.342 (1.305) 48.692⇤⇤⇤ (7.688) 13.267⇤⇤⇤ (9.948)

Jump Sum

(11)

11280 0.845

4.973 (0.999) 43.148⇤⇤⇤ (6.338) 11.071⇤⇤⇤ (8.516)

Jump Sum

(11)

where = 0 and = 1 correspond to the upper and lower panels, respectively. SPY-implied tail estimates use quoted half-spreads, depth, and volume for the SPY ETF. Tail realizations are measured in counts of minutely returns exceeding basis point or spread thresholds. The count variable sums jumps exceeding 10 basis points or 5 half-spreads, and the sum variables are a weighted sum of jump sizes exceeding 5, 10, 25, and 100 basis points or 1, 5, 10, or 25 half-spreads. Continuous variation is estimated by hour with a 2.5 standard deviation threshold on minutely price movements. VPIN is constructed using bulk-volume classification with volume bins on front-month E-mini S&P 500 futures. Regressions consist of hourly observations for the 2005–2013 sample in the baseline specification and for 2005-2011 where VPIN is included. Standard errors are HAC with monthly (126 observation) bandwidth. SP Y ⇠t,M KT , ⇠t,M KT , and VPIN are normalized by their standard deviations in both panels.

tail_realizationt

This table presents results from regressions of realized jumps against estimated tail risks,

Table I: Contemporaneous and Predictive Regressions for Jump Realizations with SPY-Implied Risks

5124 0.757

-1.425 (-0.118) 43.550⇤⇤⇤ (6.884) 12.768⇤⇤⇤ (8.285) 60.252 (1.523) 25.893⇤⇤⇤ (4.709)

(12)

6400 0.870

3.550 (0.562) 40.961⇤⇤⇤ (6.489) 9.409⇤⇤⇤ (6.800) -2.728 (-0.191) 28.516⇤⇤⇤ (6.876)

(12)

Table II: Correlations of Tail Measure with Other Volatility and Tail Measures This table reports correlations of tail and volatility measures over the 2004–2013 sample period. The spread-implied measure ⇠M KT uses the two-stage procedure to compute hourly market tail risk estimates from each cross section. VIX is the (30-day) CBOE Volatility Index. Realized volatility is the square root of the average squared one-minute SPY returns within each hour. Options-implied tails are the weekly parametric left-tail risk estimates from Figure 7 of Bollerslev and Todorov (2014). VPINBV C uses bulk-volume classification and volume bars (10 buckets) and VPINT IC uses tick rule classification and one-minute time bars. These three series are available through 2011 only. The last measure is the reversal return from Nagel (2012). For daily and weekly comparisons, this measure is constructed using CRSP end-of-day prices with lagged weights of t = 1, . . . , 5 days as in the original paper. At the hourly frequency, reversal returns are calculated using VWAP returns for the prior hour. Daily and weekly values are equal-weighted hourly values within the respective time bin, with the exception of the Nagel return reversal series previously described. (a) Weekly Correlations

⇠M KT Options-Implied Tail VIX Realized Volatility VPINBV C Nagel VPINT IC

⇠M KT

Options-Implied Tail

VIX

Realized Volatility

VPINBV C

Nagel

VPINT IC

– 0.77 0.85 0.93 0.81 0.52 0.45

0.77 – 0.89 0.80 0.65 0.61 0.14

0.85 0.89 – 0.91 0.79 0.57 0.28

0.93 0.80 0.91 – 0.83 0.50 0.49

0.81 0.65 0.79 0.83 – 0.50 0.48

0.52 0.61 0.57 0.50 0.50 – 0.09

0.45 0.14 0.28 0.49 0.48 0.09 –

(b) Daily Correlations

⇠M KT VIX Realized Volatility VPINBV C Nagel VPINT IC

⇠M KT

VIX

Realized Volatility

VPINBV C

Nagel

VPINT IC

– 0.81 0.91 0.74 0.29 0.39

0.81 – 0.88 0.75 0.28 0.23

0.91 0.88 – 0.78 0.24 0.45

0.74 0.75 0.78 – 0.26 0.47

0.29 0.28 0.24 0.26 – 0.05

0.39 0.23 0.45 0.47 0.05 –

(c) Hourly Correlations

⇠M KT VIX Realized Volatility VPINBV C Nagel VPINT IC

⇠M KT

VIX

Realized Volatility

VPINBV C

Nagel

VPINT IC

– 0.71 0.87 0.65 -0.30 0.34

0.71 – 0.82 0.74 -0.29 0.21

0.87 0.82 – 0.72 -0.26 0.40

0.65 0.74 0.72 – -0.29 0.47

-0.30 -0.29 -0.26 -0.29 – -0.09

0.34 0.21 0.40 0.47 -0.09 –

39

40

= +↵

1 tail_realizationt 1

+

1 ⇠t 1,M KT

+

1V

IXt

↵ + ⇠t,M KT + V IXt + CVt + ⇣V P INt + ⌘N agelt 1

+

1 CVt 1

+⇣

1V

P INt 1

+⌘

1 N agelt 1

+ ✏t .

11280 0.765

5.539 (4.521)

6400 0.809

4.003 (4.131) 5.328⇤⇤⇤ (6.850)

⇤⇤⇤

(2)

(1)

⇤⇤⇤

6400 0.863

3.781 (6.401) 0.874⇤ (1.889)

⇤⇤⇤

11280 0.855

4.001 (7.407)

⇤⇤⇤

(2)

6400 0.790

1.898⇤⇤⇤ (4.707)

4.757 (4.379)

⇤⇤⇤

⇤⇤⇤

11280 0.796

-23.764 (-0.564)

4.018 (3.321)

⇤⇤⇤

(4)

11280 0.866

-12.347 (-0.494)

3.187 (5.129)

Jump Count

(3)

6400 0.862

0.135 (0.718)

3.908 (6.628)

⇤⇤⇤

⇤⇤⇤

(4)

Jump Count

(3)

t-statistics are given in parentheses with stars indicating

Obs. R2

CV

V IX

N agel

V P INT B

V P INV B

⇠M KT

Variable

Obs. R2

CV

V IX

N agel

V P INT B

V P INV B

⇠M KT

Variable

(1)

p < 0.01,

⇤⇤

6400 0.830

-13.787 (-0.353)

3.109 (3.218) 4.350⇤⇤⇤ (4.316)

⇤⇤⇤

(5)

6400 0.876

-10.139 (-0.405)

2.962 (4.804) 0.946 (1.483)

⇤⇤⇤

(5)

(7)

(7)

11280 0.891

63.742 (9.429)

⇤⇤⇤

11280 0.836

⇤⇤⇤

52.704 (7.320)

p < 0.05, ⇤ p < 0.1.

6400 0.838

53.301 (1.554) 1.019⇤⇤⇤ (3.189) -2.535⇤⇤ (-2.498)

4.028 (3.876) 3.454⇤⇤⇤ (3.424)

⇤⇤⇤

SPY Spread Jumps

(6)

6400 0.888

38.191⇤ (1.765) 0.712⇤⇤⇤ (4.084) -0.563 (-0.839)

3.108 (6.414) 0.721 (1.559)

⇤⇤⇤

(8)

⇤⇤⇤

⇤⇤⇤

6400 0.865

44.891 (7.589) 30.291⇤⇤⇤ (6.412)

(8)

6400 0.902

59.005 (8.977) 22.11⇤⇤⇤ (5.672)

SPY Basis-Point Jumps

(6)

(10)

(10)

11280 0.900

-89.453 (-0.243)

54.024⇤⇤⇤ (6.697)

6400 0.857

11.291⇤⇤⇤ (4.308)

48.562⇤⇤⇤ (7.441)

11280 0.853

-38.275 (-0.104)

44.686⇤⇤⇤ (5.671)

Jump Sum

(9)

6400 0.898

6.171⇤⇤⇤ (2.991)

61.293⇤⇤⇤ (8.919)

Jump Sum

(9)

6400 0.880

144.953 (0.409)

39.641⇤⇤⇤ (6.006) 26.097⇤⇤⇤ (3.836)

(11)

6400 0.911

-1.543 (-0.004)

50.506⇤⇤⇤ (6.793) 20.924⇤⇤⇤ (3.953)

(11)

Tail realizations are measured in counts of minutely returns exceeding basis point thresholds (top panel) or spread thresholds (bottom panel). The count variable sums jumps exceeding 10 basis points or 5 half-spreads, and the sum variables are a weighted sum of jump sizes exceeding 5, 10, 25, and 100 basis points or 1, 5, 10, or 25 half-spreads. Continuous variation is estimated by hour with a 2.5 standard deviation threshold on minutely price movements. VPIN is constructed using bulk-volume classification with volume bins (VB) and minutely time bins (TB) on front-month E-mini S&P 500 futures. Regressions consist of hourly observations for the 2005–2013 sample in the baseline specification and for 2005-2011 otherwise. Standard errors are HAC with monthly (126 observation) bandwidth. ⇠t,M KT and VPIN are normalized by their standard deviations in both panels.

tail_realizationt

This table presents results from regressions of realized jumps against contemporaneous estimated tail risks with lagged copies of each variable,

Table III: Contemporaneous Jump Tails and Realized Market Jumps with Lags — Extended Control Set

6400 0.884

386.327 (1.352) 7.447⇤⇤⇤ (3.381) 1.600 (0.194)

40.288⇤⇤⇤ (5.746) 20.675⇤⇤⇤ (3.081)

(12)

6400 0.922

459.930 (1.468) 12.306⇤⇤⇤ (6.398) 12.156 (1.344)

46.249⇤⇤⇤ (6.339) 21.723⇤⇤⇤ (4.735)

(12)

41

= +↵

1,M KT

1

1 +

+ V IXt

1 tail_realizationt

↵ + ⇠t 1 ⇠t

+ CVt +

1 2 +

1

1 CVt

+ ⌘N agelt

1 V IXt

+ ⇣V P INt

2,M KT

1 2

+⇣

1V

P INt 2

+⌘

1 N agelt 2

+ ✏t .

9036 0.611

9.022 (5.478)

⇤⇤⇤

(1)

9036 0.731

5.737 (9.463)

5124 0.706

6.140 (4.467) 5.257⇤⇤⇤ (8.322)

⇤⇤⇤

(2)

5124 0.75

5.387 (7.737) 0.211 (0.468)

⇤⇤⇤

(2)

-1.254⇤⇤⇤ (-5.244)

5.809 (8.133)

5124 0.654

-0.434 (-1.101)

8.072 (5.040)

⇤⇤⇤

⇤⇤⇤

5124 0.701

-10.269 (-0.137)

5.445 (3.070)

⇤⇤⇤

(4)

5124 0.787

-116.836⇤⇤ (-2.270)

3.899 (5.362)

⇤⇤⇤

Jump Count

(3)

5124 0.744

(4) Jump Count

⇤⇤⇤

(3)

t-statistics are given in parentheses with stars indicating

Obs. R2

CV

V IX

N agel

V P INT B

V P INV B

⇠M KT

Variable

Obs. R2

CV

V IX

N agel

V P INT B

V P INV B

⇠M KT

Variable

(1)

p < 0.01,

⇤⇤

5124 0.752

-2.994 (-0.040)

4.240 (2.730) 2.198⇤⇤ (1.998)

⇤⇤⇤

(5)

5124 0.804

(7)

(7)

9036 0.761

89.067 9.37

5124 0.763

101.374⇤ (1.789) 1.161⇤⇤⇤ (2.625) -1.550 (-0.819)

4.348 (2.607) 1.970⇤⇤ (1.977)

⇤⇤⇤

9036 0.673

77.240 (7.791)

⇤⇤⇤

(8)

⇤⇤⇤

⇤⇤⇤

5124 0.737

61.240 (6.962) 26.387⇤⇤⇤ (6.107)

(8)

5124 0.783

80.237 (7.785) 12.272⇤⇤ (2.218)

SPY Spread Jumps

(6)

5124 0.835

-38.461 (-0.930) 1.019⇤⇤⇤ (4.202) 3.595 (1.524)

2.661 (4.226) -0.493 (-0.622)

⇤⇤⇤

SPY Basis-Point Jumps

(6)

p < 0.05, ⇤ p < 0.1.

-147.223⇤⇤⇤ (-2.880)

3.669 (4.728) -0.664 (-0.718)

⇤⇤⇤

(5)

-15.734⇤⇤⇤ (-4.085)

(10)

5124 0.814

-1666.975⇤ (-1.906)

60.183⇤⇤⇤ (4.854)

5124 0.708

-8.412⇤⇤⇤ (-2.722)

72.059⇤⇤⇤ (7.182)

5124 0.746

-557.013 (-0.771)

53.374⇤⇤⇤ (4.186)

Jump Sum

(9)

5124 0.772

(10) Jump Sum ⇤⇤⇤

87.334 (8.038)

(9)

5124 0.785

-443.532 (-0.610)

46.589⇤⇤⇤ (3.989) 2.297 (0.205)

(11)

5124 0.828

-2160.416⇤⇤ (-2.391)

55.392⇤⇤⇤ (4.515) -3.798 (-0.268)

(11)

Tail realizations are measured in counts of minutely returns exceeding basis point thresholds (top panel) or spread thresholds (bottom panel). The count variable sums jumps exceeding 10 basis points or 5 half-spreads, and the sum variables are a weighted sum of jump sizes exceeding 5, 10, 25, and 100 basis points or 1, 5, 10, or 25 half-spreads. Continuous variation is estimated by hour with a 2.5 standard deviation threshold on minutely price movements. VPIN is constructed using bulk-volume classification with volume bins (VB) and minutely time bins (TB) on front-month E-mini S&P 500 futures. Regressions consist of hourly observations for the 2005–2013 sample in the baseline specification and for 2005-2011 otherwise. Standard errors are HAC with monthly (126 observation) bandwidth. ⇠t,M KT and VPIN are normalized by their standard deviations in both panels.

tail_realizationt

This table presents results from predictive regressions of realized jumps against lagged estimated tail risks with lagged copies of each variable,

Table IV: Forecasting Realized Market Jumps with Lagged Jump Tails — Extended Control Set

5124 0.797

200.304 (0.351) 11.636⇤⇤⇤ (3.034) 44.818⇤⇤ (2.566)

35.135⇤⇤ (2.554) 1.513 (0.141)

(12)

5124 0.858

-724.143 (-1.180) 16.958⇤⇤⇤ (3.764) 81.745⇤⇤⇤ (3.565)

34.158⇤⇤⇤ (2.740) 2.852 (0.222)

(12)

42

= +↵

,M KT

+ V IXt

1 tail_realizationt 1

↵ + ⇠t +

1 ⇠t 1

+ CVt ,M KT

+

+ ⇣CVtd 1V

IXt 1

+

1 CVt 1

+⇣

d 1 CVt 1

+ ✏t ,

(4)

11280 0.904

⇤⇤⇤

51.954 (12.786) 22.610 (1.080)

⇤⇤⇤

9036 0.813

(5)

11280 0.904

⇤⇤⇤

51.918 (12.800) 20.216 (1.007) 1.324 (1.118)

⇤⇤

9036 0.823

⇤⇤⇤

9036 0.830

40.802⇤⇤⇤ (2.892)

42.754⇤⇤⇤ (6.021) 15.573 (0.532)

(6)

11280 0.904

-2.655 (-1.086)

51.697 (12.003) 30.113 (1.386)

(6)

9036 0.648

6.508⇤⇤⇤ (4.869) 0.292 (0.083)

(7)

11280 0.781

⇤⇤⇤

(7)

5.842 (6.467) -4.200⇤⇤⇤ (-3.213)

p < 0.05, ⇤ p < 0.1.

44.851⇤⇤⇤ (5.864) 55.436⇤ (1.878) 18.843⇤⇤ (2.554)

p < 0.01,

46.051⇤⇤⇤ (5.572) 125.201⇤⇤⇤ (3.048)

t-statistics are given in parentheses with stars indicating

9036 0.791

9036 0.795

2.731⇤⇤⇤ (6.444) 2.702 (1.077)

Obs. R2

2.831⇤⇤⇤ (6.280) 4.686⇤ (1.731) 1.209⇤ (1.933)

(5)

SPY Basis-Point Jumps Jump Sum

(3)

2.398⇤⇤ (2.209)

9036 0.781

2.922⇤⇤⇤ (5.977) 9.114⇤⇤ (2.383)

Jump Count

(2)

(1)

CVhybrid

CVday

CVhour

⇠M KT

Variable

11280 0.868

11280 0.867

11280 0.867

3.362 (11.580) 1.144 (0.639)

⇤⇤⇤

Obs. R2

3.389 (12.571) 0.399 (0.230) 0.160 (1.214)

⇤⇤⇤

(4)

SPY Basis-Point Jumps Jump Sum

(3)

-0.154 (-0.920)

3.394 (12.509) 0.687 (0.362)

⇤⇤⇤

Jump Count

(2)

CVhybrid

CVday

CVhour

⇠M KT

Variable

(1)

9036 0.651

6.442⇤⇤⇤ (4.950) -3.050 (-1.203) 0.913 (1.273)

Jump Count

(8)

11280 0.781

5.843 (6.485) -4.133⇤⇤⇤ (-3.591) -0.037 (-0.197)

⇤⇤⇤

Jump Count

(8)

(10)

(10)

11280 0.844

49.056 (9.389) -1.449 (-0.092)

⇤⇤⇤

9036 0.652

1.732 (1.233)

6.407⇤⇤⇤ (5.054) -4.663⇤ (-1.676)

9036 0.711

49.641⇤⇤⇤ (5.008) 59.696⇤ (1.880)

SPY Spread Jumps

(9)

11280 0.781

-0.855⇤⇤ (-2.348)

5.955 (6.622) -2.174⇤ (-1.918)

⇤⇤⇤

SPY Spread Jumps

(9)

9036 0.718

48.708⇤⇤⇤ (5.137) 11.634 (0.540) 13.090⇤⇤ (2.051)

Jump Sum

(11)

11280 0.844

49.048⇤⇤⇤ (9.393) -1.948 (-0.134) 0.277 (0.256)

Jump Sum

(11)

where = 0 and = 1 correspond to the upper and lower panels, respectively. Tail realizations are measured in counts of minutely returns exceeding basis point or spread thresholds. The count variable sums jumps exceeding 10 basis points or 5 half-spreads, and the sum variables are a weighted sum of jump sizes exceeding 5, 10, 25, and 100 basis points or 1, 5, 10, or 25 half-spreads. Continuous variation (CV ) is estimated by hour with a 2.5 standard deviation threshold on minutely price movements. Daily CV aggregates continuous variation to the daily frequency, and hybrid CV scales daily CV by hourly average CV relative to average CV. Hour lags of daily CV are excluded. Regressions consist of hourly observations for the 2005–2013 sample. Standard errors are HAC with monthly (126 observation) bandwidth. I normalize ⇠t,M KT by its standard deviations in both panels, and I suppress coefficients on VIX.

tail_realizationt

This table presents results from regressions of realized jumps against estimated tail risks,

Table V: Contemporaneous and Predictive Regressions for Jump Realizations for Several Estimates of Continuous Variation

9036 0.725

28.949⇤⇤ (2.418)

47.473⇤⇤⇤ (5.208) -19.821 (-0.904)

(12)

11280 0.845

-5.135⇤⇤ (-2.019)

49.763⇤⇤⇤ (9.382) 9.881 (0.664)

(12)