Abstract
Prior research shows that noisy prices can introduce biases in returns causing equal-weighted portfolios to outperform value-weighted portfolios. In this paper, we reevaluate the superiority of EW portfolios in the presence of market frictions. We find that trading costs have limited impact on the performance of EW portfolios, while taxes cause the performance of both EW and VW portfolios to deteriorate by a similar magnitude leaving the superiority of EW portfolios intact. Besides mispricing, the results for small cap portfolios may also be affected by the size effect. Overall, EW portfolios earn annualized risk-adjusted returns between 1.23% and 1.79% even after accounting for trading costs and taxes.
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\(E\left[ {HPR_{EW} } \right] = \mathop \prod \limits_{t = 1}^{T} E\left[ {1 + r_{EW,t} } \right] = \mathop \prod \limits_{t = 1}^{T} \left( {\mathop \sum \limits_{i = 1}^{N} w_{i} E\left[ {1 + r_{i,t} } \right]} \right)\). As \(E\left[ {1 + r_{i,1} } \right] = 1 + r^{*}\) and \(E\left[ {1 + r_{i,T} } \right] = \left( {1 + r^{*} } \right)e^{{\sigma_{z}^{2} }}\), we have \(E\left[ {HPR_{EW} } \right] = \left( {1 + r^{*} } \right)\left( {\mathop \prod \limits_{t = 2}^{T - 1} \left( {\mathop \sum \limits_{i = 1}^{N} \frac{1}{N}\left( {1 + r_{t}^{*} } \right)e^{{\left( {1 - \rho } \right)\sigma_{z}^{2} }} } \right)} \right)\left( {1 + r^{*} } \right)e^{{\sigma_{z}^{2} }} = \left( {1 + r^{*} } \right)\left( {\left( {1 + r^{*} } \right)e^{{\left( {1 - \rho } \right)\sigma_{z}^{2} }} } \right)^{T - 2} \left( {1 + r^{*} } \right)e^{{\sigma_{z}^{2} }} = \left( {1 + r^{*} } \right)^{T} e^{{\left( {\left( {T - 2} \right)\left( {1 - \rho } \right) + 1} \right)\sigma_{z}^{2} }}\).
The results are similar when value and growth style indices are considered.
Liu and Strong (2008) points out that the traditional method of portfolio return estimation used by a number of prior studies (i.e. applying constant weights at the beginning of each month during a multi-month holding period) would generate inaccurate portfolio returns for EW or VW portfolios, as that approach ignores the fact that the weight attached to each stock in the portfolio actually depends upon the stock's performance over previous holding-period months. Our study, however, is not subject to this bias as the weight of a stock in our EW or VW portfolio at the beginning of a non-rebalancing month is fully adjusted by the performance of the stock in the prior months.
Reported index returns are based on trade prices that ignore the bid-ask bounce. The bid-ask bounce mechanically creates an upward bias in returns due to Jensen’s inequality. In the paper, however, we remove the bias due to bid-ask bounce by using quote midpoints to computer returns. As a result, the true Russell 2000 returns are significantly lower than reported returns, which show up as negative alphas in Table 3 and other tables.
See Appendix A for detailed estimation process.
Though both are commonly used, the measures differ significantly in magnitude. For example, Petersen and Fialkowski (1994) find that effective spreads are only half of the quoted spreads and that increase in the quoted spread results in only 10%-22% increase in the effective spreads. The quoted spread may overestimate the transaction costs because many trades are executed inside the quoted spreads, while the effect spread may underestimate the transaction costs since trades may time market liquidity. Therefore, both RES and RQS are used in our tests.
The use of 30-min interval gives very similar results.
Such results are expected since actively-managed portfolios have more flexibility to choose when and what securities to trade in order to lower trading costs.
With $500 million TNA at the beginning of 2002 and ignoring taxes, an EW S&P 500 portfolio with monthly rebalancing should have approximately $1.92 billion TNA at the end of 2016, an EW Russell 1000 portfolio should have approximately $2.18 billion TNA at the end of 2016, and an EW Russell 2000 portfolio should have approximately $1.70 billion TNA at the end of 2016.
The top tax rate was reduced from 39.6% to 37% effective January 1, 2018. However, overall tax liability remained essentially unchanged at the higher tax brackets due to elimination of certain deductions.
A total of four different tax avoidance approaches were tested: (1) minimize the capital gains tax incurred by each sale; (2) always sell the shares with the highest purchase price; (3) sell shares that were purchased more than one year ago before selling any shares that were purchased within one year, (4) first-in-last-out. The first approach provides the best results, while first-in-last-out provides the worst results. All results reported are based on the first approach.
As reported in revised Table 7, EW portfolios outperform VW portfolios by 1.23%, 1.35%, 1.79% and 1.59% per year based on the risk-adjusted returns after deduction of trading costs and taxes. Typically, the tracking error of indexed mutual funds cannot exceed 0.05% a year—for example, compare VFINX and SPY with the S&P 500, all including dividends. Excess returns of 1.23% are large and economically significant for investment managers. Alternatively, consider the expense ratios of mutual funds as reported in Figs. 6.7 and 6.8 of the 2020 ICI Factbook: for our sample period of 2002–2016, average expense ratios of active equity funds, passive equity funds, and equity ETFs are 0.95%, 0.17%, and 0.28%, respectively. The excess returns of EW portfolios will more than compensate for the expense ratios of even most of the active equity funds. In summary, we consider the EW outperformance to be economically significant as it could give a fund significant advantage when competing with peers.
As the magnitude of mispricing and return volatility may be significantly different during the 2008 financial crisis, results of our analysis may be significantly influenced by the crisis period. To address this issue, we repeat the tests in Tables 5 and 7 based on a subsample where the crisis period (2008–2009) is excluded and report the results in Table 10. The results are qualitatively and quantitatively similar, thus excluding the crisis period does not change our conclusions. Results for Table 3 are also qualitatively and quantitatively similar when the crisis period was removed (not tabulated due to limited space).
VMA lags beyond five are assumed to have little impact and are ignored to simplify the estimation process. As mentioned by Hasbrouck (1993), more lags generally increase the estimates of pricing error variance but make little change in the cross-sectional ranking.
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Acknowledgements
We thank participants at the 2015 Financial Management Association meetings, and seminar participants at Virginia Tech for comments and suggestions.
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Appendix A: Estimation of Hasbrouck’s (1993) pricing error
Appendix A: Estimation of Hasbrouck’s (1993) pricing error
Intraday trade and quote data obtained from the NYSE TAQ database are used for estimation of \(\sigma_{s}\). Following Boehmer and Kelly (2009), we use quotes and trades that are within the regular trading hours (9:30AM-4:00PM) and ignore overnight price changes. A quote is removed if the ask price is lower than the bid price, if the bid price is lower than $0.10, or if the bid-ask spread is higher than 25% of the quote midpoint. To be eligible for estimation, a trade is required to have a value of zero in TAQ’s CORR field, marked as ‘*’, ‘@’, ‘@F’, ‘F’, ‘B’, ‘E’, ‘J’, ‘K’, or blank in TAQ’s COND field, and have a positive trade size and price. A trade is removed if its price differs by more than 30% from the previous trade. We ignore the natural times but view transactions as untimed sequences. This approach is preferable since it gives more weights to periods with heavier price discovery activities, represented by more transactions, and uses information delivered from every single transaction. Following Hasbrouck (1993), we estimate the lower bound for \(\sigma_{s}\) using a vector autoregression (VAR) model with five lags over the four-variable set \({\varvec{X}}_{t} = \left\{ {r_{t} , {\varvec{x}}_{t} } \right\}^{^{\prime}}\), where \(r_{t} = p_{t} - p_{t - 1}\) and \({\varvec{x}}_{t}\) is a \(3 \times 1\) vector of the following trade variables: (1) sign of trading direction that takes value of 1 if the transaction is buyer-initiated value of ‒1 if it is seller-initiated, and value of 0 for a quote midpoint transaction, (2) signed trading volume, and (3) the signed square root of trading volume. Following Harris (1989) and Lee and Ready (1991), we classify a trade as buyer-initiated (seller-initiated) if the transaction price is above (below) the prevailing quote midpoint. The inclusion of square root of trading volume aims to allow for concave dependencies in both \(m_{t}\) and \(s_{t}\). In each month, we estimate V(s) for stocks that have at least 100 trades over that month. Specifically, the joint process of \({\varvec{X}}_{t}\) is described by a five-lag VAR model:
where \({\varvec{B}}_{k}\) is the \(4 \times 4\) coefficient matrix for lag k; \({\varvec{u}}_{t}\) is a \(1 \times 4\) vector of zero-mean error terms with \(E\left( {u_{i,t} ,u_{j,t} } \right) = 0\). The VAR model is then transformed into a five-lag approximation of vector moving average (VMA) representationFootnote 14:
Variance of pricing error is expressed by:
where
and \(Cov\left( {\varvec{u}} \right)\) is the residual covariance matrix from the VAR model. We use the average of the monthly estimates of \(\sigma_{s}^{2}\) in a quarter as the pricing error variance of that quarter ( See Tables 9 and 10).
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Qin, N., Singal, V. Equal-weighting and value-weighting: which one is better?. Rev Quant Finan Acc 58, 743–768 (2022). https://doi.org/10.1007/s11156-021-01008-w
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DOI: https://doi.org/10.1007/s11156-021-01008-w