Smart Rebalancing

Abstract

The sometimes vast gap between live results and paper portfolio performance is caused in part by trading costs, discontinuous trading, and missed trades or other frictions, along with asset management fees. Smart beta and factor strategies are not exempt from this sort of “implementation shortfall.” This paper provides new evidence on the efficacy of prioritizing transactions so as to focus portfolio turnover on the trades that offer the strongest signals and hence the highest potential performance impact. Rebalancing filters of this sort can capture much of the factor premia for a long-only paper portfolio while cutting turnover and trading costs relative to a fully rebalanced portfolio.

Keywords:

• alpha
• factor premia
• implementation shortfall
• portfolio turnover
• rebalancing methods
• risk premia
• trading costs
• trading signals

PL Credits:

• 2.0

Introduction

Hundreds of articles, in the top academic journals alone, have introduced new factors that may deliver higher returns or improve portfolio risk attributes. From an academic standpoint, these are important and useful advances that can provide insights into equilibrium prices, help us understand asset price behavior, and offer the possibility of better performance. Typically, this research tacitly presumes continuous markets and free trading with boundless liquidity and ignores revaluation alpha.¹ Indeed, many factors rely on illiquid microcap stocks, which cannot be used on an institutional scale, for their statistical significance.² For the investor, this research is useful only if performance is improved by more than the resulting trading costs, which are typically ignored in the research.

Far less research has focused on thoughtful portfolio trading that might help investors to capture more of the risk premium or alpha associated with a factor or strategy. While academic factor research is happy to trade every stock with every twitch in a factor signal, a signal that is rendered worthless by trading costs and other market frictions is not particularly interesting to the practitioner, even if the backtests look good. Research on portfolio construction techniques may not be as exciting as finding the latest new statistically significant factor with a brilliant backtest, but it is arguably just as important. Better implementation can potentially help us capture more of a factor strategy’s premia, net of trading costs and other forms of implementation shortfall.³ The benefits are measurable and economically significant.

In this study, we first examine the performance and related turnover of a few popular factor strategies. We draw a distinction between long-short factors and long-only factor-based strategies; the latter are the focus of this paper. We present the results for the same strategies after applying three different turnover reduction methods to the periodic portfolio rebalancing trades. Each begins with a target turnover limit, ranging from a 10% annual turnover limit to a 100% limit.

We call the first rebalancing method priority-best, and it is the focus of much of this paper. Priority-best buys the stocks with the most attractive signals and sells the stocks with the most unattractive signals until the turnover limit is reached. For example, suppose a strategy uses stocks’ book-to-price (B/P) ratios to construct a value portfolio. Suppose the model portfolio sells stocks A, B, and C and buys stocks D, E, and F, which have B/P ratios of 1, 2, 3, 4, 5, and 6, respectively. Selling stock A and buying stock F will improve the portfolio valuation multiple the most, whereas selling stock C and buying stock D will make only a marginal difference. If the strategy has a restrictive turnover limit, the largest impact would come from trading stocks A and F, and perhaps some of stocks B and E, and likely forgoing trades in stocks C and D.

The second and third rebalancing methods are proportional rebalancing and priority-worst rebalancing. Priority-worst is similar to the priority-best method, except that we deliberately sort the buying and selling queues in the “wrong” order. We buy the stocks that seem least useful (i.e., the most marginal in terms of their signals) and sell those that just barely fall out of the target portfolio, saving the strongest buy or sell signals to trade last. In the previous example, this would mean that we first sell stock C and buy stock D and likely forego trading stocks A and F. One might view this trading rule as deliberately maximizing Black’s (1986) “noise” in the rebalancing process. With proportional rebalancing, we trade all stocks pro rata to meet the turnover limit. For example, if the unconstrained strategy has turnover that is twice our turnover limit, this method trades half of every indicated trade in the portfolio.

While proportional rebalancing is not uncommon in the quantitative practitioner community, no investor will likely choose priority-worst. Both are included as a robustness test for the priority-best approach. The priority-best method typically outperforms proportional rebalancing, which typically outperforms the priority-worst approach. When priority-best does not outperform, we trace the reason for its failure to a non-monotonic relationship between the signals and expected returns.

In the Appendix, we also test a non–calendar-based rebalancing methodology, specifically for momentum, the factor with the shortest half-life and the highest turnover. Instead of forcing the portfolios to rebalance on a fixed schedule, such as at the end of every month or once a year, we consider a rule in which we rebalance when the distance between the current and target portfolios exceeds a preset threshold. We then rebalance a prespecified proportion of the deviations using one of the three rules: (a) priority-best, (b) priority-worst, or (c) proportional rebalancing, up to but not exceeding our turnover limit. We compare the performance and turnover rates to highlight how to improve factor investing through the lens of smart rebalancing. We show that the priority-best rule also generally outperforms the other two rules in the context of non–calendar-based rebalancing.

Literature Review

The literature concerning trading costs is somewhat limited, particularly in light of the extensive literature exploring novel factors. Here, we provide a selection of key papers in this important, albeit gradually expanding, domain.

Novy-Marx and Velikov (2016, 2019) study how anomalies perform after accounting for transaction costs. The authors’ starting point is the observation that, because trading costs reduce a strategy’s profitability and the associated statistical significance, the degree to which a real-world investor can benefit from a factor depends on its turnover. They examine three common transaction-cost-mitigation techniques: (1) limiting trading to stocks with low expected trading costs, (2) using lower rebalancing frequencies, and (3) using a buy/hold spread that lowers turnover by delaying trades until they are compelling.

Another way of reducing trading costs, which is popular in asset management firms, is to apply a penalty when incorporating factor portfolios into a mean-variance efficient portfolio: the optimal portfolio is computed with a penalty for trading costs. The penalties are the weights minus the original optimal weights multiplied by trading costs. In the absence of the trading costs, we trade fully to our new target portfolio. This approach trades optimally toward our new efficient portfolios while being cognizant of trading costs. The practitioner community is split between those who prefer utility-maximizing with a mean-variance quadratic programming optimizer and those who trade using a heuristics approach. In this paper, we examine turnover reduction mechanisms that can be applied to simple heuristics-based portfolio construction, and we analyze how trades can be prioritized toward securities with the strongest signals.⁴

Detzel, Novy-Marx, and Velikov (2021) highlight the importance of considering trading costs in model selection. They show that the Fama and French (2015) five-factor model has a significantly higher squared Sharpe ratio (the metric favored by Barillas and Shanken 2018) than other alternatives after accounting for trading costs, whereas the gross squared Sharpe ratio—before trading costs—indicates otherwise. DeMiguel et al. (2020) illustrate the importance of considering trading costs by showing how the number of characteristics, which are jointly significant in an investor’s optimal portfolio, increases with trading costs. The trades based on different characteristics cancel out when considered simultaneously. Their research highlights the importance of real-life frictions when determining an investor’s optimal portfolio in a multifactor setting. DeMiguel, Martin-Utrera, and Uppal (2022) present an alternative method for mitigating price impact during trading while also enhancing the effectiveness of underlying factor-based strategies. This approach involves concurrent trading with other investors who are capitalizing on distinct factors.

Rattray et al. (2020) introduce the concept of “strategic rebalancing.” Strategic rebalancing differs from our approach in ways that are highly complementary. In rebalancing a balanced stock/bond mix, Rattray et al. find that results are improved by allowing trends to persist and then rebalancing the asset mix once trends show signs of reversing. Our focus is on rebalancing within the equity portfolio. Assuming that our chosen factors continue to add value in the years ahead, this approach should capture more return per unit of risk from our stock market holdings.

Data Analysis

We start with some standard academic factors, use them to create long-only factor strategies,⁵ and then measure the efficacy of different rebalancing rules for preserving as much of the factor premiums as possible. We therefore use the same data, rules, and universes as other academic papers for the construction of long-only factor strategies as well as the market portfolio (such as Fama and French 1993).

We use data on stocks listed on the NYSE, AMEX, and Nasdaq from the Center for Research in Securities Prices (CRSP). We include ordinary common shares (share codes 10 and 11) and use CRSP delisting returns. If a stock’s delisting return is missing and the delisting is performance-related, we impute a return of −30% for NYSE and AMEX stocks and −55% for Nasdaq stocks (Shumway and Warther 1999).

We obtain accounting data from annual Compustat files. We follow the Fama and French (1993) convention, lagging accounting information by six months. For example, if a firm’s fiscal year ends in December in year t, we assume that this information is available to investors at the end of June in year t + 1. Our data begin in July 1963 and end in December 2020. When computing factor returns and turnover, we start the sample in July 1964; in other words, we ignore the initial portfolio formation turnover (100%) that occurred in July 1963 because any investor entering the same long-only portfolio would incur the same transaction-related costs.

Regularly Rebalanced Factor Portfolios

We construct value, profitability, investment, and momentum factors. We use the Fama and French (1993, 2015) definitions for the signals underlying the value, profitability, and investment factors: value sorts stocks into portfolios by their B/P ratios, profitability by their operating profits-to-book value of equity, and investment by the annual growth in total assets. The signal underlying the momentum factor is that of Carhart (1997): a stock’s return from month t-12 through t-2, excluding t-1.

Our benchmarks, which we call the reference factors, are long-only factors based on these signals. We sort stocks into quartiles using NYSE breakpoints. We create four reference factors with annual rebalancing, and three are rebalanced monthly. We rebalance the value, profitability, and investment factors annually at the end of June. These factors are long the stocks in the top signal quartile with weights proportional to their end of June market capitalization.⁶ We do not trade these annual factors at all from July to the next May; the weights evolve based on the stocks’ monthly returns excluding dividends. We assume that we reinvest any proceeds from dividends, corporate actions (e.g., mergers and acquisitions), or delisting evenly across all the remaining stocks in the portfolios so that the relative portfolio weights remain unchanged, apart from price drift.⁷

In addition to the value, profitability, and investment factors, we also create an annual composite factor based on the three signals. We construct this composite signal by computing first a cross-sectional robust z score for each individual signal. The robust z score is the stock’s signal minus the median signal across all stocks in the same month, divided by the mean absolute deviation around the median. The composite signal is the average of these three robust z scores. We rebalance this composite factor annually. The composite signal is the only one in which trade “crossing” can happen. For example, if the value factor wants to buy a stock and the profitability factor wants to sell, the composite factor will trade only the net signal.

We rebalance the momentum factor monthly. This factor is also long the stocks in the top signal quartile, and the weights are proportional to the stocks’ current market capitalization. We also create a monthly version of the value factor following Asness and Frazzini (2013). The signal underlying this factor divides the book value of equity not with the December market value of equity (as in Fama and French 1993) but with the current market value of equity. This signal therefore changes monthly as the portfolio stocks’ market values change.

In the long-only factor investing arena, investors are not only interested in getting exposure to a specific factor, one that they believe can deliver excess return over a cap-weighted benchmark through an entire market cycle. They are also keen to diversify the risk associated with investing in a single factor. A multifactor strategy offers a one-stop solution for investors seeking excess returns associated with several factors. The diversifying aspect of combining factors with different risk and return characteristics and low correlations ostensibly helps investors “weather the storm” during adverse market conditions for any single factor. Crossed trades can also reduce turnover, relative to the sum of the single-factor turnover. We also construct a monthly composite factor based on the monthly value and momentum signals. For this monthly version, we use the same z score method we described earlier for the annual composite.

In Table 1 we report average returns, standard deviations, Sharpe ratios, capital asset pricing model (CAPM) alphas, and annual turnovers for the seven long-only factors we study. We also report the market portfolio’s average return and Sharpe ratio for reference. The market factor’s Sharpe ratio over the 1964–2020 period is 0.41. The long-only factors, which hold various segments of the market, earn higher Sharpe ratios, with no exceptions, ranging from 0.60 for the monthly-rebalanced composite factor to 0.47 for the monthly-rebalanced value factor. All factors, except for the monthly value factor, earn CAPM alphas that are statistically significant at the 5% level; most are significant at the 1% level.⁸

Table 1. Performance and Turnover of Reference Factors, Jul 1964–Jun 2020

Rebalancing frequency	Factor	Average return – R0	Standard deviation	Sharpe ratio	CAPM attributes			Turnover
					β	⍺	t(⍺)
None	Market	6.3%	15.5%	0.41
Annual	Value	9.0%	17.5%	0.51	1.00	2.7%	2.49*	39%
	Profitability	7.5%	15.2%	0.49	1.04	1.4%	3.08**	19%
	Investment	8.4%	15.8%	0.53	1.03	2.3%	3.35**	69%
	Compositeann	8.6%	16.4%	0.52	1.05	2.6%	2.64**	51%
Monthly	Value	9.4%	19.9%	0.47	0.92	2.6%	1.79	149%
	Momentum	9.9%	17.5%	0.57	0.97	3.4%	3.61**	304%
	Compositemo	11.0%	18.4%	0.60	0.94	4.3%	3.91**	319%

Notes. The turnover numbers include all trading, including the non-June, corporate event–induced trading for the annual factors.

*Significant at the 5% level.

**Significant at the 1% level.

CAPM = capital asset pricing model.

Source: Research Affiliates using data from CRSP/Compustat.

These Sharpe ratios and alphas are based on the portfolios’ gross returns.⁹ The extent to which an investor falls short of this idealized paper-portfolio performance depends on the turnover the factor strategies incur and how much the underlying stocks cost to trade. Value, profitability, and investment entail slowly changing signals, so much of their alpha can be retained. Table 1 shows that an annually rebalanced value portfolio has an annualized return 40 bps lower than a monthly rebalanced one but incurs only one-third of its turnover rate; net of the incremental trading costs, the monthly rebalancing adds little. For other strategies, up to half of the alpha is dissipated in trading costs.

We should also note that these results are based strictly on the factor returns, without adjusting for revaluation alpha. While it is well beyond the scope of this paper to delve deeply into the concept of revaluation alpha, completeness requires a brief description of this important nuance. If the valuation multiples of a strategy have risen relative to the market, then the past returns will greatly improve, while subsequent returns will (at best) not. It is dangerous to assume that revaluation alpha will persist, because revaluation is presumably a nonrecurring source of alpha, either a random walk or a mean-reverting random walk. Unless upward repricing persists, this source of alpha should disappear. Moreover, mean reversion in the strategy’s relative valuation could take a strong historical alpha and deliver a negative future alpha.¹⁰ The same phenomenon reciprocally plays out the opposite way for factors and strategies that have become meaningfully cheaper relative to the market. Historical alpha will likely be understated, and any mean reversion could lead to a much higher future alpha than that earned in the past.¹¹

In Table 1 and Figure 1, in which we report average turnover by calendar month, we measure a factor’s turnover as one-half of the absolute sum of the differences between the old weights and the new ones. As discussed earlier, we measure active turnover, excluding trades induced by corporate actions (typically mergers and acquisitions), delistings, and dividends. The annual factors, by construction, trade only once a year. If we had a factor that each year switched from one set of stocks to another non-overlapping set of stocks, the factor would incur a maximum one-way turnover of 100%. For monthly factors, by the same logic, the maximum annual turnover is capped at 1200%.

Figure 1. Average Turnover by Calendar Month for Reference Factors, Jul 1964–Jun 2020

Source: Research Affiliates using data from CRSP/Compustat.

Actual turnover, while far less than these limits, varies across both factors and time. Although the signal that drives the value factor, for example, updates once a month, most of the new information arrives at the end of June: because most companies have December fiscal year-ends, this new accounting information starts to be used in June when the six-month “embargo” of Fama and French (1993) expires.¹² When a firm’s book value of equity does not get updated, only its market value of equity changes, and this change is often too small to induce the stock to transition from the top quartile to a lower quartile or vice versa. For the momentum strategy, because the price information updates constantly, its trades spread out fairly evenly across all months.

The turnover for the monthly composite portfolio is usually lower than for the momentum portfolio. Averaging the slower-changing monthly value signal with the fast-changing momentum signal will also cancel some of the trades whenever the signals shift in opposite directions in two or more of the single-factor–based portfolios. We surmise that this same interaction with momentum may play a role in creating a peculiar “kink” in the CAPM alpha deciles that we explore shortly.

Do We Want to Prioritize Trades Based on Signal Values?

We seek to construct a turnover-constrained factor that retains as much of the reference factor’s performance as possible. We can constrain turnover in multiple ways: (1) from the set of proposed trades, select a random subset to hit the turnover limit; (2) complete all trades, but only partially, to hit the target; or (3) prioritize some trades over others based on some criteria.

An intuitive rule for prioritizing trades is based on stocks’ signal values. For example, if two new stocks enter the top quartile and our turnover capacity is sufficient to trade into just one of them, it might make sense to trade the one with the more attractive signal. The signals are, of course, the indication as to which stocks are most or least favored within the context of the particular factor strategy. If expected returns increase in signals, at least near-monotonically, the investor should prefer buying the stocks with the most attractive signals and removing from the portfolio the least attractive stocks.¹³ We place every stock the investor would like to trade in two queues, then sort the queues by the signal values. The buy queue is sorted in descending order and the sell queue in ascending order. The investor then begins to process the trades in the order of the respective queues, buying the most attractive stocks in the buy queue and selling the least attractive in the sell queue, stopping the process at the turnover limit.

In Table 2, we report CAPM alphas and t statistics associated with these CAPM alphas for each decile, for each of our seven factors, sorted by signal strength. We use this computation to assess whether returns are monotonic on the signal strength. We sign the signals in the same way as we sign them when constructing the factors; the high decile represents stocks with (1) high B/P ratios (value firms), (2) high profitability (profitable firms), (3) low asset growth (in the Fama–French nomenclature, this is the “investment” factor, for firms that reinvest profits conservatively), and (4) high recent returns (momentum). The seven sets of portfolios are similar to those underlying the reference factors; whereas the reference factor holds a long position in the top quartile of stocks, here we divide stocks into deciles and report CAPM alphas for all of them.

Table 2. Average Excess Returns and t Statistics for Decile Portfolios, Jul 1964–Jun 2020

Rebalancing frequency	Factor	Decile
		Low	2	3	4	5	6	7	8	9	High
Annual	Value	−0.6%	0.7%	0.7%	0.1%	0.4%	2.1%	0.9%	2.1%	2.7%	2.6%
		(-0.63)	(0.96)	(1.18)	(0.16)	(0.51)	(2.45)*	(0.92)	(1.94)	(2.42)*	(1.80)
	Profitability	−3.1%	−2.0%	−2.1%	0.6%	−0.2%	0.2%	−0.9%	0.9%	2.1%	1.4%
		(-2.40)*	(-1.96)*	(-2.58)**	(0.87)	(-0.21)	(0.28)	(-1.50)	(1.53)	(3.29)**	(1.91)
	Investment	−3.2%	−1.6%	−0.1%	1.2%	0.6%	1.5%	1.5%	1.7%	2.3%	2.0%
		(-3.42)**	(-1.78)	(-0.17)	(1.90)	(0.85)	(2.13)*	(2.20)*	(2.21)*	(2.70)**	(1.98)*
	Compositeann	−11.0%	−4.1%	−1.1%	−0.4%	−0.3%	0.5%	0.5%	2.3%	2.5%	6.0%
		(-3.25)**	(-2.73)**	(1.71)	(1.50)	(1.04)	(2.11)*	(1.52)	(2.54)*	(2.34)*	(2.57)*
Monthly	Value	0.7%	−0.6%	−0.2%	−0.1%	0.4%	1.7%	1.0%	2.3%	2.4%	2.5%
		(0.76)	(-0.83)	(-0.23)	(-0.19)	(0.51)	(1.76)	(1.02)	(1.84)	(1.62)	(1.23)
	Momentum	−11.0%	−4.1%	−1.1%	−0.4%	−0.3%	0.5%	0.5%	2.3%	2.5%	6.0%
		(-4.77)**	(-2.58)**	(-0.84)	(-0.42)	(-0.36)	(0.58)	(0.62)	(2.93)**	(2.66)**	(3.95)**
	Compositemo	−3.2%	−0.9%	0.8%	1.6%	0.9%	1.4%	2.2%	4.2%	4.0%	4.9%
		(-3.72)**	(-1.24)	(1.32)	(2.23)*	(1.24)	(1.90)	(2.71)**	(4.59)**	(3.81)**	(3.12)**

Notes: The long-only, top-quartile portfolios hold the stocks in deciles 10 and 9 and half of decile 8. For the priority-best rebalancing rule to be superior to other schemes, average returns are approximately monotonic when you go from decile 8 into 9 and 9 into 10. If so, it makes sense to "prioritize" trades that alter the portfolio’s signal the most.

*Significant at the 5% level.

**Significant at the 1% level.

Source: Research Affiliates using data from CRSP/Compustat.

The estimates in Table 2 indicate that expected returns are not entirely monotonic for most of the factors’ signals.¹⁴ For example, the alphas associated with the three annual signals—value, profitability, and investment—are better in the ninth decile than in the top decile. Although these differences are not statistically significant, a trading rule that mostly buys stocks in that top decile may not always add value. The decile portfolios associated with the investment factor best illustrate this issue. The portfolio of stocks represented by the top decile associates with a lower t statistic (proportional to information ratio) than deciles six through nine. This implies that if we are contemplating trading into new stocks that now look best based on their investment characteristic, we would prefer stocks that “marginally” fit our investment rule rather than those in the “best” decile of the signal distribution.

It would be easy to dismiss this “kink” in the CAPM alpha deciles in Table 2 as mere noise. We think noise is probably not the sole culprit for this non-monotonicity. If it were, we would also expect to find similar non-monotonicity in the distribution’s left tail: stocks in the bottom decile of some factors should not always underperform those in decile two. Unlike in the right-tail of the distribution, the stocks in the bottom decile are, except for the monthly value factor, markedly worse investments than those in decile two.¹⁵

The estimates in Table 2 suggest that a simple trading rule that prioritizes buying stocks with high signals (and disposing of those with low signals) should work best with the momentum and composite signals; the non-monotonicity likely impairs the rule’s efficacy with the investment factor; and the remaining factors fall between these extremes.

Priority Rebalancing, Calendar-Based

In the real world, in the presence of market frictions and trading costs, investors may find that the expense of fully rebalancing their portfolio exceeds the benefits. If potential trades can be ranked by their attractiveness, in terms of their expected contribution to the portfolio’s alpha, the marginal benefit of trading diminishes but the costs associated with execution do not, as we relax our turnover limit and execute 100% of the indicated trades.

There are many ways to trade “smartly.” In this study, we apply turnover limits to the reference factor portfolios and prioritize trades based on how high the stock’s signal ranks. For example, when we buy stocks, we rank all stocks in the universe by the attractiveness of the strategy’s corresponding signal (e.g., the B/P or other valuation ratios, for value) from high to low and buy the stocks at the top of the list first. When we sell stocks, we first sell those ranked at the bottom. We match the dollar amounts of buys and sells to ensure that the trades are dollar-neutral. The total amount to buy and sell is determined by the turnover limit: when it is high, we buy and sell more stocks; when it is low, we buy and sell fewer stocks. We call this trading rule, which prioritizes purchases with the highest signals and sales with the lowest signals, the priority-best rule.

To make the comparison interesting, we also create the opposite rebalancing rule: with priority-worst trading, we rank stocks by the strategy’s signals but reverse the order in which we trade. When we process purchases, we start with the weakest buy signals—that is, the most marginal stocks, which often just barely made it onto our buy list—and we start our selling with the stocks that barely make it onto the sell list. We compare the priority-best and priority-worst rules to measure two different ways an investor may benefit (and how much) by trying to rank trades based on their attractiveness. If expected returns increase approximately monotonically with the strategy’s signals, the priority-best rule should earn more of the alpha, per unit of turnover, than the unconstrained portfolio. With the priority-worst rule, by contrast, we expect to see minimal benefits until we accept relatively loose turnover limits (i.e., trading most of the buy and sell lists) because this rule leaves the seemingly most attractive trades to the end.

We also consider the simplest way to meet the turnover requirement: the proportional trade rule cuts the trade size across all stocks pro rata. For example, if the unconstrained rebalance requires selling or buying 20% of the portfolio and the turnover limit is 10%, then we execute half of each of the proposed buys and sells.

Only trades with sufficient conviction can generate a post-trading-cost benefit to investors. If the signals were to convey perfect information about the stocks’ future performance, a fully rebalanced portfolio would deliver the best outcome, though not necessarily net of trading costs. In the real world, where the signals are noisy and imperfect predictors of expected returns and trades are costly, a full-fledged rebalance is not likely to be the best solution. We now impose turnover limits and measure how the resulting turnover-constrained portfolios perform relative to each other and the unconstrained reference factors. We set the annual one-way turnover limit to 10%, 20%, 50%, and 100%.¹⁶

In Table 3, we report average annual returns, CAPM alphas, the corresponding t statistics associated with these alphas, annualized turnover, and net-of-turnover alphas and t statistics for portfolios managed using the priority-best rule to satisfy the turnover limit. Panel A shows that the value portfolio earns its highest return when turnover is constrained to a mere 20% per annum (not counting forced turnover due to corporate actions). Panel B shows that the value portfolio’s CAPM alpha reaches its highest value (2.9% per year) under the same 20% annual turnover limit. The additional 19% turnover for the unconstrained value strategy actually erodes the end result. Note that this is before subtracting any estimates of trading costs! As we loosen the constraint, the strategy’s turnover increases and its alpha converges to match the unconstrained strategy (2.7% per year).

Table 3. CAPM Alphas and Average Turnover Using Priority-Best Trading Rule to Control Turnover of Long-Only Factors, Jul 1964–Jun 2020

		Panel A: Average annual returns					Panel E: CAPM alpha/average turnover
Rebalancing frequency	Factor	None	Annual turnover limit				None	Annual turnover limit
			10%	20%	50%	100%		10%	20%	50%	100%
Annual	Value	9.0%	8.7%	9.2%	9.0%	9.0%	7.0%	23.3%	14.6%	7.2%	7.0%
	Profitability	7.6%	7.3%	7.6%	7.6%	7.6%	8.2%	14.6%	9.2%	8.2%	8.2%
	Investment	8.6%	7.2%	8.0%	8.7%	8.6%	3.6%	10.8%	8.8%	5.2%	3.6%
	Compositeann	8.6%	8.2%	8.9%	8.8%	8.6%	4.8%	20.2%	13.3%	5.8%	4.8%
Monthly	Value	9.5%	8.5%	8.8%	8.7%	8.9%	1.8%	21.0%	11.6%	3.9%	2.2%
	Momentum	10.0%	7.2%	8.2%	10.6%	10.8%	1.1%	2.1%	5.4%	6.6%	4.0%
	Compositemo	11.1%	8.7%	8.6%	9.7%	10.3%	1.4%	22.3%	10.9%	5.6%	3.4%
		Panel B: Annual CAPM alphas					Panel F: Annual Net CAPM alphas
Rebalancing frequency	Factor	None	Annual turnover limit				None	Annual turnover limit
			10%	20%	50%	100%		10%	20%	50%	100%
Annual	Value	2.7%	2.3%	2.9%	2.7%	2.7%	2.2%	2.0%	2.5%	2.2%	2.2%
	Profitability	1.5%	1.5%	1.6%	1.5%	1.5%	1.3%	1.2%	1.4%	1.3%	1.3%
	Investment	2.5%	1.1%	1.8%	2.6%	2.5%	1.9%	0.7%	1.4%	2.0%	1.9%
	Compositeann	2.6%	2.0%	2.7%	2.7%	2.6%	2.0%	1.7%	2.3%	2.2%	2.0%
Monthly	Value	2.7%	2.1%	2.3%	1.9%	2.1%	1.5%	1.7%	1.9%	1.3%	1.2%
	Momentum	3.4%	0.2%	1.1%	3.3%	4.0%	1.7%	−0.1%	0.7%	2.7%	3.1%
	Compositemo	4.4%	2.2%	2.2%	2.8%	3.4%	2.3%	1.9%	1.8%	2.1%	2.5%
		Panel C: CAPM t(⍺)s					Panel G: Net CAPM t(⍺)s
Rebalancing frequency	Factor	None	Annual turnover limit				None	Annual turnover limit
			10%	20%	50%	100%		10%	20%	50%	100%
Annual	Value	2.46*	2.23*	2.73**	2.51*	2.46*	1.97*	1.89	2.34*	2.02*	1.97*
	Profitability	3.19**	2.42*	3.28**	3.19**	3.19**	2.65**	2.06*	2.76**	2.65**	2.65**
	Investment	3.63**	1.50	2.49*	3.60**	3.63**	2.68**	1.04	1.92	2.81**	2.68**
	Compositeann	2.58**	2.26*	2.66**	2.71**	2.58**	2.04*	1.93	2.29*	2.20*	2.04*
Monthly	Value	1.81	1.84	1.94	1.46	1.46	1.01	1.51	1.56	0.98	0.85
	Momentum	3.64**	0.32	1.30	3.23**	3.77**	1.80	−0.16	0.82	2.66**	2.99**
	Compositemo	3.90**	2.29*	2.56**	2.75**	3.10**	2.04*	1.94	2.06*	2.12*	2.27*
		Panel D: Average turnover					Panel H: Average annual trading cost
Rebalancing frequency	Factor	None	Annual turnover limit				None	Annual turnover limit
			10%	20%	50%	100%		10%	20%	50%	100%
Annual	Value	39%	10%	20%	38%	39%	0.5%	0.4%	0.4%	0.5%	0.5%
	Profitability	19%	10%	18%	19%	19%	0.3%	0.2%	0.3%	0.3%	0.3%
	Investment	70%	10%	20%	50%	70%	0.7%	0.3%	0.4%	0.6%	0.7%
	Compositeann	53%	10%	20%	47%	53%	0.5%	0.3%	0.4%	0.5%	0.5%
Monthly	Value	151%	10%	20%	50%	93%	1.2%	0.4%	0.5%	0.6%	0.9%
	Momentum	305%	10%	20%	50%	100%	1.7%	0.3%	0.4%	0.6%	0.8%
	Compositemo	322%	10%	20%	50%	100%	2.1%	0.3%	0.4%	0.6%	0.9%

Notes. *Significant at the 5% level.

**Significant at the 1% level.

Bold type denotes the highest alphas and t statistics.

CAPM = capital asset pricing model.

Source: Research Affiliates using data from CRSP/Compustat.

While this result is surprising—with less turnover to match a model portfolio, we would expect the alpha to erode, at least modestly—it bears mention that it is not statistically significant and may easily result from random noise in market returns. While it is beyond the scope of this paper to diagnose a statistically insignificant finding, if it is not random noise, the answer may well lie in the nonlinearity of returns.

Consider the value signal, where the best constrained alpha of 2.9% exceeds the unconstrained alpha of 2.7%. In Table 2, we can see that returns for the most favored purchases in deciles 9 and 10 are about the same, but the bottom two are very different. If the unconstrained value factor is (hypothetically) equally buying the top three deciles and equally selling the bottom three, then we are buying stocks with a blended average alpha of 2.5% and selling stocks with a blended average alpha of 0.3%, for a spread of 2.2%. If the constrained factor is buying the top decile and selling the bottom decile, we are buying stocks with an alpha of 2.6% and selling stocks with an alpha of −0.6%, for a wider spread of 3.2%. As we will get around to the other trades in the following year, if they then become compelling, it is easy to see how we might capture more alpha with less turnover.

It bears mention that the average turnover under a “binding” turnover limit can be less than the limit itself. For example, when the annual constraint for the composite factor is 50%, the average realized turnover is 47% because in some years the target turnover is less than the 50% constraint, so the average turnover is strictly less than 50%. This same mechanism—the constraint binds in some years but not in all—is also the reason the average turnover for the annual version of profitability is 18% (not the unconstrained turnover of 19%), when the turnover limit is 20%. There are a few years when the portfolio would trade more than 20%, and this constraint cuts the turnover in those few years down to 20%.

The investment portfolio tends to deliver a higher alpha when we loosen the turnover limit. The investment strategy incurs the highest turnover (70%) when left unconstrained, but setting the constraint at 50% is apparently enough to capture nearly all of the trading benefits. Partial rebalancing for the annual strategy based on the composite signal seems to be sufficient in capturing most of the premium, because when we increase the annual turnover limit from 10% to 20%, the alpha increases from 2.0% to 2.7%, and the t statistic associated with this alpha increases from 2.26 to 2.66, very near the peak significance which occurs at a looser 50% turnover limit.¹⁷

Perhaps surprising to the casual observer, the composite has unconstrained turnover that markedly exceeds the average turnover of the three constituent strategies. This is because, if any of the strategies wants to trade any particular stock, the composite will typically want to do the same. The exception is the occasional (and infrequent) crossed trade. If value wants to buy a stock and profitability wants to sell, these trades cancel. So, the annual composite has turnover more closely matching that of investment, the highest-turnover strategy, while capturing very nearly the full benefit of unconstrained trading with a mere 20% turnover limit.

For the monthly-rebalanced strategies, the alpha for the momentum strategy increases monotonically as we increase the turnover limit. This behavior is consistent with our earlier observation from Table 2 that its decile portfolios show a continuous, near-monotonic upward trend in performance. However, because the unconditional turnover for momentum is very high at 305%, this strategy’s CAPM alpha does not become statistically significant at the 5% level until the annual turnover limit reaches 50%. By the time we have relaxed the limit to 100% per year, the constrained and unconstrained strategies are close to each other for the alphas (4.0% and 3.4%, respectively) and the t statistics associated with the alphas (3.77 and 3.64, respectively).

The monthly value factor does not show as clear a pattern and performs distinctly worse than the annual value factor. Its alpha starts at 2.1% per year when the turnover limit is very tight (i.e., 10%), rises modestly to 2.3% with a more relaxed 20% turnover limit, then decreases to 1.9% for the 50% turnover limit and never quite converges to the unconstrained value of 2.7% even when the turnover limit is 100%. Why is it worse than the annual value factor, and why do results not converge toward the unconstrained return even with liberal turnover limits of 50% or 100%? The monthly value factor uses the most recently recorded market value of equity, relative to a book value-of-equity signal that updates only once a year. So, the monthly value factor, in all months except June, will have a signal that is essentially an anti-momentum signal. That is, the value signal alone favors stocks that have recently performed poorly, so most of the monthly turnover will be trading against momentum. The signal therefore would be more reliable with a control for the momentum part of the signal, buying when the value signal is strong and the momentum signal is no longer dire.¹⁸

The monthly composite factor, which is a blend of the monthly value and momentum signals, approximates this approach. The monthly-rebalanced composite strategy’s alpha increases monotonically as we loosen the constraint. When the annual turnover limit is 100%, this strategy’s annual alpha is fully 100 bps below that of the unconditional version. Not shown here, to earn the next 50 bps of alpha, an investor needs to incur another 50 percentage points of annual turnover.¹⁹ But, as we will see shortly, at a 100% turnover limit, the alpha net of trading costs already exceeds that of the unconstrained strategy, at barely one-third of the turnover.

Trading Costs and Net Alphas

The estimates in Panels A, B, and C of Table 3 are gross of trading costs. They indicate that, particularly relative to the annually rebalanced strategies, an investor could have earned all of an unconstrained strategy’s gross alpha, and in some cases a bit more, by constraining turnover using the priority-best rule. However, the primary benefit of imposing turnover limits in trading is to lower trading costs. We quantify the trading costs of the priority-best rule in Table 3, Panel H, just to the right of the turnover itself in Panel D. We use low-frequency (monthly) trading cost estimates provided by Chen and Velikov (2023). Because their sample ends in December 2017, the data presented in Panels E through H are 2.5 years shorter than those in Panels A through D.²⁰

While we measure the net-of-trading cost alphas in Panels F through H, on the presumption that the Chen and Velikov trading costs are correct, there is an alternative way to finesse this challenge. Suppose we take the ratio of alpha to turnover, which we see in Panel E. This measure, perhaps first introduced in Arnott, Hsu, and Moore (2005), tells us how high trading costs would need to be in order to wipe out our alpha. If the figure is small, at 1% or less, there is a real risk that trading costs will eliminate our alpha; if the ratio is large, at perhaps 10% or more, there is essentially no risk that trading costs could wipe out our gross alpha.

Would the priority-best rule have benefited an investor, net of trading costs? The results in Panels F and G of Table 3 indicate that the answer is an emphatic “yes.” The annual trading costs associated with the unconstrained reference factors align with their respective turnover. The lowest-turnover factor, profitability, incurs an average trading cost of 0.3% per year. By contrast, the highest turnover strategy, the monthly rebalanced composite factor, incurs a cost of 2.1% when left unconstrained. Turnover restrictions reduce trading costs very roughly in proportion to the realized reduction in turnover. For example, the momentum factor’s trading cost increases monotonically from 0.3% to 0.8% as we relax the constraint from 10% per year to 100% per year. Note that, when the constraint is 50%, we already capture more alpha net of trading costs than the unconstrained result, with turnover reduced by five-sixths, indeed because of that reduced turnover.

Importantly, because the turnover restriction is not random, the data do not guarantee that this is a linear relationship. That is, the trades preferred by the priority-best could be disproportionately more expensive to trade than more marginal stocks. For example, deep-value stocks’ bid–ask spreads could be significantly higher than those of “normal”-value stocks. This is likely the reason that trading costs for value rise only minimally—from 0.4% to 0.5%—as we quadruple the turnover from the 10% turnover limit to the 39% unconstrained turnover, which means that the most attractive deep-value stocks are also the most expensive to trade.²¹

Panels B and F show the alpha gross and net of the trading costs shown in Panel H. Value, traded with a 20% turnover limit, delivers a return of 9.2%, which works out to a CAPM alpha of 2.9%. That 20% turnover limit causes us to incur 0.4% in annual trading costs. The resulting alpha net of trading costs is 2.5%, well above the 2.2% alpha for the unconstrained portfolio. And the 2.34 t statistic for the net alpha is well above that of the unconstrained value strategy, which sports a barely significant 1.97 t statistic.

The alphas net of trading costs in Panel F confirm that the realized benefit of constraining turnover using the priority-best rule is often substantial. Indeed, there is not a single factor, annual or monthly, where the alpha for the unconstrained strategy, net of trading costs, cannot be improved by imposing some turnover limit. Similarly, while the t statistic for the CAPM alpha is best for the unconstrained strategy in two instances (the annual investment factor and the monthly composite), the t statistic for the net-of-trading costs CAPM alpha is always better for some level of constrained turnover than for the unconstrained, with no exceptions.

It will come as no surprise that turnover management becomes even more compelling in the small-cap arena. Table 4 shows essentially the same comparisons that we just reviewed in Table 3, when the identical long-only factor strategies are run in the small-cap arena (using the median market cap of the NYSE as the cutoff for defining large- and small-cap). The practitioner will not be surprised to find that, if we choose the best-performing turnover limit for each factor, the alphas are higher, sometimes much higher in the small-cap arena than in large-cap, for each of the factors. Likewise, the turnover and estimated trading costs are higher for the small-cap strategies. Net of these elevated trading costs, the best net alphas are also generally higher than in the large-cap arena. The t statistics of the factor alphas, at the ex-post best turnover limit, are also generally higher. For gross returns the investments factor is an exception, while for net returns (net of trading costs) the profitability factor and monthly composite are exceptions.

Table 4. Returns, CAPM Alphas, Turnover, and Trading Costs Using Priority-Best Trading Rule to Control Turnover of Small-Cap Long-Only Factors, Jul 1964-Dec 2017

		Panel A: Average annual returns					Panel E: CAPM alpha/average turnover
Rebalancing frequency	Factor	None	Annual turnover limit				None	Annual turnover limit
			10%	20%	50%	100%		10%	20%	50%	100%
Annual	Value	10.8%	11.0%	11.1%	11.0%	10.8%	8.4%	45.6%	21.7%	9.0%	8.4%
	Profitability	9.9%	11.6%	11.2%	9.9%	9.9%	5.9%	56.0%	21.3%	6.1%	5.9%
	Investment	10.1%	10.1%	10.1%	10.5%	10.1%	3.6%	32.9%	14.5%	5.6%	3.6%
	Compositeann	11.6%	11.7%	12.3%	12.0%	11.6%	7.6%	53.2%	28.0%	10.1%	7.6%
Monthly	Value	11.5%	10.5%	10.9%	9.4%	10.5%	2.0%	37.2%	20.0%	3.8%	2.5%
	Momentum	13.5%	11.2%	11.4%	11.2%	13.3%	2.2%	39.7%	18.5%	6.5%	5.1%
	Compositemo	15.0%	8.9%	9.0%	12.6%	13.1%	2.2%	19.4%	9.3%	11.0%	5.5%
		Panel B: Annual CAPM alphas					Panel F: Annual Net CAPM alphas
Rebalancing frequency	Factor	None	Annual turnover limit				None	Annual turnover limit
			10%	20%	50%	100%		10%	20%	50%	100%
Annual	Value	3.8%	4.6%	4.3%	4.0%	3.8%	2.5%	3.7%	3.4%	2.7%	2.5%
	Profitability	2.6%	5.6%	4.3%	2.7%	2.6%	1.7%	5.0%	3.5%	1.7%	1.7%
	Investment	2.5%	3.3%	2.9%	2.8%	2.5%	0.8%	2.4%	1.8%	1.2%	0.8%
	Compositeann	4.6%	5.3%	5.6%	5.0%	4.6%	3.2%	4.6%	4.7%	3.7%	3.2%
Monthly	Value	3.3%	3.7%	4.0%	1.9%	2.5%	0.6%	2.8%	2.9%	0.4%	0.5%
	Momentum	5.9%	4.0%	3.7%	3.2%	5.1%	2.9%	3.4%	3.0%	2.1%	3.4%
	Compositemo	7.4%	1.9%	1.9%	5.5%	5.5%	3.5%	1.3%	1.1%	4.3%	3.7%
		Panel C: CAPM t(⍺)s					Panel G: Net CAPM t(⍺)s
Rebalancing frequency	Factor	None	Annual turnover limit				None	Annual turnover limit
			10%	20%	50%	100%		10%	20%	50%	100%
Annual	Value	2.37*	2.63**	2.84**	2.46*	2.37*	1.57	2.16*	2.20*	1.67	1.57
	Profitability	1.98*	3.51**	3.00**	2.01*	1.98*	1.26	3.16**	2.49*	1.30	1.26
	Investment	1.57	1.99*	1.70	1.65	1.57	0.47	1.47	1.04	0.69	0.47
	Compositeann	2.89**	3.33**	3.58**	3.16**	2.89**	2.01*	2.87**	2.98**	2.33*	2.01*
Monthly	Value	1.46	1.96*	2.27*	1.00	1.18	0.27	1.49	1.66	0.19	0.22
	Momentum	4.08**	1.67	1.62	2.12*	3.14**	1.97*	1.43	1.30	1.35	2.10*
	Compositemo	5.05**	0.64	0.64	3.11**	3.11**	2.40*	0.44	0.38	2.43*	2.09*
		Panel D: Average turnover					Panel H: Average annual trading cost
Rebalancing frequency	Factor	None	Annual turnover limit				None	Annual turnover limit
			10%	20%	50%	100%		10%	20%	50%	100%
Annual	Value	45%	10%	20%	44%	45%	1.3%	0.8%	1.0%	1.3%	1.3%
	Profitability	44%	10%	20%	44%	44%	0.9%	0.6%	0.7%	0.9%	0.9%
	Investment	71%	10%	20%	50%	71%	1.8%	0.9%	1.1%	1.6%	1.8%
	Compositeann	60%	10%	20%	50%	60%	1.4%	0.7%	0.9%	1.3%	1.4%
Monthly	Value	167%	10%	20%	50%	100%	2.7%	0.9%	1.1%	1.5%	2.0%
	Momentum	274%	10%	20%	50%	100%	3.1%	0.6%	0.7%	1.2%	1.7%
	Compositemo	338%	10%	20%	50%	100%	3.9%	0.6%	0.8%	1.2%	1.8%

Notes. *Significant at the 5% level.

**Significant at the 1% level.

Bold type denotes the highest alphas and t statistics.

CAPM = capital asset pricing model.

Source: Research Affiliates using data from CRSP/Compustat.

Trading costs are difficult to measure with any precision, because we are measuring a counterfactual: the difference between the share price at which we trade and the share price if we had chosen not to trade. As in our large-cap study, we can alternatively consider the ratio of alpha to turnover in Panel E. This simple measure is higher in small-cap stocks than in large-cap stocks for all the factors except the monthly composite factor. While turnover and trading costs are elevated, the alpha improves more than proportionally and so more than covers these incremental costs. This means that the factor strategies are more robust in small-cap portfolios than in large-cap portfolios.

Subsample Analysis of the Priority-Best Rule

The estimates in Table 3 indicate that an investor would have benefited from priority-best trading over the 1964–2020 sample period, not only from lower trading costs but also, for some factors, from higher gross alphas, before subtracting trading costs. In Table 5, we measure how the optimal turnover threshold and the benefits from turnover limits vary over time. We split the overall sample period into three segments: July 1964 to June 1983, July 1983 to June 2002, and July 2002 to June 2020. For each of the segments, we report gross and net alphas together with their t statistics for two versions of each factor: an unconstrained factor and an ex-post optimally constrained factor. We select the optimal threshold based on factors’ in-sample net-of-trading costs alphas.

Table 5. Subsample Results for Using Priority-Best Trading Rule to Control Turnover of Long-Only Factors, Jul 1964-Jun 2020

Panel A: July 1964 to June 1983
	Factor	Objective	Optimal limit	Turnover	Gross alphas		Net alphas
					⍺gross	t(⍺gross)	⍺net	t(⍺net)
Annual	Value	Unconstrained	None	40.5%	6.1%	3.30	5.6%	0.41
		Max ⍺net	45%	38.1%	6.2%	3.33	5.6%	3.05
	Profitability	Unconstrained	None	16.9%	0.7%	0.78	0.4%	0.50
		Max ⍺net	20%	16.0%	0.7%	0.83	0.5%	0.55
	Investment	Unconstrained	None	69.5%	3.2%	2.78	2.5%	2.21
		Max ⍺net	20%	20.0%	4.1%	3.29	3.7%	2.97
	Composite	Unconstrained	None	52.9%	4.5%	2.70	4.0%	2.40
		Max ⍺net	35%	35.0%	4.8%	2.92	4.4%	2.66
Monthly	Value	Unconstrained	None	164.7%	7.6%	3.33	6.4%	2.80
		Max ⍺net	265%	155.9%	7.8%	3.43	6.6%	2.91
	Momentum	Unconstrained	None	304.6%	6.7%	3.84	5.0%	2.85
		Max ⍺net	125%	125.0%	7.5%	3.81	6.5%	3.34
	Composite	Unconstrained	None	330.9%	8.1%	4.60	5.9%	3.39
		Max ⍺net	225%	219.1%	8.3%	4.54	6.8%	3.70
Panel B: July 1983 to June 2002
					Gross alphas		Net alphas
	Factor	Objective	Optimal limit	Turnover	⍺gross	t(⍺gross)	⍺net	t(⍺net)
Annual	Value	Unconstrained	None	40.4%	4.1%	2.58	3.5%	2.21
		Max ⍺net	15%	15.0%	5.2%	3.24	4.8%	2.98
	Profitability	Unconstrained	None	19.8%	2.4%	2.65	2.1%	2.30
		Max ⍺net	20%	18.0%	2.4%	2.61	2.1%	2.27
	Investment	Unconstrained	None	69.2%	3.6%	2.72	2.9%	2.15
		Max ⍺net	45%	45.0%	3.8%	2.83	3.2%	2.35
	Composite	Unconstrained	None	53.4%	5.6%	3.28	5.0%	2.94
		Max ⍺net	20%	20.0%	6.2%	3.74	5.8%	3.51
Monthly	Value	Unconstrained	None	155.1%	3.7%	1.60	2.4%	1.03
		Max ⍺net	20%	20.0%	4.4%	2.51	3.8%	2.21
	Momentum	Unconstrained	None	290.8%	2.7%	1.78	0.8%	0.55
		Max ⍺net	135%	134.7%	3.4%	2.20	2.3%	1.48
	Composite	Unconstrained	None	332.1%	6.0%	2.56	3.7%	1.57
		Max ⍺net	215%	213.7%	6.4%	2.71	4.7%	2.00
Panel C: July 2002 to June 2020
	Factor	Objective	Optimal limit	Turnover	Gross alphas		Net alphas
					⍺gross	t(⍺gross)	⍺net	t(⍺net)
Annual	Value	Unconstrained	None	34.6%	−3.0%	−1.55	−3.5%	−1.80
		Max ⍺net	10%	10.0%	−2.3%	−1.42	−2.6%	−1.61
	Profitability	Unconstrained	None	20.2%	1.9%	2.77	1.7%	2.45
		Max ⍺net	10%	10.0%	2.7%	2.96	2.6%	2.77
	Investment	Unconstrained	None	70.6%	0.8%	0.76	0.2%	0.20
		Max ⍺net	90%	70.6%	0.8%	0.76	0.2%	0.20
	Composite	Unconstrained	None	52.3%	−2.9%	−1.91	−3.4%	−2.25
		Max ⍺net	10%	10.0%	−1.8%	−1.35	−2.1%	−1.54
Monthly	Value	Unconstrained	None	131.0%	−4.8%	−1.86	−5.7%	−2.24
		Max ⍺net	5%	5.0%	−1.8%	−1.02	−2.1%	−1.19
	Momentum	Unconstrained	None	319.6%	1.2%	0.76	−0.4%	−0.25
		Max ⍺net	5%	5.0%	3.4%	2.70	3.2%	2.56
	Composite	Unconstrained	None	303.0%	−1.2%	−0.78	−3.0%	−1.91
		Max ⍺net	10%	10.0%	0.3%	0.22	0.0%	0.01

Notes. Bold type denotes the highest alphas and t statistics.

CAPM = capital asset pricing model. Source: Research Affiliates using data from CRSP/Compustat.

To illustrate the computation, consider the annual value factor in the middle part of the sample that runs from July 1983 to June 2002. In this sample, the value factor realizes a turnover of 40.4% per year and earns gross and net alphas of 4.1% and 3.5%. During this sample period, an investor implementing the priority-best rule with a turnover limit set at 15% would have realized turnover of 15% (that is, the constraint binds throughout the sample) and gross and net alphas of 5.2% and 4.8%, respectively. The 15% threshold is the ex-post optimal threshold in that this threshold maximizes the net-of-trading cost alpha during this sample period. We search for the optimal threshold via a grid search, considering all thresholds in 5–percentage point increments starting at a 5% threshold.²²

The estimates in Table 5 show that the priority-best rule would have been consistently beneficial to an investor in an intuitive way whenever the underlying reference factor has delivered a premium. During these times, an investor would typically have saved at least something in trading costs by constraining the turnover (while making smaller sacrifices, if any, in gross alphas). However, when the reference factor does not perform well, the results are, at first glance, counterintuitive. For example, if the unconstrained monthly value factor earns a gross alpha of −4.8% in the last subsample with a turnover of 131.0%, why is the optimal turnover threshold 5%? The reason is that, with such a negative performance, it would have been ideal not to trade value during this period; the algorithm gets closest to this point in the data by limiting the turnover as much as possible. At the 5% threshold, which is the lowest threshold we consider, the investor realizes a turnover of just 5%, a gross alpha of −1.8%, and a net alpha of −2.1%; the portfolio is still a value portfolio and therefore loses during this subsample. With a 5% turnover limit, the investor would have saved money both by making the portfolio very stale (and therefore less value-like) and by spending less on trading costs.²³

The anemic and often negative alphas of the most recent subsample are noteworthy. It is beyond the scope of this paper to delve deeply into these outcomes, but two observations bear mention. First, the earlier subsamples were partially or entirely in-sample for some of these factors. These good in-sample returns were at least a contributing reason for the subsequent identification or “discovery” of these factors. Basu (1977) is often credited as the first to publish a formal study of the value factor, midway through the first subsample. Jegadeesh and Titman (1993) are often credited with the first formal examination of momentum, midway through the second subsample. Finally, profitability and investment—Fama and French (2006, 2015)—were arguably discovered during the third subsample, meaning that the first two subperiods were entirely in-sample backtests of factors discovered and popularized far later. Of course the in-sample returns were brilliant, else the factors would never have been published.

Second, as documented in Arnott et al. (2021), the value factor owes its negative return in the latest subperiod to revaluation alpha, specifically a vast widening in the valuation spread between growth and value stocks. Absent this negative revaluation alpha (i.e., if relative valuations of value relative to growth were the same at the end of the subsample as at the beginning), the alpha would have been positive. Because value is part of both the annual and monthly composites, the negative composite alphas in the most recent subperiod—both annual and monthly—are also a direct result of the downward revaluation of value relative to growth. In other words, absent downward revaluation, factors in the latest subperiod would all have been positive, albeit lower than in the previous subperiods.

In Figure 2, we illustrate how the benefits from optimal turnover limits accrue over time. In this analysis, similar to that presented in Table 5, we compare the unconstrained factor to the ex-post optimally turnover-constrained factor. Net of trading costs, the latter wins in every case. That is, we identify the rule which would have benefited the investor the most over the 1964–2017 sample period, for which we have Chen–Velikov trading cost estimates. The purpose of this analysis is not to measure the absolute achievable benefits from turnover limits—after all, the rules we apply are ex-post optimal—but instead to measure whether they accrue evenly over time. We estimate for both the unconstrained and constrained factors the market betas and use these betas to remove the market component from the factor returns. We then graph the performance of the unconstrained factors; this difference measures the monthly factors.

Figure 2. Cumulative Abnormal Return (Net of Market Beta, Net of Trading Costs) from Priority-Best Turnover Limits vs. Unconstrained, Jul 1964–Dec 2017

Source: Research Affiliates using data from CRSP/Compustat.

In all cases, priority-best trading limits offer the opportunity of improving the returns. Panel A focuses on the annual factors, and Panel B focuses on the monthly factors. These busy graphs are simpler than they seem. Consider the green lines in Panel A. The solid green line finishes at 1.72, meaning that, net of all trading costs, an investor in the long-only profitability factor would be 72% richer after 53½ years than an investor in the S&P 500. The dashed green line finishes at 1.79, which means that—with an optimal level of priority-best turnover control—we can do about 4% better. After over a half-century, this is hardly worth the bother, though the slender difference is unsurprising for a factor with just 19% unconstrained turnover.

By contrast, the purple lines in Panel B reflect the results for the monthly momentum factor. Unconstrained monthly implementation of momentum, with average turnover of 305% per annum, leaves us with relative wealth of 222%, better than twice as wealthy as the S&P 500 index fund investor, net of all trading costs. Even this worthy gain leaves ample room for improvement. Optimal turnover control cuts our trading by two-thirds, to 100% per annum, and leaves us more than twice as wealthy again net of all trading costs, 450% as wealthy as the S&P 500 index fund investor.

Sometimes priority-best trading helps only a little, as with the investment factor. In other cases, priority-best trading helps a lot, as with the monthly momentum factor. Other results are between these extremes, in rough correlation with the unconstrained turnover. For investment, in brown, and for the annual composite, in red, the improvement is substantial at 36% and 29% more final wealth respectively than for the unconstrained factors, because the high unconstrained turnover is expensive. And, for value, in blue, the benefit is moderate, at 21% more eventual wealth, despite the relatively low turnover.

The same observation applies to the monthly factors in Panel B. For value, again in blue, the benefit is moderate, at 39% more eventual wealth, likely partly constrained because that trading is inherently anti-momentum (which is also the reason that the monthly value factor is less profitable than annual value factor). For the monthly composite (again in red) and especially for momentum (in purple), priority-best turnover control helps by a substantial margin, of 46% and 102%, respectively. The volatility of the spread for a factor such as momentum is 3.4%, indicating that the turnover limit significantly alters the portfolio.

From Figure 2, we can also visually glean whether the gains from optimal turnover control are steady or episodic. In Panel A, we can see that the benefits of optimal turnover control for the annual long-only value factor (in blue) were almost wholly absent until the 1990s, with the entire gain earned in the second and third subperiod. By contrast, the benefits of optimal turnover control for investments (in brown, Panel A), for momentum (purple, Panel B), and for the monthly composite (red, Panel B) have accrued reasonably steadily over the full span. Indeed, these incremental returns are significant at the 1% level.

The important result in Figure 2 is that, for most long-only factors, the benefits from an optimal turnover limit accrue reasonably evenly over time. The highest turnover strategies benefit the most from turnover limits. Momentum and the monthly composite factors benefit the most from priority-best turnover limits, leaving the investor 50% to 100% wealthier after 54 years than the unconstrained investor, closely followed by the investment factor and the annual composite factor. Figure 2 reveals other matters of some interest. For example, monthly value (blue, Panel B) trades as an anti-momentum strategy, and momentum (the purple line in Panel B) has not fared well this century to date, a fact not widely explored in the factor literature. So, the anti-momentum nature of monthly value rebalancing no longer impedes our results, and the short half-life of the momentum alpha amplifies the incremental benefits of priority-best turnover control. As a result, priority-best trading is far more helpful for the monthly long-only value factor than for annual value. While it is easy to find in-sample factors that, before trading costs, double our wealth over long backtests, it is striking that optimal trading can double our long-term outcomes, net of trading costs.

Priority-Best Rebalancing vs. Other Rules for Prioritizing Trades

As an alternative to priority-best, we also consider two alternative rules which we call priority-worst and proportional trading. We do not advocate either approach. Rather, this is a robustness test for priority-best trading. With priority-worst, we buy the stocks with the weakest buy signals and sell the stocks with weakest sell signals, up to the turnover limit. With proportional trading, we do every trade that the unconstrained strategy indicates, scaled to match the turnover limit. For example, if the unconstrained turnover is twice as large as our turnover limit, then we will do half of each of the indicated trades.

In Table 6, we report the t statistics associated with the strategies’ net-of-trading cost CAPM alphas. These t statistics are useful measures for the efficacy of the rules, because they are proportional to information ratios, which tell us the consistency of the alpha.²⁴ To ease comparison with the results in Table 3, we also report the comparable t statistics from the priority-best analysis, which are drawn from Table 3, Panel F. For each factor strategy and each turnover rule, we boldface the turnover limit that maximizes our net-of-trading cost t statistic, hence also maximizing our net-of-trading cost information ratio (and often maximizing our net alpha as well).

Table 6. Alternative Turnover Control Methods: Priority-Worst and Proportional Rebalancing, Jul 1964-Jun 2020

Panel A: t values for alpha, net of trading costs, for different trading rules and turnover limits, annual factors
Factor	Rebalancing rule	None	Annual turnover limit
			10%	20%	50%	100%
Value	Priority worst	1.97*	0.62	1.49	1.77	1.97*
	Priority best	1.97*	1.89	2.34*	2.02*	1.97*
	Proportional	1.97*	1.77	2.05*	2.03*	1.97*
Profitability	Priority worst	2.65**	1.22	2.49*	2.65**	2.65**
	Priority best	2.65**	2.06*	2.76**	2.65**	2.65**
	Proportional	2.65**	2.04*	2.66**	2.65**	2.65**
Investment	Priority worst	2.68**	1.69	3.27**	2.52*	2.68**
	Priority best	2.68**	1.04	1.92	2.81**	2.68**
	Proportional	2.68**	2.25*	2.91**	2.88**	2.68**
Compositeann	Priority worst	2.04*	1.22	1.99*	1.57	2.04*
	Priority best	2.04*	1.93	2.29*	2.20*	2.04*
	Proportional	2.04*	1.66	2.02*	2.01*	2.04*
Panel B: t values for alpha, net of trading costs, for different trading rules and turnover limits, monthly factors
		0.00	Annual turnover limit**
Factor	Rebalancing rule	None**	0.10	0.20	0.50	1.00
Value	Priority worst	1.01	1.12	0.91	0.73	0.07
	Priority best	1.01	1.51	1.56	0.98	0.85
	Proportional	1.01	1.84	1.92	1.37	0.97
Momentum	Priority worst	1.80	1.64	2.45*	1.75	0.63
	Priority best	1.80	−0.16	0.82	2.66**	2.99**
	Proportional	1.80	1.10	1.00	1.40	1.93
Compositemo	Priority worst	2.04*	1.87	1.32	1.97*	1.96*
	Priority best	2.04*	1.94	2.06*	2.12*	2.27*
	Proportional	2.04*	1.24	1.20	1.27	1.65

Notes. Bold type denotes strongest t statistic for each factor and rebalancing rule.

*Significant at the 5% level.

**Significant at the 1% level.

CAPM = capital asset pricing model.

Source: Research Affiliates using data from CRSP/Compustat.

Priority-best dominates the alternative methods of restraining turnover in most cases. A compelling takeaway is that priority-best turnover limits can improve the t statistic for the CAPM alpha net of trading costs for every strategy, most typically at a modest 20% or 50% turnover. For the two highest-turnover monthly strategies—momentum and the monthly composite—the best results are achieved with a more generous 100% turnover limit. Even at this more relaxed turnover limit, trading is cut by two-thirds relative to the unconstrained factor turnover.

As expected, priority-worst and proportional trading generally do not help the net-of-trading cost CAPM alpha relative to unconstrained factor trading. The priority-worst method for constraining turnover tends to deliver the poorest results, and proportional trading typically falls in the middle. These alternative trading rules win by a useful margin only for the factors with non-monotonicity at the extremes. To the extent that any turnover limit will reduce turnover and trading costs, priority-worst and proportional trading can still help, but generally not by much. As these improved returns may be mere noise, from an in-sample optimized turnover limit, we are comfortable generalizing our conclusions: we see no compelling evidence that priority-worst turnover control will be useful in any live applications.

Momentum is an interesting special case. The momentum signal, with a very short half-life of about three months, rolls over and dies—gives up its entire accumulated alpha—on average within two years (Arnott et al. 2017). If a long-term buy-and-hold investor tries to capture momentum (an oxymoron), the alpha can evaporate. Only priority-best helps in any meaningful way, and only with a fairly liberal turnover limit of 50% to 100%.

Summary and Future Directions

In the practice of investment management, trading incurs a very real cost which is loosely but directly associated with turnover (see, for example, Li et al. 2019). The more we trade, the more trading costs we incur. To overcome this trading cost erosion, most practitioners impose turnover limits. For example, in the index world, an externally imposed turnover limit may be used to limit the transactions occurred at each rebalance.²⁵ In this paper, we compare three simple ways to rebalance a portfolio with a turnover limit, in which the trades are “rationed.” Results are generally helped the most with priority-best turnover control.

The rules that we test are simple and not at all nuanced.

It is a natural extension, however, to study the efficacy of a rebalancing method by including a more holistic trading cost model. Future research should perhaps explore this particular topic. We show that even a very simple trading rule, like priority-best turnover control, can sharply reduce our portfolio turnover and boost our performance net of trading costs (and sometimes even capture more gross return before those costs). Perhaps results would be better if, in addition to signal strength, we also take account of the risks and the expected trading costs of each asset, though added complexity brings with it the obligation to gauge our expectations for each added metric carefully.

Different signals have different speeds of decay. For example, momentum decays quickly and then rolls over, giving back all of its alpha within about two years. The rewards for profitability and for size enjoy a long half-life but can be episodic, failing for long periods of time. And the rewards from a value signal start slowly, because of its anti-momentum nature, and then build, eventually decaying with a long half-life. As a result, investors cannot capture more of the value premium by continuing to trade beyond some optimal point; they would merely erode performance by incurring unnecessary trading costs. For these and many other reasons, how we can best rebalance a portfolio, selecting and weighting securities based on a composite signal, is far from a trivial question.

We have focused this study on portfolios based on individual long-only factor signals (or a single composite signal). This merely touches the surface of the puzzle. A deeper method for rebalancing our portfolios should consider how different signals interact, how diverse payoff patterns can best be integrated, and how the optimal trading method can be tuned to these very different strategies. We hope the academic community dives as deeply into these implementation challenges as it has into identifying a myriad of factors and strategies that are too often embraced based on impressive in-sample backtests with little regard for implementation niceties.

Acknowledgments

We are deeply grateful for the insights offered by Cam Harvey, Chris Brightman, Vitali Kalesnik, Que Nguyen, and the FAJ referees, all of whom offered valuable insights and suggestions as this paper took shape.

Disclosure Statement

No potential conflict of interest was reported by the author(s).

Additional information

Notes on contributors

Rob Arnott

Rob Arnott is Chairman and Founder of Research Affiliates, Newport Beach, California.

Feifei Li

Feifei Li is CIO of Kavout Algo Financial and Partner at Research Affiliates, Newport Beach, California.

Juhani Linnainmaa

Juhani Linnainmaa is Professor of Finance at the Tuck School of Business, Dartmouth, and Co-Director of Research with Kepos Capital, New York, New York.

Notes

1 If a factor or strategy soars in its valuation multiples, relative to the market, its past return will be artificially elevated; there is no reason to expect revaluation alpha to persist. Worse, if there is any subsequent mean reversion, its future return may be seriously compromised. There is shockingly little research on revaluation alpha and its possible contribution to performance-chasing in the factor investing community.

2 Hou, Xue, and Zhang (2020) show that the vast majority of the factors that they tested lose much of their statistical significance if they are constructed without owning the illiquid, thinly traded microcap stocks, specifically the bottom 2% of the market as measured by market capitalization. These stocks may be important in helping us understand the markets, and they may be where the most profitable inefficiencies reside. But an investor may have difficulty extracting incremental returns in these very small stocks, net of their trading costs.

3 See Arnott (2006) for a short exploration of the elements of implementation shortfall.

4 Optimization has an important vulnerability: its indicated portfolios and trades are exquisitely sensitive to the return and covariance assumptions. The perils of using historical return and risk metrics are well documented. That said, mean-variance optimization is theoretically superior, but only if the forecasts of risk and return resemble the future reality.

5 We primarily emphasize long-only factor strategies here to provide a straightforward illustration of portfolio rebalancing rules. Academia prefers to explore long-short strategies. But the practitioner community overwhelmingly favors long-only strategies, partly as a consequence of costs and partly due to customer demand. Investors may encounter similar trading costs when trading long-only strategies, yet they could potentially confront significantly divergent borrowing costs when shorting is required. Moreover, long-only strategies are far more widely available to investors through mutual funds and ETFs (and on a larger scale, at far lower fees), compared to long-short strategies.

6 Without sacrificing generality, we present results based on quartile portfolios with market capitalization weights. Equal weighting is not considered in this context, as it tends to result in excessive trading driven solely by price movements, potentially masking the desired trading signals. That said, partitioning the universe into quintile portfolios and equally weighting the stocks within those portfolios does not significantly alter the results.

7 This definition of annual rebalancing is different from the one used in academic studies. The Fama and French value factor (HML or high minus low, in their preferred parlance), for example, reassigns stocks into portfolios once a year but trades monthly to keep the 50/50 division between small- and large-cap stocks and to set weights each month proportional to stocks’ current market capitalization. For example, if a stock issues or repurchases equity, the HML factor is traded to reflect the change in the stock’s market value. We do not do this. We limit our study to large-cap, and apart from corporate actions and delistings, we do no intra-year trades, saving turnover for the midyear rebalance.

8 This result is consistent with the findings of Asness and Frazzini (2013). They note that, by using the most recent market capitalization, the denominator picks up part of the momentum effect: a stock is more likely a value stock if its recent return has been low, but this also implies, as Jegadeesh and Titman (1993) state, that its average return going forward is low. Asness and Frazzini find that the monthly value factor significantly outperforms the standard annual value factor when controlling for the momentum factor.

9 These are, of course, in-sample results, including spans of time that predate the first identification and publication of each factor. As such, these past performance results are a major contributing factor in the subsequent embrace of the factor.

10 The profitability factor is an example of upward revaluation. High-margin businesses always command a premium relative to low-margin businesses. If they are repriced from a 32% premium to a 160% premium, as happened from year-end 2006 through year-end 2020, then a meaningful share of the historical alpha will have been caused by this upward revaluation, without which the alpha during those 14 years would have been as much as 5% lower per year. Worse, if there is mean reversion toward historical relative valuations, then strong past alpha could presage negative future alpha.

11 Value is a countervailing example, as downward repricing for the standard B/P variant was the norm from 2007 until mid-summer of 2020 and was especially brutal during the value crash from 2017 to 2020. Value stocks are always (by definition) cheaper than growth stocks. Between March 2007 and August 2020, value repriced downward from a 66% discount to an 87% discount relative to growth; this downward revaluation contributed −6.6% per year to the value factor alpha. If there is mean reversion in the relative valuation of value stocks or high-profitability stocks, the future alpha should be far better for value and far worse for profitability than history might lead us to expect. These results are based on the long-short factors, not the long-only factors that we study in this paper. See, for example, Arnott et al. (2021), which showed that the entire drawdown for value from 2007 until the summer of 2020 was entirely explained by downward revaluation of the value factor. In other words, while value stocks were suffering their worst underperformance ever, the underlying value companies were doing fine. Indeed, if the relative valuation multiples for the value factor were the same in August of 2020 as they were in 2007, the Fama–French value portfolio would have outpaced the growth portfolio—meaning that the value factor, instead of losing more than 50%, would have been profitable during this 13½-year span.

12 From a practitioner perspective, this choice not to rely on the latest data makes no sense. For companies with a December fiscal year-end, data are lagged six months (or, more accurately, around four to five months after the fiscal year is reported). For companies with a February fiscal year-end—as is common in the retailing world—data are lagged 20 months. This is a shortcut to simplify the academic research. We use the same shortcut in this paper but would advise practitioners to use the latest-available data.

13 This rule, although intuitive, implicitly assumes that future average returns are monotonic in the signal. That is, if we have stocks A, B, and C with signals 1.0, 1.5, and 2.0, we would expect a trading rule that prioritizes trades based on signal values to outperform other trading rules if E[ra] < E[rb] < E[rc].

14 The average realized returns in Table 2 are typically not monotonic relative to the factor signals. But because realized returns equal expected returns plus noise, the non-monotonicity of the sample averages does not imply that the expected returns were (or, going forward, will be) non-monotonic.

15 Further study should allow us to discern whether an interplay with momentum may explain the non-monotonic kink at the top of these decile ranges. For example, the best B/P decile will include some stocks where the price has tanked, but the book value has not yet done so. The cheapest decile of stocks will also include most of the value traps, where the second decile will have very few value traps. Another important implication of the monotonicity of returns for the most negative deciles likely merits further study: for these factors—with non-monotonic returns for the buy signals and monotonic returns for the sell signals—more of the alpha probably comes from the stocks we avoid than from the stocks we choose to buy.

16 In computing turnover, as in Table 1, we do not measure turnover resulting from dividends or delistings. We assume that any proceeds from dividends or delistings are reinvested in the entire portfolio in amounts proportional to the current portfolio weights. We tested other turnover limits (30%, 150%, 200%), but for compactness, we limit our reporting to these turnover limits. The other turnover limits delivered results very similar to their adjacent turnover limits, so they were not particularly interesting.

17 We should note that, for actual factor portfolio management with live assets, we do not advocate cherry-picking the highest return variant of constrained partial rebalancing. That would be data-mining in its crudest form (using a backtest to maximize the backtest performance). However, as our present goal is to examine which rebalancing protocol is best, these results have relevance.

18 In Arnott et al. (2017), we documented that if an investor buys the long-short momentum factor and then holds—i.e., does not trade each month to the next monthly momentum factor portfolio—the alpha peters out after about eight months, then gives it all back over the subsequent year, finishing with a negative alpha before the two-year mark. In unpublished research, we find that under the same buy-and-hold approach with the long-short value factor an investor’s alpha starts slow, gains strength in years two to four, then fades. Given that return trajectory, it makes complete sense to trade the value factor patiently, averaging into our positions, rather than rushing to implement the strategy based on a fresh signal.

19 The careful reader will observe that, for the monthly composite strategy, turnover has the peculiar property that unconstrained turnover is higher than the highest-turnover individual strategy. The reason is rooted in the fact that value and momentum are negatively correlated. If we ignore the midyear trade, when the book value of equity updates, a stock’s price increasing makes it (a) more likely to be a momentum stock that we want to hold and (b) less likely to be a value stock we want to hold. So, outside the midyear trade, when book value of equity updates, stocks often flicker in and out of the composite portfolio because, based on price movements alone, a stock can cross the buy/sell threshold in either signal, and often the trades do not cancel. This flickering shows up as additional turnover. Neither the monthly value signal nor the momentum signal, when taken in isolation, has this issue.

20 Because the sample periods differ by 2½ years between Tables 3 and 4 (53½ years vs. 56 years), we compute net alphas in Panel F by subtracting the average trading costs from the Panel B alphas. We then compute the t values by dividing this estimate by the standard error of the alpha estimate from Panel B. This approach assumes that the inclusion of the trading costs in the missing 2½-year period would be similar to the prior 53½ years and that the month-to-month variation in trading costs would not affect the market beta estimates. Had we shortened the data span for Panels B and C by the 2½ years, we would have results that differ only minimally from the figures shown.

21 Not shown in Table 3, we measured the average idiosyncratic volatility (relative to the CAPM) and market beta for the average stock included in our portfolios at different turnover limits. Momentum and value have high idiosyncratic volatility, while profitability has less idiosyncratic volatility. Momentum tends toward high beta stocks and profitability toward low beta stocks. These resulting portfolios, regardless of turnover limit, are nearly identical in these attributes.

22 It is important to note that this is data-mining, as is much of the factor literature. We should not assume that the alpha-maximizing historical turnover limit will be the alpha-maximizing turnover limit in the future, nor should we expect to capture the full historical alpha boost in the future. If this kind of research is applied to product development, the reader should take care to avoid overfitting the past data.

23 Here, we again ignore the effects of revaluation alpha, which was starkly negative for value during this subsample. As we previously observed, the entire value rout, when the Russell 1000 Value index underperformed the broader Russell 1000 index by 3700 basis points between early 2007 and the summer of 2020, was a consequence of value becoming cheaper relative to growth. Value stocks were getting crushed, while value companies were collectively doing fine. A worthy extension of this research would examine how robust priority-best constrained trading fares net of revaluation alpha, which is by its very nature a transitory phenomenon.

24 An information ratio is a variant of the Sharpe ratio, based on a strategy’s excess returns relative to benchmark, divided by its tracking error relative to benchmark: $\mathit{\text{SR}}=\raisebox{1ex}{$\overline{r_e}$}\!\left/ \!\raisebox{-1ex}{$\sigma $}\right.,$ where $\overline{r_e}$ is the strategy’s average excess return. The t-value associated with the average excess return is $t=\raisebox{1ex}{$\overline{r_e}$}\!\left/ \!\raisebox{-1ex}{$\sigma $}\right.\times \sqrt{T},$ where T is the number of observations. Because the time series are of the same length, the ordering of the t-values is the same as the ordering of the Sharpe ratios. The same argument applies to the relationship between strategies’ information ratios and t-values associated with CAPM alphas.

25 MSCI (2017, Section 3.2), for example, reviews the MSCI Minimum Volatility Index semiannually and constrains the maximum turnover to 10%.

26 This daily momentum strategy is a translation of the standard monthly momentum strategy (see Table 1) to a daily frequency. The two versions are not the same, and we do not suggest that the daily version is necessarily better, even gross of trading costs, than the monthly version. We construct the daily version of the momentum strategy to create a setting that can give a natural impetus to intra-month rebalancing.

27 We say that the amount traded is “at least” 5% in this example because SAD evolves in discrete steps from day to day. For example, SAD might increase from 9.8% to 10.2% from one day to the next, breaching the 10% SAD threshold and inducing 10.2% × 50% = 5.1% of turnover.

28 This strategy’s alpha associates with a relatively high t value of 3.44 because it is far less active than the other two strategies that use the priority-best and proportional rebalancing rules. That is, its alpha is low but the standard error of the alpha is low. Why? By virtue of trading stocks close to the threshold, the portfolio that the priority-worst rule holds is more stable over time and its tracking error relative to the market portfolio is therefore much lower. In the CAPM regressions, this strategy’s R² is 98.3%; for the priority-best and proportional rebalancing rules, the R²’s are 82.1% and 88.4%, respectively.

Appendix A.

Non–Calendar-Based Rebalancing

In the preceding analysis, we focused on the situation in which an investor rebalances the portfolio on a fixed calendar-based schedule. We considered either annually or monthly rebalanced factors and assumed that investors trade these factors either once a year at the end of June or monthly at the end of each month. There is no reason, however, to limit the analysis to strategies that rebalance only at the end of the month. Many quantitative strategies are moving in the direction of rebalancing whenever a market movement is sufficient to justify a trade, rather than hewing to the artificiality of calendar rebalancing.

The reason to incur turnover is to reduce the deviation between the weights of the portfolio we hold and the target portfolio. If this deviation grows large enough in the middle of a tumultuous month, it may be better to rebalance the portfolio mid-month (when the large deviation emerges) rather than wait until the end of the month. In this section, we analyze such conditional rebalancing rules and, in this context, measure the efficacies of the priority-best, priority-worst, and proportional rebalancing rules.

Note that this does not mean daily rebalancing, but daily assessment of whether rebalancing is warranted. Any strategy can incur vast turnover if we rebalance daily, benefiting our brokers and the HFT (high-frequency trading) communities handsomely. Indeed, one reason the quantitative community has been slow to embrace event-based rebalancing is a fear that the incremental trading costs will devour the incremental alpha. So, the natural question is whether priority-best rebalancing can profitably improve our net alphas.

Not all strategies benefit from intra-month rebalancing, at least if we ignore any benefits gained by splitting trades to reduce trading costs. Specifically, if the reference portfolio (1) is built from signals that update at monthly or lower frequency and (2) uses weights proportional to the market capitalization, then the weights of the portfolio we hold and that of the target portfolio evolve hand-in-hand as long as we do not reassign stocks in or out of this portfolio. That is, if the capitalization-weighted reference portfolio holds stocks A and B and the price of A increases and the price of B decreases, we would want to hold more in stock A and less in stock B. If we already hold the reference portfolio, then our portfolio weights automatically track the target portfolio.

In practice, however, investors’ portfolios often do not hold stocks with weights proportional to their market capitalization. As one example, fundamental weighting schemes, drawing on Arnott, Hsu, and Moore (2005), assign weights in proportion to companies’ revenues, book values of equity, dividends, profits, numbers of employees, or combinations thereof. More importantly, signals in the real world rarely update at a monthly frequency. A value investor who chooses stocks based on firms’ B/P ratios, for example, would likely prefer to update signals continuously, which means they might consider rebalancing the portfolio whenever a company announces its latest balance sheet, so that its B/P signal reflects the latest fresh data.

We study the efficacy of the different rebalancing rules in the non–calendar-based setting by considering a signal that updates daily. Specifically, we translate the monthly momentum signal to daily frequency. The monthly momentum signal is typically the return from month t-12 to t-2 (the prior year, skipping a month). The momentum signal at the daily frequency is the stock’s cumulative return, with dividends reinvested, from trading day d-252 to d-21. We rebalance this portfolio at the end of each trading day by selecting the stocks in the top quartile of the NYSE distribution and hold the resulting set of stocks with weights proportional to the market value of equity. This portfolio updates only if the signal changes from one day to the next. That is, if all stocks were to earn identical returns on the day one month ago and one year ago, the momentum signal would not change as the 11-month window rolls forward by one day, and the portfolio assignments would therefore remain unchanged.

This daily momentum strategy is profitable in the historical data but also incurs very high turnover. Its average turnover is 3,162% per year and earns an annualized CAPM alpha of 3.9% (t statistic = 4.11).²⁶ These numbers represent the benchmark against which we compare the performance of the different turnover control methods: how much can we lower the strategy’s turnover and how much of this alpha can we retain while doing so?

A conditional rebalancing rule requires two parameters:

1.  What is the event that triggers rebalancing?

2.  Conditional on rebalancing, how much do we trade toward the target portfolio?

These two rules together determine how much turnover the portfolio incurs. In addition, as before, if we choose not to rebalance fully, we need to choose a rule that determines how we choose the trades we execute. This is the step in which the priority-best, priority-worst, and proportional rebalancing rules come into play.

We define the triggering event endogenously by measuring and updating daily the distance in the weight space between the portfolio we hold and the target portfolio. We take the sum of the absolute differences (SAD) between the current and target weights and rebalance the portfolio at the end of the day if this difference exceeds a preset threshold. We consider different values for this threshold, ranging from 5% to 100%. This limit is akin to a measure of tracking error in the weights space: the higher the threshold, the more we permit the current portfolio to deviate from the target portfolio. We define the size of the intended trade toward target as a fraction of SAD. We let this fraction range from 1% to 20%. For example, if the SAD threshold that triggers the rebalancing event is 50% and the proportion we rebalance, conditional on rebalancing, is 10%, we incur turnover of at least 10% × 50% = 5% on the days we rebalance.²⁷

In Table A1, focusing specifically on the daily momentum factor, we report average annual turnover, CAPM alphas, and t statistics associated with these alphas for the three rebalancing methods: priority-best, priority-worst, and proportional rules. As with calendar-based rebalancing, the priority-best rule typically outperforms the other two rules as a method for managing turnover while retaining high alpha. This rule retains much of the daily momentum strategy’s alpha, even when the investor cuts turnover quite aggressively. The investor can lower turnover the most by combining a high SAD limit with a small amount to rebalance, conditional on rebalancing. For example, when the SAD limit is 70% and the investor rebalances only 1% of the deviations conditional on hitting this threshold, the investor incurs an average annualized turnover of 196%—which is approximately one-sixteenth of the unconstrained strategy’s turnover—while earning an annualized alpha of 4.0% (t statistic = 3.70).

The priority-worst rule cannot reduce turnover to the same extent. This rule always favors buying stocks that are just barely above the threshold and selling those just barely below the threshold. These stocks are the most likely to re-cross the threshold, perhaps multiple times, which is problematic. That is, an investor might begin to trade into a stock that just barely entered the top quartile because it earned a positive return. If this stock earns a low return tomorrow, however, it may fall out of the portfolio, and under the priority-worst scheme the investor would begin to trade out of this position. The priority-worst rule, by focusing on the most marginal stocks, trades the stocks that are least likely to contribute much to the portfolio’s alpha in the first place. An investor who uses the 70%/1% rule, but does so using the priority-worst scheme, incurs an average annual turnover of 674% and earns an alpha of 0.9% (t statistic = 3.44).²⁸

The proportional rebalancing rule also is not as good as the priority-best rule at managing turnover because of the same mechanism that causes the priority-worst rule’s failure. Whereas the priority-best rule always trades stocks the farthest away from the threshold—these are likely the most stable holdings—the priority-worst rule trades stocks just at the threshold. The proportional rebalancing rule falls between these two extremes because it trades, in equal amounts, stocks both far away and close to the threshold. The turnover spent trading stocks close to the threshold is often wasteful. An investor following the 70%/1% scheme under the proportional rebalancing rule incurs an average turnover of 230% and earns an alpha of 2.7% (t statistic = 3.48). Priority-worst rebalancing loses two-thirds of this alpha, while nearly tripling the turnover, and priority-best boosts this alpha by 50% while lightly trimming the turnover.

The estimates in Table A1 suggest that the priority-best rule is even better for controlling turnover in a non–calendar-based rebalancing setting than in a calendar-based setting. Its superior performance in controlling turnover relative to the two alternatives—the priority-worst and proportional rebalancing rules—comes as no surprise when we recognize that the priority-best rule, by virtue of prioritizing trades in stocks that are the farthest removed from the portfolio selection threshold, is likely to minimize the expected need for additional trading. Moreover, if expected returns are approximately monotonic in signals, this rule also has the benefit of spending the turnover on stocks that make the greatest difference in terms of expected return.

Table A1. Average Annual Turnover for Daily Rebalancing Rules on the Momentum Factor: Priority-Best and Proportional Rebalancing Rules, Jul 1964-Jun 2020

		Turnover (%/year)					Alpha (%/year)					t statistics
		Proportion to rebalance conditional on rebalancing					Proportion to rebalance conditional on rebalancing					Proportion to rebalance conditional on rebalancing
		1%	2.5%	5%	10%	20%	1%	2.5%	5%	10%	20%	1%	2.5%	5%	10%	20%
		Priority-best
Sum of Absolute Deviations Threshold	5%	286	509	780	1172	1751	4.5	4.5	4.3	4.2	4.1	4.08**	4.21**	4.15**	4.11**	4.09**
	10%	286	509	779	1170	1714	4.5	4.5	4.3	4.2	4.1	4.08**	4.21**	4.15**	4.12**	4.12**
	20%	286	508	760	1049	1342	4.5	4.5	4.3	4.2	4.0	4.08**	4.21**	4.10**	4.07**	3.92**
	30%	285	487	655	796	938	4.5	4.5	4.2	4.2	4.0	4.11**	4.21**	3.97**	3.99**	3.87**
	40%	280	422	510	578	676	4.4	4.6	4.2	4.2	3.9	4.03**	4.22**	3.92**	3.93**	3.79**
	50%	261	341	383	430	502	4.3	4.4	4.3	4.2	4.1	3.99**	4.06**	3.99**	3.91**	3.90**
	60%	230	272	293	324	389	4.2	4.1	4.1	4.0	3.9	3.85**	3.85**	3.80**	3.77**	3.71**
	70%	196	219	232	257	309	4.0	3.9	3.9	3.9	3.8	3.70**	3.64**	3.65**	3.66**	3.58**
	80%	166	180	190	210	257	3.6	3.6	3.8	3.4	3.5	3.49**	3.46**	3.57**	3.28**	3.37**
	90%	140	149	157	174	215	3.3	3.3	3.3	3.2	3.4	3.27**	3.24**	3.25**	3.09**	3.26**
	100%	116	122	130	146	181	3.0	3.1	2.9	3.2	3.0	3.22**	3.28**	2.98**	3.23**	2.97**
		Proportional
Sum of Absolute Deviations Threshold	5%	299	504	750	1124	1704	3.0	3.4	3.6	3.8	3.9	3.51**	3.74**	3.86**	3.97**	4.06**
	10%	299	504	750	1118	1658	3.0	3.4	3.6	3.8	3.9	3.51**	3.74**	3.86**	3.96**	4.04**
	20%	299	502	730	1005	1288	3.0	3.4	3.6	3.7	3.7	3.51**	3.73**	3.83**	3.91**	3.94**
	30%	298	485	645	780	921	3.0	3.4	3.5	3.6	3.6	3.50**	3.72**	3.79**	3.85**	3.80**
	40%	294	441	524	590	675	3.0	3.3	3.5	3.5	3.5	3.49**	3.73**	3.79**	3.78**	3.75**
	50%	282	376	418	459	518	2.9	3.2	3.4	3.4	3.4	3.50**	3.70**	3.78**	3.78**	3.78**
	60%	260	314	339	366	412	2.9	3.1	3.2	3.2	3.2	3.52**	3.66**	3.70**	3.66**	3.64**
	70%	230	260	277	300	338	2.7	2.9	2.9	3.1	3.1	3.48**	3.59**	3.60**	3.67**	3.64**
	80%	197	217	229	245	278	2.5	2.6	2.7	2.8	3.0	3.41**	3.49**	3.51**	3.56**	3.58**
	90%	166	179	189	203	232	2.3	2.4	2.4	2.5	2.6	3.34**	3.35**	3.34**	3.31**	3.30**
	100%	135	145	153	169	197	2.1	2.1	2.2	2.3	2.5	3.24**	3.22**	3.25**	3.23**	3.38**
		Priority-worst
Sum of Absolute Deviations Threshold	5%	674	1351	1885	2278	2609	0.9	1.2	1.7	2.4	3.4	3.46**	3.20**	3.08**	3.39**	3.99**
	10%	674	1351	1885	2278	2606	0.9	1.2	1.7	2.4	3.4	3.46**	3.20**	3.08**	3.39**	3.97**
	20%	674	1351	1885	2276	2507	0.9	1.2	1.7	2.4	3.2	3.46**	3.20**	3.08**	3.34**	3.84**
	30%	674	1351	1885	2249	2198	0.9	1.2	1.7	2.4	3.0	3.46**	3.20**	3.05**	3.45**	3.79**
	40%	674	1351	1880	2163	1840	0.9	1.2	1.7	2.4	2.9	3.46**	3.20**	3.15**	3.43**	3.78**
	50%	674	1351	1860	1992	1528	0.9	1.2	1.8	2.4	2.7	3.46**	3.20**	3.30**	3.56**	3.78**
		Priority-worst
	60%	674	1347	1816	1762	1279	0.9	1.2	1.8	2.3	2.4	3.46**	3.26**	3.39**	3.61**	3.38**
	70%	674	1335	1741	1530	1061	0.9	1.3	1.7	2.3	2.3	3.44**	3.44**	3.39**	3.83**	3.48**
	80%	672	1313	1614	1294	879	0.9	1.3	1.6	2.2	2.2	3.44**	3.33**	3.21**	3.79**	3.53**
	90%	664	1272	1437	1083	733	1.0	1.4	1.5	2.2	2.0	3.53**	3.57**	3.18**	3.91**	3.36**
	100%	650	1190	1208	878	589	1.0	1.3	1.7	1.7	1.9	3.30**	3.23**	3.71**	3.31**	3.26**

Notes. ** Significant at the 1% level.

CAPM = capital asset pricing model.

Source: Research Affiliates using data from CRSP/Compustat.