Extreme Downside Risk and Expected Stock Returns

Wei Huang Shidler College of Business, University of Hawaii at Manoa

Qianqiu Liu*

Shidler College of Business, University of Hawaii at Manoa

S. Ghon Rhee SKKU Business School (Korea)

and Shidler College of Business, University of Hawaii at Manoa

Feng Wu

Shidler College of Business, University of Hawaii at Manoa

* Corresponding author. Address correspondence to Qianqiu Liu, Department of Financial Economics and Institutions,

Shidler College of Business, University of Hawaii at Manoa, 2404 Maile Way, Honolulu, Hawaii, 96822; telephone:

(808) 956-8736; fax: (808) 956-9887; e-mail: qianqiu@hawaii.edu.

Extreme Downside Risk and Expected Stock Returns

Abstract This paper proposes a measure for the extreme downside risk (EDR) to investigate whether bearing such a risk can be rewarded by higher expected stock returns. By constructing an EDR measure with the left tail index in the classical generalized extreme value distribution, we find a significant positive premium on firm-specific EDR in cross-section of stock returns even after we control for size, value, return reversal, momentum, and liquidity factors. EDR serves as a good indicator of extreme market plunges. High-EDR stocks generally exhibit high idiosyncratic volatility, large value-at-risk, large negative co-skewness, and high bankruptcy risk. We also controlled for these characteristics to find that the EDR premium remains robust. Furthermore, the EDR effect exhibits long-run persistence and is not subsumed by business cycles.

JEL Classification: G10; G12; and G33 Keywords: Extreme Downside Risk, Generalized Extreme Value Distribution,

Idiosyncratic Volatility, Bankruptcy Risk

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“Much of the real world is controlled as much as the ‘tails’ of distributions as by means or averages: by the exceptional, not the mean; by the catastrophe, not the steady drip; by the very rich, not the ‘middle class’. We need to free ourselves from ‘average’ thinking.”

-- Philip Anderson, 1977 Nobel Laureate in physics

In financial markets, the stability and sustainability of future payoffs of an investment are

largely determined by extreme changes in financial conditions rather than typical movements.

It has been well documented in the literature that investors generally shun positions with

which they would be subject to catastrophic losses however slight probability these losses

carry. Such a “disaster avoidance motive” (Menezes, Geiss, and Tressler (1980)) implies that

investors care about extreme negative scenarios in investment and are averse to the risk of

sharp price plunges. Hence, the potential loss from extreme undesirable returns, denoted as

extreme downside risk (EDR), should become a significant factor in asset pricing. In this

study, focusing on the downside risk at the extreme level, we investigate whether EDR can

indeed be priced. Specifically, we explore how EDR can be appropriately measured, as well

as its ability in explaining the cross-sectional differences in expected stock returns.

EDR deserves much attention in finance because extreme losses are encountered far more

frequently than predicted by traditionally assumed return distributions (such as normal

distribution). The Great Depression of the 1930s, the stock market crash in 1987, the Dotcom

“bubble” burst at the turn of the century, the failure of large corporations such as Long Term

Capital Management (LTCM), Enron, Barings, and the collapse of companies like Fannie

Mae, Freddie Mac, and Lehman Brothers in the wake of the sub-prime mortgage crisis all

remind us that seemingly impossible catastrophes can occur with a striking probability. The

influence of extreme losses can be substantial to the stock market. For example, for all the

common stocks traded on NYSE, AMEX, and NASDAQ during 1963 through 2005, if we

exclude the bottom 1% returns for each stock within each year, the average daily return is

more than doubled, jumping from 0.09% to 0.23%.

Given the fact that extreme losses are detrimental to investors’ overall welfare and the

probability of extraordinary losses is not negligible in financial markets, investors are

concerned with the extreme tail of the return distribution. Thus, it is natural to expect higher

returns from holding stocks with high EDR. Despite the intuitively appealing idea, there has

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been little study examining whether and how the EDR is priced in asset returns. One

challenge is to find a good EDR proxy. Though the definition of EDR is conceptually

straightforward, it requires estimation of the probability of rare events, sometimes outside the

range of available data. That is, “estimates are often required for levels of a process that are

much greater than have already been observed” (Coles (2001)). In this study, we conduct

investigation on EDR measure and its asset pricing implications through an extreme value

approach which is specifically designed to describe unusual, extreme events. We draw on the

extreme value theory (EVT) and utilize maximum likelihood estimation (MLE) of the left tail

index in the classical generalized extreme value distribution (GEV) as a proxy for EDR.

Using U.S. daily stock return data over a 42-year period, we estimate the EDRs of stocks

from the abnormal returns relative to the Fama and French (1993) three-factor model. This

procedure concentrates exclusively on the far-end left tail of the return distribution and

specifically describes the extreme downside movement of the stock itself, after the market

return as well as size and value factors are controlled for. We find that firm-specific EDR

shows a significant positive relation with expected stock returns, reflecting a premium for

bearing risks associated with extreme losses. This finding is robust even after controlling for

widely used variables known to explain cross-sectional variation in stock returns, such as beta,

size, book-to-market ratio, liquidity, momentum, and lagged return, confirming a strong

EDR-return tradeoff. The EDR premium remains significant for a longer period window up to

one year and is not sensitive to business cycles captured by such indicators as interest rates,

inflation, default and term spreads, and GDP growth. Furthermore, a zero-investment strategy

that holds a long position in high-EDR stocks and a short position in low-EDR stocks

generates significant profits during 27 out of 38 years between July 1967 and June 2005.

Our EDR measure incorporates both downside risk and fat-tail risk. There is a large set of

literature that studies these risks beyond the Gaussian paradigm. In contrast to the

mean-variance framework of Markowitz (1952), which assumes normal distribution and

entails equal risks for both positive and negative deviations from the mean, a menu of risk

measures adopted by these studies varies. Starting from the safety-first rule by Roy (1952),

various downside measures have been developed, such as semi-variance (Markowitz (1959)),

lower partial moment (Bawa (1975) and Fishburn (1977)), value-at-risk (VaR), conditional

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tail expectation, and the lower partial standard deviation, among others. Some researchers

adopt leptokurtic distribution such as student’s t (Liesenfeld and Jung (2000)) and

higher-order moments (Chung, Johnson, and Schill (2006)) to capture information endowed in

the fat tails. The efforts made in the literature in modifying the traditional risk measures

significantly expand the understanding of risk. In this paper, we focus on the downside risk at

the extreme level directly. Unlike mean-centered risk measures which describe the deviations

from the mean or VaR which needs a subjective confidence level, our EDR measure comes

directly from the EVT which provides accurate risk assessment of extreme outcomes. In

addition, the EDR measure is different from traditional risk measures in that it is extracted

from the market-adjusted returns, which captures the information not included in common

factors.

Recent empirical asset pricing studies have examined some non-traditional risk measures.

Harvey and Siddique (2000) document a significant premium for conditional skewness.

Dittmar (2002) introduces the fourth moment (kurtosis) into the asset pricing model. Both

measures are based on the relation between individual stock returns and market returns. Ang,

Chen, and Xing (2006) show that downside beta helps explain the cross-sectional variation of

average stock returns, where they calculate the security beta when the market is down. Bali,

Demirtas, and Levy (2008) measure downside risk with VaR from the empirical distribution

of stock returns, and find a positive risk-return tradeoff for several stock market indices.

Similarly, Bali, Gokcan, and Liang (2007) combine the empirical distribution VaR and

higher-order moments, and document a positive relation between VaR and expected returns on

hedge funds. Our results in this paper show that EDR is related to those risk measures.

High-EDR stocks normally exhibit larger VaR, smaller downside beta, and more negative

co-skewness and co-kurtosis. However, we find that the EDR premium persists after

controlling for VaR and the co-skewness. Because stocks with higher downside beta and

higher co-kurtosis with the market command higher returns (Ang, Chen, and Xing (2006) and

Dittmar (2002)), our EDR measure thus captures virtually different information components

in stock returns. Because our analysis utilizes a different approach to measure extreme tail risk

with asymmetric characteristics, the EDR premium documented in this paper serves as an

additional compensation for holding risky assets, which is beyond the rewards for exposures

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to standard downside or high-order moment risks.

Our empirical analysis clearly highlights the role of the EVT-based risk measure in

capturing catastrophic risks. We find that EDR exhibits time-variation that reflects the market

downtrends. Furthermore, there is a “smoothing effect” on firm-level volatility after

individual stock returns are adjusted with their corresponding EDR portfolio returns. In

particular, we decompose aggregate volatility following the methodology in Campbell, Lettau,

Malkiel, and Xu (2001) and observe a dramatic reduction in return variation when EDR is

controlled for as compared with the market-adjusted return. EDR-adjusted volatility becomes

much more stable, especially during the periods of market distress.

Since firms with financial distress typically face severe downside risk, we examine the

links between EDR and firm distress measures. Significant relation is observed between EDR

and Ohlson’s (1980) O-score, implying higher EDR stocks carry higher likelihood of

bankruptcy. Moreover, stocks with higher leverage, lower profitability, lower past returns,

larger volatility, and lower prices tend to have higher EDRs. Since these variables are firm

distress indicators according to Campbell, Hilscher, and Szilagyi (2008), EDR is also a good

indictor of distress risk. Nevertheless, after we control for bankruptcy risk, high-EDR stocks

still outperform low-EDR stocks on average. Furthermore, the EDR premium remains robust

after controlling for idiosyncratic volatility, higher moment risk measures such as

co-skewness, other downside risk measures including VaR. Overall, the results indicates

EDR’s robustness in predicting expected stock returns.

The remainder of this paper is organized as follows: In Section I, we introduce EVT and

GEV distribution. We then present detailed discussion of sample selection, the estimation

method of EDR measures, and summary statistics. In Section II, we examine the relation

between EDR and expected stock returns, as well as the robustness of this relation when

controlling for other characteristics variables. We also analyze the long-run EDR effect. In

Section III we document time variation of EDR premiums, and show EDR’s volatility

reduction effect. The relations between EDR and traditional downside risk, high-order

moment risk measures, idiosyncratic volatility, bankruptcy risk, and other characteristics

variables are explored in Section IV. Section V concludes.

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I. Extreme Value Theory and Extreme Downside Risk Estimation

A. Classical Extreme Value Theory: The Generalized Extreme Value Distribution

EVT plays an increasingly important role in describing the probabilistic attributes of

extraordinary events. It has been used broadly in engineering and the environmental sciences

for over 50 years. EVT is specifically designed to assess the shape of the extreme end of a

random process and provides the best description of the tail behavior for the distribution

function among all existing statistical fat tail estimation tools. It plays a role in the sample

extrema (maxima or minima) parallel to that played by the central limit theorem in the sample

means. Classical EVT depicts the asymptotic distribution of the extrema (or the tail variables)

in a GEV with a parameter called tail index, which indicates the thickness of distribution tail.

This result is very elegant and robust since it is independent of the original (parent)

distributions of the random process. This left tail index becomes an appropriate proxy for

EDR because it specifically focuses on the far-end tails of return distribution, providing a

more accurate risk assessment of extreme outcomes. A higher EDR is revealed consistently by

a higher left tail index. Compared to traditional mean-centered risk measures, EDR avoids

potential distribution misspecification or higher-order moments to capture information

endowed in the thickness of tails. It also considers the distribution asymmetry of stock

returns.

The application of EVT in the finance area is demonstrated by the pioneering work of

Longin (1996) who examines the asymptotic distribution of extreme stock market returns.

Previous applications, however, have been largely limited to documenting fat tail evidence for

stock market indices (Longin (1996)) or to establishing the relation between tails of different

stock indices (Longin (2000), Longin and Solnik (2001), Jondeau and Rockinger (2003), and

Poon, Rockinger, and Tawn (2003, 2004)). EVT is also utilized to improve the estimations of

VaR or expected shortfall which is influenced by the shape of the tail distribution function

(Danielsson and de Vries (1997), McNeil and Frey (2000), and Bali (2003)). Surprisingly,

there has been no research directly incorporating EVT into asset pricing studies. By applying

a classical EVT model to financial market data, our paper investigates the role of EDR in

asset pricing.

Suppose we have an independent and identically distributed (i.i.d.) random variable

6

series { }nXXX ,...,, 21 , EVT shows the limiting distributions for the sample maxima or

minima of the normalized random variable. We take the minima case as an example.1 Let Mn

denote the minimum of the sample, then for a location parameter μ and a scale parameterσ ,

Fisher and Tippett (1928), and Gnedenko (1943) prove that the non-degenerate limiting

distribution of the normalized random variable σ

μ−nMmust fall into one of three types: the

Frechet type; the Weibull type; and the Gumbel type.2 Jenkinson (1955) combines the three

types into a generalized formula, which is the GEV distribution mentioned above:

,0,01,)1(exp1)(1

≠>−

−⎥⎥⎦

⎤

⎢⎢⎣

⎡ −−−−=

−

γσμγ

σμγ γ xxxH (1)

where γ is the shape parameter, also known as the tail index which indicates the thickness

of the left tail or the probability of extreme negative outcomes. The larger the tail index, the

thicker the tail, and the higher the extreme event probability. In the GEV distribution, 0>γ

corresponds to the Frechet type, which is heavy (fat)-tailed; 0<γ corresponds to the

Weibull distribution, which is short-tailed; 0→γ corresponds to the Gumbel type, which is

a thin-tailed process. Normal distribution, with its exponentially decaying tail, leads to

Gumbel distribution for extremes, so it is thin-tailed as suggested by the tail index. Fat-tailed

distribution like Student’s t provides Frechet type extreme value distribution, whose tail

decays polynomially. An example leading to the Weibull type for extremes is uniform

distribution. For fat-tailed case, i.e., Frechet type, the inverse of shape parameter γ1

corresponds to the maximal order moment. Therefore only a limited number of moments are

finite for very heavy-tailed distributions.

Although the asymptotic GEV distribution is derived under the i.i.d. assumption of

random variables, Smith (1985) suggests that dependence of the data does not constitute a

major obstacle to attaining the limiting distribution of large samples; De Haan, Resnick,

1 Similar results hold for the maxima case since }{ }{ nn XXXXXX −−−−= ,...,,min,...,,max 2121

. 2 Non-degenerate distribution does not put all its mass at a single point.

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Rootzen, and De Vries (1989) prove that the maximum or minimum of an ARCH process has

a Frechet distribution. Furthermore, data property can be improved by building a series of

non-overlapping block (period) extrema in the sampling process, which is the basic statistical

procedure we use in this paper.

In our empirical analysis, we pick the minima out of each non-overlapping block (in this

paper, one month) from a random sample. Specifically, the time-series observations can be

grouped into a sequence of sub-periods, and from each sub-period, we select the minimal

observation. The pool of all sub-period minima constitutes the extrema sample and follows

the GEV distribution asymptotically with the expansion of the sub-period length. This simple

approach can largely reduce possible interdependence of extrema, which is one of the major

advantages of block-extrema sampling over alternative methods.3

MLE can be applied to the extrema sample to estimate the parameters in GEV, with tail

index being the focus of interest. MLE provides an unbiased estimate with the minimum

variance among all the estimates. For processes with not-too-short tails (i.e., tail index > -1/2),

MLE estimate is distributed asymptotically normal, which makes it very convenient for

statistical inference. In the following empirical analyses, we employ the MLE method to the

GEV distribution and estimate the tail indexes for per-month minima of daily return processes.

These tail indexes, as proxy measures of EDRs, are used in our asset pricing tests.

B. Data and Sampling

The sample consists of all common stocks traded on the NYSE, NASDAQ, and AMEX.

To be included in the sample, a stock must have at least two years’ daily information records

in the CRSP database during the period between July 1963 and June 2005.4 Daily and

monthly market data are obtained from CRSP, with the corresponding accounting data from

Standard & Poor’s COMPUSTAT yearly files. Consistent with most other asset pricing papers,

we match CRSP-COMPUSTAT data following Fama and French (1992).5 We use daily data

for the EDR (tail index) estimation and monthly data for asset pricing tests. 3 Another well-known statistical methodology is to fit observations higher (or lower) than an extreme threshold into a generalized Pareto distribution. 4 The minimum-two-year requirement is to allow enough sample observations for our empirical analysis based on monthly extrema. 5 The monthly CRSP data from current year’s July to following year’s June are merged with COMPUSTAT data for the latest fiscal year ending in the preceding calendar year. This guarantees a minimum six-month gap between fiscal yearend and stock returns.

8

In order to capture EDR for different risk components, we extract the abnormal returns or

residuals from the Fama-French (1993) three-factor model:

,,,,,,i

dtdtiHMLdt

iSMBdt

iMKT

it

idt HMLSMBMKTr εβββα +⋅+⋅+⋅+= (2)

where idtr , is stock i’s daily excess return (raw return minus risk free rate, approximated by

U.S. one-month Treasury bill rate) at day d in month t, dtMKT , is the market excess return,

dtSMB , and dtHML , represent the returns on portfolios formed to capture the size and

book-to-market effects respectively, and idt ,ε is the resulting residuals. Daily risk factor data

and U.S. one-month Treasury bill rates are from Kenneth French’s data library.6 These

residuals show return variations that are not reflected by common factors, and the EDR

measure based on them indicates extreme downside movements for individual stocks when

the effect of systematic risks have been filtered out. Statistically, as Diebold, Schuermann, and

Stroughair (1998) and McNeil and Frey (2000) indicate, there is an advantage in applying

EVT to model residuals rather than raw returns since the residuals are approximately

independent over time.

C. Estimation of EDR Measures

Each month in our sample period, we obtain the Fama and French (1993) three-factor

daily residuals for each stock according to equation (2) using all up-to-date available data, and

choose the minimal value from each month to construct extrema sample. Based on GEV

distribution, we use MLE from the sample extrema to compute pre-ranking tail index

estimates each month for each stock. In this first-stage process, we set 24 months as the

minimum length of estimation window.7

Since getting tail index estimates involves using extreme daily returns, the reported daily

data from CRSP may be subject to severe microstructure biases, especially for small firms, as

emphasized by Blume and Stambaugh (1983) and Bessembinder and Kalcheva (2007). To

mitigate the liquidity biases and other error-in-variables problems, we conduct the second

stage of estimation. In this stage, we first sort stocks into 100 portfolios according to their

6 We are grateful to Kenneth French for keeping these data sets updated and available. 7 We base the MLE on an expanding window by utilizing all previous data because normally a large sample size is needed for reliable tail index estimation.

9

pre-ranking tail index estimates at the end of each month. We next compute weighted average

daily excess returns in the following month for each percentile portfolio, where the weights

are stock returns in the portfolio formation period. Such a “return weighting” scheme in

computing the buy-and-hold portfolio returns is proven to be effective in eliminating the

microstructure biases in daily returns (Bessembinder and Kalcheva (2007)). We then apply

Fama and French (1993) three-factor decomposition to generate daily residuals for the

portfolio excess return series each month starting from July 1965.8 We also conduct

per-month minima sampling to the 100 portfolio residual series utilizing all up-to-date

available data as in individual stock case. MLE is applied again to these minimum return

residual samples (with at least 24 observations used). We thus obtain post-ranking tail index

estimate for each portfolio in each month.9 Finally, we assign these portfolio-level EDR

measures to each stock contained in the portfolios on a monthly basis. They serve as the final

individual EDR measures that will be used in the portfolio or regression analyses.10 Note that

a particular stock’s EDR estimate may change over time since the stock may switch between

different portfolios due to the pre-ranking of its tail index estimation each month.

D. Summary Statistics of EDR Measures

Panel A of Table I reports basic summary statistics of the EDR estimates. Average EDR

for each stock is calculated based on the full period. The mean, standard deviation, and

median of EDRs for all sample stocks are reported. The last column of Panel A shows that the

abnormal return distributions of most stocks (88% of all the stocks in the full sample) do have

fatter left tails than implied by normal distribution, even after the market risks have been

excluded. Moreover, the mean of the tail indexes has positive two-digit t-statistic (75.24),

suggesting that the sample stocks have significantly heavy tails on average.

We show more firm characteristics across EDR quintiles in Panel B, where both mean

and median statistics are reported. Apparently, the range of EDR is large. The highest-EDR

quintile has a mean and median of 0.1324 and 0.1201 respectively, while the mean and

8 This series starts from July 1965 because the first-stage tail index estimation for individual stocks begins from June 1965. 9 Note that the first portfolio-level tail index estimation begins from June 1967 since at least 24 monthly minima are used to generate MLE estimate. . 10 Our EDR measures are based on per-month minimum sampling. Loretan and Phillips (1994) and Longin (1996) report that tail index is very stable over different sub-period lengths.

10

median values in the lowest-EDR quintile are -0.0620 and -0.0324 respectively.11 The high

EDR quintile portfolio takes a smaller market share of about 15%, measured as the average

fraction of total market capitalization, while the market shares of the three middle quintiles

are slightly higher than 21%. Overall, there is no considerable variation in market share across

EDR quintiles. The systematic risk measure of beta is largely flat across the portfolios.12 The

mean value ranges from 1.31 to 1.34, and the median value from 1.3107 to 1.3385. This

suggests that EDR and beta are not highly correlated. Finally, stocks in the high-EDR quintile

portfolio exhibit lower share price than those in the low-EDR portfolio.13

Insert Table I Here

The time-series behavior of EDR is illustrated in Figure 1. The top panel shows the mean

and median EDRs in each month during the study period. Clearly, the mean and median EDRs

show upward trends over time. EDRs are below zero (short-tailed distribution) prior to early

1970s and are above zero (fat-tailed distribution) afterwards, and increase rapidly over time.

Also, the mean EDR exceeds median EDR after mid-1980s, which means more extreme fat

tails exist in recent two decades. This phenomenon can be further confirmed by the bottom

panel, where we plot the time variation of the proportion of the number of stocks that fall into

the top decile of EDRs. The proportion is obtained by dividing the number of stocks

belonging to highest-EDR decile by the total number of stocks every month. From the bottom

panel, we observe a sharp increase in the high-EDR proportion after mid-1980s. The

proportion in the mid-1990s is more than four times larger than that in the end of 1980s.14

Hence in the past two decades, fat-tailed distributions have become more common and the

magnitude of the left tail has increased. This implies that stocks are subject to more frequent

and severe massive losses in recent years, especially after mid-1980s. This trend coincides

11 Negative value implies that stock returns are short-tailed according to GEV. 12 We estimate beta following the method in Fama and French (1992). In particular, pre-ranking beta is estimated as the sum of coefficients of contemporaneous and lagged market returns in an OLS regression with monthly individual stock returns over the past two to five years as the dependent variable. Post-ranking beta is computed in a similar way from data in the full sample period. Market return is approximated by CRSP’s value-weighted NYSE/AMEX/NASDAQ index return. 13 In order to capture the particularity of low-priced stocks more precisely, the closing prices in the data set are winsorized above $15. 14 Mathematically, reliable tail index estimation requires a large enough sample size. The large high-EDR proportion before 1970 could be due to the inaccurate EDR estimates from small samples in the MLE. In an unreported figure, we observe that before 1970 there are also a large proportion of low-EDR stocks. Besides, the range of EDR is more than two times wider before 1970 than after 1970.

11

with the series of extreme market downturns including the Black Monday in October 1987,

the failure of LTCM in August 1998, and the NASDAQ “bubble” burst in early 2000s. As

illustrated in the bottom panel, the EDR measure effectively highlights the market crashes. A

jump in the high-EDR proportion captures each of the major crises.

Insert Figure 1 Here

II. Extreme Downside Risk and the Cross-Section of Stock Returns

A. EDR Premium: Portfolio Analysis

To examine whether EDR commands a risk premium, we sort all sample stocks into

quintiles portfolios based on their prior-month EDR measures each month between July 1967

and June 2005, and report portfolio average returns in Table II. The last column indicates that

the return difference between high-EDR and low-EDR portfolios is significantly positive. On

the monthly basis, the spreads is 21.64 basis points with the Newey-West (1987) robust

t-statistic of 3.04. Since investors generally do not want to suffer from massive losses, they

require a significantly high expected payoff to hold high-EDR stocks. Our evidence illustrates

that downside risks at extreme level are priced in the market, even though such risks are

derived from the firm-level idiosyncratic information.

Insert Table II Here

One may argue that the above results may be attributed to different risk factor loadings

other than EDR. To filter out the influence of other risk factors, we run time-series regressions

with EDR-sorted portfolio returns as dependent variable and commonly used risk factors as

the explanatory variables. We report portfolio alphas from the intercepts of these regressions.

The difference of intercepts between the high- and low-EDR quintiles reflects the information

components not captured by existing asset pricing models. To confirm the robustness of the

results, we use three asset pricing models, which include the CAPM, the Fama and French

(1993) three-factor model, and the Carhart (1997) four-factor model. The last column from

rows 2 to 4 of Table II show clear evidence of return premium associated with EDR in each

asset pricing specification. All three alphas for the “High-Low” EDR return spreads are

significantly positive: the CAPM alpha spread is 20.81 basis points per month; the Fama and

French (1993) three-factor alpha spread is 20.10 basis points; lastly, the four-factor alpha

12

shows a monthly spread of 31.38 basis points. Overall, the results from Table II show a

significant EDR-return tradeoff which cannot be subsumed by other risk factors, suggesting

that bearing high EDR will be rewarded by higher expected returns in stock market.

B. Persistence of EDR Premium

To examine whether the EDR premium is a persistent phenomenon or is caused by a few

outliers by chance, we construct a zero-investment trading strategy that takes a long (short)

position on stocks in the highest (lowest) EDR quintile. We compute profit on the

zero-investment strategy for each of the 38 twelve-month sub-periods during the period of

July 1967 and June 2005, where the zero-investment profit is the average monthly return

difference between the highest and the lowest EDR quintile portfolios. The average monthly

returns on the zero-investment portfolios are depicted in Figure 2. The figure indicates that

high-EDR firms outperform low-EDR firms in 27 out of the 38 twelve-month sub-periods.

This means that a zero-investment strategy is profitable during most of the sample periods.

The evidence here suggests that the market has not properly priced the extreme downside risk

adequately, and the EDR effect persists throughout most of the sample periods. Consistent

with Table I, the average zero-investment profit over the full period (not reported in Figure 2)

is significantly positive, implying investors require more premiums for holding high-EDR

stocks.

Insert Figure 2 Here

C. Do Firm Size and Value Factors Subsume EDR Premium?

Considering EDR may be captured by some known firm characteristics such as size and

book-to-market ratio, we control for size and book-to-market effects in the EDR portfolios in

Table III. In particular, following Fama and French (1993), each month, we sort all stocks into

two size groups (Small and Big) according to market capitalization of NYSE-listed stocks of

the preceding June; similarly, we sort all stocks into three book-to-market (BM) groups

(Low-bottom 30%, Medium-middle 40%, and High-top 30%) according to book-to-market

ratio of last fiscal year ending in the preceding calendar year. Within each size or BM group,

stocks are further sorted into quintiles based on prior-month EDR. Table III reports these

double-sorted portfolio returns. Also reported are the intercepts of the time-series regressions

13

represented by alphas from the four-factor model (4-factor Alpha).15 “High-Low” in the last

column is the mean difference in monthly returns or alphas between the highest and lowest

EDR quintile stocks. The last row “Average” in each panel refers to the average monthly

return or alpha of each EDR quintile across all size or BM groups.

Insert Table III Here

Panel A of Table III shows that small stocks have higher EDR premium on the average.

Conditioning on firm size, the highest EDR quintile portfolio still has an average return

higher than that of the lowest EDR quintile portfolio, and this positive relationship is more

pronounced among the smaller stocks. On the average, we observe a spread of 14.87 basis

points per month between the highest EDR quintile stocks and the lowest EDR quintile stocks

across the size groups. Similar pattern are observed from the four-factor alphas. Alphas from

the highest EDR quintile are significantly greater than those from the lowest EDR quintile in

both size portfolios. Also, the difference in alphas is larger in smaller size group. Therefore,

size cannot explain the EDR premium, even though the positive EDR-return relation becomes

stronger for small firms. This is understandable since firm-specific EDR has a more

substantial risk manifestation among small-cap stocks which are generally subject to extreme

price drops more frequently.

Panel B indicates that the positive EDR premium is still significant after controlling for

book-to-market ratios. In each book-to-market sorted portfolio, the highest EDR quintile has a

higher return than the other quintiles. The average spread between the highest quintile EDR

stocks and the lowest EDR quintile stocks across three book-to-market portfolios is 23.01

basis points. Interestingly, the spread increases monotonically from the highest BM quintile

portfolio (15.64 basis points) to the lowest BM quintile portfolio (37.22 basis points).

Therefore the positive relation between EDR and expected stock returns is stronger among

growth stocks. Alphas from the four-factor model show the same pattern. Overall, results in

Panel B suggest that book-to-market effect is not an explanation for EDR premium either.

D. Do Arbitrage Opportunities Persist over Longer Holding Periods?

In this subsection, we examine whether the EDR premium persists over a longer holding

15 We only report four-factor alphas in Table IV to conserve space. The results from CAPM and Fama and French (1993) three-factor alphas are qualitatively similar.

14

period beyond one month. In Table IV, we extend the portfolio holding periods ranging from

two to twelve months in order to check long-run arbitrage opportunities.

Insert Table IV Here

In Panel A of Table IV, we group stocks into EDR quintiles each month and calculate

average monthly returns for each portfolio during post-formation period of two to twelve

months. From the “High-Low” column which reports the mean difference between the returns

on portfolios with highest and lowest EDRs, we observe that the positive relation between

EDR and expected stock returns exists in at least one year after the portfolio is formed. The

resulting “High-Low” spread is significantly positive throughout all time periods, and

becomes only a little smaller along the time line. For example, the monthly spread decreases

from 19.74 basis points two months after portfolio formation to 17.25 basis points one year

after, resulting in a trivial difference of less than 2 basis points over ten months. Interestingly,

the significance level of the EDR premium becomes even higher as time passes, as indicated

by the monotonically increasing t-statistics along the time line (from 3.71 two months after

portfolio formation to 4.71 twelve months after portfolio formation). Hence returns on

high-EDR stocks continue to be significantly higher than returns on low-EDR stocks for at

least one year, which indicates that EDR is a very stable characterization for stocks and the

results of EDR premium are robust in a longer holding period. Similarly, Panel B reports the

long-run EDR effect on four-factor alphas. Not surprisingly, the results from alphas are very

similar to those from the monthly returns. Table IV also implies that stock prices do not fully

absorb all underlying risk information for a long time and investors may not be able to take

advantage of the arbitrage opportunities.

E. EDR Premium in Cross-Sectional Regressions

To further explore the relation between EDR and expected stock returns, we run Fama

and MacBeth (1973) cross-sectional regressions with multiple explanatory variables

introduced. This allows us to examine the robustness of the EDR premium when these

variables are controlled at the same time. Each month during the sample period, we use all

individual stock returns as the dependent variable and prior-month EDR as the explanatory

variable, while controlling for various firm characteristics that are known to influence

cross-sectional asset returns. Time-series averages of the cross-sectional regression

15

coefficients for explanatory variables are reported in Table V with the Newey-West (1987)

robust t-statistics.

In addition to EDR, other explanatory variables include beta, size, BM ratio, momentum,

previous monthly return, and liquidity. Following existing literature, we take logarithms of

size and BM ratio. Beta is the systematic risk proxy suggested in CAPM. Following

Jegadeesh and Titman (1993), we calculate momentum as past year return of the stock,

skipping the most recent month. We also add past month return to the independent variable

list, controlling for return reversal as suggested by Jegadeesh (1990). Finally, liquidity is

measured by liquidity beta from Pastor and Stambaugh (2003).16

Insert Table V Here

The overall findings summarized in Table V indicate that the coefficients of EDR are

significantly positive in each model, even after controlling for those known factors. The

coefficient estimates are higher than those associated with other explanatory variables such as

beta, size, and book-to-market ratio, and the t-statistics indicate that the EDR coefficients are

significant at the 1% level. In models 1 through 3 which include EDR and different control

variables, the EDR coefficient is 1.1697 (t-value = 3.08), 0.9321 (t-value = 3.25), and 0.7832

(t-value = 3.12) respectively. This confirms our results from portfolio analysis that the

explanatory power of EDR is not subsumed by the size and book-to-market effects. In

addition, after controlling for past one-month return and momentum, a significant EDR

premium remains. Furthermore, adding EDR in the regressions does not change the predictive

relations and explanatory powers of conventional variables such as size, book-to-market,

momentum, and short-term return reversal. They all have significant coefficients with signs

consistent with the results in the extant literature. Hence, we conclude that EDR captures

different risk components from these traditional systematic risks.

III. Extreme Downside Risk and Time-Series Variation in Returns

A. EDR Premium and Business Cycles

In this subsection, we investigate the impact of business cycles and economic activities

16 Liquidity beta is the OLS regression coefficient of the innovation in liquidity measure after controlling for Fama and French (1993) three factors, as in Pastor and Stambaugh (2003).

16

on the EDR premium. This allows us to examine whether the EDR premium exhibits variation

related to business cycles, which might explain its existence. For this purpose, we run

time-series regressions of EDR premium on a number of business cycle indicators. Table VI

reports the intercepts (Alphas) and regression coefficients. The dependent variable is the EDR

premium from row 1 of Table II, i.e., the monthly return difference between highest and

lowest EDR quintile stocks. We first include a business cycle dummy variable which takes the

value of 1 for contractions and 0 for expansions, according to the NBER’s indicator of

business cycles. This dummy variable tests whether the EDR premium in economic downturn

is different from that in business boom. Following Harvey (1988) and Fama and French

(1989), additional variables are introduced to track business cycle fluctuations, among which

“Tbill” is U.S. three-month Treasury bill rate; “CD-Tbill” is the difference between

three-month CD rate and three-month Treasury bill rate; “Term” is term spread measured as

the difference between ten-year and one-year Treasury bond yields with constant maturity;

“Default” refers to the default spread calculated as the difference between BAA- and

AAA-rated corporate bond yields; “Inflation” is the monthly percentage change of U.S.

consumer price index; and finally, “GDP” refers to current quarter real GDP growth rate. All

macroeconomic data are obtained from the Federal Reserve Economic Database (FRED)

maintained by the Federal Reserve Bank of St. Louis.

Insert Table VI Here

A common observation from all models is that the EDR premium cannot be explained by

business cycle indicators. In fact, none of the business cycle indicators has a significant

coefficient. The regression alphas are all positive and significant in most cases, which is not

surprising given that the average EDR premium is positive in most of the sample periods. The

adjusted R-squares in the last column indicate that out of the seven models, only one has

positive adjusted R-squared value, and these business cycle variables can explain at most

0.19% of the total variation in monthly EDR premium. Since EDR is estimated from the

residuals of stock returns, the market effects have been filtered out. The evidence from Table

VI further shows that EDR contains substantially different information about cross-sectional

stock returns than what is reflected in business conditions.

B. Extreme Downside Risk and Extreme Market Downturns

17

During a stock market crisis, most stocks experience extreme downside movements that

usually could not be explained by conventional asset pricing models. The resulting pricing

errors will then be reflected in their residuals, which consist of the basis of our EDR measure.

We therefore conjecture that the EDR measure may capture the extreme downside movements

in financial markets. Figure 1 has shown that dramatic jumps in magnitude and

extraordinarily large EDR premium are associated with the extreme market downturns, which

is good indication of EDR’s ability of detecting market distresses. Here we go one step further

to check whether controlling for EDR can reduce the firm-level return volatility, especially

during the extreme market down times.

Campbell et al. (2001) use a disaggregated approach to study the volatility of common

stocks at the market, industry, and firm levels. They express the returns on industry portfolios

as the market-adjusted returns and also express returns on individual stocks as the

industry-adjusted returns. An important advantage of this methodology is that beta estimations

are not needed in these models at the aggregate level. The weighted average of the total return

volatility of all firms is the sum of market return volatility, average industry-level volatility,

and the weighted average of firm-level volatility across all firms. In order to examine the role

of EDR in the time-variation of firm-level volatility, we follow the Campbell et al. (2001)

approach and replace industry portfolios with EDR-sorted portfolios. This beta-free variance

decomposition allows us to express the weighted average of the total return volatility of all

firms as

222)( ttmtijt

jiijt

jjt rVarww ηε σσσ ++=∑∑

∈

(3)

where j denotes different EDR-sorted portfolios, jtw is the weight of EDR portfolio j in the

total market, and ijtw is the weight of firm i in EDR portfolio j. 2mtσ is the market return

volatility; 2tεσ is the average volatility of EDR-sorted portfolios, with market return adjusted;

2tησ is the weighted average of firm-level volatility across all firms, when firm returns are

adjusted with EDR-sorted portfolio returns. It represents the firm-level idiosyncratic volatility

that cannot be captured by market and EDR factors.

We use the 100 EDR-sorted portfolios each month during the sample period, which are

18

constructed in Section I. To calculate EDR-adjusted firm-level volatility, we first subtract

value-weighted daily EDR portfolio returns from individual stock returns to get EDR-adjusted

returns, then compute the variance from these adjusted returns each month for each stock.17

We calculate value-weighted average variance within each EDR portfolio, where the weight

depends on the value of individual stock in its corresponding portfolio. At the last step, we

obtain the value-weighted average of the previous series across all portfolios, where the

weight is measured as the ratio of each portfolio in the total market. This process is repeated

every month to estimate time-varying firm-level volatility series.

We compare this estimate with the average firm-level volatility when only market factor

is considered, where we use CRSP’s NYSE/AMEX/NASDAQ value-weighted index as

market proxy. The time series of the average firm-level volatility difference between

EDR-adjusted and market-adjusted schemes are shown in Figure 3.

Insert Figure 3 Here

Compared with the market-adjusted return model, controlling for EDR factor can reduce

firm-level volatility, especially during financial depression periods when the overall return is

extremely volatile. Since the total volatility increases significantly during extreme market

downside movements, the pricing error is more severe with extreme downside events.

However, when firm returns are adjusted with their associated EDR portfolio returns,

idiosyncratic volatility has a smoothing effect during extreme market downturns. For example,

during 1987 stock market crash, EDR-adjusted firm-specific volatility is much lower than that

in the market-adjusted returns. This reduction is almost twice as large during the period of

NASDAQ bubble burst period. This indicates that EDR-adjusted idiosyncratic volatility will

be more stable and the stock return is much less drastic during market distresses after EDR is

accounted for. In other words, controlling EDR will reduce firm-level volatility and the

reduction is more substantial during extreme market downside movements. This reinforces

the important role of EDR in capturing the risk in massive shortfall events.

17 We use value-weighted scheme here to be consistent with the market-adjusted return decomposition below, where the market return is represented by CRSP’s NYSE/AMEX/NASDAQ value-weighted index return. In fact, Campbell et al. (2001) report that this variance decomposition approach is still valid with different weighting schemes and different estimation methods of mean returns. We have also tried equal-weighted case and got similar results.

19

IV. Extreme Downside Risk and Other Traditional Risk Measures

So far, we have gained much insight into distinct properties of EDR as a risk factor that

captures special and different risk components other than those embedded in existing risk

proxies. We have documented in Section II that even after controlling for beta, size,

book-to-market ratio, momentum, and liquidity, high EDR stocks still outperform low EDR

stocks. In this section we examine the relation between EDR and other risk measures to

directly study the characteristics of stocks with different EDR levels. Specifically, we

establish the relation between EDR and traditional risk measures that also reflect downside

risk or fat tail risk (but not extreme risk). Since our EDR estimates are obtained from the

firm-specific return series after controlling for Fama and French (1993) factors, this motivates

us to examine the relation between EDR and idiosyncratic volatility. Finally, firms in financial

distress typically carry a high probability of extreme negative returns. It is thus interesting to

examine whether the EDR-return tradeoff is robust after controlling for the bankruptcy risk.

These analyses are useful in identifying the source of EDR premium.

A. Correlations between EDR and Other Traditional Risk Measures

In order to directly examine the characteristics of stocks with different EDR levels, we

compute the correlation coefficients between EDR and traditional risk measures and other

characteristic variables each month and take time-series averages of the correlations. The

results are reported in Table VII.

Insert Table VII Here

Among the traditional risk variables, we first consider idiosyncratic volatility calculated

as the standard deviation of the residuals from the Fama and French (1993) three-factor model,

which is a second moment measure. The second one is the downside beta as in Ang, Chen,

and Xing (2006). They calculate the security beta when the market is down and find that

downside beta is positively priced. The third one is high-order moment risk measures such as

the CAPM-based co-skewness which represents a third moment measure constructed

according to Harvey and Siddique (2000), who show that more negative co-skewness is

rewarded with higher average return cross-sectionally. We also include a co-kurtosis measure,

following Ang, Chen, and Xing (2006). Dittmar (2002) indicates that kurtosis captures the

probability of large negative outcomes and it should have a positive premium in theory.

20

Finally, we adopt the approach in Bali et al. (2008) to estimate monthly VaR based on the

actual empirical distribution at the 4.76% confidence level.18 We select VaR since it is

another commonly used downside risk indicator, especially among the practitioners. A

positive VaR-return relation has been documented in the literature.

It can be seen from Table VII that EDR is positively related to idiosyncratic volatility and

VaR, and negatively related to co-skewness, meaning high EDR stocks generally have high

idiosyncratic volatility, high VaR, and large negative co-skewness with the market. Moreover,

EDR, as an extreme tail risk indicator, has a closer connection with the tail-based downside

risk measure, as confirmed by the higher correlation level with VaR: EDR-VaR has the

highest correlation coefficient among all reported correlation values. Somewhat surprisingly,

EDR has a negative relation with both downside beta and co-kurtosis, which indicates that

these two measures cannot explain the positive premium of our extreme downside risk

measure.19

Also included in the variable list are bankruptcy risk measure and a set of firm distress

indicators. The direct bankruptcy risk measure is represented by Ohlson’s (1980) O-score.20

A higher O-score implies higher bankruptcy possibility. The distress indicators are from

Campbell et al. (2008), who show that firms with higher leverage, lower profitability, lower

past stock returns, higher return volatility, and lower stock prices suffer more from the risk of

going bankruptcy, being delisted from exchanges, or receiving a D-rating. Among these

variables that are related to distress risk, past stock return is the monthly return in the previous

month; past return volatility refers to standard deviation of daily stock returns in the previous

month; leverage is the ratio of total liabilities to total assets; profitability is the ratio of net

income to total assets; price is defined as the logarithm of last month closing price winsorized

above $15. To directly check the delisting risk, we also include a dummy variable which

18 Since the estimation is at one-month horizon, and there are about 21 trading days each month, the confidence level of monthly VaR based on empirical distribution is 4.76%(=1/21). 19 The negative relation could be caused by two facts: First, our EDR is measured from the residuals of stock returns while downside beta and co-kurtosis in Ang, Chen, and Xing (2006) are mean-centered measures from raw returns; Second, Ang, Chen, and Xing (2006) evaluate the contemporaneous relation between realized returns and realized co-kurtosis in a year, while we measure downside beta and co-kurtosis using a rolling window of the previous 60 months. 20 We do not adjust the GNP price-level index since in monthly regression analysis the index is unchanged. Furthermore, because Ohlson (1980) model is based on industrials, we exclude stocks with COMPUSTAT Standard Industrial Classification (SIC) codes between 4000 and 4999, and those above 6000.

21

equals 1 when the particular stock is delisted from NYSE, AMEX, or NASDAQ, and 0

otherwise. It should be noted that O-score and most of the distress indicators are derived from

accounting data, while EDR is measured using pure market data. As stated by Hillegeist,

Keating, Cram, and Lundstedt (2004) and Vassalou and Xing (2004), the effectiveness of

accounting-based measure is impaired by the going-concern principle guiding financial

statement construction, the downside-biased asset valuation due to conservatism principle,

and the failure to incorporate asset volatility. Checking the links between EDR and those

bankruptcy or distress indicators is interesting, because they are measured in different ways

but they all deliver similar information on large-loss risk.

Not surprisingly, EDR is closely related to bankruptcy probability as it is significantly

positively related to O-score. Table VII also indicates that high EDR is associated with firms

with high leverage, low profitability, bad and more volatile past return performances, and low

stock prices. A positive and significant correlation coefficient on the delisting dummy variable

suggests that stocks delisted from exchanges generally show thicker tails. In summary, the

evidence here suggests that high EDR stocks are more likely to be stocks of financially

distressed companies or companies with high bankruptcy risk.

Notably, even though most of the correlations documented above are statistically

significant, the reported correlations are relatively low. The highest coefficient (in absolute

value) is less than 0.05 (i.e., -0.0438 of EDR-price correlation). This feature implies that the

EDR effect cannot be subsumed by other existing risk measures. In the following subsection,

we further explore this issue.

B. EDR Premium after Controlling for Other Risk Measures

We have seen from Table VII that high EDR stocks are more likely to have high

idiosyncratic volatility, large negative co-skewness with the market, high VaR, and high

bankruptcy risk (represented by O-score), but the correlation coefficients are not large. In this

subsection, we investigate whether the EDR premium still exists after controlling for these

variables by utilizing Fama and MacBeth (1973) regression approach. We add one of these

four existing risk measures to the regression model as explanatory variable along with EDR

and commonly used asset pricing factors (beta, size, book-to-market), and report time-series

averages of the cross-sectional regression coefficients for explanatory variables. The results

22

are summarized in Table VIII.

Insert Table VIII Here

Panel A of Table VIII indicates that controlling for idiosyncratic volatility cannot

eliminate the EDR premium since all the EDR coefficients in both models are significantly

larger than zero. This suggests that high idiosyncratic volatility cannot account for the EDR

premium. The coefficients of idiosyncratic volatility are negative, which is consistent with the

findings by Ang, Hodrick, Xing, and Zhang (2006). Interestingly, both constructed from Fama

and French (1993) residuals, EDR and IV are correlated but have exactly opposite effect,

suggesting that they basically reflect different risk characteristics in stock returns.

We have seen a significantly negative relation between EDR and co-skewness, indicating

that a higher EDR implies a more negative co-skewness. Harvey and Siddique (2000) show

that stocks with more negative co-skewnesses with the market have higher returns. The

evidence in Panel B, however, indicates that co-skewness is not a reason for EDR premium:

in each of the regression models, EDR always has a significantly positive coefficient.

Co-skewness, on the other hand, also shows its negative association with stock returns.

Therefore, co-skewness, like idiosyncratic volatility, cannot capture the risk embedded in

EDR.

Since EDR and VaR are positively correlated and both require a premium in expected

stock returns, it is necessary to specifically check whether the EDR-return tradeoff is merely a

reflection of the VaR effect. Our results show that VaR cannot explain the EDR premium in

asset pricing. After controlling for VaR in regression models in Pane C, we still observe

significantly positive EDR coefficients. This further reinforces the robustness of EDR

premium.

Similarly, in Panel D, the evidence once again emphasizes that the EDR premium is very

robust, since controlling for bankruptcy risk cannot lower the significance of the positive

EDR coefficients. Although high EDR stocks normally have high bankruptcy risk, the latter

does not explain for the EDR premium. Moreover, Dichev (1998), Griffin and Lemmon

(2002), and Campbell et al. (2008), among others, have documented that firms with high

bankruptcy risk are not rewarded by higher returns. However, our results show that the effect

of bankruptcy risk on stock returns becomes negligible in both models. This suggests that

23

EDR has distinctively different risk implications from traditional bankruptcy and distress risk

measures.

V. Conclusion

In this paper, we propose an appropriate downside risk measure associated with

catastrophic losses and examine how this extreme downside risk (EDR) is related to average

expected stock returns on a cross-sectional basis. We use the maximum likelihood estimation

of the tail index from classical GEV distribution under the extreme value theory framework to

measure EDR. We apply the method to the return residuals from the Fama and French (1993)

three-factor model to estimate the EDR for individual stocks.

We find a significant return premium for the EDR even after we control for firm size,

book-to-market ratio, return reversal, momentum, and liquidity effects. Moreover, the EDR

premium remains significant one year after the EDR-sorted portfolios are formed. Our results

suggest that EDR serves as a good indicator of extreme market downside movements. When

stock returns are adjusted with their corresponding EDR portfolio returns, firm-level

idiosyncratic volatility becomes much smoother, especially during extreme market downturns.

Although high EDR stocks generally have high idiosyncratic volatility, more negative

co-skewness, higher VaR, and high bankruptcy risk, the EDR premium remains robust after

these effects are controlled.

Since EDR is a measure beyond the traditional mean-variance framework, it is important

to understand the difference between EDR and existing downside risk measures. EDR derived

from the EVT model benefits from its ability in detecting potential extreme risks; on the other

hand, EVT concentrates on the extremes and does not provide inferences for the sample

means. In addition, our EDR is constructed from residuals after adjusting common factors.

For all these reasons, we claim that EDR renders a new risk component of stock returns. A

direction in the future study is to explore the mechanism through which the extreme downside

movement of stock returns can be captured in general asset pricing process. An EDR-based

asset pricing model will contribute to the current literature on risk-return tradeoff and will

help a more comprehensive understanding of various risk components.

24

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67-90. Vassalou, Maria, and Yuhang Xing, 2004, Default risk in equity returns, Journal of Finance 59,

831-868.

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Figure 3. Volatility decomposition via market and EDR-portfolio returns. 100 EDR portfolios are constructed monthly for the period of July 1967 to June 2005 and value-weighted portfolio returns are calculated. Following the Campbell et al. (2001) method, we decompose total volatility, and obtain the EDR-adjusted and market-adjusted (using CRSP value-weighted market return) firm-level volatilities for each month. This figure plots the time variation of volatility reduction when the EDR-adjusted volatility is compared with market-adjusted volatility at the firm level. Vertical axis is the volatility reduction (market-adjusted volatility minus EDR-adjusted volatility), and horizontal axis represents the time.

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Table I Summary Statistics of Extreme Downside Risk Measures

The sample includes all NYSE, AMEX, and NASDAQ-listed common stocks with at least 24 months’ return data in CRSP daily and monthly files and COMPUSTAT yearly file from July 1963 to June 2005. Panel A presents full-sample summary statistics of EDRs. Average EDR for each stock is calculated from the full period. The mean, standard deviation (SD), and median of EDRs over the sample of all stocks are reported. The “Fat tail” column shows the proportion of stocks with positive tail indexes. Panel B reports mean and median statistics of more characteristics for quintile portfolios sorted by EDR. “Market share” is measured as the average fraction of total market capitalization for each quintile. “Beta” is estimated following Fama and French (1992). “Price” is the logarithm of past month’s market closing price winsorized above $15. Also reported are the mean and median EDR values of each quintile. The Newey-West (1987) robust t-statistics are reported in brackets.

Panel A: EDR Summary in Full Sample Mean SD Median Fat Tail Full sample 0.0470 0.0582 0.0359 88% [75.24]

Panel B: Characteristics of EDR Sorted Portfolios EDR Market Share Beta Price Mean Median Mean Median Mean Median Mean Median 1 Low -0.0620 -0.0324 19.77% 19.32% 1.3155 1.3178 10.34 10.06 2 -0.0064 0.0116 22.25% 22.17% 1.3100 1.3107 10.54 10.40 3 0.0280 0.0385 21.44% 21.44% 1.3064 1.3093 10.36 10.12 4 0.0646 0.0647 21.64% 21.48% 1.3141 1.3206 10.37 10.17 5 High 0.1324 0.1201 14.90% 14.65% 1.3400 1.3385 9.77 9.74

31

Table II Average Returns and Alphas of Extreme Downside Risk-Sorted Portfolios

Each month between July 1967 and June 2005, all stocks in the full sample are grouped into quintile portfolios based on their prior-month EDRs. Row 1 reports average monthly returns in percentage for each portfolio. Rows 2 through 4 report EDR-sorted portfolio alphas measured by the intercepts in the time series regressions from the CAPM, the Fama and French (1993) three-factor model and the Carhart (1997) four-factor model. The column “High-Low” represents the mean difference in monthly returns or alphas between the highest and lowest EDR quintile stocks. The Newey-West (1987) robust t-statistics are reported in brackets for the “High-Low” portfolios. EDR Portfolios 1 Low 2 3 4 5 High High-Low

1 Return 1.2778 1.3397 1.3143 1.3717 1.4942 0.2164

[3.04]

2 Alpha-CAPM 0.3039 0.3693 0.3514 0.3978 0.5120 0.2081

[2.89]

3 Alpha-3 factor 0.0143 0.0935 0.0773 0.1127 0.2153 0.2010

[2.95]

4 Alpha-4 factor 0.1621 0.2432 0.2216 0.3031 0.4758 0.3138

[3.44]

32

Table III Extreme Downside Risk Effects in Different Size and Book-to-Market Portfolios

Each month between July 1967 and June 2005, all sample stocks are sorted into size and book-to-market (BM) groups following the Fama and French (1993) approach. In Panel A, all stocks are sorted into two size groups (Small and Big) according to market capitalization of NYSE-listed stocks of the preceding June. In Panel B, all stocks are sorted into three book-to-market groups (Low-bottom 30%, Medium-middle 40%, and High-top 30%) according to book-to-market ratio of last fiscal year ending in the preceding calendar year. Within each size or book-to-market group, stocks are further sorted into quintiles based on prior-month EDR. Monthly average returns and the intercepts of the time-series regressions from four-factor model (4-factor Alpha) are reported in the left-hand side and right-hand side respectively for each size-EDR or BM-EDR sorted portfolio. “High-Low” in the last column is the mean difference in monthly returns or alphas between the highest and lowest EDR quintile stocks. The last row in each panel refers to the average monthly return or alpha of each EDR quintile across all size or book-to-market groups. The Newey-West (1987) robust t-statistics are reported in brackets for the “High-Low” portfolios.

Panel A: EDR Effects in Different Size Groups Returns 4-factor Alphas EDR Quintiles EDR Quintiles 1 Low 2 3 4 5 High High-Low 1 Low 2 3 4 5 High High-Low Small 1.3567 1.4098 1.3699 1.4565 1.5664 0.2097 0.2004 0.2804 0.2452 0.3584 0.5219 0.3215 [2.68] [3.42] Big 1.0518 1.0425 1.0428 1.0806 1.1395 0.0877 0.0693 0.0653 0.0633 0.1322 0.2461 0.1768 [1.34] [1.91] Average 1.2042 1.2261 1.2063 1.2686 1.3530 0.1487 0.1348 0.1728 0.1543 0.2453 0.3840 0.2491 [2.44] [3.74]

Panel B: EDR Effects in Different BM Groups Returns 4-factor Alphas EDR Quintiles EDR Quintiles 1 Low 2 3 4 5 High High-Low 1 Low 2 3 4 5 High High-Low Low 0.7130 0.8856 0.9231 0.9753 1.0851 0.3722 -0.1768 -0.0097 0.0220 0.1511 0.3211 0.4979 [3.81] [3.79] Medium 1.3034 1.3768 1.3343 1.3290 1.4651 0.1617 0.1575 0.2587 0.2114 0.2099 0.3992 0.2417 [1.91] [2.53] High 1.8181 1.7245 1.7611 1.8146 1.9745 0.1564 0.5004 0.4295 0.4944 0.5506 0.7243 0.2238 [1.63] [2.29] Average 1.2782 1.3290 1.3395 1.3730 1.5083 0.2301 0.1604 0.2262 0.2426 0.3039 0.4815 0.3211 [3.39] [4.94]

33

Table IV Long-run Extreme Downside Risk Effects on Average Stock Returns

Each month between July 1967 and June 2005, all stocks are grouped into quintiles according to their prior-month EDRs and such portfolios are held for periods longer than one month, represented by two, three, six, nine, and twelve months, respectively. For each portfolio of different holding periods, average monthly returns are reported in Panel A and the intercepts of the time-series regressions from four-factor model (4-factor Alpha) are reported in Panel B. “High-Low” column indicates the mean difference in monthly returns or alphas between the highest and the lowest EDR quintile stocks. The Newey-West (1987) robust t-statistics are provided in brackets for the “High-Low” portfolios.

Panel A: Returns for EDR-sorted Portfolios 1 Low 2 3 4 5 High High-Low

Post 2-month return 1.2875 1.3418 1.3270 1.3640 1.4849 0.1974 [3.17] Post 3-month return 1.2873 1.3331 1.3369 1.3684 1.4857 0.1984 [3.46] Post 6-month return 1.2948 1.3322 1.3194 1.3609 1.4786 0.1838 [3.75] Post 9-month return 1.3028 1.3376 1.3202 1.3564 1.4779 0.1752 [4.19] Post 12-month return 1.3117 1.3395 1.3297 1.3561 1.4841 0.1725 [4.71]

Panel B: 4-factor Alphas for EDR-sorted Portfolios 1 Low 2 3 4 5 High High-Low

Post 2-month return 0.4017 0.4703 0.4501 0.5086 0.6316 0.2298 [4.07] Post 3-month return 0.5507 0.6111 0.6085 0.6549 0.7753 0.2245 [4.23] Post 6-month return 0.7169 0.7703 0.7520 0.8008 0.9351 0.2181 [4.49] Post 9-month return 0.7633 0.8128 0.7912 0.8307 0.9611 0.1978 [4.66] Post 12-month return 0.7822 0.8222 0.8088 0.8381 0.9718 0.1896 [4.96]

34

Table V Average Coefficients from Cross-Sectional Regressions with

Extreme Downside Risk as One Explanatory Factor Fama and MacBeth (1973) cross-sectional regressions are run each month between July 1967 and June 2005, with individual stock return as the dependent variable. The explanatory variables are prior-month EDR and systematic risk measures including beta, size, and book-to-market ratio (BM), as well as past month return, momentum, and liquidity beta (Liqu. Beta). “Beta” is estimated following Fama and French (1992), “Size” is the logarithm of the market capitalization of the preceding June, and “BM” is the logarithm of book-to-market ratio of the latest fiscal year ending in the preceding calendar year. “Lagret” denotes past month return. “Momentum” is measured as the past twelve-month return, skipping the most recent month. Liquidity beta is constructed following Pastor and Stambaugh (2003). Time-series averages of the slopes are reported along with the Newey-West (1987) robust t-statistics in brackets. EDR Beta Size BM Lagret Momentum Liqu. Beta 1 1.1697 [3.08] 2 0.9321 -0.1228 -0.1490 0.3208 [3.25] [-0.47] [-3.34] [3.94] 3 0.7832 -0.2442 -0.1371 0.3375 -0.0636 0.0664 0.4060 [3.12] [-0.91] [-3.06] [4.08] [-12.91] [3.57] [0.37]

35

Table VI Time-Series Variation of the Extreme Downside Risk Premium

Time-series regressions of the EDR premium on business cycle indicators are run each month between July 1967 and June 2005. The dependent variable is EDR premium represented by monthly return difference between highest and lowest EDR quintile stocks (refer to row 1 in Table II). Among the independent variables, “Recession” is a dummy variable which is equal to 1 for contractions and 0 for expansions according to NBER’s dating of business cycles. “Tbill” is U.S. three-month Treasury bill rate. “CD-Tbill” is the difference between three-month CD rate and three-month Treasury bill rate. “Term” (i.e., term spread) is the difference between ten-year and one-year treasury constant maturity rates. “Default” (i.e., default spread) is the difference between BAA- and AAA-rated corporate bond yields. “Inflation” is monthly percentage change of US consumer price index. “GDP” refers to current quarter real GDP growth rate in percentage. All macroeconomic data are obtained from the Federal Reserve Economic Database (FRED) maintained by the Federal Reserve Bank of St. Louis. Regression intercepts (Alpha), slope coefficients, and the adjusted R-square values (Adj. R2) are reported, with the Newey-West (1987) robust t-statistics in brackets. Alpha Recession Tbill CD-Tbill Term Default Inflation GDP Adj. R2 1 0.2058 0.0739 -0.002 [2.58] [0.45] 2 0.3517 -0.02251 -0.0007 [1.81] [-0.92] 3 0.2054 0.0145 -0.0022 [1.78] [0.16] 4 0.1310 0.0903 0.0019 [1.76] [1.30] 5 0.1887 0.0261 -0.0022 [1.12] [0.20] 6 0.2886 -0.1862 -0.0010 [2.02] [-0.76] 7 0.1744 0.0669 -0.0006 [2.53] [1.38]

36

Table VII Correlations between Extreme Downside Risk and Other Risk Measures

Correlation coefficients between EDR and other risk measures are computed each month between July 1967 and June 2005. Among the traditional risk variables, “IV” represents idiosyncratic volatility measured as the standard deviation of prior-month’s daily residuals from the Fama and French (1993) three-factor model. “Dbeta” is the downside beta estimated using monthly returns in the previous twelve months, following Ang, Chen, and Xing (2006). “Coskew” is the co-skewness defined in Harvey and Siddique (2000). “Cokurt” is the co-kurtosis constructed as in Ang, Chen, and Xing (2006). “VaR” is monthly value-at-risk measure based on the actual empirical distribution at the 4.76% confidence level. “O-score” is bankruptcy risk proxy defined according to Ohlson (1980), and higher O-score indicates higher bankruptcy risk. “Lev” is leverage measure defined as total liabilities over total assets, “Profit” is profitability measure defined as net income over total assets, and “Price” is the logarithm of past month’s market closing price winsorized above $15. “Lagret” indicates past performance represented by prior-month realized return. “Vol” refers to past month’s return volatility (standard deviation of daily returns). “Delist” is exchange delisting dummy variable that equals 1 when the particular stock is delisted from NYSE, AMEX, or NASDAQ, and 0 otherwise. Accounting data are obtained from COMPUSTAT yearly dataset, and accounting-related variables are calculated using data of the latest fiscal year ending in the preceding calendar year. Time-series averages of the monthly correlations are reported, with t-statistics in brackets.

Correlation t-statistic 1 IV 0.0331 [9.72] 2 Dbeta -0.0032 [-1.21] 3 Co-skew -0.0094 [-8.57] 4 Co-kurt -0.0563 [-17.63] 5 VaR 0.0381 [10.72] 6 O-score 0.0295 [20.06] 7 Lev 0.0067 [6.39] 8 Prof -0.0018 [-2.12] 9 Price -0.0438 [-13.94] 10 Lagret -0.0044 [-3.20] 11 Vol 0.0319 [9.21] 12 Delist 0.0233 [11.04]

37

Table VIII Average Coefficients from Cross-Sectional Regressions

Controlling for More Risk Variables Fama and MacBeth (1973) cross-sectional regressions are run each month between July 1967 and June 2005, with individual stock returns as the dependent variable. Among the explanatory variables, “EDR” refers to prior-month extreme downside risk measure, “Beta” is estimated following Fama and French (1992), “Size” is the logarithm of the market capitalization of the preceding June, and “BM” is the logarithm of book-to-market ratio of the latest fiscal year ending in the preceding calendar year. Idiosyncratic volatility measure (IV) is measured as the standard deviation of prior-month’s daily residuals from the Fama and French (1993) three-factor model, “Coskew” is the co-skewness defined in Harvey and Siddique (2000). “VaR” is monthly value-at-risk measure based on the actual empirical distribution at the 4.76% confidence level. “O-score” is bankruptcy risk proxy defined according to Ohlson (1980). “IV”, “Coskew”, “VaR”, and “O-score” are added in Panel A through D respectively as independent variables. Time-series averages of the slopes are reported, with the Newey-West (1987) robust t-statistics in brackets.

Panel A: Controlling for Idiosyncratic Volatility EDR Beta Size BM IV 1 0.6408 -4.5929 [2.11] [-0.80] 2 0.6879 -0.0407 -0.1511 0.3037 -10.1045 [2.47] [-0.18] [-3.77] [3.97] [-2.71]

Panel B: Controlling for Co-skewness EDR Beta Size BM Coskew 1 1.1316 -0.3963 [2.85] [-2.45] 2 0.8080 -0.1484 -0.1145 0.2874 -0.1497 [2.81] [-0.56] [-2.92] [3.64] [-1.10]

Panel C: Controlling for VaR EDR Beta Size BM VaR 1 0.5286 0.0573 [1.84] [2.35] 2 0.5780 -0.2369 -0.0849 0.3599 0.0525 [2.23] [-0.99] [-2.07] [4.59[ [3.60]

Panel D: Controlling for Bankruptcy Risk EDR Beta Size BM O-score 1 1.6774 0.0300 [3.43] [0.82] 2 1.1904 -0.2401 -0.1441 0.3239 -0.0173 [3.18] [-0.84] [-3.09] [3.37] [-0.65]