Weekday variation in the leverage effect: A puzzle

Weekday variation in the leverage effect: A puzzle

Finance Research Letters 17 (2016) 193–196 Contents lists available at ScienceDirect Finance Research Letters journal homepage: www.elsevier.com/loc...

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Finance Research Letters 17 (2016) 193–196

Contents lists available at ScienceDirect

Finance Research Letters journal homepage: www.elsevier.com/locate/frl

Weekday variation in the leverage effect: A puzzle Geoffrey Peter Smith∗ Department of Finance, WP Carey School of Business, Arizona State University, Tempe, AZ 85287, USA

a r t i c l e

i n f o

Article history: Received 25 January 2016 Accepted 5 March 2016 Available online 12 April 2016 JEL Classification: G12 G17 G19

a b s t r a c t There is large variation in the leverage effect on each weekday. In the past 15 years, the average difference between the impact of negative and positive stock return innovations on future volatility in the S&P 500 Index is 45% on Monday, 14% on Tuesday, 60% on Wednesday, 6% on Thursday, and 28% on Friday. This variation is not predicted by any prevailing hypothesis on why there is a leverage effect. © 2016 Elsevier Inc. All rights reserved.

Keywords: Leverage effect Volatility feedback EGARCH

1. Introduction Two prevailing hypotheses explain negative correlation between stock returns and stock return volatility. First, the leverage effect hypothesis of Black (1976) argues that stock price declines increase the debt-to-equity ratio and thus trigger automatic increases in the ensuing stock return volatility. Due to the popularity of Black’s hypothesis, the term “leverage effect” is now synonymous with the amount of negative correlation itself. Second, the volatility feedback hypothesis of Pindyck (1984), French et al. (1987), and Campbell and Hentschel (1992) argues that an increase in current volatility raises expectations of future volatility and the expected return on the stock. This amplifies the effect of bad news and dampens the effect of good news on the stock price. Whether either hypothesis is correct is an open question.1 This study introduces a new wrinkle by comparing the leverage effect on different weekdays in the S&P 500 Index. I study the S&P 500 Index because it is widely-considered to represent the overall stock market. Option contracts on the S&P 500 Index are also some of the most actively-traded in the world and their valuation depends critically on accurate estimates of future volatility. Well-known studies by French et al. (1987) and Schwert (1989) also look at volatility in the S&P 500 Index. This study should therefore have broad appeal to anyone interested in developing a better understanding of the behavior of stock return volatility in the overall stock market. My sample period runs from January 1, 20 0 0 to December 31, 2014 for topicality and also to balance the benefits of a large sample size against the potential for structural breaks.



Tel.: (480) 965-8623; Fax: +1 480 965 8539. E-mail address: [email protected] 1 See, for example, Christie (1982), Cheung and Ng (1992), Duffee (1995), Koutmos and Saidi (1995), Bekaert and Wu (20 0 0), Wu (20 01), Bollerslev et al. (2006), Daouk and Ng (2011), Hasanhodzic and Lo (2013), and Smith (2015). http://dx.doi.org/10.1016/j.frl.2016.03.001 1544-6123/© 2016 Elsevier Inc. All rights reserved.

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G.P. Smith / Finance Research Letters 17 (2016) 193–196 Table 1 Descriptive statistics for daily S&P 500 Index returns. Reported are descriptive statistics for daily log returns on the S&P 500 Index from January 1, 20 0 0 to December 31, 2014 in basis points. Day

Mean

Median

Std. dev.

Min.

Max.

N

Monday Tuesday Wednesday Thursday Friday

−1.90 5.72 −1.09 4.16 −2.66

1.38 4.87 4.60 12.09 7.38

140.91 132.22 125.71 129.77 112.47

−935.36 −591.08 −946.95 −792.24 −600.45

1095.72 1024.57 557.33 669.23 613.28

710 772 775 759 757

For the purpose of this study, I define the term “leverage effect” to mean the average percent difference between the impact of negative and positive stock return innovations on future volatility. Stock returns are said to exhibit a leverage effect when this percent is positive and significant. To estimate the leverage effect, I apply the EGARCH (exponential generalized autoregressive conditional heteroskedasticity) model of Nelson (1991) to the S&P 500 Index returns. Unlike the GARCH (generalized autoregressive conditional heteroskedasticity) model of Engle (1982) and Bollerslev (1986), the EGARCH model allows for an asymmetric effect between negative and positive stock return innovations on future volatility. Closely-related studies by Cheung and Ng (1992), Koutmos and Saidi (1995), and Smith (2015) also use the EGARCH model to calculate the leverage effect. My results show there is large variation in the leverage effect on each weekday. The average difference between the impact of negative and positive stock return innovations on future volatility is 45% on Monday, 14% on Tuesday, 60% on Wednesday, 6% on Thursday, and 28% on Friday. This variation is not predicted by either the leverage effect hypothesis or the volatility feedback hypothesis. In fact, it is hard to imagine a scenario where such a large amount of weekday variation is plausibly linked to changes in the debt-to-equity ratio or in expected volatility. Nevertheless, this study contributes toward the important goal of developing a better understanding of the behavior of stock return volatility over time. The remainder of this study proceeds as follows. I describe the S&P 500 Index return data in Section 2. I describe the EGARCH model and empirical method in Section 3. I discuss the results in Section 4. Section 5 concludes. 2. S&P 500 data The data I study are the daily log returns on the S&P 500 Index from January 1, 20 0 0 to December 31, 2014 from CRSP (Center for Research in Security Prices). I study the returns on S&P 500 Index for a number of reasons. First, the returns on the S&P 500 Index are more likely to exhibit a leverage effect than individual stocks.2 Second, the returns on the S&P 500 Index are less likely to be influenced by illiquidity or nonsynchronous trading because the stocks in the index are some of the most liquid and most actively-traded in the world. Third, option contracts on the S&P 500 Index are also some of the most actively-traded in the world and their valuation depends critically on accurate estimates of future volatility. Finally, the S&P 500 Index is widely-considered to represent the overall stock market and well-known studies by French et al. (1987) and Schwert (1989) also look at volatility in the S&P 500 Index. I study the current 15-year sample period in order to balance the benefits of a large sample size against the potential for structural breaks. The potential for structural breaks is a real concern based on the evidence in Robins and Smith (2016) who show that mean stock index returns on each weekday are not constant over long sample periods. I split the S&P 500 Index returns into five separate time series in order to calculate a leverage effect for each day Monday to Friday. Table 1 reports descriptive statistics for the S&P 500 Index returns by weekday. Mean daily returns are negative on Monday, Wednesday, and Friday and positive on Tuesday and Thursday. This is consistent with French (1980) and Gibbons and Hess (1981) who are the first to document large variation in average stock returns on each weekday. Median daily returns are positive each day Monday to Friday. The number of observations ranges from 710 on Monday to 775 on Wednesday due to market closings and holiday observances. 3. EGARCH model and the leverage effect 3.1. The model I estimate the EGARCH model of Nelson (1991) on the above-described S&P 500 Index returns. Specifically, let rt be the percent log return on the S&P 500 Index on day t for each day Monday to Friday. Then define the EGARCH conditional mean equation as:

rt = μ + φ rt−1 + at , 2

See, for example, Kim and Kon (1994), Tauchen et al. (1996), and Andersen et al. (2001).

(1)

G.P. Smith / Finance Research Letters 17 (2016) 193–196

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where μ is an intercept term and at is a serially uncorrelated, but dependent stock return innovation. The rt−1 is the percent log return on the same weekday one observation prior. The EGARCH conditional variance equation for the stock return innovation, at , is:

at = σt t ,

2 ln(σt2 ) = ω + αt−1 + γ (|t−1 | − E |t−1 | ) + β ln(σt−1 ),

(2)

where I follow the recommendation in Nelson (1991) and let the standardized stock return innovation,  t , follow the GED (generalized error distribution). The EGARCH conditional variance evolves in a nonlinear manner depending on the sign of the stock return innovation, at , because:



2α t−1

σ =σ 2 t

exp(α∗ )

 

 if at−1 ≥ 0,  |at−1 |

exp (γ + α ) σat−1 t−1

exp (γ − α ) σt−1

if at−1 < 0.

(3)

Eq. (3) shines a spotlight on the asymmetric effect between negative and positive stock return innovations on future volatility under the EGARCH model. By taking the difference in volatility, σt2 , between when at−1 < 0 and when at−1 ≥ 0, it is easy to see that for a standardized stock return innovation with magnitude two:

σt2 (t−1 = −2 ) = exp(−4α ) − 1 = λ, σt2 (t−1 = +2 )

(4)

where λ × 100 is the average percent difference between the impact of a size two standard deviation negative stock return innovation and a size two standard deviation positive stock return innovation on future volatility. It is therefore appropriate to refer to λ as the “leverage effect” with the expectation that α is negative in the actual S&P 500 Index return data. I estimate the EGARCH model on each of the five time series of S&P 500 Index returns in order to obtain estimates for μ, φ , ω, α , β , and γ for each day Monday to Friday.3 Tests of the significance of the EGARCH parameters are based on robust standard errors calculated via the method of White (1982). 3.2. Regression diagnostics Two concerns arise when building the EGARCH model. The first is serial correlation in the residuals of the EGARCH mean equation. I test for serial correlation by applying the weighted Ljung–Box test of Fisher and Gallagher (2012) to the standardized residuals of the EGARCH mean equation. The Lag[5] test statistics do not reject the null hypothesis of no serial correlation in each of the five time series. The second is the validity of the assumption that the EGARCH parameters are constant for the entire 15-year sample period. I test whether the EGARCH parameters are constant for the entire 15-year sample period by applying the joint test for parameter stability of Nyblom (1989). The Nyblom (1989) test does not reject the null hypothesis of joint constancy of the parameters in each of the five time series. 4. Results Table 2 reports the EGARCH estimates and the leverage effect for each weekday. The main parameter of interest in Table 2 is α . The EGARCH conditional variance evolves in a nonlinear manner if α = 0. There is a leverage effect if α is negative and significant. I find that α is negative and significant on Monday, Wednesday, and Friday and not significant on Tuesday and Thursday. This variation in α is not predicted by either the leverage effect hypothesis or the volatility feedback hypothesis. Other significant EGARCH parameters are β and γ . The β captures the well-known behavior of volatility clustering in financial time series and the γ captures the effect of the size of the standardized stock return innovation on future volatility. The significant β is noteworthy because it means volatility is serially correlated at the weekly level in the S&P 500 Index. The significant γ shows that larger stock return innovations have a larger impact on future volatility than smaller stock return innovations. The significant β and γ give credence to my application of the EGARCH model to stock returns on each weekday. The intercept term in the mean equation, μ, is not significant except on Thursday. Serial correlation in the returns is weak as expected. The AR(1) parameter, φ , is not significant except on Thursday. In order to compare the leverage effect on different weekdays, I insert each of the α into Eq. (4) to produce a λ for each day Monday to Friday. I find λ is 45% on Monday, 14% on Tuesday, 60% on Wednesday, 6% on Thursday, and 28% on Friday.4 This variation in λ is not predicted by any prevailing hypothesis on why there is a leverage effect. Why there is such a large amount of weekday variation in the leverage effect is a puzzle.

3 I estimate the EGARCH model with computer software provided by the R Project for Statistical Computing in conjunction with the rugarch package of Ghalanos (2015). 4 I also find large variation in the leverage effect on each weekday in other sample periods. Wednesday is most commonly the weekday with the largest leverage effect in these other sample periods.

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G.P. Smith / Finance Research Letters 17 (2016) 193–196 Table 2 EGARCH estimates and leverage effect by weekday. Reported are parameters estimated via the EGARCH model: rt = μ + φ rt−1 + at , at = σt t , ln(σt2 ) = ω + 2 αt−1 + γ (|t−1 | − E |t−1 | ) + β ln(σt−1 ), where rt is the log return on the S&P 500 Index on day t from January 1, 20 0 0 to December 31, 2014 in percent. The leverage effect for each weekday is calculated as λ = exp(−4α ) − 1. Standard errors reported below the parameter estimates in parentheses are calculated using the method of White (1982). ∗∗ and ∗ indicate significance at the 0.01 and 0.05 levels, respectively. Parameter

Monday

Tuesday

Wednesday

Thursday

Friday

μ

0.0230 (0.0148) 0.0171 (0.0222) −0.0 0 04 (0.0116) −0.0930∗ (0.0373) 0.9604∗∗ (0.0202) 0.2837∗∗ (0.1012)

0.0505 (0.0448) −0.0376 (0.0399) 0.0066 (0.0135) −0.0337 (0.0330) 0.9461∗∗ (0.0320) 0.3169∗∗ (0.0810)

0.0513 (0.0468) −0.0043 (0.0373) 0.0011 (0.0118) −0.1181∗∗ (0.0367) 0.9460∗∗ (0.0212) 0.2584∗∗ (0.0570)

0.0876∗∗ (0.0297) 0.0336∗ (0.0137) 0.0113 (0.0179) −0.0153 (0.0454) 0.9433∗∗ (0.0358) 0.3120∗∗ (0.1179)

0.0299 (0.0429) −0.0096 (0.0347) −0.0058 (0.0072) −0.0618∗ (0.0304) 0.9719∗∗ (0.0153) 0.1934∗∗ (0.0477)

45.06%

14.43%

60.38%

φ ω α β γ λ

6.31%

28.04%

5. Conclusion I find puzzling new evidence of large variation in the leverage effect on each weekday. In the past 15 years, the average difference between the impact of negative and positive stock return innovations on future volatility in the S&P 500 Index is 45% on Monday, 14% on Tuesday, 60% on Wednesday, 6% on Thursday, and 28% on Friday. This variation is not predicted by the leverage effect hypothesis of Black (1976) or the volatility feedback hypothesis of Pindyck (1984), French et al. (1987), and Campbell and Hentschel (1992). Other leading hypotheses on why there is a leverage effect, such as Schwert (1989), who attributes the leverage effect to asymmetry in the volatility of macroeconomic news, and Avramov et al. (2006), who attribute the leverage effect to differences in trading activity between uninformed and informed investors, also do not suppose weekday variation in the leverage effect. Nevertheless, this study contributes toward the development of a better understanding of the behavior of stock return volatility over time, which is vital to asset pricing and portfolio management, and also to stock option valuation, which depends critically on accurate estimates of future volatility. References Andersen, T.G., Bollerslev, T., Diebold, F.X., Ebens, H., 2001. The distribution of realized stock return volatility. J. Financ. Econ. 61, 43–76. Avramov, D., Chordia, T., Goyal, A., 2006. The impact of trades on daily volatility. Rev. Financ. Stud. 19 (4), 1241–1277. Bekaert, G., Wu, G., 20 0 0. Asymmetric volatility and risk in equity markets. Rev. Financ. Stud. 13 (1), 1–42. Black, F., 1976. Studies of stock price volatility changes. In: Proceedings of the 1976 Meetings of the American Statistical Association, Business and Economic Statistics Section. Bollerslev, T., 1986. Generalized autoregressive conditional heteroskedasticity. J. Econom. 31, 307–327. Bollerslev, T., Litvinova, J., Tauchen, G., 2006. Leverage and volatility feedback effects in high-frequency data. J. Financ. Econom. 4 (3), 353–384. Campbell, J.Y., Hentschel, L., 1992. No news is good news: an asymmetric model of changing volatility in stock returns. J. Financ. Econ. 31, 281–318. Cheung, Y.-W., Ng, L.K., 1992. Stock price dynamics and firm size: an empirical investigation. J. Financ. 47 (5), 1985–1997. Christie, A.A., 1982. The stochastic behavior of common stock variances: value, leverage and interest rate effects. J. Financ. Econ. 10, 407–432. Daouk, H., Ng, D., 2011. Is unlevered firm volatility asymmetric? J. Empir. Financ. 18, 634–651. Duffee, G.R., 1995. Stock returns and volatility: a firm-level analysis. J. Financ. Econ. 37, 399–420. Engle, R.F., 1982. Autoregressive conditional heteroscedasticity with estimates of the variance of United Kingdom inflation. Econometrica 50 (4), 987–1007. Fisher, T.J., Gallagher, C.M., 2012. New weighted portmanteau statistics for time series goodness of fit testing. J. Am. Stat. Assoc. 107 (498), 777–787. French, K.R., 1980. Stock returns and the weekend effect. J. Financ. Econ. 8, 55–69. French, K.R., Schwert, G.W., Stambaugh, R.F., 1987. Expected stock returns and volatility. J. Financ. Econ. 19, 3–29. Ghalanos, A., 2015. rugarch: Univariate GARCH models. https://cran.r-project.org/web/packages/rugarch/citation.html. Gibbons, M.R., Hess, P., 1981. Day of the week effects and asset returns. J. Bus. 54 (4), 579–596. Hasanhodzic, J., Lo, A.W., 2013. Black’s leverage effect is not due to leverage. Unpublished working paper. Kim, D., Kon, S.J., 1994. Alternative models for the conditional heteroscedasticity of stock returns. J. Bus. 67 (4), 563–598. Koutmos, G., Saidi, R., 1995. The leverage effect in individual stocks and the debt to equity ratio. J. Bus. Financ. Account. 22 (7), 1063–1075. Nelson, D.B., 1991. Conditional heteroskedasticity in asset returns: a new approach. Econometrica 59 (2), 347–370. Nyblom, J., 1989. Testing for the constancy of parameters over time. J. Am. Stat. Assoc. 84 (405), 223–230. Pindyck, R.S., 1984. Risk, inflation, and the stock market. Am. Econ. Rev. 74 (3), 335–351. Robins, R.P., Smith, G.P., 2016. No more weekend effect. Crit. Financ. Rev. 6. Schwert, G.W., 1989. Why does stock market volatility change over time? J. Financ. 44 (5), 1115–1153. Smith, G.P., 2015. New evidence on sources of leverage effects in individual stocks. Financ. Rev. 50, 331–340. Tauchen, G., Zhang, H., Liu, M., 1996. Volume, volatility, and leverage: a dynamic analysis. J. Econom. 74, 177–208. White, H., 1982. Maximum likelihood estimation of misspecified models. Econometrica 50 (1), 1–25. Wu, G., 2001. The determinants of asymmetric volatility. Rev. Financ. Stud. 14 (3), 837–859.