On Black’s Leverage Effect in Firms with No Leverage

Abstract

One of the most enduring empirical regularities in equity markets is the inverse relationship between stock prices and volatility. Also known as the leverage effect, this relationship was first documented by Black, who attributed it to the effects of financial or operating leverage. This article documents that firms that had no debt (and thus no financial leverage) from January 1973 to December 2017 exhibit Black’s leverage effect. Moreover, the authors find that the leverage effect of firms in this sample is not driven by operating leverage. On the contrary, in this sample the leverage effect is stronger for firms with low operating leverage as compared to those with high operating leverage. Interestingly, the firms with no debt from the lowest quintile of operating leverage exhibit a leverage effect that is on par with or stronger than that of debt-financed firms.

TOPICS: Fundamental equity analysis, equity portfolio management, accounting and ratio analysis

Key Findings

• The inverse relationship between stock prices and volatility is often attributed to the effects of leverage and is known as the leverage effect.

• We find that firms with no debt (and thus no financial leverage) exhibit the leverage effect.

• Interestingly, among the firms with no debt, those with the lowest levels of operating leverage exhibit the strongest leverage effect.

One of the most enduring empirical regularities of equity markets is the fact that stock-return volatility increases after price declines, with larger declines inducing greater volatility spikes. In a seminal paper, Black (1976) provided a compelling explanation for this phenomenon in terms of the firm’s financial leverage: A negative return implies a drop in the value of the firm’s equity, thus increasing its leverage, which, in turn, leads to higher equity-return volatility. Black further argued that this inverse return/volatility relationship may arise even if the firm has almost no debt because of the presence of so-called operating leverage: Because of fixed costs, when income falls, it does so by less than expenses; hence, a firm’s equity value falls and its volatility increases. Black’s explanations have been so tightly coupled with empirical regularity that the inverse relation between equity returns and volatility is now commonly known as the leverage effect.

This effect, and the leverage-based explanation, have been empirically confirmed by a number of studies since that by Black (1976) (e.g., Christie 1982, Cheung and Ng 1992, and Duffee 1995). These studies focused on typical firms (all firms in their respective time frames). They used linear regressions of returns on subsequent changes in volatility for individual stocks and stock portfolios to argue that these relationships become stronger as the firms’ debt-to-equity ratios increase. Daouk and Ng (2011) also used a large sample of US firms. By carefully unlevering the levered firms, they showed that almost all of the inverse relationship between returns and volatility at the firm level can be explained by financial leverage and, to a smaller extent, operating leverage. In a theoretical model, Aydemir, Gallmeyer, and Hollifield (2006) showed that, in an economy that is consistent with the assumptions of Black (1976) and Christie (1982) (their benchmark model), the leverage effect hypothesis holds at the firm level, but not at the market level.

However, a growing literature has called into question the validity of the leverage-based explanation. For example, Figlewski and Wang (2000) documented several anomalies associated with it, including the fact that the inverse relation becomes much weaker when positive returns reduce leverage. Hens and Steude (2009) found evidence of the leverage effect in data generated by students who were asked to trade artificial securities with each other using an electronic trading system with no leverage. LeBaron (2014) developed a relatively simple, computational agent-based model that was successful in replicating several long-range empirical features for aggregate stock returns, including the leverage effect. Finally, Kogan (2004) developed a production economy model that implies that market-to-book ratio, or q, rather than financial leverage, might lead to the leverage effect.

The debate is far from settled, and this article adds to this literature by testing for Black’s leverage explanation using a novel estimation approach: Rather than focusing on typical firms, we consider the extreme case of all-equity-financed (AE) firms (i.e., a sample of firms that have no debt, and thus no financial leverage, in their capital structures). Using the linear regression specifications considered by prior studies, we check for the presence of an inverse relationship between returns and the subsequent volatility changes in this sample.

We find that AE firms from January 1973 to December 2017 exhibit Black’s leverage effect. Moreover, we find that the leverage effect of AE firms is not driven by operating leverage. On the contrary, among the AE firms, the leverage effect is stronger for those with low operating leverage as compared to those with high operating leverage. Interestingly, the AE firms from the lowest quintile of operating leverage exhibit a leverage effect that is on par with or stronger than that of debt-financed (DF) firms.

We next provide a review of the literature in which the stock-return/volatility relationship is documented, focusing on a few key regression-based studies that we replicate using the sample of AE firms. Subsequently, we describe our data and empirical strategy and present our results. Finally, we conclude with a discussion of some possible interpretations of our findings.

LITERATURE REVIEW

Black (1976) is widely credited as the originator of the leverage-effect literature. In this pioneering paper, Black used daily data from 1964 to 1975 of a sample of 30 stocks (mostly Dow Jones Industrials) to study the relationship between volatility changes and returns in individual stocks and the portfolio of those stocks. For each stock, Black constructed 21-day summed returns and estimated volatility over these intervals with the square root of the sum of squared returns. The portfolio-level equivalents of these estimates, which he called the summed market return and the market volatility estimate, are obtained by averaging the summed returns and the volatility estimates, respectively, across the sample of stocks. He then defined the volatility change as the difference between the volatility estimate of the current and the previous period, divided by the volatility estimate of the previous period, and regressed the volatility change at time t + 1 on the summed return at time t. His results suggested a strong inverse relationship between the two: A 1% summed-return decline implies a more than 1% volatility increase.

Black (1976) proposed two possible explanations for this relationship. The first explanation, which he termed the direct causation effect, refers to the causal relation between stock returns and volatility changes. A drop in the value of the firm’s equity will cause a negative return on its stock and will increase the leverage of the stock (i.e., its debt/equity ratio); this increase in the debt/equity ratio will lead to a rise in the volatility of the stock. A similar effect may arise even if the firm has almost no debt because of the presence of so-called operating leverage (fixed costs that cannot be eliminated, at least in the short run; hence, when expected revenues fall, profit margins decline as well). The second explanation, which Black (1976) called the reverse causation effect, refers to the causal relationship between volatility changes and stock returns. Changes in tastes and technology lead to an increase in uncertainty about the payoffs from investments. Because of the increase in expected future volatility, stock prices must fall so that the expected return from the stock rises to induce investors to continue to hold the stock.

Using a sample of 379 stocks from 1962 to 1978, Christie (1982) estimated the following linear relationship between changes in volatility from one quarter to the next and the return over the first quarter for each stock:

[Formula omitted: see PDF.]

where σ_t−1 and r_t−1 are the volatility estimate and return of the stock over quarter t − 1. He found the cross-sectional mean elasticity to be −0.23, consistent with the leverage effect, and then derived testable implications of various volatility models and efficient estimation procedures to investigate the implications of risky debt and interest-rate changes on this relationship. He found a significant positive association between equity volatility and financial leverage, but the strength of this association declines with increasing leverage. Christie found that, contrary to the predictions stemming from the contingent claims literature, the riskless interest rate and financial leverage jointly have a substantial positive impact on the volatility of equity. Finally, he tested the elasticity hypothesis, which says that the observed negative elasticity of equity volatility with respect to the value of equity is, in large measure, attributable to financial leverage. For this purpose he used the constant elasticity of variance model for equity prices, according to which σ_S = λS^θ, and estimated the linear regression:

[Formula omitted: see PDF.]

where σ_S,t is the volatility estimated over quarter t, and S_t is the stock price at the beginning of that quarter. He performed two separate tests of the hypothesis that θ_S is a function of financial leverage—one based on leverage quartiles and the other on subperiods—and found evidence in support of this hypothesis from both.

Cheung and Ng (1992) examined the inverse relation between future stock volatility and current stock prices using daily returns of 252 NYSE-AMEX stocks with no missing returns from 1962 to 1989, under the assumption of an exponential generalized autoregressive conditional heteroskedasticity (GARCH) model for stock prices (to control for heteroskedasticity and possible serial correlation in their returns). In this model, the conditional variance equation has the logarithm of the lagged stock price on the right-hand side; hence, the corresponding coefficient θ is a measure of the leverage effect. Applying the Spearman rank correlation test, the authors found a strong positive correlation between θ and firm size and explained this pattern by arguing that the smaller the firm, the higher the debt/equity ratio. Their subsample analysis showed that the strength of this relationship changes over time. In particular, conditional variances of stock returns on average have become less sensitive to changes in stock prices; the authors suggested this may be due to an increase in the firms’ liquidity over the sample period.

Duffee (1995) provided a new interpretation for the negative relationship between current stock returns and changes in future stock return volatility at the firm level by arguing that it is mainly due to a positive contemporaneous relationship between returns and volatility. In addition to the usual test of leverage effect based on lagged returns, Duffee estimated the following two contemporaneous regressions:

[Formula omitted: see PDF.]

[Formula omitted: see PDF.]

and observed that the usual lagged stock-return coefficient is simply the difference, λ₀ ≡ λ₂ − λ₁. Using data for 2,500 firms traded on the AMEX or NYSE from 1977 to 1991 (not necessarily continuously for the entire sample),¹ Duffee found that, for a typical firm, λ₁ is strongly positive, λ₂ is positive at the daily frequency and negative at the monthly frequency, and that, regardless of the sign of λ₂, λ₁ > λ₂, implying that λ₀ < 0. Duffee then tested the theory behind the leverage effect, according to which highly leveraged firms should have a stronger negative relation between stock returns and volatility than less highly leveraged firms. Prior to his study, researchers documented that the inverse relation between period t stock returns and changes in the stock return volatility from period t to period t + 1 is stronger for firms with larger debt/equity ratios (e.g., Christie 1982 and Cheung and Ng 1992) and that this relation is stronger for smaller firms (Cheung and Ng 1992). Using the Spearman rank correlations between the individual-firm regression coefficients (λ₀, λ₁, and λ₂) and debt/equity ratios and market capitalizations, Duffee obtained the following three findings. First, the negative relation between λ₀ and the debt/equity ratio found by Christie (1982) and Cheung and Ng (1992) was confirmed only with monthly data for the subset of continuously traded firms. When a larger sample of firms without survivorship bias is used, the correlation between λ₀ and the debt/equity ratio turns positive. Second, highly leveraged firms exhibit stronger negative relations between stock returns and volatility than less highly leveraged firms (the rank correlations between λ₁ and the debt/equity ratio, and λ₂ and the debt/equity ratio, are both negative). And third, because the leverage effect theory has no implications for the strength of the contemporaneous relation between stock returns and volatility, Duffee argued that something other than the leverage effect causes at least part of the correlation between firm debt/equity ratios and the regression coefficients and determines the negative correlation between the firm debt/equity ratio and λ₁. Furthermore, in accordance with Cheung and Ng (1992), he found that λ₀ is positively correlated with size, at both monthly and daily frequencies, for all firms and the subsample of continuously traded firms.

Other explanations for the inverse relation between stock volatility and lagged returns have been proposed, each developing into its own strand of literature. The most prominent of these strands is the time-varying risk premium literature, according to which an increase in return volatility implies an increase in the future required expected return of the stock, and hence a decline in the current stock price. In addition, owing to the persistent nature of volatility (large realizations of either good or bad news increase both current and future volatility), a feedback loop is created: The increased current volatility raises expected future volatility and thus expected future returns, causing current stock prices to fall. The time-varying risk premiums explanation, also known as the volatility feedback effect, has been considered by Pindyck (1984), French, Schwert, and Stambaugh (1987), and Campbell and Hentschel (1992), typically using aggregate market returns within a GARCH framework.

Yet another explanation for the inverse relation between volatility and lagged returns, first proposed by Schwert (1989), involves the apparent asymmetry in the volatility of the macroeconomic variables. Empirical evidence suggests that real variables are more volatile in recessions than in expansions; hence, if a recession is expected but not yet realized (i.e., gross domestic product growth is forecasted to be lower in the future), stock prices will fall immediately, and this will be followed by higher stock-return volatility when the recession is realized.

Disentangling these effects has proved to be a challenging task. For example, using the data for the portfolios of Nikkei 225 stocks and a conditional capital asset pricing model with a GARCH-in-mean parametrization, Bekaert and Wu (2000) rejected the leverage model and found support for the volatility feedback explanation. In contrast, examining the relationship between volatility and past and future returns using the S&P 500 futures high-frequency data, Bollerslev, Litvinova, and Tauchen (2006) found that the correlations between absolute high-frequency returns and current and past high-frequency returns are significantly negative for several days, lending support to the leverage explanation; the reverse cross correlations are negligible, which is inconsistent with the volatility feedback story.

DATA

Our sample consists of daily stock returns from the University of Chicago’s Center for Research in Security Prices (CRSP), quarterly fundamental data from the CRSP/COMPUSTAT Merged Database (Fundamentals Quarterly), and annual accounting data from COMPUSTAT Fundamentals Annual. We select only those stocks with zero total debt for all quarters from December 1972 to December 2017, where total debt is defined as the sum of total long-term debt, debt in current liabilities (short-term debt), and total preferred stock.² Furthermore, in keeping with Black (1976), Christie (1982), and Duffee (1995), we use a 21-day interval to estimate volatilities and returns, and we impose a minimum of 40 daily observations for each regression. These filters yield 275 firms, which comprise our AE sample.

We also construct a complementary sample of firms with positive levels of total debt in their capital structure in every quarter from December 1972 to December 2017. This yields a considerably larger sample of DF firms. From this universe, we select a smaller subset of 275 firms to match the size distribution and equity duration of the AE sample.³ The rationale for matching the size distributions of the two samples is to control for size effects—we have learned from the prior literature that they may matter for the results. For example, Cheung and Ng (1992) showed that the leverage effect is much stronger for small stocks. Similarly, Aydemir, Gallmeyer, and Hollifield (2006) showed that leverage contributes more to the dynamics of stock return volatility for a small firm. The rationale for matching the samples by duration is Chen’s (2011) finding that the expected life duration of a stock affects both its returns and the subsequent change in volatility, and we do not want this effect to be stronger in one sample than in the other. Finally, for each of the two samples (AE and DF), we obtain daily returns from the CRSP Daily Database.

EMPIRICAL STRATEGY

We test for the leverage effect using linear regression models in which the dependent and independent variables of the regression are those proposed by Black (1976), Christie (1982), and Duffee (1995); use of all three specifications serves as a robustness check on our results. To obtain the dependent and independent variables used in the regression equations, we first split the daily stock-returns data for each firm or the portfolios of firms into non-overlapping periods of a certain length and compute the volatility and total returns over those periods. Then, for each firm or portfolio, we regress the change in volatility between the current period and the previous period on the total return over the previous period; we also regress the change in volatility between the current period and two periods ago on the return in the previous period. The latter regression has the advantage of no data being common to both sides of the regression equation at the same time, which, as Black (1976, p. 181) observed, reduces the chance that the coefficient estimates are biased by errors in the volatility estimates. More formally, we estimate the following eight specifications—each a variation of the same linear regression model—where period t − 1 returns are related to changes in volatility between periods t and t − 1 in the first four specifications and to the changes in volatility between periods t and t − 2 in the four remaining specifications:

1. BlackLag1 is given by [Formula omitted: see PDF.], where σ_t is the square root of the sum of the squared daily simple returns over period t multiplied by the ratio of 252 to the number of days in period t, and r_t is the sum of daily simple returns over that period.

2. LogBlackLag1 is given by [Formula omitted: see PDF.], where σ_t and r_t are the same as in BlackLag1.

3. ChristieLag1 is given by [Formula omitted: see PDF.], where σ_t is the square root of the sum of squared daily simple returns over the period, and r_t is the sum of daily log-returns over that period.

4. DuffeeLag1 is given by σ_t – σ_t–1 = α + λr_t−1 + ε_t, where σ_t is the square root of the sum of squared daily log-returns multiplied by the ratio of 252 to the number of days in period t, and r_t is the sum of daily log-returns over that period.

5. BlackLag2 is given by [Formula omitted: see PDF.], where σ_t and r_t are the same as in BlackLag1.

6. LogBlackLag2 is given by [Formula omitted: see PDF.], where σ_t and r_t are the same as in BlackLag1.

7. ChristieLag2 is given by [Formula omitted: see PDF.], where σ_t and r_t are the same as in ChristieLag1.

8. DuffeeLag2 is given by σ_t − σ_t−2 = α + λr_t−1 + ε_t, where σ_t and r_t are the same as in DuffeeLag1.

RESULTS

We start by comparing the main characteristics of firms in the two samples to ensure there are no major discrepancies that could be driving the results. We next present results of the leverage effect regressions for the two samples. Finally, we study the strength of the leverage effect for different quintiles of operating leverage in the AE sample.

Summary Statistics

Exhibit 1 presents summary statistics of the AE and DF samples of firms. This exhibit shows that the distributions of the firms’ average market capitalization and equity duration are virtually indistinguishable for the two samples. For example, the mean, median, and standard deviation of the average lifetime market capitalization across firms, measured in billions of dollars, are 0.49, 0.15, and 0.92, respectively, in the AE sample. The corresponding values in the DF sample are 0.49, 0.14, and 0.93. The maximum value of average lifetime market capitalization in the DF sample—5.85—is somewhat smaller but still comparable to its AE counterpart of 6.75. Similarly, the mean, median, maximum, and standard deviation of equity duration across firms, measured in days, are 1,966, 1,397, 11,352, and 1,524 in the AE sample, respectively. The corresponding values in the DF sample are 1,965, 1,397, 11,352, and 1,522. Although we did not match firms based on their start date, the average difference in start dates between the two samples is only three years.

[Figure omitted: see PDF.]

Financial Leverage

To test whether the leverage effect is due to firms’ financial leverage (i.e., the presence of debt in its capital structure), we estimate each of the preceding regression models on the AE and DF samples from January 2, 1973, to December 31, 2017. As mentioned before, in keeping with Black (1976), Christie (1982), and Duffee (1995), we use a 21-day interval to estimate volatilities and returns, and we impose a minimum of 40 daily observations for each regression.

In Exhibit 2 we report the distributional summary statistics of the individual-firm regression estimates and goodness-of-fit statistics from the AE and DF datasets. For example, according to the top-left panel of Exhibit 2, corresponding to the BlackLag1 regression specification, the average across 275 firms in the AE dataset of the firm-by-firm leverage coefficients λ is −0.49, which is comparable to the average λ of −0.65 for the 275 firms of its DF counterpart. The corresponding median λ values are −0.47 and −0.59 for the AE and DF samples, respectively. The average t-statistics are relatively low (−1.12 and −1.49 for the AE and DF datasets, respectively), a finding consistent with that of Christie (1982), who obtained an average t-statistic of −1.01. The adjusted R² statistic tends to be small, ranging across specifications from 1.4% to 5.9% on average, which is also consistent with Christie (1982), who obtained the average adjusted R² of 1%.

[Figure omitted: see PDF.]

Similar results are observed in two of the three remaining left-hand-side panels of Exhibit 2, where we regress the change in volatility between the current period and the previous period on the return in the previous period according to the various regression specifications under consideration. Namely, the average λ values are −0.57 (AE) versus −0.71 (DF) in LogBlackLag1 and −0.42 (AE) versus −0.54 (DF) in ChristieLag1. The final left-hand-side specification, DuffeeLag1, exhibits the largest λ values for both samples. Specifically, the average λ is −0.09 in the AE sample, which is twice as large as its −0.18 counterpart in the DF sample. However, such a large average λ in the AE sample seems to be driven by a very large maximum λ of 2.43 in that sample. Indeed, the median λ values are closer in magnitude: −0.13 (AE) versus −0.20 (DF).

For the right-hand-side panels of Exhibit 2, where we regress the change in volatility between the current period and two periods ago on the return in the previous period,⁴ there is a larger gap between the average λ values of the two datasets; however, the median λ values are still comparable: −0.38 (AE) versus −0.62 (DF) for BlackLag2, −0.34 (AE) versus −0.47 (DF) for LogBlackLag2, −0.36 (AE) versus −0.50 (DF) for ChristieLag2, and −0.22 (AE) versus −0.30 (DF) for DuffeeLag2.

These results show that AE firms capture a significant portion of the leverage effect of DF firms. Specifically, considering average λ values, across all specifications, the leverage effect in the AE sample captures over 50% of that in the DF sample (i.e., the two samples’ average λ values are within at least 50% of each other). For five out of eight specifications, it captures over 65%. Considering median λ values, the corresponding values are over 60% across all specifications and over 70% for six out of eight specifications.

In Exhibit 3, we separate firms into size quintiles based on each firm’s average lifetime market capitalization and compute the means and medians of the firm-level leverage coefficients for each quintile. This exhibit reveals that, in the AE sample, the average leverage coefficient is at least twice as small for the smallest quintile (Q1) compared to the largest quintile (Q5) when the regressor is defined as the change in volatility between the current period and the previous period: −0.82 (Q1) versus −0.39 (Q5) for BlackLag1, −0.95 (Q1) versus −0.47 (Q5) for LogBlackLag1, −0.78 (Q1) versus −0.34 (Q5) for ChristieLag1, and −0.38 (Q1) versus 0.01 (Q5) for DuffeeLag1. In the DF sample, the average coefficients also increase from Q1 to Q5 for those same regression specifications; however, the difference in much less pronounced. For the remaining regression specifications, the story is reversed: Now the average coefficients are smaller for Q5 than for Q1, and much more so in the DF sample. Specifically, in the DF sample we have −0.55 (Q1) versus −1.11 (Q5) for BlackLag2, −0.39 (Q1) versus −0.94 (Q5) for LogBlackLag2, and −0.44 (Q1) versus −0.91 (Q5) for ChristieLag2. The corresponding difference in the AE sample is much less pronounced. We draw two main conclusions from these observations. First, the pattern of increasing or decreasing leverage effect across size quintiles can depend dramatically on the regression specification considered. Second, for a given specification, the pattern is much more pronounced for one sample versus another (e.g., for ChristieLag2 the average Q5 coefficient is over two times smaller than its Q1 counterpart in the DF sample and only 36% smaller in the AE sample). This implies that drawing conclusions from regressions using value-weighted portfolios of AE and DF firms may be misleading; hence, in our portfolio-level analysis we focus on equal-weighted portfolios.

[Figure omitted: see PDF.]

Exhibit 4 presents results from running leverage effect regressions on the equal-weighted portfolios of firms in the AE and DF samples.⁵ It reveals that the estimated λ for the equal-weighted portfolio of AE firms is close to that of the equal-weighted portfolio of DF firms for each of the eight regression specifications considered: −0.73 (AE) versus −0.93 (DF) for BlackLag1, −1.64 (AE) versus −2.05 (DF) for BlackLag2, −0.66 (AE) versus −0.75 (DF) for LogBlackLag1, −1.32 (AE) versus −1.53 (DF) for LogBlackLag2, −0.62 (AE) versus −0.70 (DF) for ChristieLag1, −1.32 (AE) versus −1.52 (DF) for ChristieLag2, −0.13 (AE) versus −0.12 (DF) for DuffeeLag1, and −0.31 (AE) versus −0.36 (DF) for DuffeeLag2. Again we see that a significant portion of the leverage effect in the DF sample is captured by AE firms. Specifically, the leverage effect in the AE sample is at least 78% as strong as that in the DF sample across all regression specifications considered, and over 85% as strong for six out of eight specifications.

[Figure omitted: see PDF.]

Estimation errors aside,⁶ we conclude that the inverse relationship between a firm’s stock return and the resulting change in volatility is present in the AE sample, where it cannot be attributed to the firm’s financial leverage.

Operating Leverage

In this section, we study whether the leverage effect observed in the AE sample of firms is driven by operating leverage, as suggested by Black (1976). Recall that the idea behind the operating leverage explanation for the leverage effect is as follows: Because of fixed costs, when income falls, it does so by less than expenses; hence, a firm’s equity value falls and its volatility increases. We follow the literature and define operating leverage as the ratio of operating costs to book assets.⁷ Operating costs is computed as the sum of costs of goods sold (COGS) and selling, general, and administrative expenses (XSGA) items from the COMPUSTAT Fundamentals annual database. The COGS item represents all costs directly allocated by the company to production, such as material, labor, and overhead. The XSGA item represents all commercial expenses of operation (i.e., expenses not directly related to product production) incurred in the regular course of business pertaining to the securing of operating income. Book assets is measured by the total assets item, which represents the total assets of a company at a point in time. When we combine the annual accounting data with the daily stock returns data, we make sure that the accounting data were available at the time the stock return was observed. Namely, the accounting data for the fiscal year ending in calendar year t − 1 are considered to be available as of June of year t. We compute the operating leverage quintiles for the AE sample and sort firms from that sample into those quintiles based on the average operating leverage over their lifetime.

In Exhibits 5–9, we report summary statistics of firm-level regressions for each operating leverage quintile, from the first (lowest) to the fifth (highest). If the leverage effect in the AE sample were driven by operating leverage, we would expect the leverage coefficients for that sample to be more negative in the higher operating leverage quintiles than in the lower ones. However, these exhibits reveal that in seven out of eight regression specifications λ is more pronounced in the lowest (Q1) than in the highest (Q5) operating leverage quintile: −0.61 (Q1) versus −0.52 (Q5) for BlackLag1, −0.67 (Q1) versus −0.33 (Q5) for BlackLag2, −0.66 (Q1) versus −0.62 (Q5) for LogBlackLag1, −0.46 (Q1) versus −0.34 (Q5) for LogBlackLag2, −0.51 (Q1) versus −0.47 (Q5) for ChristieLag1, −0.46 (Q1) versus −0.34 (Q5) for ChristieLag2, and −0.27 (Q1) versus −0.19 (Q5) for DuffeeLag2. Only for DuffeeLag1 is this relationship reversed: −0.09 (Q1) versus −0.16 (Q5).

[Figure omitted: see PDF.]

[Figure omitted: see PDF.]

[Figure omitted: see PDF.]

[Figure omitted: see PDF.]

[Figure omitted: see PDF.]

[Figure omitted: see PDF.]

[Figure omitted: see PDF.]

Finally, as Exhibit 10 reveals, for equal-weighted portfolios of firms, the leverage effect is stronger for Q1 than for Q5 for all eight regression specifications and at least 1.5 times as strong for seven of them: −1.18 (Q1) versus −0.76 (Q5) for BlackLag1, −1.57 (Q1) versus −0.53 (Q5) for BlackLag2, −1.32 (Q1) versus −0.74 (Q5) for LogBlackLag1, −1.31 (Q1) versus −0.48 (Q5) for LogBlackLag2, −1.20 (Q1) versus −0.69 (Q5) for ChristieLag1, −1.33 (Q1) versus −0.49 (Q5) for ChristieLag2, −0.29 (Q1) versus −0.25 (Q5) for DuffeeLag1, and −0.34 (Q1) versus −0.19 (Q5) for DuffeeLag2.

[Figure omitted: see PDF.]

These results show that the presence of the leverage effect in AE firms, which again are free from financial leverage, cannot be attributed to operating leverage either, thus ruling out Black’s leverage explanation for the leverage effect in this sample.

CONCLUSION

The inverse relationship between equity returns and subsequent volatility changes is one of the most well-established empirical regularities in stock-market data. Traditionally, this relationship is considered to be the result of leverage. This article documents the leverage effect in firms with no leverage and further shows that in the sample the leverage effect cannot be explained by operating leverage.

Our analysis does not provide a clear-cut alternative to the leverage explanation. By ruling out leverage as the source of the return/volatility relationship, our results may be interpreted as supportive of the time-varying expected return hypothesis of Pindyck (1984); French, Schwert, and Stambaugh (1987); and Campbell and Hentschel (1992). Our findings provide support for the volatility feedback model of Danielsson, Shin, and Zigrand (2009), in which asset-market volatility is endogenously determined in equilibrium by a combination of leverage constraints, feedback effects, and market conditions.

However, our results are also consistent with a behavioral interpretation in which investors’ behavior is shaped by their recent experiences, which alter their perceptions of risk and, consequently, give rise to changes in their demand for risky assets. Such biased perceptions of risk have been modeled by Gennaioli and Shleifer (2010), who reported that individuals make judgments by recalling the past experiences and scenarios that are the most representative of the current situation and combining these experiences with current information. Such judgments will be biased not only because the representative scenarios that come to mind depend on the situation being evaluated but also because the scenarios that first come to mind tend to be stereotypical ones. In our context, the first memories that come to mind for an investor who has experienced significant financial loss are ones of despair; as a result, emotions take hold, prompting the investor to quickly reverse positions, rather than continuing with a given investment policy. Because, as documented by Strebulaev and Yang (2013), AE firms tend to have higher chief executive officer ownership, have smaller and less independent boards, or be family run—all of which may imply fewer checks and balances—such path-dependent cognitive perception of risk may be especially pronounced in our sample.

The view that our recent experiences can have substantial effects on our future behavior is also backed by Lleras, Kawahara, and Levinthal (2009). In their research, they showed that memories of past experiences affect the kinds of information we pay attention to today. In particular, they compared the effects on the attention system of externally attributed rewards and penalties to the memory-driven effects that arise when subjects repeatedly perform a task. They found that in both cases the attention system is affected in analogous ways. This led them to conclude that memories are tainted (positively or negatively) by implicit assessments of our past performance.

Additional support for a behavioral interpretation of the leverage effect may be found in the aforementioned experimental evidence of Hens and Steude (2009). Recall that in their study, 24 students were asked to trade artificial securities with each other using an electronic trading system. The authors found that returns generated by these trades were negatively correlated with changes in future volatility estimates. Clearly, in this experimental context, neither leverage nor time-varying expected returns can explain the inverse return/volatility relationship.

Finally, our finding that, among the AE firms, the leverage effect is the strongest for firms with the lowest levels of operating leverage also points to the behavioral explanation. The management of these firms seems to be extremely risk averse in not taking leverage of any kind (perhaps because they are ill-equipped to manage it, which again would be consistent with Strebulaev and Yang’s characterization), and yet they cannot avoid the leverage effect.

To distinguish among these competing explanations, further empirical and experimental analysis—with more explicit models of investor behavior and market equilibrium—is required. We hope to pursue these extensions in future research.

ACKNOWLEDGMENTS

Jasmina Hasanhodzic was partially supported by the Babson Faculty Research Fund Summer Stipend Award. Andrew Lo gratefully acknowledges research support from the MIT Laboratory for Financial Engineering. The views and opinions expressed in this article are those of the authors only, and do not necessarily represent the views and opinions of any institution or agency or any of their affiliates or employees.

ENDNOTES

¹ It is worth noting that, unlike Black (1976), Christie (1982), and Cheung and Ng (1992) who included only firms that were continuously traded throughout their sample periods, Duffee’s (1995) sample is broader, including both continuously traded firms and those that exit the sample, which greatly reduced survivorship bias. He found that continuously traded firms are, on average, much larger and have lower debt/equity ratios than the typical firm.

² We first eliminate observations for which any of the long-term debt, current liabilities, or preferred stock is missing. Note that our definition of total debt differs slightly from Christie’s (1982), in which short-term debt is measured by the accounting variable current liabilities (COMPUSTAT mnemonic LCT). This variable has four components: accounts payable (AP), current liabilities − other − total (LCO), debt in current liabilities (DLC), and income taxes payable (TXP). Because our filter is intended to separate firms with and without debt in their capital structures, we use DLC as our measure of short-term debt.

³ Specifically, we first sort the AE firms into size quintiles using the average market capitalization of each firm. For each AE size quintile, we take all the DF firms that would fall in it (again, there will be more such DF firms than the corresponding AE firms). Finally, for each AE firm, we select the DF firm from the corresponding size quintile that is closest to it in equity duration.

⁴ Recall that Black’s (1976) motivation for this procedure was to mitigate the volatility estimation errors by not allowing any data common to the dependent and independent variables at the same time.

⁵ The equal-weighted portfolio return, R_p,t, is computed as [Formula omitted: see PDF.], where R_i,t is the total return (including dividends) of firm i at time t and N is the number of firms that exist at the end of time t − 1.

⁶ Recall that coefficient estimates are biased by errors in the volatility estimates, especially when dependent and independent variables are based on the same data at the same time, as is the case here.

⁷ See, for example, Novy-Marx (2011) and Garcia-Feijoo and Jorgensen (2010).