Previously, I wrote about how investors can simulate leverage via concentrated stock selection. That’s still true as far as I can tell. However, I also wrote something that I now believe to be false: concentrated equal-weighted factor portfolios have alpha on top of value-weighted factor portfolios. The numbers I found before were not wrong per se. However:

  • The alpha came primarily from small-cap and micro-cap stocks. That alpha may not be feasible to capture, or it may be defeated by trading costs; and historical estimates of micro-cap returns are biased upward because closing prices do not accurately represent the average investor’s trade price (Blume & Stambaugh (1983)1).

    When I constructed hypothetical factor portfolios that had high concentration but screened out small-caps, the results did simulate leverage—they had higher returns and volatility than diversified factor portfolios—but alphas were not consistently positive.

  • In the United States (where the data goes back the furthest), the alpha only shows up over the full data series (1927–2025). When restricting to 1964 onward, the alphas are close to zero.
  • Concentrated value and momentum had positive alpha; but when I tested two new factors, profitability and investment, they each had negative alpha.

Contents

Methodology

I examined four factors: value (HML), momentum (UMD), profitability (RMW), and investment (CMA).2 Those are the factors from the standard Fama-French five-factor model, minus the market factor and size, plus momentum.3 All factors use the standard Fama-French definitions with data pulled from the Ken French Data Library.

To compare diversified vs. concentrated factors, I defined “diversified” as the top 40% of stocks (ranked by the factor of interest), cap-weighted; and “concentrated” as the top 20%, equal-weighted. All factors were defined as long-only rather than long/short, to better represent how most investors invest.

For each factor, I ran a factor regression with the concentrated factor of interest as the dependent variable, against three independent variables: market beta, the long/short size factor (SMB), and the diversified factor of interest minus the market4. I included SMB on the hypothesis that the outperformance of equal-weighted stocks is partially driven by the size factor, but this turned out not to be relevant (SMB coefficients were close to zero).5

To make the strategies realistically tradable, I excluded the smallest 20% of companies. As of 2025, this would have excluded stocks with a market cap less than $1 billion or so. That’s probably more conservative than necessary, but looking at narrower slices (e.g. excluding the bottom 10% instead of 20%) would pose methodological challenges.6

My original article defined “diversified” as top 50% value-weighted and “concentrated” as top 10% equal-weighted. Based on the data available in the Ken French Data Library, I couldn’t construct realistically tradable strategies using 50% and 10% cutoffs, so I used 40% and 20% instead.

Results for long-only factors

The tables below show alphas (t-stats in parentheses) for concentrated factors over diversified factors. “All-cap” does not filter based on market cap; “realistic” excludes the smallest 20% of companies.

United States value, momentum, profitability, and investment start in 1927, 1928, 1964, and 1964, respectively (rounded up to the nearest year). Developed ex-US factors start in 1991.

Table 1: Comparing factor alphas, United States
factor all-cap alpha realistic alpha
Value (B/M) 0.70% (1.69) -0.07% (-0.18)
Momentum 0.92%* (2.22) 0.27% (0.71)
Profitability -0.87%* (-2.20) -0.80%* (-2.36)
Investment -3.18%*** (-4.95) -3.21%*** (-5.25)

*, **, and *** indicate significance at p < 0.05, 0.01, and 0.001, respectively.

Table 2: Comparing factor alphas, Developed ex-US
factor all-cap alpha realistic alpha
Value (B/M) 1.68%*** (4.03) 0.62% (1.44)
Momentum 1.33%*** (3.36) 0.98%** (2.71)
Profitability 0.21% (0.55) -0.03% (-0.08)
Investment -1.35%** (-2.78) -1.34%** (-2.85)
Full factor regression results

US (all-cap)

  beta SMB factor annual alpha t-stat
Value (B/M) 1.06 -0.05 1.28 0.70% 1.69
Momentum 1.00 -0.01 1.23 0.92%* 2.22
Profitability 1.06 0.00 1.17 -0.87%* -2.20
Investment 1.11 -0.05 1.31 -3.18%*** -4.95

US (realistic)

  beta SMB factor annual alpha t-stat
Value (B/M) 1.08 -0.02 1.26 -0.07% -0.18
Momentum 1.01 -0.00 1.34 0.27% 0.71
Profitability 1.07 -0.01 1.29 -0.80%* -2.36
Investment 1.12 -0.00 1.31 -3.21%*** -5.25

Developed ex-US (all-cap)

  beta SMB factor annual alpha t-stat
Value (B/M) 1.05 -0.07 1.26 1.68%*** 4.03
Momentum 1.07 -0.03 1.32 1.33%*** 3.36
Profitability 1.03 0.02 1.07 0.21% 0.55
Investment 1.06 -0.29 1.69 -1.35%** -2.78

Developed ex-US (realistic)

  beta SMB factor annual alpha t-stat
Value (B/M) 1.06 -0.05 1.29 0.62% 1.44
Momentum 1.06 0.01 1.32 0.98%** 2.71
Profitability 1.04 0.01 1.15 -0.03% -0.08
Investment 1.06 -0.15 1.62 -1.34%** -2.85

Excluding small-caps reduced alphas to near zero for concentrated equal-weight value and momentum factors. However, even when excluding small-caps, concentrated portfolios had factor exposures greater than 1 across the board (see under “Full factor regression results”). This suggests that concentrated portfolios offer synthetic leverage, but no alpha.

The two new factors (profitability and investment) had negative alphas. The relatively large t-stats suggest that the variance in alphas is better explained by heterogeneity than by random chance. That is, equal-weighted all-cap value and momentum had genuine positive alpha, while investment had genuine negative alpha.

International momentum maintained a statistically significant alpha when excluding small-caps (p = 0.007). After a Bonferroni correction, this p-value becomes 0.054, which corresponds to a likelihood ratio of 6.4:1.

The data series for value and momentum start in 1927/1928, while profitability and investment start in 1964. Is that fact relevant? The next table shows concentrated factor alphas when restricting the data series to 1964 onward:

Table 3: Comparing factor alphas, United States (1964–2025)
factor all-cap alpha realistic alpha
Value (B/M) 0.45% (0.92) -0.18% (-0.37)
Momentum 0.20% (0.47) -0.13% (-0.29)
Profitability -0.87%* (-2.20) -0.80%* (-2.36)
Investment -3.18%*** (-4.95) -3.21%*** (-5.25)

*, **, and *** indicate significance at p < 0.05, 0.01, and 0.001, respectively.

Full factor regression results

US (all-cap), 1964–2025

  beta SMB factor annual alpha t-stat
Value (B/M) 1.07 -0.02 1.19 0.45% 0.92
Momentum 1.06 0.04 1.28 0.20% 0.47
Profitability 1.06 0.00 1.17 -0.87%* -2.20
Investment 1.11 -0.05 1.31 -3.18%*** -4.95

US (realistic), 1964–2025

  beta SMB factor annual alpha t-stat
Value (B/M) 1.09 0.01 1.22 -0.18% -0.37
Momentum 1.06 0.06 1.34 -0.13% -0.29
Profitability 1.07 -0.01 1.29 -0.80%* -2.36
Investment 1.12 -0.00 1.31 -3.21%*** -5.25

Can small investors capture factor premiums in small-caps?

Collver (2014)7 looked at bid-ask spreads for US stocks in the year 2013. It found approximately the following average bid-ask spreads for stocks at various market caps (see Collver’s Table 4 and Figure 28):

Table 4: Bid-ask spreads by market cap
Market Cap Avg Spread
$500M – $1B 0.174%
$250M – $500M 0.287%
$100M – $250M 0.606%
< $100M 1.631%

The value, profitability, and investment factors have low turnover. If we assume that small investors have near-zero trading impact, then those three factors’ alphas appear to survive trading costs down to market caps of $250 million, and maybe as low as $100 million, but probably not much lower. (I don’t want to make a strong claim because I haven’t done careful calculations, and I don’t know how reliable these data are.) Momentum has a much higher turnover, so its alpha is less likely to survive trading costs.

(Remember: we’re not talking about the factors surviving trading costs; we’re talking about the alpha of concentrated factors minus diversified factors. Whether factors themselves survive trading costs is a separate question that has already been addressed by a number of publications; for example, see Frazzini et al. (2012)9.)

An important caveat: the market was much more liquid in 2013 than it was for most of the data sample. In 1940 or 1970, equal-weighted strategies were harder to trade, and the positive alphas may have represented compensation for trading costs. If so, then we would expect the alphas to shrink as trading costs decline.

Indeed, if we restrict the sample to the post-2000 period, concentrated value and momentum portfolios had weak or near-zero alphas, with only developed ex-US value retaining a significantly positive alpha. (However, note that the reduced sample size makes it more difficult to establish statistical significance.)

Table 5: US (all-cap), 2000–2025
  beta SMB factor annual alpha t-stat
Value (B/M) 1.12 -0.07 1.16 0.23% 0.23
Momentum 1.05 0.14 1.21 -0.81% -1.01
Profitability 1.11 -0.05 1.14 -1.50% -1.86
Investment 1.19 -0.18 1.44 -4.24%** -3.07
Table 6: Developed ex-US (all-cap), 2000–2025
  beta SMB factor annual alpha t-stat
Value (B/M) 1.05 -0.07 1.28 1.50%** 3.19
Momentum 1.08 -0.01 1.29 0.39% 0.93
Profitability 1.04 -0.01 1.12 0.49% 1.12
Investment 1.06 -0.41 1.88 -1.36%* -2.38

Trading costs are always difficult to assess. My current best guess is that a small investor can feasibly trade concentrated equal-weighted factors with a minimum market cap of $100M to $250M, but they should expect the alpha to be close to zero.

Results for long/short factors

The previous section examined long-only portfolio constructions, but the results for long/short factors may also be of interest. For long/short, I defined diversified factors as top 40% minus bottom 40% cap-weighted, and concentrated factors as top 20% minus bottom 20% equal-weighted.

The all-cap results look similar to the results for long-only portfolios: concentrated long/short value and momentum factors had positive alpha, profitability had near-zero (slightly negative) alpha, and investment had strong negative alpha. However, unlike the long-only portfolios, value and momentum maintained their positive alphas when excluding small-caps.

That doesn’t necessarily mean those concentrated long/short factors have positive alpha in practice, because short-selling stocks introduces additional costs.

Results for long/short factors

*, **, and *** indicate significance at p < 0.05, 0.01, and 0.001, respectively.

US (all-cap), 1964–2025

  beta SMB factor annual alpha t-stat
Value (B/M) -0.01 0.02 1.43 1.72%*** 3.54
Momentum 0.01 0.02 1.49 0.90%* 2.18
Profitability -0.04 0.01 1.45 -0.52% -0.82
Investment 0.01 -0.01 1.39 -2.83%*** -6.77

US (realistic), 1964–2025

  beta SMB factor annual alpha t-stat
Value (B/M) -0.01 0.01 1.50 1.21%* 2.37
Momentum 0.00 0.05 1.51 1.78%*** 4.01
Profitability -0.05 0.03 1.55 -0.09% -0.16
Investment 0.01 0.00 1.40 -2.32%*** -5.04

Developed ex-US (all-cap), 1991–2025

  beta SMB factor annual alpha t-stat
Value (B/M) -0.00 0.00 1.39 2.13%*** 4.79
Momentum 0.01 -0.01 1.44 0.58% 1.55
Profitability -0.04 -0.06 1.21 0.13% 0.33
Investment 0.00 -0.01 1.33 -2.43%*** -6.72

Developed ex-US (realistic), 1991–2025

  beta SMB factor annual alpha t-stat
Value (B/M) -0.01 -0.02 1.48 1.81%*** 3.65
Momentum 0.01 0.00 1.47 1.35%** 3.21
Profitability -0.02 -0.03 1.22 1.31%** 3.19
Investment 0.01 0.02 1.38 -1.26%** -3.09

I also regressed the long sides of each concentrated factor against their respective long/short diversified factors, and similarly for the short sides. The long sides had consistent positive alphas with exceptional t-stats, and the short sides had consistently strong negative alphas.

Long-side concentrated factors, regressed against long/short diversified

US (all-cap), 1964–2025

  beta SMB factor annual alpha t-stat
Value (B/M) 1.03 0.59 0.89 3.96%*** 13.20
Momentum 1.01 0.52 0.30 6.14%*** 14.30
Profitability 1.01 0.56 0.63 4.61%*** 12.42
Investment 1.03 0.57 0.23 4.50%*** 14.72

US (realistic), 1964–2025

  beta SMB factor annual alpha t-stat
Value (B/M) 1.03 0.41 0.91 4.39%*** 12.80
Momentum 1.03 0.39 0.31 5.85%*** 13.64
Profitability 1.03 0.40 0.54 4.64%*** 12.07
Investment 1.05 0.42 0.27 4.73%*** 14.26

Developed ex-US (all-cap), 1991–2025

  beta SMB factor annual alpha t-stat
Value (B/M) 1.01 0.55 0.62 2.59%*** 15.22
Momentum 1.00 0.54 0.43 4.02%*** 15.77
Profitability 1.00 0.54 0.43 3.79%*** 18.08
Investment 1.01 0.55 0.40 2.94%*** 15.79

Developed ex-US (realistic), 1991–2025

  beta SMB factor annual alpha t-stat
Value (B/M) 1.02 0.42 0.64 2.51%*** 10.55
Momentum 1.02 0.43 0.43 3.46%*** 12.64
Profitability 1.01 0.42 0.39 3.21%*** 13.10
Investment 1.03 0.44 0.39 2.47%*** 10.76
Short-side concentrated factors, regressed against long/short diversified

US (all-cap), 1964–2025

  beta SMB factor annual alpha t-stat
Value (B/M) -1.11 -0.67 0.38 -2.52%*** -3.66
Momentum -1.07 -0.69 1.01 -4.90%*** -8.20
Profitability -1.12 -0.64 0.76 -4.21%*** -5.53
Investment -1.12 -0.69 0.92 -4.43%*** -7.10

US (realistic), 1964–2025

  beta SMB factor annual alpha t-stat
Value (B/M) -1.14 -0.51 0.38 -2.82%*** -4.38
Momentum -1.12 -0.54 1.00 -3.58%*** -6.11
Profitability -1.15 -0.47 0.87 -3.86%*** -5.39
Investment -1.16 -0.53 0.89 -4.10%*** -7.14

Developed ex-US (all-cap), 1991–2025

  beta SMB factor annual alpha t-stat
Value (B/M) -1.07 -0.64 0.61 -2.07%*** -3.57
Momentum -1.07 -0.70 0.84 -4.86%*** -10.34
Profitability -1.07 -0.66 0.82 -4.00%*** -7.24
Investment -1.08 -0.67 0.62 -4.15%*** -8.22

Developed ex-US (realistic), 1991–2025

  beta SMB factor annual alpha t-stat
Value (B/M) -1.09 -0.52 0.64 -1.23%* -1.98
Momentum -1.08 -0.57 0.87 -3.10%*** -6.93
Profitability -1.08 -0.52 0.87 -2.14%*** -3.82
Investment -1.10 -0.54 0.71 -2.29%*** -4.34

The concentrated long-only portfolios had stronger alphas when regressed against long/short diversified factors than when regressed against long-only diversified factors. This suggests that there’s a component of the long side that can’t be captured by a long/short portfolio. This result has the same flavor as Blitz et al. (2020)10, which found that the short sides of (cap-weighted) factors were generally subsumed by the long sides.

Discussion

To recap the findings:

  • Concentrated long-only factor portfolios did replicate leverage: they behaved like a levered-up version of a diversified portfolio.
  • However, after excluding small-caps, concentrated factors didn’t outperform a levered-up diversified factor portfolio—they did not have alpha.
  • Even including small-caps, alpha was only positive for two of the four factors, and was much smaller post-1964 than over the full sample (1927–2025).
  • Concentrated long/short value and momentum portfolios had alpha even when excluding small-caps, but concentrated portfolios might not survive trading costs on the short side.

Estimated returns for equal-weighted portfolios are biased upward by the variance between a stock’s daily close price and its actual tradable price (Blume & Stambaugh (1983)1; see Asparouhova et al. (2013)11 for more recent data on the magnitude of the bias). Backtests determine factor returns using the daily close price, but this is not the price real-world investors trade at. This effect is most pronounced for micro-cap stocks and stocks with low prices because they have highest price variance—Asparouhova et al. found that eliminating stocks with prices under $5 reduced bias by ~90%. Among micro-caps, the bias could be as large as multiple percentage points per year for a monthly-rebalanced strategy. We would expect the momentum factor to have approximately a 12x larger bias than other factors because it rebalances monthly rather than annually, but the difference in alpha between all-cap and realistic was smaller for momentum than for value. That suggests that the bias in micro-caps only partially explains the difference in alphas.

The investment factor had consistently negative alpha, even for all-cap portfolios. Why? It doesn’t look like a statistical fluke because the absolute t-stats were large (for the US long-only factor, t = -4.95; p < 1e-6).

One possible explanation is that, even though aggressive investment is associated with worse returns on average, the smallest stocks tend to benefit from aggressive investment because startups or new companies need to raise capital. However, this explanation is inconsistent with Fama & French (2008)12, which found that the investment factor was not weaker or reversed in small-caps—in fact it only appeared in small stocks, not large stocks. Given this inconsistency, I could not find any explanation that fit the facts.

How I’ve changed my personal investments

When I wrote that concentrated factors had positive alpha, I concluded that concentrated factor investing looked particularly promising for retail investors. In accordance with that, I was investing my personal equity portfolio in concentrated long-only ETFs (primarily the AlphaArchitect funds). However, if a realistic implementation of a concentrated factor portfolio doesn’t have alpha, then there is less justification for preferring concentrated long-only funds over diversified funds.

Last month, I decreased my position in long-only factor ETFs from 125% to 100%, and added 25% in a long/short AQR fund (specifically QRPIX, although that choice of fund was somewhat arbitrary; AQR has other long/short funds with different tradeoffs13). The impetus for this change was a logistical issue—my account had new margin requirements that made my old portfolio riskier to hold. But a big part of why I bought an AQR long/short factor fund rather than something else is that I’ve come to believe that concentrated funds have less of an advantage than I previously thought.

Source code

Replication code is available on GitHub.

Notes

  1. Blume, M. E., & Stambaugh, R. F. (1983). Biases in computed returns: An application to the size effect. doi: 10.1016/0304-405x(83)90056-9 ↩ ↩2

  2. For those unfamiliar, here is how the four factors are defined:

    • Value: Stocks with high book-to-market ratios (B/M).
    • Momentum: Stocks with high past 12-month returns, excluding the most recent month.
    • Profitability: Stocks with high operating profitability (= operating profit divided by book equity).
    • Investment: Stocks with low year-over-year asset growth.

    ↩

  3. I excluded the market factor because the concept of “concentrated” vs. “diversified” doesn’t make sense for it. I excluded size because the size effect is weak. I included momentum because it’s a strong factor. ↩

  4. That is, rather than using the standard factor definition of top 30% minus bottom 30%, the factor regressions defined the factor as the top 40% minus the total market. This allows for a direct comparison between the (long-only) diversified and concentrated factor portfolios.

    Another reasonable approach would be to regress against the top 40% (without subtracting the market). This regression gives a near-zero exposure to market beta (rather than near-one) and qualitatively similar values for SMB, factor exposure, and alpha. ↩

  5. When looking at a cap-weighted index vs. an equal-weighted index, the equal-weighted historically outperformed, and this outperformance was indeed primarily driven by the size factor. However, it appears that the outperformance of equal-weighted long-only factors over cap-weighted has little if anything to do with the size factor. ↩

  6. To filter stocks according to size plus another factor, I need two-way portfolio sorts. The Ken French Data Library offers 5x5 sorts on the four factors I tested. It offers 10x10 sorts on size x value and size x investment, but not on momentum or profitability, so I only could’ve tested two out of four factors using deciles; and the 10x10 sorts have some missing data, which introduces more researcher degrees of freedom on how to handle that. ↩

  7. Collver, C. (2014). A characterization of market quality for small capitalization US equities. ↩

  8. Table 4 reported average spreads bucketed by price; Figure 2 reported the number of stocks in each price bucket. Based on Figure 2, I made the coarse assumption that the overall average spread was the equal-weighted average of the spreads of the four price buckets up to $39.99. I excluded the >= $40 bucket because, according to Figure 2, few stocks had prices exceeding $40. ↩

  9. Frazzini, A., Israel, R., & Moskowitz, T. J. (2012). Trading Costs of Asset Pricing Anomalies. ↩

  10. Blitz, D., Baltussen, G., & van Vliet, P. (2020). When Equity Factors Drop Their Shorts. doi: 10.1080/0015198x.2020.1779560 ↩

  11. Asparouhova, E., Bessembinder, H., & Kalcheva, I. (2013). Noisy Prices and Inference Regarding Returns. ↩

  12. Fama, E. F., & French, K. R. (2008). Dissecting Anomalies. ↩

  13. For example, QRPIX includes non-equity factors, while QMNIX and QLEIX include equities only. Historical evidence suggests that non-equity factors have had positive returns (Baltussen et al. (2019)14; Ilmanen et al. (2019)15), but with much lower returns and volatility than equity factors, so it’s not clear that they survive trading costs; it’s also not clear that they provide good diversification. Therefore, I’m not fully convinced that they’re worth including in a portfolio. The exception is the trend factor, which has had strong historical performance and diversification benefits in all asset classes. I have a large allocation to trend in my portfolio (I have about as much trend risk as equity beta risk). ↩

  14. Baltussen, G., Swinkels, L., & van Vliet, P. (2019). Global Factor Premiums. ↩

  15. Ilmanen, A. S., Israel, R., Moskowitz, T. J., Thapar, A. K., & Wang, F. (2019). Factor Premia and Factor Timing: A Century of Evidence. ↩