crash sensitivity and the cross-section of expected stock returns

8/13/2019 Crash Sensitivity and the Cross-Section of Expected Stock Returns

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Crash Sensitivity and the Cross-Section of ExpectedStock Returns

Stefan Ruenzi and Florian Weigert

This Version: December 2012

AbstractWe examine whether investors receive a compensation for holding crash-sensitivestocks. We capture the crash sensitivity of stocks by their lower tail dependence withthe market based on copulas. Stocks with strong contemporaneous crash sensitivityclearly outperform stocks with weak crash sensitivity and a trading strategy based onpast crash sensitivity delivers positive abnormal returns of about 4% p.a. This effectcannot be explained by traditional risk factors and is different from the impact of beta,downside beta, and coskewness. Our ndings are consistent with results from the em-pirical option pricing literature and support the notion that stock market investors arecrash-averse.

Keywords: Asset Pricing, Downside Risk, Tail Risk, Crash Aversion, Asymmetric depen-dence, Copulas.

JEL Classication Numbers: C12, G01, G11, G12, G17.

Stefan Ruenzi is from the Chair of International Finance at the University of Mannheim, Address:L9, 1-2, 68131 Mannheim, Germany, Telephone: ++49-621-181-1640, e-mail: [email protected] .Florian Weigert is from the Chair of International Finance and CDSB at the University of Mannheim,Address: L9, 1-2, 68131 Mannheim, Germany, Telephone: ++49-621-181-1625, e-mail: [email protected] . The authors thank Andres Almazan, Tobias Berg, Joseph Chen, Fousseni Chabi-Yo, Knut

Griese, Allaudeen Hameed, Maria Kasch, Alexandra Niessen-Ruenzi, Alexander Puetz, Sheridan Titman,and seminar participants at the 2011 European Economics Association Meeting, 2011 European FinanceAssociation Meeting, 2011 German Finance Association Meeting, 2011 Inquire Europe Autumn Seminar,2012 EFM Asset Management Symposium, 2012 Swiss Financial Market Research (SGF) Conference, 2012FMA European Conference, University of Mannheim and University of Texas at Austin for their helpfulcomments. This paper was previously circulated under the title Extreme Dependence Structures and theCross-Section of Expected Stock Returns. All errors are our own.


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Crash Sensitivity and the Cross-Section of ExpectedStock Returns

Abstract

We examine whether investors receive a compensation for holding crash-sensitivestocks. We capture the crash sensitivity of stocks by their lower tail dependence withthe market based on copulas. Stocks with strong contemporaneous crash sensitivityclearly outperform stocks with weak crash sensitivity and a trading strategy based onpast crash sensitivity delivers positive abnormal returns of about 4% p.a. This effectcannot be explained by traditional risk factors and is different from the impact of beta,downside beta, and coskewness. Our ndings are consistent with results from the em-pirical option pricing literature and support the notion that stock market investors arecrash-averse.

Keywords: Asset Pricing, Asymmetric dependence, Copulas, Coskewness, Downside Risk,Tail Risk, Crash Aversion

JEL Classication Numbers: C12, G01, G11, G12, G17.


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1 Introduction

Rare-disaster risk has caught a lot of attention in the nance literature in recent years (e.g.,Barro (2006) and (2009)). Bollerslev and Todorov (2011) nd that most of the aggregate

equity risk premium is a compensation for the risk of extreme events and Gabaix (2012)

shows that time-varying rare-disaster risk can explain the equity premium puzzle (as well as

several other puzzles in macro-nance). Similarly, there is now a small number of recent

papers that examine the time-series relationship between tail risk and aggregate stock market

returns (e.g., Bali, Demirtas, and Levy (2009), Kelly (2012), Bollerslev and Todorov (2011)).

They nd that proxies for tail risk can predict aggregate market returns.

Surprisingly, the potential impact of rare disasters and crash aversion has not caught

much attention in the empirical literature on the pricing of the cross-section of individual

stocks.1 If investors are crash-averse, they derive disproportionately large disutility from

large losses. Then, stocks that do particularly badly when the market performs badly, i.e.,

crash-sensitive stocks, are unattractive assets to hold. In this case, crash-sensitive stocks

should bear a premium. In this paper, we document that crash-sensitive stocks indeed

deliver higher returns than crash-insensitive stocks.

While crash aversion is not a feature of standard expected utility theory, there is sub-

stantial evidence that individuals are particularly averse to large losses. The accumulated

evidence from experiments designed to verify the cumulative prospect theory by Kahne-

man and Tversky (1979) shows that individuals distort the probabilities of low-probability

outcomes like market crashes heavily upwards (e.g., Tversky and Fox (1995), Bleichrodtand Pinto (2000) and Abdellaoui (2002)). 2 More recently, Polkovnichenko and Zhao (2012)

1 Notable exceptions are the contemporaneous papers by Kelly (2012) and Cholette and Lu (2011) thatwe will discuss in more detail below.

2 He and Zhou (2012) show that this can have important implications for investors optimal portfolio

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conrm this pattern using market data from traded nancial options to derive empirical

probability weighting functions. These results are also consistent with earlier ndings from

the empirical option pricing literature that deep out-of-the-money puts, i.e., instruments

that offer protection against extreme market downturns, have a very high implied volatility

and are relatively expensive, 3 which is typically interpreted as investors being crash-averse

or showing signs of crash-o-phobia (Rubinstein (1994), Bates (2008)).

To measure crash sensitivity at the individual asset level, we need a dependence concept

that allows us to focus on joint extreme events. Standard asset pricing models since Sharpe

(1964) and Lintner (1965) argue that the joint distribution of individual stock returns and the

market portfolio return determines the cross-section of expected stock returns. According to

the empirical interpretation of the traditional CAPM, a stocks expected return only depends

on its beta, i.e., its scaled linear correlation with the market, without any focus on tail events.

However, the correlation only describes the full dependence structure in the case of bivariate

normally distributed variables. It can not characterize the full dependence structure of non-

normally distributed random variables such as realized stock returns (Embrechts, McNeil,

and Straumann (2002)). Particularly, it can not capture clustering in the lower tail of

the bivariate return distribution between individual securities and the market, which is

of foremost importance if investors are crash-averse. Thus, we develop a novel proxy for

stock-individual crash sensitivity using copula methods based on extreme value theory. 4

choice and leads to demand for portfolio insurance.3 See, e.g., Rubinstein (1994), Jackwerth and Rubinstein (1996), At-Sahalia and Lo (2000), Bates (2000),

Jackwerth (2000), Rosenberg and Engle (2002), Broadie, Chernov, and Johannes (2009). Garleanu, Pedersen,

and Poteshman (2009) show that this effect is driven by high demand for out-of-the-money puts. Bollenand Whaley (2004) also show that buying pressure for index puts increases their price. Grossman and Zhou(1996) suggest an equilibrium model where a group of investors buys portfolio insurance, which can explainthe high prices of out-of-the-money put options, too.

4 An intuitive alternative approach would be to look at the correlation between individual stock andmarket returns conditional on the market return being below a certain threshold. This approach is similarto the downside beta used in Ang, Chen, and Xing (2006). However, to really capture crash sensitivity, one

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We capture stock individual crash sensitivity based on the extreme dependence between

individual stock returns and market returns in the lower left tail of their joint distribution

(also called lower tail dependence) and then investigate its inuence on the cross-section

of individual stock returns. 5 All else being equal, securities that exhibit strong lower tail

dependence (LTD) are unattractive assets for crash-averse investors to hold, because they

realize their lowest payoffs when investors wealth is already very low (given that one accepts

the market as a proxy for investor wealth). Thus, investing in stocks with only weak or no

LTD serves as insurance against extreme negative portfolio returns (similar to buying out-

of-the-money index puts) as these stocks are unlikely to realize their worst returns when

the market realizes its worst return. Consequently, investors who are sensitive to extreme

downside losses will require a return premium for holding stocks with strong LTD.

Based on daily return data for all US stocks from 1963 to 2009, we calculate LTD coef-

cients for each stock and year. 6 Aggregate LTD peaks around the 1987 crash as well as

during the recent nancial crisis starting in 2008. We then relate individual LTD to con-

temporaneous returns on an annual basis. In doing so, we follow Lewellen and Nagel (2006)

who suggest using short, non-overlapping periods and daily data in asset pricing exercises

when risk exposures might be time varying. Our empirical results using portfolio sorts and

multivariate regression analysis on the individual rm level show a strong positive impact

of LTD on contemporaneous returns. Top quintile LTD stocks outperform bottom quintile

LTD stocks by more than 10% p.a. Our results from Fama-MacBeth (1973) regressions show

has to focus on the really bad market outcomes (not just below average market outcomes as in their paper)

and it is not possible to estimate such conditional betas reliably (see Section 3.4).5 A positive inuence of LTD on returns is expected (but not empirically shown) in Poon, Rockinger, andTawn (2004): If tail events are systematic as well, one might expect the extremal dependence between theasset returns and the market factor returns to also command a risk premium. (p. 586).

6 To do so, we rst determine the convex combination of basic parametric copulas that best explains theempirical bivariate distribution of an individual stocks return and the market return. Then we compute therespective tail dependence coefficients.

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that an increase of one standard deviation in LTD is associated with an average return pre-

mium of about 5 .28% p.a.7 In contrast, weak LTD stocks have signicantly higher returns

than strong LTD stocks during extreme market downturns, i.e., weak LTD stocks indeed

offer some protection against extreme market downturns. Finally, we nd that an investible

trading strategy consisting of buying a portfolio of stocks with the strongest past lower tail

dependence and selling a portfolio of stocks with the weakest past lower tail dependence over

the previous twelve months delivers a signicantly positive return of 4 .4% p.a. The Carhart

(1997) four-factor alpha of this strategy amounts to 4.6% p.a.

The impact of LTD has to be distinguished from the impact of downside beta documented

in Ang, Chen, and Xing (2006) as well as from the impact of other higher co-moments.

Downside beta focuses on individual securities exposure to market returns conditional on

below-average market returns. Thus, it is different from LTD, as it places no particular em-

phasis on tail events. 8 Consequently, downside beta captures general downside risk aversion

rather than crash aversion. Lower tail dependence is a fundamentally different concept and

captures the dependence in the extreme left tail of return distributions, i.e., it focuses on how

individual securities behave during the worst market return realizations within in a given pe-

riod. We nd a strong impact of LTD even after controlling for the impact of the Ang, Chen,

and Xing (2006) downside beta. Adding lower tail dependence as an explanatory variable

drives out most of the impact of downside beta in our multivariate analysis. Thus, crash

aversion has an additional impact on the cross-section of returns that can be distinguished

7 Additionally, we conduct several robustness tests including various risk- and industry-adjustments of

returns, changes in the weighting scheme and analysis of the temporal stability of our results. We also showthat our results are very similar if we do not calibrate the optimal copula structure for each stock and yearbut instead just estimate tail dependence coefficients based on some xed ad-hoc copula combinations. Thisresult might be useful for future research, because selecting the right copula combination is computationallycostly.

8 Downside betas conditional on very low market returns (instead of just below mean market returns)are intuitively more related to LTD. However, they can not be estimated reliably, as we show in Section 3.4.

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from the impact of downside risk aversion. We can also show that the risk premium associ-

ated with LTD cannot be captured by coskewness (Harvey and Siddique (2000)), cokurtosis

(Fang and Lai (1997)), idiosyncratic volatility (Ang, Hodrick, Xing, and Zhang (2006)), or a

stocks lottery characteristics (Bali, Cakici, and Whitelaw (2011a)), and holds after control-

ling for a host of systematic risk factors suggested in the literature. Furthermore, we nd

that most of the abnormal returns of the Jegadeesh and Titman (1993) momentum strategy

are due to its exposure to a contemporaneous LTD factor which suggests that momentum

prots are highly crash-sensitive.

Taken together, the main contribution of our study is twofold: (1) We document that the

lower tail dependence between individual stock and market returns has an impact on the

cross-section of stock returns. (2) We apply copula methods to capture LTD and use it in

an asset pricing context for the rst time.

Besides its pertinence to the literature on rare-disaster risk cited above, our study is

related to the literature on downside risk and loss aversion. Downside risk aversion is already

discussed in Roy (1952), who argues that investors display safety-rst preferences, and in

Markowitz (1959), who suggests using the semi-variance as a measure of risk. 9 Kahneman

and Tversky (1979) argue that individuals evaluate outcomes relative to reference points and

show that individuals are loss-averse. Although aversion to losses and downside risk aversion

is discussed extensively in the literature, only a few papers investigate the effect of loss or

disappointment aversion on expected asset returns. 10 However, these papers (as well as the

9 Many subsequent contributions analyze the impact of higher co-moments on expected returns. Exten-

sions of the basic CAPM that allow for preferences for skewness and lower partial moments of security andmarket returns are developed by Kraus and Litzenberger (1976) and Bawa and Lindenberg (1977). Krausand Litzenberger (1976), Friend and Westereld (1980), and Harvey and Siddique (2000) document thatinvestors dislike negative coskewness of stock returns with the market return. Fang and Lai (1997) andDittmar (2002) show that stocks with high cokurtosis earn high average returns.

10 Barberis and Huang (2001), in one of their model variants, study equilibrium rm-level stock returnswhen investors are loss averse over the uctuations of the individual stocks in their portfolio. They predict

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study by Ang, Chen, and Xing (2006) discussed above) are concerned with general downside

risk aversion rather than crash aversion.

Crash aversion has still caught relatively little attention in the cross-sectional asset pricing

literature. One exception is Berkman, Jacobsen, and Lee (2011) who examine whether in-

dustries that are sensitive to a real crises index deliver higher returns and nd some evidence

for this to be the case. The only other papers we are aware of that investigate whether crash

sensitivity (or tail risk exposure) has an impact on the cross-section of individual stock re-

turns are two contemporaneous papers by Kelly (2012) and Cholette and Lu (2011). 11 These

papers predict aggregate tail risk based on individual extreme events by applying the tail

risk estimator of Hill (1975) to the cross-section of all daily stock returns in a given month.

Consistent with our results, Kelly (2012) also documents that a long-short portfolio that

is based on individual stocks exposure to an aggregate tail risk factor that hedges against

tail events delivers signicantly negative returns. Similar results are obtained by Cholette

and Lu (2011). Our paper differs from these two papers conceptually: We capture crash

sensitivity using lower tail dependence between a stock and the market. Thus, our proxy for

crash sensitivity of an individual stock is directly based on the joint distribution of its return

and the market return. This approach offers the advantage that we only need a time series

of an individual stocks return and the market return (while Kellys implementation of the

Hill (1975) estimator requires the whole cross-section of all individual daily stock returns at

a large value premium in the cross-section. Other models with loss-averse investors include Benartzi andThaler (1995) and Barberis, Huang, and Santos (2001), while Ang, Bekaert, and Liu (2005) model investors

with disappointment aversion.11 A related approach is followed in Bali, Cakici, and Whitelaw (2011b). They extend the Bawa andLindenberg (1977) mean-lower partial moment CAPM and look at co-lower partial moments of stocks withthe market conditional on the individual stock being below a specied threshold. Conditioning on theindividual stock (rather than the market) being below a specic threshold is motivated by the argumentthat many investors are not diversied. They nd that stocks with such high co-lower partial momentsoutperform stocks with low co-lower partial moments.

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each point in time).

Our paper is also related to the literature on the application of extreme value theory in

nance and contributes to the nance literature methodologically. Despite its long history

in statistics, multivariate extreme value theory has been applied to the analysis of nancial

markets only recently. 12 It is used to describe dependence patterns across different markets

and assets (Longin and Solnik (2001)). However, to the best of our knowledge, ours is the

rst paper to investigate extreme dependence structures in the bivariate distribution of the

returns of individual stocks and the market based on copulas. Our application details how

to t exible combinations of basic parametric copulas to this bivariate distribution and

how to derive the corresponding tail dependence coefficients. The copula approach has the

advantage that extreme dependence is not estimated based on a small number of observations

in the tail exclusively, but that information from the whole joint distribution can be used.

The rest of this paper is organized as follows. Section 2 gives a short overview of copula

theory and presents the estimation procedure for tail dependence coefficients. Section 3

demonstrates that stocks with strong lower tail dependence have, at the same time, high

average returns. In Section 4, we examine the persistence of tail dependence and evaluate a

trading strategy based on lower tail dependence. Section 5 concludes and briey discusses

the implications of our results.

12 Longin and Solnik (2001) and Rodriguez (2007) use extreme value theory to model the bivariate returndistributions between different international equity markets. Focusing on risk management applications, Ane

and Kharoubi (2003) propose modeling the dependence structure of international stock index returns viaparametric copulas and derive their tail dependence behavior. Poon, Rockinger, and Tawn (2004) present ageneral framework for identifying joint-tail distributions based on multivariate extreme value theory. Theyargue that the use of traditional dependence measures could lead to inaccurate portfolio risk assessment.Patton (2004) uses copula theory to model time-varying dependence structures of stock returns and assessesthe importance of skewness and asymmetric depependence for asset allocation. Patton (2009) applies copulafunctions to assess different denitions of market neutrality for hedge funds.

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2 Copula Methodology and Data

As copula concepts are not yet regularly used in standard asset pricing applications, wewill rst give a short intuitive introduction into the concept (Section 2.1) and explain how

we compute measures of tail dependence based on copulas (Section 2.2). We then describe

our data and the development of aggregate tail dependence over time (Section 2.3). 13

2.1 Copulas

Most of the standard empirical asset pricing literature focuses on risk factors based onlinear correlation coefficients. However, this measure of stochastic dependence is not typically

able to completely characterize the dependence structure of non-normally distributed random

variables (Embrechts, McNeil, and Straumann (2002)). It is now widely recognized that

many nancial time series, including stock returns, are non-normally distributed. 14 For

example, they are often characterized by leptokurtosis. This is problematic because when

we are dealing with a fat-tailed bivariate distribution F (x1, x 2) of two random variables

X 1 and X 2, the linear correlation fails to capture the dependence structure in the extreme

lower left and upper right tail. As an example, consider the following illustrations of 2,000

simulated bivariate realizations based on different dependence structures between ( X 1, X 2)

shown in Figure 1.15

In all models, X 1 and X 2 have standard normal marginal distributions and a linear cor-

relation of 0.8, but other aspects of the dependence structure are clearly different. For

comparison, in Panel A we rst show an example where we did not allow for clustering in13 A more precise technical, but still accessible, treatment of copula concepts is contained in Nelsen (2006).14 For some early evidence, see Mandelbrot (1963) and Fama (1965).15 The realizations plotted in Figure 1 are based on simulations of different popular copula functions. The

statistical models used to simulate these realizations are the Gauss-copula (Panel A), the Gumbel-copula(Panel B), the Clayton-copula (Panel C), and the Student t-copula (Panel D), all as dened in Table A.1.

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either tail of the distribution. Panels B to D show examples of increased dependence in the

upper right tail, in the lower left tail, and symmetric increased dependence in both tails.

Still, all of these bivariate distributions are characterized by a linear correlation coefficient of

0.8. These examples show that it is not always possible to describe the dependence structure

by the linear correlation alone.

Copulas offer an elegant way to describe the complete dependence structure between ran-

dom variables. Every distribution function of two random variables X 1 and X 2 (e.g., an

individual asset return and the market return) implicitly contains both, a description of

the marginal distribution functions F 1(x1) and F 2(x2) and their dependence structure. The

copula approach allows us to isolate the description of the dependence structure from the

univariate marginal distributions of the bivariate distribution. Sklar (1959) shows that all

bivariate distribution functions F (x1, x 2) can be completely described based on the univari-

ate marginal distributions and a copula C : [0, 1]2 [0, 1]. Sklars Theorem explicitly statesthat all bivariate distributions can be decomposed into copulas and that the marginal distri-

butions and bivariate distributions can be constructed by combining the univariate marginal

distributions using copulas (McNeil, Frey, and Embrechts (2005)). Formally, Sklars Theo-

rem states:

Theorem 1 (Sklar 1959). Let F be a bivariate distribution function with margins F 1 and

F 2. Then there exists a copula C : [0, 1]2 [0, 1] such that, for all x1, x 2 in R = [, ],

F (x1, x 2) = C (F 1(x1), F 2(x2)) . (1)

If the margins are continuous, then C is unique. Conversely, if C is a copula and F 1 and

F 2 are univariate distribution functions, then the function F dened in (1) is a bivariate

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distribution function with margins F 1 and F 2.

There are many different parametric copulas C that can model different tail dependencestructures. In this study, we use combinations of simple basic parametric copulas that either

show no tail dependence (the Gauss-, the Frank-, the FGM-, and the Plackett-copula), lower

tail dependence (the Clayton-, the Rotated Gumbel-, the Rotated Joe-, and the Rotated

Galambos-copula), or upper tail dependence (the Gumbel-, the Joe-, the Galambos-, and

the Rotated Clayton-copula). By using a convex combination of one copula from each class,

we allow for maximum exibility in modeling dependence structures (see Section 2.2).

We will later use copulas to estimate coefficients of upper and lower tail dependence (Sibuya

(1960)), i.e., measures of the strength of dependence in the tails of a bivariate distribution.

2.2 Computation of Tail Dependence Coefficients

Intuitively, the lower (upper) tail dependence coefficient between two variables reects the

probability that a realization of one random variable is in the extreme lower (upper) tail

of its distribution conditional on the realization of the other random variable also being in

the extreme lower (upper) tail of its distribution. Formally, lower tail dependence (LTD) is

dened as

LTD := LTD( X 1, X 2) := limu 0+

P (X 1 F 11 (u)|X 2 F 12 (u)) , (2)

where u

(0, 1) is the argument of the distribution function, i.e., lim u 0+ indicates the limit

if we approach the left tail of the distribution from above. Analogously, we can dene the

upper tail dependence coefficient (UTD) as:

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UTD := UTD( X 1, X 2) := limu

1

P (X 1 > F 11 (u)

|X 2 > F 12 (u)) . (3)

If LTD (UTD) is equal to zero, the two variables are asymptotically independent in the

lower (upper) tail. Simple expressions for LTD and UTD in terms of the copula C of the

bivariate distribution can be derived based on

LTD = limu 0+

C (u, u )u

(4)

andUTD = lim

u 1

1 2u + C (u, u )1 u

, (5)

if F 1 and F 2 are continuous (McNeil, Frey, and Embrechts (2005)). The coefficients of tail

dependence have closed form solutions for the basic parametric copulas used in this study

(see Table A.1 in Appendix A).

However, basic copulas do not allow us to model upper and lower tail dependence simul-

taneously. Thus, we obtain exible copula structures by allowing for convex combinations of

these basic copulas. To allow for the maximum possible exibility, we consider all 64 possible

convex combinations of the afore mentioned basic copulas from Table A.1 that each consist of

one copula that allows for asymptotic dependence in the lower tail, C LTD , one copula that is

asymptotically independent, C NTD , and one copula that allows for asymptotic dependence

in the upper tail, C UTD :

C (u1, u 2, ) = w1 C LTD (u1, u 2; 1)+ w2 C NTD (u 1, u 2; 2) + (1 w1 w2) C UTD (u1, u 2; 3), (6)

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where denotes the set of the basic copula parameters i , i = 1 , 2, 3 and the weights w1 and

w2. We pick the copula C (

,

; ) that best ts the data (as described in the appendix) for

each stock and year and compute the tail dependence coefficients LTD and UTD implied by

the estimated parameters of this copula. As this procedure is repeated for each stock

and year, we end up with a panel of tail dependence coefficients LTD i,t and UTD i,t at the

year-rm level. For a more detailed description of the estimation and selection method, we

refer the reader to the Appendix.

2.3 Data and the Evolution of Aggregate Tail Dependence

Our sample consists of all common stocks (CRSP share codes 10 and 11) from CRSP

trading on the NYSE and AMEX between January 1, 1963 through December 31, 2009.

Copulas and tail dependence coefficients are estimated for each rm and each year separately.

To estimate the yearly tail dependence coefficients, we use daily data for all days on which

the stocks price at the end of the previous trading day was at least $2. We retain all stocks

that have at least 100 valid daily return observations per year. Overall, there are 96 , 676rm-year observations. The number of rms in each year over our sample period ranges from

1, 489 to 2, 440. Summary statistics are provided in Table 1.

The rst four columns show the mean as well as the 25%, the 50%, and the 75% quantile

for key variables. The mean (median) yearly excess return over the riskfree rate of all stocks

in our sample is nearly 10.98% (3.73%) and the mean (median) LTD coefficient is 0.13 (0.11).

We also observe considerable variation in LTD, with an interquartile range of nearly 0.18.

The mean (median) of UTD is 0.08 (0.05) and is signicantly lower than the mean (median)

of LTD. The general tendency for stronger asymptotic dependence in the left tail than in

the right tail of the distributions is consistent with the well-documented nding that return

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correlations generally increase in down markets. 16 The rest of the table provides information

on the summary statistics regarding other rm characteristics and return patterns that we

later use in our empirical analysis. All variable denitions are contained in Appendix B.

The last three columns of Table 1 show the average characteristics of stocks with an above

and below, respectively, value of LTD in the respective year as well as the difference between

the two. Contemporaneous excess returns for high LTD stocks are 13.28% p.a., while they

are only 8.68% p.a. for low LTD stocks. The difference amounts to 4.6% and is statistically

signicant on the 1%-level. At the same time, high LTD stocks also have signicantly higher

regular betas ( ) and particularly downside betas ( ), tend to be somewhat larger and have

lower book-to-market ratios. Cross-correlations between the independent variables used in

this study are shown in Table 2 and conrm these patterns.

The correlation between LTD and UTD is relatively moderate at 0 .12. The low correlation

shows that rms with strong tail dependence in one tail of the distribution do not necessarily

exhibit strong tail dependence in the other tail. This nding also justies our exible model-

ing approach for tail dependence which allows for asymmetric tail dependence in the upper

and lower tail. LTD is correlated with downside beta with a correlation coefficient of 0 .49

and with regular beta with a correlation coefficient of 0 .38. There is also a positive (negative)

correlation between LTD and size as well as cokurtosis (illiquidity as well as coskewness).

We carefully take into account the impact of these correlations in our later analysis.

To get some idea about the temporal variation of tail dependence, we investigate the time

series of aggregate LTD. We dene aggregate LTD of the market, LTD m,t , as the yearly

cross-sectional, equal-weighted, average of LTD i,t over all stocks i in our sample. In Figure

16 In unreported tests, we also look at 5-year subperiods and nd that UTD is signicantly weaker thanLTD in each period. Increased extreme dependence among international markets during bear markets is alsodocumented in Longin and Solnik (2001) and Poon, Rockinger, and Tawn (2004).

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2, we plot the time series of LTD m,t .

There is no particular time trend in LTD m,t .17 However, the graph does exhibit occasional

spikes in LTD m,t that roughly correspond to worldwide nancial crises. The highest value in

LTD m,t corresponds to 1987, the year of Black Monday, with the largest one-day percentage

decline in US stock market history. Another spike in market LTD occurs during the years

2007 through 2009, the years of the recent worldwide nancial crisis. This pattern suggests

that LTD m,t similar to return correlations increases in times of nancial crises. 18 Figure

2 also plots aggregate UTD, UTD m,t , dened as the yearly cross-sectional, equal-weighted,

average of UTD i,t . The time series of LTDm,t and UTD m,t are positively correlated with a

linear correlation coefficient of 0.29. We nd that in most of the years of our sample LTD m,t

clearly exceeds UTDm,t .

3 Crash Sensitivity and Realized Returns

We start our empirical investigation by looking at the contemporaneous relationship be-

tween extreme dependence and returns in univariate sorts and double-sorts (Section 3.1). In

Section 3.2, we conduct Fama-MacBeth (1973) regressions to control for various other po-

tential determinants of returns. In Section 3.3 we examine the impact of a stocks riskiness

on our results, and in Section 3.4 we examine the relationship between variants of the Ang,

Chen, and Xing (2006) downside beta and LTD in more detail. In Section 3.5 we analyze

17 Performing an augmented Dickey-Fuller test rejects the null hypothesis that LTD m,t contains a unit

root with a p-value smaller that 2%.18 The time-series correlation between LTD and the market return is -0.21 and the time-seriescorrelation between LTD and market volatility is 0.59. The time-series correlation between theYale Crash Condence Index (based on data from October 1989 till December 2009 obtained fromhttp://icf.som.yale.edu/stock-market-confidence-indices-united-states-crash-index-data )based on institutional and individual survey participants is roughly -0.5, suggesting that if condence thatno crash will happen is low, aggregate LTD is high.

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how LTD and momentum are related. Furthermore, we investigate whether we get similar

results if we use a simplication of the estimation procedure for tail dependence coefficients

in Section 3.6 and nally, results from a battery of robustness tests are presented in Section

3.7.

In the empirical analysis in this section we relate realized tail dependence coefficients to

portfolio and individual security returns over the same period (while we relate returns to

lagged LTD in Section 4). Looking at the contemporaneous relationship closely follows papers

like Ang, Chen, and Xing (2006) and Lewellen and Nagel (2006), and implicitly assumes that

realized returns are on average a good proxy for expected returns. This procedure is mainly

motivated by the fact that several studies document that risk exposures (like regular beta)

are time-varying (see, e.g., Fama and French (1992), and Ang and Chen (2007)). It is likely

that a stocks LTD is also time-varying and past LTD might not necessarily be a good

predictor of future LTD. 19 Thus, as advocated by Lewellen and Nagel (2006), we use daily

return data for non-overlapping intervals of one year and over each annual period t calculate

a stock is LTD-coefficient. 20 Using an annual horizon trades off two concerns: First, we need

a sufficiently large number of observations to get reliable estimates for our tail dependence

coefficients. Second, by investigating contemporaneous relationships over relatively short

horizons we are able to account for time-varying extreme dependence.

19 In unreported tests we nd evidence for limited predictive power of past LTD on current LTD (see alsoSection 4.1).

20 The estimation procedure of the LTD- and UTD-coefficients for each stock is performed according toSection 2.2 and Appendix A.

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3.1 Portfolio Sorts

3.1.1 Average Returns and Alphas of LTD-sorted Portfolios

To examine whether stocks with strong lower tail dependence earn a premium, we rst

look at simple univariate portfolio sorts. For each year t we sort stocks into ve quintile

portfolios based on their realized LTD in the same year. Results are reported in Table 3.

The rst column shows average LTD coefficients of the stocks in the quintile portfolios.

There is considerable cross-sectional variation in LTD: average LTD ranges from 0.01 in the

weakest LTD quintile up to 0.29 in the strongest LTD quintile.In the second column, we report the annual equal-weighted average excess return over

the risk free rate of these portfolios as well as differences in average excess returns between

quintile portfolio 5 (strong LTD) and quintile portfolio 1 (weak LTD). 21

We nd that stocks with strong LTD have signicantly higher average returns than stocks

with weak LTD. Stocks in the quintile with the weakest (strongest) LTD earn an annual

average excess return of 3.99% p.a. (19.70% p.a.). The return spread between quintile port-

folio 1 and 5 is 15.71% p.a., which is statistically signicant at the 1% level. These results

are consistent with investors being crash averse and requiring a premium for holding stocks

with strong LTD. However, these ndings are only univariate and LTD is correlated with

several other variables that are contemporaneously related to returns such as regular beta

or size (see Table 2). Thus, we also compute the alphas generated by the quintile as well as

the difference portfolios based on the one-factor CAPM, the three-factor Fama and French

(1993), as well as the four-factor Carhart (1997) model.22

Results presented in the last threecolumns show that alphas always increase monotonically from the weakest to the strongest

21 Results are stable if we use value-weighting instead of equal-weighting (see Section 3.7).22 Alphas are estimated based on monthly portfolio and factor returns over the whole sample period.

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LTD quintile portfolios. The CAPM-alpha (3-factor, 4-factor alpha) of the difference portfo-

lio is economically large and amounts to 13.53% (16.19%, 14.06%) p.a., which is statistically

signicant at the 1% level.

In summary, results from Table 3 suggest that LTD determines the cross-section of stock

returns. Stocks with strong LTD earn high average contemporaneous raw- and risk-adjusted

returns. This nding is consistent with the view that stocks with weak LTD offer protection

against extremely low returns in crisis periods and thus on average earn lower returns.

3.1.2 Returns of LTD-sorted Portfolios During Crises

As a consistency check and to check whether weak LTD stocks really offer relatively good

returns during crises, we now also compute average returns of stocks in different LTD quintiles

on the most relevant nancial crisis days in our sample period. We examine Black Monday

(October 19, 1987), the Asian Crisis (October 27, 1997), the burst of the dot-com bubble

(April 14, 2000), and the recent Lehman crises (October 15, 2008). If LTD really captures

crash sensitivity, we should see an underperformance of strong LTD stocks as compared to

weak LTD stocks on these days. Results are presented in Table 4.

As expected (and opposite to what we nd in the overall sample; see Table 3), strong

LTD stocks strongly underperform weak LTD stocks on each of these individual days. The

differences are economically large: the daily return of the weak LTD portfolio is from 4.4%

to 9.2% higher than that of the strong LTD portfolio. To assess the statistical signicance,

we also jointly analyze all days on which the excess return of the market over the riskfree

rate is less than 5%, which was the case on 16 days in our sample. Results are presentedin the last column of Table 4 and show that the weak LTD portfolio outperforms the strong

LTD portfolio by nearly 5% per day on those days. The effect is statistically signicant at

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the 1% level. These ndings show that weak LTD stocks can indeed serve as an insurance

for loss-averse investors and might explain why overall they earn lower returns than strong

LTD stocks, as documented above.

3.1.3 Double-Sorts

Our univariate result of higher average returns of strong LTD stocks from Section 3.1.1

(Table 3) could be driven by differences in beta, downside beta or differences with respect to

other return characteristics like coskewness across the different LTD quintiles. This possibil-

ity is again suggested by the results from the correlation Table 2, where downside beta and

coskewness are the variables most strongly correlated with LTD. Thus, as a next step, we

conduct double sorts based on LTD as well as regular beta, downside beta, and coskewness. 23

We rst form quintile portfolios sorted on .24 Then, within each quintile, we sort stocks

into ve portfolios based on LTD.

Panel A of Table 5 reports equal-weighted contemporaneous portfolio excess returns over

the risk-free rate of the 25

LTD portfolios. In all LTD quintiles, high stocks outperform

low stocks. More importantly, in all quintiles, we document a nearly monotonic increase

in returns from the weak LTD to the strong LTD portfolio. The return difference between the

weakest LTD quintile and the strongest LTD quintile within all quintiles is economically

large and statistically signicant at the one percent level. It ranges from 5 .23% p.a. in the

lowest quintile to 18.29% p.a. in the highest quintile. The average spread in excess

returns amounts to 10 .40% p.a. Hence, regular risk cannot account for the reward earned

by holding stocks with strong LTD. This nding is consistent with our earlier ndings of a

23 Although we already control for linear beta exposure of our portfolios by looking at CAPM alphasabove, we now additionally analyze dependent portfolio double-sorts on LTD and regular , which allows usto also control for a possible non-linear impact of .

24 Our results (not reported) are stable if we reverse the sorting order or conduct independent sorts.

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highly positive CAPM-alpha of the difference portfolio from Table 3.

LTD is also related to downside beta ( ) as dened in Ang, Chen, and Xing (2006) as a

stocks conditional on the market return being below its mean (see Appendix B). Thus,

as above, in a rst step we form quintile portfolios sorted on . Then, within each

quintile, we sort stocks into ve portfolios based on LTD. Panel B of Table 5 reports equal-

weighted average excess returns of the 25 LTD portfolios. We nd that the returnsof the low portfolios tend to be smaller than those of the high portfolios. Overall,

these ndings generally conrm the results of Ang, Chen, and Xing (2006). Turning to the

impact of LTD, we nd that stocks in the weak LTD portfolios have an average (across all

quintiles) excess return of 7 .09% p.a., while stocks in the strong LTD portfolios have an

average excess return of 15.57% p.a. The spread is signicant at the 1% level. Amounting

to 8.48% p.a., it is also economically large. We nd the strongest effect of LTD on returns in

the highest quintile, but it is highly statistically signicant and economically large within

each individual quintile (ranging from 5 .16% p.a. to 12.98% p.a.). Hence the impact of

LTD on returns does not seem to be driven by , either. 25

Harvey and Siddique (2000) show that lower coskewness, coskew , is associated with higher

expected returns. Thus, to explicitly control for the impact of coskewness, we rst form

quintile portfolios sorted on coskew . Then, within each coskew quintile, we sort stocks into

ve portfolios based on LTD. Panel C of Table 5 shows equal-weighted average excess returns

of the 25 coskew LTD portfolios. In most coskew quintiles, we document a monotonicincrease of returns with LTD. The return difference between the weakest LTD quintile and

the strongest LTD quintile ranges from 9 .74% p.a. up to 13.73% p.a., with an average spread

of 12.34% p.a. This spread is again highly signicant within each individual coskew quintile,

25 We show later that our results can also not be explained by alternative denitions of downside betathat condition on more extreme bad market outcomes (see Section 3.4).

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which indicates that coskewness risk also cannot account for the reward earned by holding

stocks with strong LTD. At the same time, we can conrm the negative impact of coskewness

on returns documented in Harvey and Siddique (2000).

To summarize, based on double sorts we provide strong evidence that the risk associated

with LTD is different from risks associated with regular market beta, downside beta, and

coskewness. Double sorts offer the advantage that they allow us to control for any potential

non-linear impact. However, in double sorts we can only control for one return characteristic

at a time. Thus, we now turn to a multivariate approach that allows us to examine the joint

impact of different return and other characteristics of the rm that might have an impact

on the cross-section of stock returns.

3.2 Multivariate Evidence

We run Fama-MacBeth (1973) regressions on the individual rm level in the period from

1963 - 2009 using non-overlapping data.26 Panel A of Table 6 presents the regression results

of yearly excess returns on realized LTD and various combinations of control variables in therst six columns.

In regression (1), we only include LTD as explanatory variable. It has a highly statistically

as well economically signicantly positive impact. A one standard deviation increase in

LTD leads to additional returns of 6.26% p.a. In regression (2), we add the stocks UTD

coefficient. It has a signicantly negative impact on returns, but the economic magnitude

is much smaller than the impact of LTD. To check whether we can conrm the results

26 This econometric procedure has the disadvantage that risk factors are estimated less precisely in com-parison to using portfolios as test assets. However, Ang, Liu and Schwarz (2010) show analytically anddemonstrate empirically that the smaller standard errors of risk factor estimates from creating portfoliosdoes not necessarily lead to smaller standard errors of cross-sectional coefficient estimates. Creating portfo-lios destroys information by shrinking the dispersion of risk factors and leads to larger standard errors.

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from Ang, Chen, and Xing (2006) in our sample, in regression (3), we replace the two tail

dependence coefficients by downside- and upside beta, and + . We nd that the impact

of is highly signicant, while there is no signicant impact of + . This nding is broadly

consistent with the analysis and results in Ang, Chen, and Xing (2006) who also nd that

is an important determinant of the cross-section of stock returns while + has much

less of an impact. In the following regressions, we expand regression model (2) and add

other rm characteristics like size, book-to-market and several other return characteristics

that might have an impact on returns. 27 Specically, in regression (4) we add coskewness,

coskew , and the Amihud (2002) illiquidity ratio, illiq , as a liquidity proxy, while regression

(5) additionally includes previous year returns, idiosyncratic return volatility, cokurtosis of

individual returns with the market return, and a stocks lottery features captured by the

maximum daily return over the past year, max , similar as in Bali, Cakici, and Whitelaw

(2011a).28 Results show that the impact of LTD is very similar to before and still highly

signicant in economic as well as statistical terms after the inclusion of the control variables.

LTD exhibits the strongest inuence of all variables in terms of statistical power (t-statistic

of 11.59 and 9.89, respectively).

Several of the control variables have a signicant impact on returns, too, that conrm

ndings from the existing literature: Firm size (book-to-market ratio) has a negative (posi-

tive) impact (e.g., Fama and French (1993)), coskewness (Harvey and Siddique (2000)) and

illiquidity (Amihud (2002)) have a positive impact, while idiosyncratic volatility (Ang, Ho-

drick, Xing, and Zhang (2006) and (2009)) and max (Bali, Cakici, and Whitelaw (2011a))

27 We winsorize all realizations of our independent variables at the 1% and 99% levels in order to avoidoutliers driving our results. Our results do not hinge on this winsorization (see Section 3.7).

28 Bali, Cakici, and Whitelaw (2011a) dene max based on a monthly horizon. Our later examinationof a trading strategy is conducted on a monthly basis. Thus, we will later include a lottery-factor in thatanalysis, which matches the time-horizon of Bali, Cakici, and Whitelaw (2011a) more closely (see Section4.2).

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have a negative impact. Consistent with Fang and Lai (1997) and Dittmar (2002), we also

nd a positive coefficient for the impact of cokurtosis. However, the effect is not statistically

signicant in our setting.

In regression (6) we replace by and + . Regressions (7) to (9) are identical to

regression (6), but we use rm-individual yearly alphas estimated based on daily data from

the CAPM, the Fama and French (1993), and the Carhart (1997) models, respectively, as

dependent variable. In all cases, our earlier ndings are conrmed: there is a very strong

positive impact of LT D and the t-statistic for the impact of LTD is always the highest t-

statistic of all explanatory variables. Interestingly, including downside beta does not reduce

the impact of LTD. Rather, in regressions (7) to (9) downside beta has no signicant impact

on returns anymore when LTD is also included. 29 The last column presents the economic

signicance based on a one-standard deviation change of each explanatory variable based on

the results from regression (9): a one-standard deviation increase of LTD leads to a increase

in the Carhart (1997) four factor alpha of 5.01% p.a. This is the largest effect in terms of

economic magnitude of all variables included. In contrast, the economic impact of is

economically small, amounting to 0.21% p.a. only.

3.3 Impact of LTD in Risk-, VaR-, and Conditional VaR-sorted

Samples

So far, we have focused on the impact of LTD, dened in (2) as the limit of the probability

of a joint left tail realization of the market and an individual rms return. We use this

29 In unreported tests, we also run variants of regressions (6) to (9) where we include and + andthe other controls, but not LT D and U T D . In this case, we nd an economically meaningful and typicallystatistically signicant coefficient for the impact of . This nding shows that does not loose itssignicant impact due to the inclusion of the other control variables, but due the inclusion of LT D .

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measure to capture a stocks crash sensitivity, because it tells us how likely it is that a

stock realizes its worst return exactly at the time when the market realizes its worst return.

However, it does not take into account how bad the worst return realization of the stock

really is (similar to the difference between the correlation of a stocks return with the market

and its market beta). Thus, we now want to check whether the impact of LTD is stronger if

we can expect the worst return realization of a stock to be particularly low, i.e., if we weight

the probability of the joint outcome with a proxy for the potential severity of the outcome.

We use three ad-hoc proxies to capture how bad a bad outcome would be: a stocks annual

return standard deviation based on daily returns, 30 its Value-at-Risk (VaR, dened as the

5% percentile of the daily returns of the stock), and its conditional Value-at-Risk (CoVaR,

dened as the conditional mean of all daily returns below the 5% percentile). Stocks with a

high return standard deviation (a low VaR and a low CoVaR) tend to have particularly low

worst returns (i.e., they have particularly bad return realizations in an absolute sense if they

realize their relatively worst returns). To examine whether there is a stronger impact of LTD

on returns for stocks with a high return standard deviation (a low VaR and CoVaR), we sort

stocks into two categories according to the median of the three proxies in the respective year

and repeat regressions (6) to (9) from Panel A for these subsamples. Results for the impact

of LTD are shown in Panel B of Table 6. All other explanatory variables are included in

the regressions, but suppressed in the Panel. Our ndings clearly indicate that the impact

of LTD is much stronger for rms where bad outcomes are particularly severe, i.e., for rms

with a high return standard deviation (low VaR and low CoVaR). Irrespective of which

proxy we look at, we always nd that the coefficient for the impact of LTD on returns and

alphas is about three times as large among high standard deviation (low VaR, low CoVaR)

30 We also use a stocks semi-variance (variance conditional on the return being below its mean) insteadof its variance. Findings (not reported) are very similar.

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rms than among low standard deviation (high VaR, high CoVaR) rms. These ndings

conrm our conjecture, that tail dependence matters more for returns among more risky (in

a traditional, i.e., univariate, sense) rms.

3.4 Downside Beta vs. Lower Tail Dependence

Results in Panel B of Table 5 show that our results are not driven by downside beta ( ).

Although the concepts of LTD and seem related, this is not surprising, because the latter

focuses on all market returns below the mean, while the former explicitly focuses on extreme

events. However, one could argue that alternative denitions of that focus more on the

left tail of the market return distribution capture effects similar to LTD. To analyze this

idea more closely, we repeat our LT D double sorts from Table 5 for alternative denitions in Table 7. Specically, we calculate downside betas as betas conditional on the

market return being below its 20%, 10%, 5%, 2%, and 1% quantile (rather than being below

the mean, as before and as in Ang, Chen, and Xing (2006)).

We report results on the returns of the strong minus weak LTD portfolios within eachdownside beta quintile in the rst ve columns as well as the average of this difference

portfolio return across all downside beta quintiles in the last column (like in the last row of

Panel B in Table 5) for all alternative denitions. The average difference returns range

from 9.46% (for the denition based on the 20% quantile) up to 15.23% (for the

denition based on the 1% quantile) and is signicant at the 1% level in each case. Thus,

our results not only hold after adjusting for various alternatives, they even get stronger if

we look at more restrictive denitions. 31 At rst glance, this result might seem surprising,

as more restrictive s (that focus more on extremely bad market returns) should be more31 We also replace the standard downside beta by alternative downside beta denitions in our multivariate

regressions from Table 6 (Panel A) and obtain similar results. Results are available upon request.

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related to our LTD measure. The reason we nd even stronger results for the alternative

s is that they are actually not able to reliably capture dependence in the tails because

they are estimated based on a very small number of observations (e.g., only about 12 daily

return observations per year for the 5% quintile ) and are thus very noisy. 32

These results also illustrate the advantage of the copula approach in estimating extreme

dependence: in estimating the whole dependence structure between individual and market

returns using our semi-parametric approach, we make use of all available daily return ob-

servations within a year, which allows for a relatively more precise estimation. We then

calculate LTD coefficients based on the procedure described in Section 2.2. Thus, the com-

putation of LTD ultimately does not rely on a very small number of extreme observations

only (like the more restrictive s discussed above) and consequently is much less noisy and

more informative about the true crash sensitivity of a stock.

3.5 Lower Tail Dependence and Momentum

Chen, Hong, and Stein (2001) and Boyer, Mitton, and Vorkink (2010) show that (con-ditional) skewness is related to past returns. In particular, they document that negative

skewness (which is also a proxy for the downside exposure of stocks) is most pronounced in

stocks that have expericenced high positive returns over the last 12 to 36 months. In unre-

ported tests, we also nd that past returns are a strong positive predictor of a stocks LTD. 33

32 Correlations (not reported in tables) between the alternatives and LTD actually decrease from 0.49for the standard Ang, Chen, and Xing (2006) , to 0.41 for the correlation between LTD and based

on the 20% quintile and even further to 0.17, 0.07, and 0.03 for the correlation of LTD with

based onthe 5%, 2%, and 1% quintile. Additionally, if we compute the returns of a portfolio going long in high

stocks and short in low stocks, we typically nd no positive abnormal returns using the more restrictive denitions, irrespective of whether we look at univariate sorts or double sorts based on and LTD (notreported in tables).

33 Regressing a stocks LTD in year t on its past return in year t 1 in a univariate model delivers apositive coefficient estimate of 0.029 with a high degree of statistical signicance (t-stat: 6.00).

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Thus, it is not unlikely that the LTD-premium we document is related to the Jegadeesh and

Titman (1993) momentum effect.

To examine whether such a relationship exists, we run a simple time series regression based

on yearly data where the momentum strategy return is regressed on the excess market return,

the Fama and French (1993) size and book-to-market factor as well as the LTD premium

(computed as the yearly strong LTD - weak LTD portfolio return). Results are presented in

Table 8.

In the rst column we do not include the LTD premium factor and nd evidence for

the well-documented momentum effect. The yearly alpha after controlling for the exposure

to the size- and book-to-market-factor amounts to 12.8%. In column (2) we include the

equal-weighted return of the strong LTD - weak LTD portfolio and in column (3) the value-

weighted return of the strong LTD - weak LTD portfolio as additional factors. We nd that

in both cases the momentum strategy return positively loads on LTD. The coefficient is

around 0.7 and statistically signicant at the 5%-level, showing that a momentum strategy

indeed is crash-sensitive. Interestingly, the alpha of the momentum strategy is reduced to

1.88% and 3.91%, respectively, if we include the equal- or value-weighted LTD factor, and

not statistically signicant at conventional levels anymore. Thus, our results suggest that

part of the momentum effect can be explained by the crash sensitivity of this strategy. 34

34 It should be noted that our LTD factor is not based on an implementable trading strategy, because rmsare sorted based on contemporaneous LTD. In unreported tests, we also construct a traded factor based onpast LTD. The return of the momentum factor still has a signicantly positive loading on this LTD factor,but the alpha of the momentum strategy is only reduced by about 1.5% p.a. and remains signicant in thiscase.

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3.6 Simplifying Tail Dependence Coefficient Estimation

The copula selection procedure described in Section 2.2 and Appendix A is computation-ally costly. 35 We now investigate whether this estimation procedure could be simplied.

Instead of selecting the appropriate parametric copula by minimizing the distance between

64 different convex copula combinations and the empirical copula, we ex ante choose var-

ious xed convex copula combinations. As our ad-hoc xed copula combinations, we con-

sider the Rotated Joe/F-G-M/Joe (3-D-I)-, the Rotated Galambos/F-G-M/Joe (4-D-I)-, the

Rotated Gumbel/F-G-M/Joe (2-D-I)-, the Rotated Gumbel/Frank/Gumbel (2-B-II)-, the

Rotated Galambos/Frank/Gumbel (4-B-II)-, and the Rotated Galambos/Frank/Galambos

(4-B-III)-copula out of Table A.1. We choose the copula combinations (3-D-I), (4-D-I), and

(2-D-I) because they are the copulas most often selected in the estimation procedure in Table

A.2. In contrast, the remaining three copula combinations are the copulas least often se-

lected in the estimation procedure. We perform Fama-Macbeth (1973) regressions of excess

returns on LTD (estimated based on the xed copula combinations) as well as the full set

of control variables (as in Regression (6) of Table 6, Panel A). Results on the coefficientestimates for the inuence of LTD are displayed in the left part of Table 9.

We nd that LTD is a highly signicant explanatory factor for the cross-section of expected

stocks returns independent of the specied convex copula combination. The magnitude

of the coefficients and the signicance levels of the control variables (not shown in the

table) also remain stable across the different specications. These results document that our

main ndings are not driven by the tail dependence coefficient estimation procedure. The

estimation procedure, which is computationally intensive, can be dramatically simplied by

35 The selection of the optimal copula combination and the estimation of the tail dependence coefficientson a grid cluster takes about 20 seconds per stock and year. This amounts to an entire estimation time forthe LTD coefficients used in this study of well above 500 hours.

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just picking a reasonable convex copula combination (based on the copulas in Table A.1).

This might be a helpful result for researchers working on the impact of tail dependence in

similar settings.

3.7 Additional Analysis and Robustness Checks

In this section we conduct a battery of additional robustness tests to analyze whether

our main results from above are stable. We examine the inuence of return adjustments

(Section 3.7.1) and the weighting scheme in the portfolio sorts (Section 3.7.2), the temporal

stability of our results (Section 3.7.3), and variations of the regression setup employed in the

multivariate tests (Section 3.7.4). 36

3.7.1 Industry- and DGTW(1997)-Adjusted Returns

Some extreme market downturns can mainly be driven by a few industries. Consequently,

some of our ndings might be driven by industry effects. Thus, we repeat our multivariate

regressions with the full set of controls (i.e., Regression (6) from Table 6, Panel A) butuse industry-adjusted returns instead of raw returns as dependent variable. Results on the

estimates for the impact of LTD are presented in the right part of Table 9 .

In the rst two lines, we use the Fama-French 12 (FF12) and 48 (FF48) industry classi-

cations. In both cases, the coefficient for the impact of LTD is statistically signicant at

the one percent level and similar in magnitude to the ndings above. As Johnson, Moore,

and Sorescu (2009) point out, tests of long-term abnormal returns based on FF48 industry

adjustments might not be well-specied if there is industry clustering (because rms in the

36 Besides the robustness tests described here, we also use weekly data instead of daily data and a longerestimation horizon to determine our tail dependence coefficients. Our results (not reported) are similar if we use a longer estimation horizon of 2 years, 3 years, or 5 years instead of 12 months.

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same FF48 industry often belong to very different SIC-3 industries). Thus, we also follow

their suggestion and use the SIC-3 digit industry classication to industry-adjust returns.

Additionally, we use the SIC 2- and 4-digit industry classication. In all cases, our prior

results are conrmed. These ndings show that our results are not just driven by industry

effects.

Furthermore, instead of controlling for the impact of stock characteristics by including

them as explanatory variables, we also adjust the return of each stock by subtracting the

return of its corresponding Daniel, Grinblatt, Titman, and Wermers (1997) characteristic-

based benchmark (DGTW). 37 Results are presented in the last line of the same table. Again,

our result of a strongly positive impact of LTD on expected returns remains unaffected.

3.7.2 Weighting Scheme

Our sorts and double sorts in Section 3.1 focus on equal-weighted portfolios. Thus, results

could be inuenced by over-weighting the importance of small or tiny stocks. 38 Therefore,

we now examine value-weighted average excess returns of the same univariate as well as

bivariate sorts as in Tables 3 and 5. Results are presented in Table 10.

Value-weighted returns for our LTD quintile portfolios are presented in Panel A. We nd

that stocks in the quintile with the weakest (strongest) LTD earn an annual average excess

return of 1.03% p.a. (9.45% p.a.). The return spread between quintile portfolios 1 and 5is 10.48% p.a. and statistically signicant at the 1% level.

In Panels B to D of Table 10 we perform value-weighted double-sorts on LTD and , ,

37 The DGTW benchmarks are available via http://www.smith.umd.edu/faculty/rwermers/ftpsite/Dgtw/coverpage.htm .

38 Similar concerns can also be raised with respect to the multivariate regression results because theregression evidence presented in Table 6 is essentially also based on equal weighting as each observationsenters the cross-sectional Fama-MacBeth (1973) regressions with the same weight.

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and coskew , respectively. As with equal-weighted portfolios, we document a clear outperfor-

mance of strong LTD stocks over weak LTD stocks in each quintile (Panel B). The return

spread ranges from 2 .75% p.a. in the lowest quintile up to 16.22% p.a. in the highest

quintile. The return spread is signicant at the 1% level within all quintiles except for the

very lowest quintile, where it is signicant at the 5% level. We nd similar patterns for

the double sorts based on and LTD (Panel C) as well as double sorts based on coskew

and LTD (Panel D).

Overall, value-weighting portfolios rather than equal-weighting does not change our main

nding of signicantly higher returns of stocks with strong LTD. Thus, our results are not

driven by the returns of very small rms.

3.7.3 Temporal Stability

As a rst test of the temporal stability of our main nding, we reproduce the results of the

univariate sorts from Table 3 for 5 subperiods of roughly equal length: 1963-1972, 1973-1981,

1982-1990, 1991-1999, and 2000-2009. Results are presented in the rst ve columns in Panel

A of Table 11.

In each subperiod we nd returns that are monotonically increasing with LTD. The return

spread between the strong and the weak LTD portfolios is highest (21.21%) in the 1991-1999

period and lowest (7.86%) in the most recent period, 2000-2009. It is always statistically

signicant at least at the 5% level. In the last two columns we split the sample in 1987

in order to check whether the results differ prior to the crash of 1987 and after. In the

option pricing literature it is sometimes argued that investors became crash-o-phobic after

the experience of the 1987 crash (Rubinstein (1994)). If investors were less crash-averse prior

to 1987, we should see a weaker compensation for LTD in the earlier subperiod.

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We document a return spread between stocks with strong LTD and stocks with weak LTD

of 16.24% p.a. in the period 1963 to 1986. Results for the later subperiod from 1987 to

2009 show that the return spread is of similar magnitude at 15.17% p.a. In both cases, the

magnitude of the spread is similar to that from the complete sample and signicant at the

1% level. This shows that investors on the stock market were compensated for holding stocks

with strong LTD prior to 1987 and suggests that they were crash-averse even prior to the

crash of 1987.39

In Panel B, we report results from the same double sorts as in Table 5 separately for the

pre- and post-1987 period. We only report the average return across the , , and coskew ,

respectively, portfolios for the 5 LTD portfolios as well as the difference portfolio. While the

dependent sorts based on and LTD as well as those based on and LTD show a more

pronounced return spread in the later period (8.45% vs. 12.45% and 6.77% vs. 10.26%),

return spreads from dependent sorts based on coskew and LTD are very similar in the two

subperiods (12.71% and 11.95%). Irrespective of these patterns, results are economically

meaningful and statistically signicant in either case.

Finally, we look at the results from our multivariate regressions with the full set of ex-

planatory variables for the same ve subperiods as in Panel A. Results of the respective

Fama-MacBeth (1973) regressions are presented in the rst ve columns in Panel C of Ta-

ble 11. Even within these relatively short subsamples, we always nd a coefficient for the

impact of LTD that is of similar magnitude to the full sample. It ranges from 0.385 in the

last subperiod of our sample up to 0.605 in the fourth subperiod. The coefficient estimate

for LTD is signicant at the 5% level in the rst subperiod and signicant at the 1% level in

39 While seemingly contradictory to some of the empirical option pricing literature it should be noted thatstudies that nd no strong crash-fear effect prior to 1987 typically rely on very short pre-87 samples dueto the lack of option data availability for earlier years.

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all other subperiods. LTD is the most consistently priced factor of all explanatory variables

across the subperiods.

We also look at subsamples based on up and down market returns. Results are presented

in the last two columns. As expected, the effect of LTD on the cross-section of stock returns

is stronger in times of up markets (market return > 0) than in down markets (market return

< 0). However, the effect is signicant at the 1% level even in the latter case. Overall, the

strong positive cross-sectional impact of LTD on stock returns is remarkably stable over time

and holds in up and down markets.

3.7.4 Alternative Regression Methods

Our previous multivariate regression evidence in Section 3.2 relies on standard Fama-

MacBeth (1973) regressions with winsorized independent variables. We now perform several

variations of this basic regression approach with the full set of independent variables. Results

are presented in Table 12.

Regression (1) repeats the baseline regression (6) from Panel A in Table 6, but we now use

Newey-West standard errors in the second stage of the Fama-MacBeth (1973) regressions

to determine statistical signicance. Regression (2) repeats the standard Fama-MacBeth

(1973) regression, but we do not winsorize the independent variables. In regression (3)

we perform a pooled OLS regression with time-xed effects and standard errors clustered

by stock. Regression (4) is identical, but we cluster standard errors by industry (based

on the Fama-French 12 industry classication). 40 Regressions (5) and (6) perform panel

data regressions with rm xed effects. In regression (6) standard errors are additionally

clustered by rm. Finally, in regression (7) we regress excess returns on the independent

40 Results are virtually unchanged whether we cluster by Fama-French 48 or SIC industries.

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variables via a random-effect panel data regression. We document that LTD is a highly

signicant explanatory factor for the cross-section of expected stocks returns independent of

the specic regression setup. The point estimate for the inuence of LTD is roughly 0.45

and very similar across regressions.

Overall, the results from our robustness checks show that our main ndings are conrmed

and do not depend on the weighting scheme or return adjustments, are stable over time, and

are not driven by the specic regression technique or by a specic dependence structure of

the error terms. 41

4 Crash Sensitivity Persistence and Trading Strategy

The previous section documents that LTD has a strong impact on contemporaneous re-

turns. We now examine whether tail dependence is persistent over time (Section 4.1) and

whether it also predicts cross-sectional return differences. Although not the focus of our

paper, we thereby want to check whether a protable trading strategy could be implemented

solely based on past information on tail dependence (Section 4.2).

4.1 Persistence of Crash Sensitivity

To examine whether LTD is time-varying, we compute the average LTD of the stocks in

the LTD quintile portfolios over time. Firms are sorted into quintiles based on their realized

LTD in year t = 1. Panel A of Figure 3 displays the evolution of the average equal weighted

LTD of these portfolios over the following four years t = 2 to t = 5.

41 In unreported tests, we also examine the impact of LTD within subsamples consisting of rms from oneFama-French 12 industry at a time. The effect is signicant within most individual industry subsamples andis strongest in Business Equipment, Whole Sales, Retail, Services, and Manufacturing.

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The Figure shows that the stocks in the strong LTD portfolio also have stronger LTD than

the stocks from the weak LTD portfolio in the following years. However, the difference quickly

shrinks considerably. In unreported results from a regression analysis, we nd evidence for

limited predictive power of past LTD on current LTD. Regressing current LTD on past LTD

in a univariate model delivers a coefficient estimate of 0.248 with a high degree of statistical

signicance (t-stat: 12.56), but a relatively low R 2 of about 7%. The patterns documented

indicate that there is some predictability in extreme dependence, but the sharply decreasing

differences shown in Figure 3 suggest that many stocks are exposed to time-varying tail

risk.42

4.2 Past LTD and Future Returns

4.2.1 Trading Strategy

We now examine whether it is possible to generate abnormal returns based on prior infor-

mation about past extreme dependence. Our strategy consists of going long in stocks with

strong past LTD and going short in stocks with weak past LTD with monthly rebalancing.

Specically, we sort stocks into ve quintile portfolios at the beginning of each month based

on past LTD estimated over the previous twelve months. Then, we examine equal-weighted

returns of these portfolios over the next month. We use data from January to December

1963 to determine the LTD estimates for the rst portfolio formation in January 1964. Thus,

our trading strategy spans the period January 1964 to December 2009.

Our empirical setup requires the estimation of LTD-coefficients for each stock for 552overlapping 12 month periods. To reduce the computational effort, we rely on the results

42 This latter nding justies our previous approach of relating returns to contemporaneous tail dependencerealizations over relatively short horizons (as suggested for time-varying risk exposures in Lewellen and Nagel(2006)).

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from Section 3.6 and simplify the estimation procedure for LTD in selecting the Rotated

Joe/F-G-M/Joe (3-D-I)-copula as our xed copula convex combination for all stocks and

periods.43

Table 13 reports the monthly average excess return over the risk free rate for all quintile

portfolios based on past LTD and the return and alpha differences between quintile portfolio

5 (strong LTD) and quintile portfolio 1 (weak LTD).

Column 1 of Table 13 shows that stocks in the strongest past LTD quintile earn an average

equal-weighted excess return over the risk free rate of 0 .862% per month, while the stocks

in the weakest past LTD quintile earn 0 .499% per month. Thus, our trading strategy of

investing in strong LTD stocks and shorting weak LTD stocks delivers an economically sig-

nicant future return of 0 .363% per month, which translates into an average return premium

of 4.36% p.a. The spread in average excess returns between quintile portfolios 5 and 1 is

statistically signicant at the 1% level. 44

To check whether exposures to systematic risk factors drives this nding, we regress the

monthly return time series of the past LTD quintile portfolios as well as the difference

portfolio returns on the monthly excess market return and other systematic risk factors.

The alphas from these regressions are shown in columns (2)-(4) of Table 13. Results from

the CAPM one-factor regressions show that a part of the premium from our trading strategy

is indeed due to its sensitivity to current market beta. However, the market factor exposure

cannot explain the full LTD return premium. The trading strategy still delivers a monthly

alpha of 0.253% (signicant at the 5% signicance level) after controlling for market beta.

43 Results based on other ad-hoc chosen copula combinations are very similar.44 In unreported tests, we also investigate a trading strategy based on UTD. We nd no signicant return

spreads based on such a strategy. Furthermore, we also analyze trading strategies that are double sorted onpast LTD as well as past beta, past downside beta, and past coskewness, respectively. In each case we nda signicant outperformance of the strong past LTD over the weak past LTD portfolio.

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When taking into account the size factor (SMB) and the book-to-market factor (HML), the

alpha of our strategy increases to 0 .454% per month. The returns of our trading strategy

load signicantly negatively on both factors. In the fourth column, we also control for the

momentum factor. We still obtain a highly signicant monthly alpha of 0 .383% per month.

Results from the four-factor Carhart model also show, that the long and the short side of the

trading strategy contribute roughly equally to the overall performance. While the portfolio

with the weakest past LTD stocks delivers an alpha of 0.20% per month, the quintileportfolio with the strongest past LTD stocks delivers an alpha of 0 .19% per month (both

different from zero with a signicance level of 5%). In the columns (5) and (6) we repeat

the same analysis, but estimate the LTD of the stocks not over the past 12 months, but over

the past 24 and 36 months, respectively. Results are generally conrmed. Finally, in the

last three columns, we present the Carhart (1997) four factor alpha based on value-weighted

portfolio rather than equal-weighted portfolios for the three estimation horizons. Results are

slightly weaker than for equal-weighted portfolios, but still positive and signicant.

As downside beta and LTD are different but related concepts (see Section 3.4), we also

want to check whether a trading strategy based on delivers similar results. In the last line

of the table, we report difference returns and alphas of a trading strategy going long in stocks

in the top quintile and going short in stocks in the bottom quintile according to their lagged

. The return of the trading strategy based on estimated over the previous 12 months

is 0.09% and not statistically signicant. The CAPM (Fama and French (1993), Carhart(1997)) alpha of the strategy is also negative at 0.319% (0.333%, 0.260%). Similarresults are obtained for longer estimation horizons for and for value-weighted portfolios.

Thus, we can conclude that a trading strategy based on information about lagged does

not deliver a positive outperformance, while a trading strategy based on information about

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lagged LTD seems protable. 45

4.2.2 Alternative Factor Models

In the following, we test the robustness of our ndings from Table 13, by evaluating our

trading strategy using alternative factor models. Results are presented in Table 14.

First, we include the Pastor and Stambaugh (2003) traded liquidity risk factor in regression

(1). The monthly alpha of our LTD trading strategy from above is now 0 .38% and remains

signicant at the 1% level. In regression (2), we replace the Pastor and Stambaugh (2003)

liquidity factor with the Sadka (2006) liquidity factor that is based on the permanent (vari-

able) component of the price impact function. In regression (3) we include the Bali, Cakici,

and Whitelaw (2011a) factor to control for exposure of our strategy to lottery-type stocks

and in regression (4) we include the Baker and Wurgler (2006) sentiment index orthogonoal-

ized with respect to a set of macroeconomic conditions. 46 In regression (5) we replace the

momentum factor with the Fama-French short- and long-term reversal factors. Finally, as

alternative factor models, in regressions (6) and (7) we use the models suggested in Cremers,

Petajisto, Zitzewitz (2010) that contain various combinations of return differences between

the S&P 500 and Russell 2000 and 3000 subindexes as well as the momentum factor. 47 In

each case, we document a signicant positive alpha of our trading strategy ranging from

0.30% up to 0.55% per month, showing that the results from our basic trading strategy are

45 Ang, Chen, and Xing (2006) also nd no positive outperformance of a simple trading strategy based oninformation about past only.

46

The lottery factor is provided by Nusret Cakici ( http://www.bnet.fordham.edu/cakici/ ) and thetime series of the sentiment factor is taken from http://people.stern.nyu.edu/jwurgler/ .47 Cremers, Petajisto, Zitzewitz (2010) propose using either four or seven factors out of the ten calculated

by them. The data on the factor returns are taken from http://www.petajisto.net/data.html . The fourfactor model contains the factors s5rf, r2s5, r3vr3g, and the momentum factor (umd), while the seven factormodel contains the factors s5rf, rms5, r2rm, s5vs5g, rmvrmg, r2vs2g, and the momentum factor (umd) (asexplained in their data library).

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crash sensitivity and the cross-section of expected stock returns

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