Is Unlevered Firm Volatility Asymmetric?

Transcription

1 Is Ulevered Firm Asymmetric? By Hazem Daouk Corell Uiversity David Ng Corell Uiversity This versio: May 2, 2008 Abstract Asymmetric volatility refers to the stylized fact that stock volatility is egatively correlated to stock returs. Traditioally, this pheomeo has bee explaied by the fiacial leverage effect. This explaatio has recetly bee challeged i favor of a risk premium based explaatio. We develop a ew, uleverig approach to documet how well fiacial leverage, rather tha size, beta, book-tomarket, or operatig leverage, explais volatility asymmetry o a firm-by-firm basis. Our results reveal that, at the firm level, fiacial leverage explais much of the volatility asymmetry. This result is robust to differet uleverig methodologies, samples, ad measuremet itervals. However, we fid that fiacial leverage does ot explai idex-level volatility asymmetry, which is cosistet with theoretical results i Aydemir, Gallmeyer ad Hollifield (2006). JEL Classificatio: G2 Keywords: asymmetry; Fiacial leverage. Both authors are i the Departmet of Applied Ecoomics ad Maagemet, Corell Uiversity, Ithaca, NY Their addresses are: HD35@corell.edu ad DTN4@corell.edu. We thak Utpal Bhattacharya, David Brow, Achada Charoerook, Tim Crack, Robert Dittmar, Michael Gallmeyer, Robert Hodrick, Craig Holde, Robert Jeigs, Da Jubiski, reeivas Kamma, Josef Lakoishok, Ju Pa, Lasse Pederse, Richard hockley, Albert Wag, Xiaoya Zhag, ad Guofu Zhou, as well as semiar participats at Amsterdam, Corell, UC Riverside, Ciciati, Illiois, Marylad, Oklahoma, Quee s, Washigto, ad York Uiversity, the America Fiace Associatio meetig, the Wester Fiace Associatio meetig, the Uiversity of Chicago-CRP forum ad the Frak Batte Youg cholars Coferece for helpful discussios ad commets. We thak Ajay Palvia ad Jiyou A for excellet research assistace. Remaiig errors are our ow.

2 I. Itroductio Asymmetric volatility refers to the stylized fact that stock volatility is egatively correlated to stock returs. Traditioally, this pheomeo has bee explaied by the fiacial leverage effect (Black (976) ad Christie (982a)). The fiacial leverage hypothesis posits that as the price of a firm s stock decreases, that firm s fiacial leverage icreases, leadig to a higher volatility of equity. Ideed, this leverage effect explaatio is so domiat that it has become syoymous with asymmetric volatility. However, Pidyck (984) ad Frech, chwert ad tambaugh (987) have challeged the fiacial leverage hypothesis with the risk premium hypothesis ad, subsequetly, this challege has bee supported by may major studies. 2 The risk premium hypothesis, also kow as the volatility feedback effect, proposes that a icrease i uexpected volatility will icrease expected future volatility. The resultig icrease i expected returs will cause prices to drop ad this will lead to volatility asymmetry. As a result, it appears that fiacial leverage does ot play much of a role i explaiig volatility asymmetry. Usig a ew, ituitive, approach that utilizes the cocept of uleverig, we examie whether the fiacial leverage explaatio should be rejected i favor of the risk premium hypothesis. Iterestigly, we fid that the leverage effect plays a importat role i explaiig volatility asymmetry. I fact, at the firm level, fiacial leverage explais much of volatility asymmetry. This result is robust to differet uleverig methodologies, differet samples, ad differet measuremet itervals, applies to firms of differet sizes ad i differet idustries, ad is the same whether we examie the pael data set or just the cross sectio of firms. We fid that fiacial leverage is more importat tha size, beta, book-tomarket, or operatig leverage i explaiig volatility asymmetry at the firm level. The reaso that we fid a sigificat role for fiacial leverage is that we coduct a firm-level aalysis rather tha the portfolio- or idex-level aalyses which have bee used i most recet studies. 3 Our uleverig approach makes it possible to examie volatility asymmetry o a firm-by-firm basis usig a broad cross sectio of firms. Equity volatility is first trasformed by strippig out the effect of fiacial leverage. The reductio i volatility asymmetry subsequet to this operatio idicates the stregth of the effect of fiacial leverage. This approach allows us to examie the reasos for volatility asymmetry amog a broad cross sectio of firms. For istace, the parameter that captures the covariace of expected returs ad volatility i GARCH is called leverage effect. 2 Gloste, Jagaatha ad Rukle (993), Campbell ad Hetschel (992), Bekaert ad Wu (2000), Tauche (2005) ad Deis, Mayhew ad tivers (2006). 3 The exceptio is Deis, Mayhew ad tivers (2006). Deis, Mayhew ad tivers (2006) examie implied volatilities o idices ad stocks ad coclude that the risk premium effect is importat. However, they do ot examie fiacial leverage.

3 The ext task is to use the uleverig approach to fid out whether fiacial leverage is as importat at the idex level as it is at the firm level. The uleverig approach allows us to remove fiacial leverage from each compoet firm before differet firms returs are aggregated ito a idex. I cotrast to the result foud at the firm level, at the idex level, a large portio of volatility asymmetry persists eve after uleverig. I other words, eve ulevered idex-level returs have higher volatility whe the idex goes dow tha whe it goes up. Hece, fiacial leverage aloe explais oly a small portio of the idex-level volatility. To sum up, we fid that the leverage effect is extremely importat i explaiig volatility asymmetry at the firm level, but it is ot importat i explaiig idex-level asymmetry. The paper makes three mai cotributios to the volatility asymmetry literature. First, we develop the uleverig approach that helps us to examie the leverage effect o a firm-by-firm basis. This approach opes up opportuities to empirically documet the effect of fiacial leverage o a broad cross sectio of firms over time. We fid that the fiacial leverage hypothesis is by far the most importat explaatio for volatility asymmetry at the firm level. ecod, as Aydemir, Gallmeyer ad Hollifield (2006) poit out, ay study of the effect of fiacial leverage o volatility should use market debt valuatios, which are difficult to obtai i practice. As a result of this difficulty, o previous work has allowed for market price of debt i computig fiacial leverage. Whe we strip out fiacial leverage, we accout for market price of debt either through a Merto-KMV model or a procedure that matches the bod prices with bods with similar credit ratigs. Third, we fid that volatility asymmetry at the market level ad at the firm level have differet causes. While fiacial leverage accouts for most of the firmlevel asymmetry, it does ot explai idex-level asymmetry. Our empirical fidig is cosistet with the theoretical results developed i Aydemir, Gallmeyer ad Hollifield (2006). While Campbell ad Hetschel (992), Wu (200), ad Tauche (2005) make importat theoretical cotributios cocerig the modelig of asymmetric volatility at a market level, the Aydemir, Gallmeyer ad Hollifield (2006) model is particularly relevat i our case because it icludes firm-level as well as market-level asymmetries. It icorporates both the leverage effect ad a time-varyig risk premium i a dyamic geeral equilibrium ecoomy with debt ad equity claims. Uder a represetative aget model with habit formatio preferece, fiacial leverage is liked edogeously to iterest rates, prices of risk ad volatility. At the market level, fiacial leverage is drive by the aggregate risk i the ecoomy. As a result, risk premium rather tha fiacial leverage explais the market-level volatility asymmetry. At the firm level, however, the equity portio of fiacial leverage is drive by idiosycratic risk shocks, i additio to aggregate risk shocks. Hece, fiacial leverage could have a substatial impact o firm-level volatility asymmetry, although it cotributes very little to marketlevel stock retur volatility. 2

4 Our paper is related to Bekaert ad Wu (2000), a major study of idex- ad firm-level volatility i the Japaese stock market. Usig the market portfolio ad portfolios with differet leverage costructed from Japaese Nikkei stocks, Bekaert ad Wu reject the leverage effect i favor of the volatility feedback hypothesis. Their result is ot cosistet with our fidig for the followig reasos. First, our samples are differet. Bekaert ad Wu use 72 Japaese stocks from 986 to 995, while we examie thousads of U stocks from 986 to ecod, Bekaert ad Wu coduct their aalysis by aggregatig stocks ito portfolios, while we examie firm-level asymmetry directly, based upo firmlevel data. As we will show i this paper, leverage effect is extremely importat o the firm level but dimiishes after stocks are aggregated ito portfolios. Our fidig is also related to the empirical pheomeo of covariace asymmetry. Erb, Harvey, ad Viskata (994), Bekaert ad Wu (2000), Ag ad Che (2005), ad Ag, Che, ad Xig (2006) fid that covariace for stocks is higher i a dow market. We fid that ulevered stock returs also have higher covariace whe the market idex goes dow. While firm-level ulevered volatility stays the same i a dow market, idex-level ulevered volatility is higher due to higher covariace betwee stocks. Other papers propose alterative explaatios of asymmetry volatility. Cheug ad Ng (992) show that volatility asymmetry is much stroger for small stocks. Bekaert ad Wu (2000) propose that the risk premium effect behid volatility asymmetry will be more proouced for firms with higher covariace with the market. Koga (2004) develops a productio ecoomy model which implies that firm ivestmet activity ad firm characteristics, particularly the market-to-book ratio, or q, might lead to volatility asymmetry. Christie (984b) proposes that operatig leverage affects volatility asymmetry. Fially, Duffee (995) proposes that volatility asymmetry at the firm-level could be due to a positive cotemporaeous relatio betwee returs ad volatility. We empirically examie these explaatios ad coclude that firm-level volatility asymmetry is due to fiacial leverage rather tha size, beta, bookto-market ratio, operatig leverage, or a positive cotemporaeous relatioship betwee returs ad volatility. The remaider of the paper proceeds as follows. ectio II presets the uleverig methodology. ectio III presets data ad summary statistics. ectios IV ad V preset results usig pooled firmmoth observatios ad a cross sectio of firms. ectio VI shows the market-level versus firm-level asymmetry. Fially, sectio VII offers coclusios. II. Methodology For Uleverig 4 Wu (200) also fids leverage play a role i the U market idex, which is differet from Bekaert ad Wu s (2000) fidig o Japaese stocks. 3

5 I this sectio, we propose a methodology that trasforms equity volatility to extract from it the effect of fiacial leverage. This is doe by usig a equatio that expresses the volatility of equity i terms of a volatility that is ot affected by fiacial leverage. A. Effect of fiacial leverage We accout for the effect of fiacial leverage through three differet methodologies. First, we compare volatility asymmetries for firms with the highest value of book debt, lowest value of book debt ad the media value of book debt. ecod, we adopt the approach developed by Merto (974) ad implemeted by KMV (Crosbie ad Boh (200)). Vassalou ad Xig (2004), Campbell, Hilscher ad zilagyi (2007) ad Bharath ad humway (2004) have also recetly applied the Merto-KMV model i the cotext of examiig firm-level default risk. Third, we ulever usig the chwert (989) formula that has bee used i this literature. To improve o this method, we use the bod idex prices for differet credit ratig classes istead of face value to approximate market value of debt. We will describe the methodologies i this sectio. First, we examie volatility asymmetry based o sub-samples of firms with differet levels of book leverage. I particular, we report the volatility asymmetry of the firm-moths with lowest % leverage, media (49-5%) leverage, ad highest % leverage. We will report the chage i raw volatilities together with the returs for firms with differet leverage levels i figure 3. To preview the results, we fid that all three methodologies provide similar coclusios. ecod, we ulever firm volatilities based o Merto-KMV models. I the Merto-KMV model, the equity of the firm is a call optio for the uderlyig value of the firm with a strike price equal to the face value of the firm s debt. The model recogizes that either the uderlyig value of the firm or its volatility are directly observable but they ca be iferred from the value of equity, the volatility of equity ad several other observable variables by solvig two oliear simultaeous equatios (see Bharath ad humway (2004) for details of the procedure ad computer code). I this paper, we are iterested i iferrig the volatility of the value of the firm. The Merto-KMV model makes two importat assumptios. The first is that the total value of a firm will follow geometric Browia motio, dv = µv dt + V V dw, () where V is the total value of the firm, µ is the expected cotiuously compouded retur o V, V is the volatility of firm value ad dw is a stadard Weier process. The secod importat assumptio of the Merto-KMV model is that the firm has issued just oe discout bod maturig i T periods. Uder these assumptios, the equity of the firm is a call optio for the uderlyig value of the firm, with a strike price equal to the face value of the firm s debt ad a time-to-maturity of T. Moreover, the value of equity as a fuctio of the total value of the firm ca be described by the Black-choles-Merto Formula. By put-call 4

6 parity, the value of the firm s debt is equal to the value of a risk-free discout bod mius the value of a put optio writte o the firm, agai with a strike price equal to the face value of debt ad a time-tomaturity of T. The Merto model stipulates that the equity value of a firm satisfies = V N(d ) e rt FN(d 2 ), (2) where is the market value of the firm s equity, F is the face value of the firm s debt, r is the istataeous risk-free rate, N( ) is the cumulative stadard ormal distributio fuctio, d is give by d 2 ( ) + ( + V ) l V F r 0.5 T =, (3) T V ad d 2 = d V T. Uder Merto s assumptios, the value of equity is a fuctio of the value of the firm ad time, so it follows directly from Ito s lemma that V = V. (4) V I the Black-choles-Merto model, it ca be show that / V = N(d ), so that uder the Merto V = I other model s assumptios, the volatility of the firm ad its equity are related by N( ). words, V = V N d ( ). d V The most sigificat step i implemetig the model is to solve equatios (2) ad (5) umerically for values of V ad V. imultaeously solvig equatios (2) ad (5) is a reasoably straightforward thig to do. However, Crosbie ad Boh (200) explai that i practice the market leverage moves aroud far too much for [equatio (5)] to provide reasoable results. To resolve this problem, we adopt a iterative procedure used i Bharath ad humway (2004). First, we use a iitial value of V = [/( + F)]. This value of V is used i cojuctio with equatio (3) to ifer the market value of each firm s assets every day for the previous year. We the calculate the implied log retur o assets each day ad use that returs series to geerate ew estimates of V ad µ. We iterate o V i this maer util it coverges (so that the absolute differece i curret ad previous value of V is less tha 0 3 ). Followig Bharath ad humway (2004), we assume a forecast horizo of oe year (T=) ad take the book value of debt as the face value of the firm s debt. Third, we also adopt a simple uleverig approach, recogizig that the Merto model has its limitatios. To compare our results with previous literature, we ulever usig the same formula that is (5) 5

7 used i this literature (see for example chwert (989)). The volatility of the retur to the assets of the firm V is V =. V where represets the fractio of the market value of the firm due to stocks, ad is the volatility of V the retur to the stock. We call this method simple uleverig. While previous papers use a firm s book debt as a proxy of its market debt, we make a improvemet by usig bod idex prices for differet credit risk categories istead of face value to approximate market value of debt. Compariso of (5) ad (6) shows that the Merto-KMV method is differet from the simple uleverig method by a factor of N(d ), which ca be iterpreted as a adjusted default probability. This is because the Merto-KMV method captures the fact that the debt may default i the future. I this paper, we examie two versios of volatility. We first report the raw stock volatility without ay adjustmet. We the examie the stock volatility after uleverig for fiacial leverage V. This is doe by usig either the Merto-KMV method i equatio (5) or the simple uleverig i equatio (6). III. Data ad Asymmetry Metrics A. Data We merge all firms i the itersectio of COMPUTAT idustrial ad research files ad the CRP database. The data period is from Jauary 986 to December Because leverage takes o a differet meaig i fiacial firms, we remove stocks of fiacial firms from the sample. For book value of debt, we use quarterly data from COMPUTAT by addig total liabilities (#54) ad preferred stock carryig value (#55). The market price of debt is implied by the optio pricig formula i the Merto-KMV model. As a proxy for equity value, we multiply mothly stock prices from CRP by commo shares outstadig. The, we compute a mothly asset-to-equity ratio. To facilitate estimatio of a robust model, we drop firms with prices below $3 per share. We also elimiate firms with egative book value (defied as commo equity), ad ay firms missig price or accoutig data that is eeded for the estimatio regressio. After these screes, the umber of firms rage from 456 i 987 to 2448 firms i 2003, for a total of firm-moth observatios. Whe we ulever usig the simple uleverig method istead of the Merto-KMV method, we use proxies for the market value of debt. To obtai such proxies, we require that the firms have credit ratigs data from COMPUTAT. We costruct market value of debt based o book value of debt alog with Lehma Brothers Corporate Bod Idex of differet credit ratigs. Returs data for the Lehma (6) 6

8 Brothers Corporate Bod Idex for these ratig classes has to be available for those years. We assume that the iitial book value of debt is evaluated at market price ad that, subsequetly, the price of the debt follows the movemet of bods with the same credit ratigs. The requiremet for the availability of credit ratig data reduces the sample substatially whe we use simple uleverig method. But, as we will see, the results are very similar whe we ulever usig the Merto-KMV method or the simple uleverig method. First, we calculate the variace of stock returs for every moth as the variace of daily returs i the moth: ( ), (7) N t 2 2 st, = rit, rt ( Nt ) i= where there are N t daily returs r i,t i moth t. Usig o-overlappig samples of daily data to estimate the mothly variace creates a estimatio error that is ucorrelated through time. We take the square root to obtai the stadard deviatio (i.e. volatility) of the stock returs. We ivestigate the robustess of our results i relatio to a firm s size, beta, book-to-market, ad operatig leverage. As metioed before, a firm s size (i.e. market capitalizatio) is calculated by multiplyig mothly stock prices by commo shares outstadig. Beta is based upo a rollig regressio of stock returs o the &P 500 idex o a firm-by-firm basis. Book-to-market ratio is computed by dividig a firm s book equity by its market capitalizatio. The degree of operatig leverage (OL) is the percetage chage i EBIT (earigs before iterest ad taxes) for a give percetage chage i sales reveue i.e. ( OL) % ΔEBIT + =. Followig Madelker ad Rhee (984), we compute operatig % Δsales leverage by ruig the followig regressio for each firm j: l EBIT = a + b l ales + ε jt j j jt jt The coefficiet b j is the operatig leverage for firm j. 5 Table reports the summary statistics for our mai variables of iterest: volatility, mothly returs, fiacial leverage, market capitalizatio, beta, book-to-market, ad operatig leverage. B. Asymmetry Metrics To examie volatility asymmetry, we adopt a regressio approach that allows both clea graphical evidece as well as statistical iferece. The geeral form of the regressio model is the followig: 5 We also coduct a robustess check, i which we estimate idustry-wide operatig leverage ad use that as a firm s operatig leverage. The results remai the same. 7

9 d kt, l = βi Di, t + εt. kt, 2 i= (8) where k = or V. These two specificatios pertai to log chages i mothly stadard deviatios of stock returs before ay trasformatio ad log chages i mothly stadard deviatios with fiacial leverage removed V. D i,t- is a dummy variable that equals if returs at t- fall withi the rage, ad d is the umber of sets of returs that are created by partitioig the space of returs. We partitio the stock returs ito te itervals (i.e. d = 0), each of which is represeted by a dummy variable i the followig way: D for returs less tha 0.20; D 2 for returs betwee 0.20 ad 0.5; D 3 for returs betwee 0.5 ad 0.0; D 4 for returs betwee 0.0 ad 0.05; D 5 for returs betwee 0.05 ad 0; D 6 for returs betwee 0 ad 0.05; D 7 for returs betwee 0.05 ad 0.0; D 8 for returs betwee 0.0 ad 0.5; D 9 for returs betwee 0.5 ad 0.20; ad D 0 for returs larger tha The idepedet variables are dummy variables, represetig returs of differet sigs ad magitude. This gives us the mea respose of the chage i volatility to returs of differig sigs ad magitudes. I this model, a asymmetric relatio would be assessed by examiig the respose of volatility to returs of the same magitude but with opposite sigs. This fuctioal form is close i spirit to the ews impact curve i Egle ad Ng (993) 7. It has the appealig features of simplicity (liear least squares) ad flexibility (allowig differet parameter values for differet retur categories). It also allows for the well kow ARCH effect, where volatility icreases followig both large egative ad large positive returs. We should ote that the leverage effect does ot mea that leverage is the mai determiat of the movemet of volatility. The large ARCH literature has show that the magitude of lagged retur has the most explaatory power for volatility chages. The respose of volatility to returs therefore has the shape of a parabola. The leverage effect tilts this parabola to the right by makig the reactio of volatility to large egative returs more proouced tha that from large positive returs. Figlewski ad Wag (2000) ivestigate the leverage effect but do ot accout for this ARCH effect. 8 Followig Peterse (2007) ad Vuolteeaho (2002), we use Rogers s (983, 993) robust stadard error methodology i order to calculate cross-sectioal correlatio cosistet stadard error. I order to ru a statistical test o the level of volatility asymmetry, we develop a ew metric for volatility 6 All of our results are robust to alterative partitios with differet d s. 7 Egle ad Ng (993) derive a procedure that plots the implied relatio betwee the coditioal variace from a Asymmetric GARCH model ad lagged residuals. They explore the curve patter of may models, ad assess how well these models fit the data. They also propose a partially o-parametric ews impact curve. 8 Figlewski ad Wag (2000) fid that volatility icreases eve whe stock price icreases. They call this a reverse leverage effect ad cosider this evidece agaist the leverage effect. However, they do ot accout for the ARCH effect. Whe positive returs are large, ARCH effect pushes volatility higher while the leverage effect pushes volatility lower. ice ARCH is the domiat effect, volatility icreases. It does ot mea that the leverage 8

10 asymmetry, which describes how much of the asymmetry is explaied by each competig hypothesis. The metric i questio is based o the differece betwee the left tail ad the right tail of the curve i the graphs. The left tail is the respose of volatility to egative shocks, ad the right tail is the respose of volatility to positive shocks. If the two tails are exactly the same, the there is o asymmetry. If the two tails are very dissimilar, the there is strog asymmetry. The sig of the asymmetry i this case will deped o which tail of the curve is higher tha the other, o average. The metric formalizes the above ituitio by summarizig the degree of asymmetry by a sigle umber. We resort to the cocept of itegratio which ca be see i Figure. Figure left pael plots the chage of stock volatility (o the y-axis) agaist stock returs (o the x-axis). Picture the two tails of the curve superimposed with absolute returs o the x-axis as i Figure, right pael. The area betwee the two tails ca be used as a measure of asymmetry. If there is o asymmetry, the two tails of the curve are idetical. I such a case, superposig the two tails provides a sigle curve with o area i it, ad the asymmetry metric has a value of zero. O the other had, if there is strog asymmetry, the the area betwee the two tails becomes very large, ad the asymmetry metric has a large value. We adopt the covetio that the area is egative (positive) if there is egative (positive) asymmetry. To compute the asymmetry metric from our coefficiets, we approximate the (siged) area described above by summig the differece betwee the two (superposed) tails of the curve for each category of absolute returs. We give stadard errors aroud the metric i the brackets followig the poit estimate. The stadard errors are derived from oe millio replicatios of Mote Carlo Itegratio of the area uderlyig the asymmetry metric. We refer to this first volatility asymmetry metric as the U-shaped metric, because a symmetric relatioship looks like a capital letter U. The secod volatility asymmetry metric is based o a simplified versio of equatio (8): = D + D + u (9) kt, t t l β, t l β+ +, t l t, kt, 2 t t where D -,t- (D +,t- ) is a dummy variable that takes o a value of whe t- returs are egative (positive), ad 0 otherwise. This model fits a liear regressio for egative returs ad positive returs separately. It is similar to equatio (8) except that it uses a lie istead of a curve to fit the volatility chages. The differece i areas betwee the egative segmet ad the positive segmet is a measure of volatility asymmetry (see Figure 2, right pael). We refer to this secod volatility asymmetry metric as the V- shaped metric, because a symmetric relatioship looks like a capital letter V. We show our results based o both the U- ad the V-shaped metrics. While equatio (8) provides clear graphical evidece ad statistical iferece for volatility asymmetry, it requires that a wide rage of returs are observed. The effect is ot importat. Our methodology makes it easy to see if the magitude of the tilt i the parabola correspods 9

11 metric based o equatio (9) has the advatage that it ca be calculated eve whe a stock does ot have returs withi a certai rage. For some of the later aalysis, we use oly the V-shaped metric because of this data limitatio. We will explai this i more detail later. I the remaider of the paper, we will preset results o volatility asymmetry. First, we will examie the role of fiacial leverage based o pooled firm-moth observatios (ectio IV). ecod, we will examie the fiacial leverage, risk premium ad other hypotheses usig a cross sectio of firm-level volatility asymmetries (ectio V). Third, we will relate firm-level volatility asymmetry to idex-level volatility asymmetry (ectio VI). IV. Results Usig Pooled Firm-Moth Observatios A. Assessig the effect of leverage o the relatio betwee volatility ad returs for stocks We first examie volatility asymmetry based o a sub-sample of firms with differet levels of book leverage. 9 I particular, we report the volatility asymmetry of the firm-moths with lowest % leverage, media (49-5%) leverage, ad highest % leverage. We plot the chage i raw volatilities together with the returs for firms with differet leverage levels. Paels A, B, ad C i Figure 3 show the results i three graphs, for firms with low, media ad high leverage, respectively. The graphs show that volatility asymmetry is substatially higher for firms with high leverage, while firms with the lowest leverage have miimal volatility asymmetry. The U- shaped volatility asymmetry metric is for firms with low leverage, -2.9 for firms with media leverage, ad -4.3 for firms with high leverage. Table 2 shows results based o the uleverig approach usig the Merto-KMV model, through equatio (5). Table 3 reports the regressio results i (6) for stock volatility by removig the fiacial leverage via the simple uleverig approach. We also report robustess checks for differet firm sizes ad idustry groups. Table 2, Pael A, shows the result for raw (i.e. utrasformed) stock volatility ad ulevered volatility V. The volatility asymmetry metric is egative ad sigificat at (t-statistic -4.00). After uleverig, the volatility asymmetry metric is substatially reduced to (t-statistic -0.3). The chage from strippig out fiacial leverage is 2.7 (2.94 mius 0.23). I other words, fiacial leverage accouts for 92% of the chage i asymmetry. 0 Table 2, Pael B shows the result usig the V-shaped volatility asymmetry metric. The result is similar to that i the first pael. asymmetry drops to what is expected give the magitude of the chage i leverage. 9 Book leverage is used istead of market leverage i this sectio because our focus is o firms with close to zero leverage. Our sample of firms with market leverage requires credit ratigs of firms. Usually oly firms with substatial leverage have credit ratigs. 0

12 from -2.4 (t-stat -8.65) to -0.9 (t-stat -.92). Agai, after uleverig, a vast majority (92%) of the volatility asymmetry is goe, although i this case the asymmetry is still statistically sigificat. Figure 4 displays graphically the results i Table 2, Pael A. The bad aroud the curves is a two stadard error cofidece iterval. The graph o the left shows the results related to raw stock volatility. The respose of volatility chage to returs exhibits a substatial egative asymmetry. Large egative returs lead to a large icrease i volatility. O the other had, positive returs i geeral lead to a small decrease i volatility. The graph o the right shows the curve related to the ulevered volatility V usig the Merto-KMV uleverig method i equatio (5). As oe ca tell, after takig out fiacial leverage, there is very little egative volatility asymmetry left. This offers strog support for the firm-level fiacial leverage effect ad mirrors the volatility asymmetry metric results i showig that most of the egative asymmetry has disappeared after accoutig for fiacial leverage. ice the results i Pael B are the same, we do ot report the graph separately. Table 3 presets results from usig the simple uleverig method i equatio (6) to accout for risky corporate debt. As ca be see, eve with a differet uleverig approach, the results are very similar to our previous results. Table 3, Pael A shows the results for the U-shaped volatility asymmetry metric. After uleverig, volatility asymmetry drops from -4.5 (t-statistics -5.65) to (t-statistics ). Aroud 83% of the volatility asymmetry is removed after fiacial leverage is take out. 2 Figure 4 shows the results from this uleverig procedure. Table 3, Pael B, shows that the V-shaped volatility asymmetry metric also drops substatially. Approximately 89% of the volatility asymmetry is removed after uleverig. B. Robustess Checks with Differet Firm izes ad Idustry Groups We ivestigate the robustess of our results for firms i differet size quitiles or idustries. Cheug ad Ng (992) examie the volatility asymmetry for idividual stocks from 962 to 989 ad show that volatility asymmetry is much stroger for small stocks. I this sectio, we check whether stocks of differet sizes or idustries behave differetly. We examie the volatility asymmetry metrics for differet size quitiles. We fid sigificat egative asymmetry across all size quitiles (ragig from to for U-shaped asymmetries). 0 As the asymmetry metric is the measure of a (siged) area, the chage i the area due to the removal of a specific effect divided by the total chage i the area will give the percetage of the asymmetry effect attributable to a specific explaatio. Eve if leverage effect is the oly explaatio for volatility asymmetry, the asymmetry curve based o KMV- Merto uleverig will ot be flat but will be U-shaped sice the pure Merto model does ot capture the ARCH effect. For the same reaso, the levered curve is dowward slopig but ot liear as i a pure Merto leverage model. 2 To idetify whether our market debt assumptio makes a big differece i the results, we repeat our aalysis i Table 3 usig book debt data. Most (84%) of the volatility asymmetry is removed after adjustig for fiacial leverage.

13 More importatly, regardless of the size quitiles ad metrics, volatility asymmetries drop dramatically after uleverig. After uleverig, volatility asymmetry is reduced by 85% (size 5 quitile) to 97% (size quitile). The results for V-shape volatility asymmetry results are broadly similar. We also check the volatility asymmetry metric for differet idustry groups. The idustry groupig is the same as i Griffi ad Karolyi (998). Agai, we fid sigificat egative asymmetry across all idustry groups. After takig out fiacial leverage, volatility asymmetries drop by 88% or more i all idustries. Thus, the fiacial leverage effect is strog i all idustry groups. C. Timig issue We preset the mai results usig the chage of volatility from t-2 to t because Duffee (995) cojectures that the egative volatility asymmetry documeted i the literature is largely due to a positive cotemporaeous relatio betwee firm stock returs ad firm stock retur volatility. Essetially, usig the chage i volatility betwee t ad t- as a depedet variable might iduce the regressio to pick up the positive cotemporaeous relatio. Give this possibility, fidig egative volatility asymmetry might be a spurious result (see Duffee (995) for more details). To address Duffee s (995) valid poit, we use a specificatio that ca show that volatility asymmetry is a pervasive pheomeo that is ot limited oly to the t- specificatio. Throughout the paper, we use the chage i volatility betwee times t ad t-2 as the depedet variable istead of that betwee times t ad t-. This specificatio is i the spirit of evet study methodologies i which a ormal period is used as a bechmark, excludig the widow of data that cotais the evet. As a robustess check, we examie results with the specificatio for chage i volatility that uses moth t- istead of t-2. Followig Duffee (995), we coduct the followig liear regressio: l = β + β l + ε. The egative asymmetry is revealed whe β is foud to be egative. We fid a st, t 0 st, t t β coefficiet of -0.4 (t-statistic of -45.9), which reveals sigificat egative asymmetry. Whe we use l s, t as the depedet variable, we fid a β coefficiet of (t-statistic of ), which cofirms st, 2 that egative asymmetry is a pervasive pheomeo that holds for more tha the t- specificatio. To examie this further, we estimate a modified versio of equatio (8) with a oe-moth lag (from t- to t) istead of a two-moth lag as before. k l, t k, t d = β i. Di i=, t + ε. t (0) 2

14 Table 4 Pael A reports the results. The raw retur volatility asymmetry is statistically sigificat, ad the egative asymmetry goes away whe we remove leverage usig the Merto-KMV model. 3 This is cofirmed i Table 4 Pael B. V. Results Usig the Cross ectio of Firm-pecific Asymmetry This sectio reports the results from the cross-sectioal aalysis of volatility asymmetry for differet firms. We ru a time-series regressio (9) for each firm i our sample ad costruct a volatility asymmetry metric for each firm. We the relate the cross sectio of firm volatility asymmetries to differet firm-level variables. This allows us to examie the fiacial leverage hypothesis, the risk premium hypothesis ad other potetial explaatios of volatility asymmetry. We focus, here, o the V- shaped volatility asymmetry istead of the U-shaped volatility asymmetry. The U-shaped volatility asymmetry is calculated based upo the regressio i equatio (8). However, ot every firm has the etire rage of returs ecessary to allow us to ru the regressio. I a case i which a stock ever showed returs i a particular iterval, the U-shaped volatility asymmetry would be udefied. A. Cross-ectioal Distributio of Asymmetries I Figure 6, we plot the distributio of the volatility asymmetry metric for our firms. Lookig at the distributio of volatility asymmetries before uleverig, it is clear that the distributio is mostly i the egative rage (solid lie). This shows that, o average, firms exhibit egative volatility asymmetry. However, after the Merto-KMV uleverig procedure, the ulevered volatility seems to be equally distributed betwee positive ad egative values (dotted lie). This agai idicates that the origial egative asymmetry is, to a large extet, accouted for by fiacial leverage. B. Asymmetry based o Differet Fiacial Leverage, Beta, Book-to-market ad Operatig Leverage Quitiles I Table 5, we assess the effect of fiacial leverage as compared to the effect of the risk premium ad other hypotheses. We compute each firm's fiacial leverage, beta, book-to-market, ad operatig leverage based o its time-series data. tocks are the sorted ito quitiles based o the cross sectio of these variables. I each quitile, the average volatility asymmetry for stocks is computed. Table 5, Pael A presets results from sortig o fiacial leverage. There is a clear egative mootoic relatio betwee the volatility asymmetry metric ad fiacial leverage. A test of differece i 3 Whe we examie the coefficiets i Pael A, we otice that large returs for either sig are associated with a large decrease i volatility. The reaso for this is selectio bias, as i Duffee (995). A moth with a large absolute retur is associated with a large stadard deviatio. ice this regressio uses the chage i stadard deviatio as the depedet variable, a abormally large retur will be associated with a decrease i volatility sice the followig moth s stadard deviatio will likely be lower. Because of this problem, we choose to use the chage i volatility betwee moth t ad moth t-2 i this paper. 3

15 the volatility asymmetry metric betwee the first ad fifth quitile yields a p-value of This shows that fiacial leverage is a driver of volatility asymmetry. Table 5, Pael B presets results from sortig o beta. As Bekaert ad Wu (2000) idicate, the risk premium effect will be more proouced for firms with higher covariace with the market. The crosssectioal implicatio is that firms with higher beta will have higher risk premia. This is because beta is defied as the covariace divided by the variace of the market (which does ot vary i the crosssectio). I cotrast to the results for fiacial leverage, there does ot seem to be ay recogizable patter that relates beta ad volatility asymmetry. The differece i the volatility asymmetry metric betwee the first ad fifth quitile is egative (p-value of 0.065). This is the opposite of the predictio from the risk premium story. This idicates that risk premium does ot explai volatility asymmetry i the cross sectio. Table 5, Pael C shows the results from sortig o book-to-market. Koga (2004) develops a model of a productio ecoomy i which real ivestmet is irreversible ad subject to covex adjustmet costs. A implicatio of his model is that firm ivestmet activity ad firm characteristics, particularly the market-to-book ratio, or q, might lead to volatility asymmetry. Also, book-to-market is ofte cosidered a factor uderlyig the risk premium that is separate from beta (Fama ad Frech (993)). We fid o discerible patter across the quitiles. The differece betwee the first ad fifth book-to-market quitile is isigificat. This shows that it is fiacial leverage, ad ot book-to-market, that drives volatility asymmetry. Fially, Table 5 Pael D presets results from sortig o operatig leverage. Operatig leverage is the degree to which a firm is committed to fixed productio costs. A firm with low (high) fixed costs will have low (high) operatig leverage. Theoretically, as Christie (984b) shows, operatig leverage ca cause volatility asymmetry. A forecast of lowered cash flows ca result i a immediate fall i stock prices. Cash flows, ad stock prices, become more volatile whe their levels decrease because fixed costs act like a lever i the sese that a small percetage chage i operatig reveue ca be magified ito a large percetage chage i operatig cash flow. There is some evidece that may aggregate ecoomic series are more volatile durig recessios (chwert (989)). However, the effect of operatig leverage o stock-market volatility asymmetry has ot bee tested empirically. Therefore, we examie operatig leverage i additio to fiacial leverage. We fid that there does ot seem to be ay recogizable patter that relates operatig leverage ad volatility asymmetry. C. Cross-ectioal Regressios of Firm-Level Asymmetry Table 6 presets cross-sectioal regressios of the firm volatility asymmetry metric o the explaatory variables used above. Followig Peterse (2007), we calculate robust stadard errors with firm clusterig to correct for cross-sectioal correlatio i all regressios. Colum shows the results for 4

16 fiacial leverage. Colum 2 shows the results for beta. Colum 3 shows the results for book-to-market. Colum 4 shows the results for operatig leverage. Colums 5, 6, ad 7 show the results whe we iclude fiacial leverage alog with the competig variable, i.e. beta, book-to-market ad operatig leverage. Colum 8 shows the multivariate regressio with fiacial leverage, beta, book-to-market ad operatig leverage. Our results i the first two colums show that the coefficiet of fiacial leverage is egative ad sigificat at the % level. However, the coefficiet of firm beta is of the wrog sig. More importatly, whe we ru multivariate regressios with fiacial leverage ad beta, the coefficiet of firm beta is isigificat ad of the wrog sig. The book-to-market factor ad operatig leverage are isigificat i both uivariate regressios ad multivariate regressios with fiacial leverage. This shows that volatility asymmetry is drive by fiacial leverage but ot by book-to-market or operatig leverage. We also replace firm beta with covariace of firm returs with the idex (results available upo request). The covariace factor is ot sigificat. Overall, our empirical results cofirm our coclusios. Firms with higher fiacial leverage have more egative volatility asymmetry. O the other had, the risk premium explaatio for idividual firm egative volatility asymmetry is ot supported. Furthermore, either the book-to-market or the operatig leverage explaatios for volatility asymmetry are supported. VI. Market-Level Versus Firm-Level Asymmetry Our results so far suggest that fiacial leverage explais most firm-level volatility asymmetry. I this sectio, we examie the impact of fiacial leverage o market-level volatility asymmetry. We fid that volatility asymmetry is more proouced at the market level tha at the firm level. Iterestigly, we fid that fiacial leverage has little effect o market-level volatility asymmetry. A. Market-Level vs. Firm-Level Asymmetry We first costruct a equal-weighted market idex based o stocks i the tadard ad Poor s (&P) mid-cap 400 ad small-cap 600 idices, as well as the &P 500 idex. We first compare the raw ad ulevered volatility asymmetry of the idex to that of the idividual stocks that compose the idex. The results are reported i Table 7 ad Figure 7. Table 7, Pael A shows the volatility asymmetry results for the idex, while Pael B shows the results for the compoet stocks. Figure 7 graphically depicts the results. We fid that both the market idex ad compoet stocks show sigificat egative asymmetry, but the asymmetry for the market is much larger tha that for the idividual stocks. The V-shaped asymmetry metric for the market idex is -4.49, compared to for the compoet stocks. I other words, the market-level volatility asymmetry is 3.84 times that of the firm-level volatility asymmetry. 5

17 Our regressios show that, o average, market volatility rises by 4.64% more whe retur drops by %, as compared to a case i which retur rises by %. 4 I cotrast, firm volatility rises by oly.2% more whe retur drops by %, as compared to a case i which retur rises by %. This fidig is qualitatively similar to the oes based o implied volatility i Deis, Mayhew ad tivers (2006), ad o retur skewess o equity optios i Bakshi, Kapadia ad Mada (2003). To uderstad what drives the differece i market-level ad firm-level asymmetry, we use the uleverig approach. The daily returs of each stock are first ulevered based o a equatio similar to equatio (6). 5 The ulevered market retur ad volatility are the computed based o the ulevered returs of the compoet stocks. Eve after uleverig, idex volatility remais highly asymmetric. The volatility asymmetry is with a t-statistic of I cotrast, the volatility asymmetry metric for the compoet stocks is with a t-statistic of Oly % of the idex volatility asymmetry is removed after uleverig, while 77% of the compoet stock volatility asymmetry is removed after uleverig. Figure 7 plots this result. Ulevered market-level volatility still exhibits cosiderable egative asymmetry (Figure 7, Pael A, dotted lie). However, ulevered firm-level volatility has o sigificat egative asymmetry (Figure 7, Pael B, dotted lie). I summary, our fidigs are as follows. asymmetry for the market idex is much higher tha firm-level asymmetry. As expected, we fid that, after fiacial leverage is removed, firmlevel volatility asymmetry is reduced to a isigificat level. However, eve after fiacial leverage is removed from the compoet firms, market-level volatility asymmetry is still very substatial. This meas that, while leverage explais firm-level asymmetry, it does ot explai market-level volatility asymmetry. Our fidig is cosistet with Aydemir, Gallmeyer ad Hollifield (2006), i which the represetative aget has a Campbell-Cochrae (999) type preferece. I their model, leverage effect is explicitly modeled ad volatility is edogeously determied alog with iterest rates ad time-varyig risk premiums. O the market level, fiacial leverage is solely drive by aggregate risk ad does ot affect volatility beyod risk premium. O the firm level, leverage is iflueced by idiosycratic risk ad may have a substatial impact o firm-level volatility asymmetry. Through simulatios Aydemir et al. fid that fiacial leverage is importat for firm-level but ot market-level volatility asymmetry. Our empirical work shows that fiacial leverage ideed accouts for a vast majority of firm-level volatility asymmetry although it does ot accout for market-level volatility asymmetry. B. Relatio to Covariace Asymmetry 4 This equals β + β + = =

18 The differece i ulevered volatility asymmetry betwee the idex ad the compoet stocks is related to the empirical fidig of covariace asymmetry. To better uderstad this, we calculate a diversificatio factor which also shows the average correlatio of firm returs. A diversificatio factor ca be defied as the ratio of the equal-weight-idex retur variace to the firm-level retur variace (see for example Elto et al. (2003) ad Vuolteeaho (2002)). Formally, the diversificatio factor is writte as Diversificatio factor = var ri i=, var( ri ) i= where r i are the firms returs, var is the variace, ad is the umber of firms i the market portfolio. There is less diversificatio i the equal weighted idex as the diversificatio factor icreases. The variace of equal-weight idex returs ca be expressed as a fuctio of the elemets of the cross-sectioal covariace matrix: var ri = 2 var (2) ( ri ) + cov( r, r ) var cov. 2 i j = + i= i= i= j=, j i where average variace ad covariace are deoted by var ad cov. Defiig the average correlatio as corr cov / var, the diversificatio factor ca the be writte as: var ri i= var i i= ( r ) = + corr. From equatio (4), it ca be see that for relatively large, the diversificatio factor equals the average correlatio. A portfolio is said to display a egative asymmetric diversificatio factor if var ri r i= var i i= m ( r r < 0) m < 0 > var ri r i= var i i= m > 0 ( ). r r > 0 This meas that the diversificatio factor is higher whe stocks are dow tha whe stocks are up. We compute the diversificatio factors based o ulevered returs. This allows us to examie the differece i asymmetry betwee ulevered volatility of idex ad ulevered volatility of firms. m (3) (4) (5) 5 Book value of debt is used here due to the absece of daily corporate bod idex prices. 7

19 We fid that the diversificatio factor is 0.38 whe lagged returs are egative, ad 0.29 whe lagged returs are positive, i.e. diversificatio is worse i bad times tha i good times. This meas that the market portfolio displays a egative asymmetric diversificatio factor. This fidig is cosistet with our fidig that market-level ulevered asymmetry is more egative tha firm-level ulevered asymmetry. This also meas that average correlatio is higher whe stocks are dow tha whe stocks are up. Because of the icrease i correlatio whe returs are lower, there is market-level volatility asymmetry, eve though there is o firm-level volatility asymmetry. Ideed, we have see earlier that, after egative returs, average firm-level ulevered volatility does ot icrease (see Figure 7, Pael B, dotted lie). However, the average ulevered correlatio betwee firms icreases, which implies a icrease i ulevered covariace (i.e. covariace asymmetry). This causes the ulevered volatility of the market-level portfolio to icrease, eve though the average firm-level ulevered volatility does ot. Notice that the covariace asymmetry we fid is based upo ulevered returs rather tha levered returs. I other words, we fid that, o a ulevered basis, var rr i m < 0 > var rr i m > 0, i = i = < = eve though var ( rr i m 0) ( rr i m > ) i i = var 0. The reductio i diversificatio whe retur is egative causes the ulevered market idex to become more volatile durig bad times eve though a average ulevered firm s volatility remais the same. Covariace asymmetry is therefore related to our fidig that idex-level volatility asymmetry still exists after uleverig, while firm-level volatility asymmetry is elimiated after uleverig. VII. Coclusio Usig a uleverig approach, this paper examies the source of volatility asymmetry i thousads of U firms. Our uleverig approach makes it possible to examie the impact of leverage o volatility asymmetry usig a firm-level, as supposed to a portfolio-level, aalysis. Usig this approach, we documet the key role of fiacial leverage i affectig volatility asymmetry at the firm level. We have doe extesive robustess checks to adjust for the effect of market leverage, icludig the use of the Merto-KMV model ad the market price of debt with similar ratigs. While we ackowledge that each method aloe may ot fully capture market leverage, cosistecy across differet sets of results oetheless suggests that fiacial leverage ideed accouts for most of the volatility asymmetry at the firm level. While fiacial leverage explais most of the firm-level asymmetry i a large sample of U firms, it explais oly a small portio of the idex-level asymmetry. This is cosistet with the geeral equilibrium model of Aydemir, Gallmeyer ad Hollifield (2006), which shows that the determiats of firm-level ad market-level volatility asymmetry are potetially differet. We hope that our result will 8