Equal or Value Weighting? Implications for Asset-Pricing Tests

Abstract

We show that an equal-weighted portfolio has a higher total return than a value-weighted portfolio. As one may expect, this is partly because the equal-weighted portfolio has higher exposure to value and size factors, but we show that a considerable part (42%) comes from rebalancing to maintain constant weights. We then demonstrate, through four applications, that inferences from asset-pricing tests are substantially different depending on whether one uses equal- or value-weighted portfolios. These four applications are tests of the: Capital Asset Pricing Model, spanning properties of the stochastic discount factor, relation between characteristics and returns, and pricing of idiosyncratic volatility.

We gratefully acknowledge comments from Andrew Ang, Elena Asparouhova, Turan Bali, Hank Bessembinder, Michael Brennan, Oliver Boguth, Ian Cooper, Victor DeMiguel, Engelbert Dockner, Bernard Dumas, Nikolae Gârleanu, Will Goetzman, Amit Goyal, Antti Ilmanen, Ivalina Kalcheva, Philipp Kaufmann, Ralph Koijen, Lionel Martellini, Stefan Nagel, Stavros Panageas, Andrew Patton, David Rakowski, Tarun Ramadorai, Paulo Rodrigues, Bernd Scherer, Norman Seeger, Eric Shirbini, Mungo Wilson, Michael Wolf, Josef Zechner, and participants of seminars at the EDHEC-Risk Days Europe Conference, Endowment Asset Management Conference at the University of Vienna, European Summer Symposium in Financial Markets at Gerzensee, Edhec Business School (Singapore), Goethe University Frankfurt, Multinational Finance Society Conference (Krakow), Norges Bank Investment Management, S&P Indices, University of Innsbruck, and University of Southern Denmark. Yuliya Plyakha is from University of Luxembourg, Faculté de Droit, d’Economie et de Finance, 4, rue Albert Borschette, L-1246 Luxembourg; e-mail: plyakha@gmx.de. Raman Uppal is from CEPR and EDHEC Business School, 10 Fleet Place, Ludgate, London, United Kingdom EC4M 7RB; e-mail: raman.uppal@edhec.edu. Grigory Vilkov is from Frankfurt School of Finance & Management, Adickesallee 32–34, 60322, Frankfurt am Main, Germany; e-mail: vilkov@vilkov.net.

Appendices

Appendix 1: Stock Characteristics

This section explains how we use CRSP and COMPUSTAT data to construct the various characteristics used in our analysis. Summary statistics for these characteristics are provided in Table 1.

1.1 Size, Book, Book-to-Market

To compute the size characteristic of the stock, we multiply the stock’s price (as given in CRSP) by the number of the shares outstanding (variable name in COMPUSTAT database: CSHOQ Common Shares Outstanding). To compute the book characteristic, we take current assets (ACTQ Current Assets, Total), subtract current liabilities (LCTQ Current Liabilities D Total), subtract preferred/preference stock redeemable (PSTKRQ Preferred/Preference Stock, Redeemable), and add deferred taxes and investment tax credit (TXDITCQ Deferred Taxes and Investment Tax Credit). The book-to-market characteristic is a ratio of the computed book characteristic and the market characteristic.

1.2 Momentum and Reversal

To compute three- and twelve-month momentum, we aggregate the returns over the past three months (months t − 4 to t − 2) and past twelve months (months t − 13 to t − 2). The stock’s reversal characteristic is the return on the stock in the previous month.

1.3 Liquidity

We compute the Amivest liquidity characteristic (Goyenko et al. 2009) as the previous month’s (22 working days) average of the ratio of the stock’s dollar volume to the absolute value of the return.

1.4 Idiosyncratic Volatility

The typical ARCH model of Engle (1982) gives a volatility prediction at the sampling frequency of the input data. Hence, when the model is fitted to daily returns, it is not very suitable for longer horizon forecasts that we need for a typical passive investor. Some recent papers (see, for example, Fu (2009)) suggest using for this purpose the EGARCH model of Nelson (1991) fitted to monthly excess returns, but in our experiments the estimation did not lead to very stable results. To increase the stability of the estimation and the amount of data available for it, we utilize the MIDAS (Ghysels et al. 2005) approach. It separates the volatilities into short-run and long-run components, and the latter can be used to predict the second moments at a slower frequency than the data.

For the conditional expectation of idiosyncratic volatility we use the long-term volatility component from the asymmetric GARCH-MIDAS model that we fit to the residual from regressing the daily stock return on the Fama and French (1993) factors. For a detailed discussion of MIDAS models for volatility modeling see, for example, Ghysels et al. (2005), Engle et al. (2008).

Specifically, excess volatilities follow the ASYGARCH-MIDAS process as follows:

$$\displaystyle \begin{aligned} \begin{array}{rcl} r_{k,t} & =&\displaystyle \alpha_{k,t}+\beta^{MKT}_{k,t}\times F^{M}_{t}+\beta^{SMB}_{k,t}\times F^{SMB}_{t}+\beta^{HML}_{k,t}\times F^{HML}_{t}+\varepsilon_{k,t} \end{array} \end{aligned} $$

(11)

$$\displaystyle \begin{aligned} \begin{array}{rcl} \varepsilon_{k,t} & =&\displaystyle \sqrt{m_{k,t}\times g_{k,t}}\xi _{k,t} \end{array} \end{aligned} $$

(12)

$$\displaystyle \begin{aligned} \begin{array}{rcl} g_{k,t} & =&\displaystyle (1-\alpha _{k}-\kappa _{k})+\alpha _{k}\frac{\varepsilon_{k,t}^{2}}{m_{k,t}}+\kappa _{k}\cdot g_{k,t-1} \end{array} \end{aligned} $$

(13)

$$\displaystyle \begin{aligned} \begin{array}{rcl} m_{k,t} & =&\displaystyle \overline{m}_{k}+ \theta^{+}_{k}\sum_{l=1}^{Lv}\varphi (\omega^{+} _{k,v})\times RV^{+}_{k,t-l}+ \theta^{-}_{k}\sum_{l=1}^{Lv}\varphi (\omega^{-} _{k,v})\times RV^{-}_{k,t-l} \end{array} \end{aligned} $$

(14)

$$\displaystyle \begin{aligned} \begin{array}{rcl} RV^{+}_{k,t}& =&\displaystyle \frac{1}{N^{+}}\sum_{\tau=0}^{20}\left(r_{k,t-\tau}*1_{r_{k,t-\tau>0}}\right) ^{2}, \ \ \ RV^{-}_{k,t}=\frac{1}{N^{-}}\sum_{\tau=0}^{20}\left(r_{k,t-\tau}*1_{r_{k,t-\tau<0}}\right) ^{2},\\ \end{array} \end{aligned} $$

(15)

where r _k,t is the (daily) return of asset k = 1, 2, …, N, $F^{J}_{t}$ is factor $J\in \left \{MKT,SMB,HML\right \}$, the short-run idiosyncratic volatility component g _k,t follows a unit GARCH process, and the long-run idiosyncratic volatility component m _k,t is the weighted sum of positive ($RV^{+}_{k,t}$) and negative ($RV^{-}_{k,t}$) mean-squared-return innovations (where N ⁺ and N ⁻ are the number of positive and negative return innovations, respectively). To aggregate the past RV ’s, we use the Beta polynomial weighting functions φ(ω ⁺), and φ(ω ⁻) with Lv = 126 lags.

We fit the above model for each underlying stock at the end of each month, using three years of daily returns. We use maximum likelihood to find simultaneously factor sensitivities ($\beta ^{J}_{k,t}$), the parameters of the short-run volatility α _k and κ _k, the parameters of the long-run volatility $\overline {m}_{k}$, $\theta ^{+}_{k}$, $\theta ^{-}_{k}$, and the optimal weights for the Beta weighting function ω ⁺ and ω ⁻. After estimating these parameters, we compute the predicted value of long-run idiosyncratic volatility m _k,t and use that as the characteristic for idiosyncratic volatility.

After we compute the predicted long-run idiosyncratic volatility, we also produce a “clean” version of it by taking out the effect of illiquidity of individual stocks as suggested by Han and Lesmond (2011). We describe this procedure in Sect. 5.4.

Appendix 2: Resampling Procedures and Monotonicity Relation Tests

From February 1967 to the end of 2009 there were 1,449 stocks that were part of the S&P 500 index. Our time series consists of 515 months, which corresponds to 43 years. From the set of 1,449 stocks, we randomly choose a sample of stocks that are constituents of the index at the time they are selected.

We work with these samples and construct portfolios of different stock numbers: N = {30, 50, 100, 200, 300} stocks. In order to reduce the selection bias, for each portfolio with N stocks, we randomly resample to select N stocks 1,000 times and construct 1,000 portfolios. To compute the portfolio performance metrics, we compute the performance metrics for each of the 1,000 portfolios and report the performance metrics averaged across these 1,000 portfolios.

In Sect. 5.3, we study the characteristics of assets that potentially drive the differences in the performance of the equal-, value- and price-based weighting rules. To perform the monotonicity test for each resampled set of assets for each portfolio would be very demanding in terms of computer power and time.

Therefore, we carry out a special procedure to create “synthetic” assets described next. For each of the 1,000 portfolios consisting of N = 100 stocks, we sort the stocks at the end of each month by a particular characteristic. From these 1,000 sorted portfolios we create 100 synthetic assets, where for each asset j = {1, …, 100} the characteristic is set equal to the mean characteristic of all stocks across the thousand portfolios with rank j after the sorting procedure, and the return of the synthetic asset j for the next period is equal to the mean return of all the stocks with the same rank j. Then, we group the sorted assets into deciles and compute the return for each decile by applying equal, value, and price weights within each decile.

We then analyze the performance of the portfolio deciles constructed from the synthetic assets. Each decile’s characteristic is the time-series mean of an average value of the characteristic of the assets in this decile. Annualized returns of each decile expressed in percentage are computed as the time-series mean of the returns of the portfolios constructed from the assets of that decile, with three different weighting rules (equal-, value- and price-weighted), for each decile. Each decile return is then weighted by the weight of the respective decile in the large portfolio of synthetic assets.

To test for an increasing (decreasing) relation we form the pairwise differences of the values of the test series, that is, the value of decile i minus the value of decile i − 1, where i = {2, …, 10}, bootstrap the differences in the time-series dimension,²⁸ find the minimum (maximum) of each bootstrapped sample, and compute the probability that the minimum (maximum) of the differences is greater (smaller) than the sample minimum (maximum) of the differences. We also perform a stronger test for a monotonic relation, in which we consider not only the pairwise differences of the adjacent data points but also the differences between all possible pairs.

Appendix 3: Robustness Tests

In this section, we briefly discuss some of the experiments we have undertaken to verify the robustness of our findings.

1.1 Different Number of Stocks

The results that we have reported are for portfolios with N = 100 stocks. In addition to considering portfolios with 100 stocks, we also consider portfolios with 30, 50, 200, and 300 stocks (again, with resampling over 1,000 portfolios). We find that our results are not sensitive to the number of assets in the portfolio. To conserve space, these results are not reported.

1.2 Different Stock Indexes

In addition to stocks sampled from the S&P 500 for large-cap stocks, we consider also stocks from the S&P 400 for mid-cap stocks, and the S&P 600 for small-cap stocks. The performance of portfolios constructed from the stocks constituents of S&P 400 and S&P 600 is reported in Tables 9 and 10, respectively. Comparing the performance metrics in these tables to those for the stocks constituents of S&P 500, we see that the main insights for the weighting rules are similar across the three indexes.

Table 9 Portfolio Performance for S&P 400 Constituent Stocks. In this table we report the performance metrics of the portfolios constructed from the constituents of the S&P 400 index. All metrics are calculated using monthly returns from July 1991 to December 2009 (222 months). The first column gives the various metrics we use to measure portfolio performance on a per annum basis. The remaining columns report the performance, before transactions costs, and net of transactions costs of fifty basis points, for portfolios formed using different weighting rules: EW denotes the equal-weighted portfolio; VW, the value-weighted portfolio; and PW, the price-weighted portfolio

Table 10 Portfolio Performance for S&P 600 Constituent Stocks. In this table we report the performance metrics of the portfolios constructed from the constituents of the S&P 600 index. All metrics are calculated using monthly returns from November 1994 to December 2009 (182 months). The first column gives the various metrics we use to measure portfolio performance on a per annum basis. The remaining columns report the performance, before transactions costs, and net of transactions costs of fifty basis points, for portfolios formed using different weighting rules: EW denotes the equal-weighted portfolio; VW, the value-weighted portfolio; and PW, the price-weighted portfolio

1.3 Different Economic Conditions

We also investigate whether the superior performance of the equal-weighted portfolio relative to the value- and price-weighted portfolios is sensitive to the date on which one invests in the portfolio. In particular, we examine whether the relative performance of these portfolios is different if one starts at the peak or trough of the business cycle.

The NBER identifies peaks of the business cycle in March 2001 and December 2007, and a trough in November 2001. In Table 11, we report the performance of the equal-, value-, and price-weighted portfolios starting at these three dates and that are held to the end of our data period, December 2009. For all three starting dates, we find that the equal-weighted portfolio has a significantly higher total mean return. For all three starting dates, the one-factor and four-factor alphas are significantly higher for the equal-weighted portfolio relative to the value-weighted portfolios. In fact, the four-factor alpha for the equal-weighted portfolio is positive for all three starting dates, while it is negative for the value-weighted portfolio for the start dates of March 2001 and December 2007. The Sharpe ratio of the equal-weighted portfolio also exceeds that of the value- and price-weighted portfolios. For instance, if one had initiated the portfolios at the peak of March 2001, the Sharpe ratio of the equal-weighted portfolio would have been 0.2639 compared to only 0.0037 for the value-weighted portfolio; if one had started at the trough of November 2001, the Sharpe ratio of the equal-weighted portfolio would have been 0.3615 rather than the 0.1252 for the value-weighted portfolio; and, if one had started at the peak of December 2007, the Sharpe ratio of the equal-weighted portfolio would have been − 0.0795 while that of the value-weighted portfolio was − 0.3780, and that for the price-weighted portfolio was − 0.2995. The certainty equivalent return for an investor with a risk aversion of γ = 2 is also higher for the equal-weighted portfolio relative to the value- and price-weighted portfolios. For a risk aversion of γ = 5, the equal-weighted portfolio outperforms the value-weighted portfolio but not the price-weighted portfolio; however, in both cases the difference is not statistically significant.

Table 11 Portfolio Performance for Different Start Dates Over Business Cycle. In this table we report the performance metrics for portfolios constructed from the constituents of the S&P 500 index. All metrics are calculated using monthly returns with different starting dates but all ending at December 2009. The three starting dates considered are: the peak of the business cycle in March 2001; the trough of November 2001; and, the peak of the business cycle in December 2007. The first column gives the various metrics we use to measure portfolio performance on a per annum basis. The remaining columns report the performance, before transactions costs and net of transactions costs of fifty basis points for portfolios formed using different weighting rules: EW denotes the equal-weighted portfolio; VW, the value-weighted portfolio; and PW, the price-weighted portfolio

1.4 Bias in Computed Returns

In this section, we examine the effect on our findings of correcting returns for the potential biases that may arise from noisy prices and liquidity differences. To make this correction, we use the approaches suggested in Blume and Stambaugh (1983), Asparouhova et al. (2010, 2013), and Fisher et al. (2010).

Asparouhova et al. (2013) show that for realistic assumptions about the noise parameter, the first-best method for reducing the bias in the estimated performance of the equal-weighted portfolio is to use prior-gross-return weighting (RW) instead of the pure equal weighting (EW):

$$\displaystyle \begin{aligned}w^{RW}_{t,i}=\frac{r_{t-1,i}+1}{\sum^{N}_{i=1}(r_{t-1,i}+1)}.\end{aligned}$$

In this case, the value- and price-weighted portfolios are still computed using the end-of-month returns reported in CRSP.

Comparing the results in Table 12 to those in Table 2, we see that using prior-gross-return weighting instead of the standard equal weighting reduces the total and nonsystematic returns only slightly and does not change our main conclusions. For example, the total return of the equal-weighted portfolio after the correction is 0.1292, instead of the previously reported 0.1319; moreover, even with the correction, the equal-weighted portfolio outperforms the value- and price-weighted portfolios at less than 1% significance level. Similarly, the systematic return of the equal-weighted portfolio after the correction is 0.1146, instead of the previously reported 0.1144; and, even with the correction, the equal-weighted portfolio outperforms value-weighted at less than 1% significance level. Finally, the four-factor alpha of the equal-weighted portfolio using the prior-gross-return weighting to construct the equal-weighted portfolio is 1.46%, instead of the previously reported 1.75% for the equal-weighted portfolio using uncorrected returns; the p-value of the difference with the alpha for the value-weighted portfolio is now 6%, instead of the earlier p-value of 2%, and for the difference with the alpha of the price-weighted portfolio, the p-value is still smaller than 1%.

Table 12 Portfolio Performance with Prior-Gross-Return Weighting for Equal-Weighted Portfolio. In this table we report the performance metrics for portfolios constructed from the constituents of the S&P 500 index. All metrics are calculated using monthly returns from February 1967 to December 2009 (515 months). The performance of the equal-weighted portfolio is computed using the prior-gross-return weighting. The first column gives the various metrics we use to measure portfolio performance on a per annum basis. The remaining columns report the performance, before transactions costs, and net of transactions costs of fifty basis points, for portfolios formed using different weighting rules: EW denotes the equal-weighted portfolio; VW, the value-weighted portfolio; and PW, the price-weighted portfolio

Thus, we conclude from Table 12 that using prior-gross-return weighting instead of pure equal weighting does not alter the main conclusions of our analysis: (i) the outperformance of the equal-weighted portfolio relative to value- and price-weighted portfolios is monotonically related to the average value of various characteristics of the stocks in each portfolio; (ii) part of the outperformance across the three weighting schemes arises from differences in systematic risk, which stems from a difference in exposure to common factors; and, (iii) the nonsystematic outperformance measured by differences in the alphas is a result of more frequent rebalancing of the equal-weighted portfolio as compared to value- and price-weighted portfolios.²⁹ Therefore, our findings about the differences in the returns of equal- and value-weighted portfolios are complementary to the findings of Asparouhova et al. (2013).

Table 13 presents the results of the Patton and Timmermann (2010) monotonicity tests when the returns of the equal-weighted portfolio are constructed using prior-gross-return weighting. Comparing the statistics in Table 13 to those reported in Table 4, we see that the results are almost unchanged, with one exception: for the liquidity characteristic, we now fail to reject both increasing and decreasing relations between liquidity of the stocks and the outperformance of the equal-weighted portfolio compared to value- and price-weighted portfolios. This is consistent with the insight in Asparouhova et al. (2010).

Table 13 Tests of Monotonicity Relations with Prior-Gross-Return Weighting for Equal-Weighted Portfolio. In this table we report the p-values of the Patton and Timmermann (2010) test for a monotonic relation between a particular characteristic listed in the first column, and the difference in performance of the equal- and value-weighted portfolios (EW−VW), and the equal- and price-weighted portfolios (EW−PW). The performance of the equal-weighted portfolio is computed using the prior gross-return weighting. We report the p-values of the null hypothesis that difference in returns is increasing with respect to a given characteristic (first row) and also that the difference in returns is decreasing with respect to that characteristic (second row). We undertake two tests: in the first we consider only the differences of neighboring pairs of data points; in the second, stronger, test, we consider also the differences between all possible pairs. The analysis is based on monthly returns from February 1967 to December 2009

There are three additional methods that one can use to correct from the potential bias arising from microstructure effects. Each of these methods requires additional information—either about bid-ask prices, or about trading volume. The first additional method is to correct the end-of-month returns for the bid-ask bias by computing returns as follows:

$$\displaystyle \begin{aligned}r_{t,i}=\tilde{r}_{t,i}-\left(\frac{ask_{t-1,i}-bid_{t-1,i}}{ask_{t-1,i}+bid_{t-1,i}}\right)^{2},\end{aligned}$$

where $\tilde {r}_{t,i}$ is the “noisy” closing return reported in CRSP for time t and stock i, and r _t,i is the return after correction. The second additional approach for correcting returns for microstructure effects is to use the midpoint of the closing bid and ask prices from CRSP. The third additional approach is to compute returns using the volume-weighted average prices (VWAP) for the last day of each month from the TAQ Database.

Note, however, that the bid and ask prices, as well as the high-frequency trading data are not available for our entire sample period, so we implement these three additional correction measures only for the period of 1995 to 2009 (180 points). In order to compare the results using the four methods for reducing the bias described above, with the results without correction for the bias, we report in Table 14 the “base-case” results, which are based on the methodology adopted in our manuscript using end-of-month CRSP returns, but for the period of 1995 to 2009. We report in Table 15 the results based on the prior-gross-return weighting for the period 1995–2009, and in Tables 16, 17, and 18, the results for the three additional correction methods described above.

Table 14 Portfolio Performance Using Returns From Closing Prices without Correction (Base Case). In this table we report the performance metrics for portfolios constructed from the constituents of the S&P 500 index. All metrics are calculated using monthly returns from January 1995 to December 2009 (180 points). The first column gives the various metrics we use to measure portfolio performance on a per annum basis. The remaining columns report the performance, before transactions costs, and net of transactions costs of fifty basis points, for portfolios formed using different weighting rules: EW denotes the equal-weighted portfolio; VW, the value-weighted portfolio; and PW, the price-weighted portfolio

Table 15 Portfolio Performance with Prior-Gross-Return Weighting for the Equal-Weighted Portfolio (Case 1). In this table we report the performance metrics for portfolios constructed from the constituents of the S&P 500 index. All metrics are calculated using monthly returns from January 1995 to December 2009 (180 points). The performance of the equal-weighted portfolio is computed using the prior-gross-return weighting. The first column gives the various metrics we use to measure portfolio performance on a per annum basis. The remaining columns report the performance, before transactions costs, and net of transactions costs of fifty basis points, for portfolios formed using different weighting rules: EW denotes the equal-weighted portfolio; VW, the value-weighted portfolio; and PW, the price-weighted portfolio

Table 16 Portfolio Performance Using Returns From Bid-Ask Prices (Case 2). In this table we report the performance metrics for portfolios constructed from the constituents of the S&P 500 index. All metrics are calculated using monthly returns from January 1995 to December 2009 (180 points). The performance of the equal-weighted portfolio is computed with returns corrected for potential biases using bid-ask prices. The first column gives the various metrics we use to measure portfolio performance on a per annum basis. The remaining columns report the performance, before transactions costs, and net of transactions costs of fifty basis points, for portfolios formed using different weighting rules: EW denotes the equal-weighted portfolio; VW, the value-weighted portfolio; and PW, the price-weighted portfolio

Table 17 Portfolio Performance Using Returns from Midpoint of Bid and Ask Prices (Case 3). In this table we report the performance metrics for portfolios constructed from the constituents of the S&P 500 index. All metrics are calculated using monthly returns from January 1995 to December 2009 (180 points). The performance of the equal-weighted portfolio is computed using the returns computed from the midpoint of closing bid and ask prices instead of closing prices. The first column gives the various metrics we use to measure portfolio performance on a per annum basis. The remaining columns report the performance, before transactions costs, and net of transactions costs of fifty basis points, for portfolios formed using different weighting rules: EW denotes the equal-weighted portfolio; VW, the value-weighted portfolio; and PW, the price-weighted portfolio

Table 18 Portfolio Performance Using Returns from Volume-Weighted Average Prices (Case 4). In this table we report the performance metrics for portfolios constructed from the constituents of the S&P 500 index. All metrics are calculated using returns computed from value-weighted average prices for the last day of each month from the TAQ Database from January 1995 to December 2009 (180 points). The performance of the equal-weighted portfolio is computed the volume-weighted average prices (VWAP) for the last day of each month. The first column gives the various metrics we use to measure portfolio performance on a per annum basis. The remaining columns report the performance, before transactions costs, and net of transactions costs of fifty basis points, for portfolios formed using different weighting rules: EW denotes the equal-weighted portfolio; VW, the value-weighted portfolio; and PW, the price-weighted portfolio

We see from these tables that: (i) For all four bias-reduction methods, the equal-weighted portfolio outperforms the value- and price-weighted portfolios, and this outperformance is statistically significant for total return, systematic return, and the one- and four-factor alphas in most of the cases. (ii) The systematic return for the base case without bias correction is almost identical to the systematic return for the four cases with bias correction. (iii) We observe some variation in factor alphas among bias-correction methods, but the difference between alphas for equal-weighted and other portfolios is stable, and it continues to be economically significant. (iv) The noise in prices and the shorter sample period reduce slightly the statistical significance in the difference between the four-factor alphas of the equal- and value-weighted portfolios, and the one-factor alphas of the equal- and price-weighted portfolios.

Based on the above analysis, we conclude that the bias in end-of-month returns is very small in the context of our analysis, and it does not change the findings of our paper. The main reason why our results are not affected by these corrections for differences in bid-ask spreads and liquidity across stocks is because these differences are much smaller in the samples with which we are working. That is, because we are looking at stocks only in the S&P 500, the heterogeneity across stocks is much smaller than it is across the entire population of stocks in the CRSP database. Moreover, the effect of the correction for differences in bid-ask spreads and liquidity across stocks is largest for small stocks.³⁰