Investors Can Simulate Leverage via Concentrated Stock Selection
Summary
Last updated 20220415; see Errata.
Confidence: Highly likely.
Some altruists are much less riskaverse than ordinary investors, and may want to use leverage. But foundations and donoradvised funds legally cannot access most forms of leverage. As an alternative approach, leverageconstrained investors could buy concentrated positions in smallcap value and momentum stocks. For example, instead of buying an ETF that holds the best half of the market as ranked by value or momentum, they could buy the top 10%.
According to backtests, when portfolio concentration increases, both return and risk increase, and return increases more than risk (so that concentrated portfolios have higher riskadjusted returns).
Large investors cannot hold concentrated portfolios without moving the market, so they probably prefer to use leverage if they can. Small investors probably prefer to buy concentrated investments because they offer higher riskadjusted returns than leveraged broad portfolios.
Disclaimer: This should not be taken as investment advice. Any given portfolio results are hypothetical and do not represent returns achieved by an actual investor.
Contents
 Summary
 Contents
 Motivation
 Known approaches
 Increasing expected return via concentration
 Fees, transaction costs, and taxes
 How does concentrated investing compare to using leverage?
 On future expectations
 Finding concentrated ETFs
 Errata
 Appendix A: Significance tests
 Appendix B: Replication on international equities
 Appendix C: Factor regression on selected portfolios
 Notes
Motivation
Some investors cannot use leverage. For example, foundations, donoradvised funds, and IRAs are prohibited from using certain types of investments, including margin loans. Other investors have access to leverage, but have to pay high fees to get it. But sometimes, investors want to increase their return and risk. Can we do that without explicitly using leverage?
Known approaches
Some stocks are more volatile than others. One way to simulate leverage might be to buy stocks with high volatility. Unfortunately, this doesn’t work: highvol stocks do not provide enough return to justify the higher risk.^{1}^{2} This may happen because leverageconstrained investors already overpay for stocks with high volatility.
As a different approach, we could tilt toward smallcap, value, and momentum stocks. These tilts all tend to increase both risk and return.^{3}^{4}^{5}^{6} (Gordon Irlam has proposed that altruists use smallcap value rather than leverage as a (sometimes) better method of increasing return.) Most funds that implement these tilts hold around half to a third of the total market, and end up with a little bit higher volatility and expected return than the market, but not much higher.
Increasing expected return via concentration
Smallcap stocks, value stocks, and especially smallcap value stocks tend to earn higher return (and higher risk) than the broad market. Smallcap and value index funds typically hold about half the market. Can we enhance our return^{7} by concentrating in a narrower corner of the market, such as 10% instead of 50%?
Let’s use the Ken French Data Library to look at portfolios formed on size, value, and momentum at various levels of concentration.
Ignoring size for now, did more concentrated value and momentum portfolios historically perform better?
The chart below shows historical returns for US stocks using three valuation metrics and one momentum metric:
 value: book to market (B/M)
 value: earnings to price (E/P)
 value: cash flow to price (CF/P)
 momentum: 12month return, excluding the most recent month (Mom)
It includes both equalweighted (EW) returns (where each stock receives the same weight) and valueweighted (VW) returns (where stocks are weighted in proportion to market cap).^{8}
For each of the four metrics, return monotonically increased with concentration (with the sole exception of B/M valueweighted moving from top 20% to top 10%).
Let’s narrow in on value stocks as measured by B/M. The next chart shows B/M returns over the total market as well as over small caps (defined as the 10% smallest stocks):^{9}
We see a similar pattern, where more concentration means higher return. Within small caps, this pattern has a steeper slope, and smallcap value stocks consistently outperformed allcap value stocks except at the weakest level of concentration. Relatedly, equalweighting outperformed valueweighting—equalweighting gives relatively more weight to small caps, so this is a different form of small cap tilt.
Most offtheshelf value and momentum funds hold about half the market, valueweighted. If instead of buying one of those funds, we held only the top 10% and used equal weighting, historically we could have increased returns by a lot:
50% VW  10% EW  Improvement  

B/M  11.0%  18.7%  7.7% 
E/P  13.8%  17.8%  4.0% 
CF/P  12.6%  18.1%  5.5% 
Mom  12.6%  19.2%  6.6% 
As return increased with concentration, volatility increased at a similar pace. Equalweighted portfolios had higher riskadjusted returns across the board than their valueweighted counterparts; but equalweighted portfolios had similar riskadjusted returns at all levels of concentration (including 50%, 40%, 30%, 20%, and 10%).
These differences in returns across concentration, market cap, and weighting are highly statistically significant, as are the differences in volatility (see Appendix A).
The following table shows the riskadjusted returns (Sharpe ratios) for value and momentum portfolios:
50% VW  10% EW  Improvement  

B/M  0.45  0.57  0.12 
E/P  0.71  0.74  0.03 
CF/P  0.63  0.71  0.08 
Mom  0.57  0.73  0.16 
We can get higher returns (accompanied by higher risk) by going even deeper into smallcap value and momentum. According to some backtests I ran on a different (nonpublic) data set, a portfolio of the top 30 value stocks (equalweighted) would have historically returned about 30% per year gross of costs, with a standard deviation of about 45%. This is similar to the historical performance of a US total market index with 3:1 leverage. However, this portfolio would mainly have consisted of very small stocks (market cap $200 million or less), so trading costs could destroy a lot of the value of this strategy in practice.^{10}
Fees, transaction costs, and taxes
So far, I have focused on gross returns. But realworld strategies have to pay various costs, including management fees, trading costs, and taxes. How much do these matter?
I did not find much published research on realworld transaction costs and I only have limited personal experience trading individual stocks, so take the following with a grain of salt.
Investors with less than $1 million probably shouldn’t worry about transaction costs. Even microcap stocks (with market caps between $50 million and $200 million) are liquid enough that small investors will probably end up paying 0.25% or less per trade. If you buy value stocks and rebalance once a year, transaction costs will probably be low enough not to matter. Momentum requires more frequent rebalancing, so even small investors might prefer to restrict themselves to some minimum market cap threshold (somewhere in the $200 to $500 million range) or take other measures to reduce costs.
Larger investors must take care not to move the market when they buy and sell stocks. Investors with substantially more than $1 million might prefer not to trade microcap stocks. Investors with much more money (perhaps around $100 million) can’t use equal weighting because they will move the market too much if they try to put substantial positions into smallcaps.^{11}
If you hire an investment manager to run a strategy like this, they will charge a fee—probably around 1% per year. This dampens the value of investing in a concentrated portfolio rather than simply buying an offtheshelf mutual fund or ETF. Or you could manage your own portfolio, avoiding the fee but giving yourself more work to do. I used to manage my own concentrated portfolio of value stocks, and it took me two hours of work per year. A momentum strategy would probably require somewhere around 10–20 hours per year. But if you manage the portfolio yourself, you need to be confident that you won’t make trading mistakes.
Value and momentum portfolios both have some turnover, and turnover means you pay taxes on gains. The more concentrated a strategy, the more turnover it has. This may be prohibitive for taxable investors, who will pay substantial taxes if they have to sell positions regularly.^{12} Taxable investors could avoid turnover by buying an ETF that uses an appropriately concentrated strategy. ETFs charge fees, and concentrated funds tend to have substantially higher fees than broad ETFs. But these fees are probably still lower than the cost of hiring an investment manager, and lower than the tax drag from trading individual stocks.
How does concentrated investing compare to using leverage?
Adding leverage to a portfolio increases both return and volatility while keeping riskadjusted return the same (before costs). In the backtests we looked at, more concentrated portfolios had both higher returns and higher riskadjusted returns.
Two other important considerations:
 Small investors have no difficulty buying an equalweighted portfolio of smallcap stocks. Large investors cannot buy smallcaps without moving the market.
 Small investors face relatively high costs of leverage. Large investors can take advantage of their size to get leverage more cheaply.
These considerations suggest that small investors should invest in concentrated portfolios while large investors should use leverage.
Under some conditions, small investors may still prefer to hold leveraged broad portfolios if they can, mainly for two reasons:
 Manually managing a basket of stocks is both less taxefficient and more timeconsuming than buying and holding an ETF (or group of ETFs) and applying leverage.
 The analysis in this essay only considered stock portfolios. Adding bonds, real assets, or other factors might improve an investor’s overall riskadjusted return.
On future expectations
So far, we have looked at simulated historical performance of different strategies. That doesn’t tell us how these strategies will perform in the future.
We care about two questions:
 Will value/momentum investing continue to outperform the market?
 Will concentrated value/momentum investing continue to beat the market, and continue to beat diversified value/momentum investing?
The first question has been addressed elsewhere in detail. For a deep dive into arguments on why value investing might not work anymore and why they’re probably wrong, see Israel, Laursen & Richardson (2020), Is (Systematic) Value Investing Dead? and Cliff Asness’ (less rigorous, easier to read) article of the same name. The future of momentum investing is much less clear. Israel & Moskowitz (2013)’s The Role of Shorting, Firm Size, and Time on Market Anomalies provides at least some reason to expect momentum’s outperformance to persist, but no strong evidence.^{13}
For the second question, if we expect value and maybe momentum to persist, then we should probably expect concentrated portfolios’ even better performance to persist for the same reasons. In fact, the argument that concentrated value/momentum will persist appears even stronger. If strategies like value and momentum become more popular among large sophisticated investors, then they will tend to perform worse. But large investors cannot buy equalweighted small cap portfolios. And small investors who do use value/momentum typically invest via broad mutual funds or ETFs, not concentrated baskets of stocks.
As a counterpoint, just because a strategy has higher risk doesn’t mean you’re compensated for that risk. For example, a basket of 30 randomlychosen stocks is riskier than an index fund, but doesn’t have higher expected return.^{14} Investing in concentrated value/momentum portfolios only makes sense if they’re able to outperform randomlychosen concentrated portfolios.
Finding concentrated ETFs
The etf.com screener can be used to identify concentrated ETFs. For example, the following table lists every ETF filtered by “Strategy: Value” and “Weighting Scheme: Equal”. This list does not include every concentrated value ETF because (a) some funds are concentrated but not equalweighted, and (b) some funds don’t have sufficiently detailed metadata on etf.com to show up in these screens. I attempted to estimate concentration by comparing the number of stocks in each fund to the underlying index, but the numbers might not be fully accurate.
ETF  Region  Market Cap  Concentration 

EEMD  emerging markets  mid+large  10% 
IVAL  developed exUS  mid+large  5% 
QVAL  US  mid+large  5% 
SPDV  US  mid+large  10% 
SVAL  US  small  12.5% 
Disclosure: I invest in QVAL and IVAL.
Errata
20220415:
The original version of this post claimed that concentrated and broad factor portfolios have similar riskadjusted returns. That is incorrect: equalweighted concentrated portfolios have higher riskadjusted returns than valueweighted broad portfolios.
Originally, I calculated the Sharpe ratio using the geometric mean, that is, (geometric mean  riskfree rate) / standard deviation
. I should have calculated it using the arithmetic mean: (arithmetic mean  riskree rate) / standard deviation
. The geometric Sharpe ratio understates the difference in riskadjusted return between (relatively) lowvolatility and highvolatility portfolios.
This change suggests that small investors should favor concentrated portfolios over leveraged broad portfolios even if they have access to cheap leverage.
I added a new table to report Sharpe ratios for various portfolios. I also corrected the table in Appendix B and updated it to include historical data through 202202.
Appendix A: Significance tests
This table shows pvalues for the excess return of top 10% equalweighted over top 50% valueweighted for various metrics.^{15}^{16}
 Tested using monthly data.
 Calculated over log returns on the assumption that log returns follow a normal distribution.
 Using a twosided Ttest on the null hypothesis that the excess return of the concentrated portfolio is zero.
metric  mean  stdev  n  pval 

B/M  0.66  5.401  1132  3.2e5 
E/P  0.32  3.294  832  6.8e3 
CF/P  0.43  3.536  832  6.4e4 
Mom  0.53  3.615  1119  9.6e7 
Interestingly, among the two null hypotheses
 a concentrated factor portfolio has the same return as a broad factor portfolio, and
 a concentrated factor portfolio has the same return as the broad market,
the first hypothesis is rejected much more strongly. This happens because the two factor portfolios are highly correlated, so their difference has a relatively low standard deviation.
The excess returns of (a) top 10% VW over top 50% VW, (b) top 10% EW over top 50% EW, (c) top 50% EW over top 50% VW, and (d) top 10% EW over top 10% VW were generally statistically significant, but to a lesser degree. The next table gives pvalues for these four portfolio differences using each of the four factors:
B/M  E/P  CF/P  Mom  

10% VW  50% VW  0.28  0.08  0.03  2e4 
10% EW  50% EW  2e3  2e3  3e3  2e4 
50% EW  50% VW  2e5  0.06  3e3  1e4 
10% EW  10% VW  2e5  0.11  0.03  0.03 
(9 out of 16 are significant at p=0.01, and 5 are significant at p=0.001.)
This suggests that both methods of increasing concentration (moving from 50% to 10%, and from valueweight to equalweight) by themselves increased return.
The next table shows pvalues from significance tests for standard deviation over B/M (equalweighted). Differences in standard deviations between portfolios were highly significant.
 Tested using monthly data, n=1132.
 Differences in standard deviations were tested using a twosided Ftest.
pval  

50% BM  10% BM, small 10%  1e22 
50% BM  10% BM  1e11 
30% BM  10% BM  1e6 
10% BM  10% BM, small 10%  1e8 
30% BM  30% BM, small 10%  1e10 
Caveat #1: These ttests assume that stock returns follow a normal distribution. Although this is a common assumption, it’s not quite accurate, as stocks tend to experience tail events more frequently than a normal distribution would predict. That means the “true” pvalues are higher than what’s reported above, and the lower the reported pvalue, the more wrong it is. For example, at one point the concentrated B/M portfolio experienced an excess drawdown of 43% over a 10month period, which “should” only happen once every 21,000 years.^{17} (Similarly, US equities “should” experience an 80% drawdown only once every 70,000 years, and yet it happened during the Great Depression.)
Caveat #2: I am not particularly wellversed in significance testing, so I could have made mistakes in these calculations.
Appendix B: Replication on international equities
The Ken French Data Library only includes international equity returns back to 1990 (instead of 1926), and only reports value/size quintiles rather than deciles. So this international replication is more limited, but it illustrates the same patterns.
This table gives summary statistics for a variety of equalweighted and valueweighted portfolios on B/M and size at various levels of concentration. As with US equities, geometric return increased with higher concentration, smaller size, and equal weighting. These three relationships were all statistically significant at p<0.001, even when excluding the first portfolio in the table (which had a much higher return than the others).^{18}^{19}
Sharpe ratio  Return  Standard Deviation  

20% BM, small 20% EW  0.97  18.1%  16.3% 
20% BM EW  0.50  10.2%  17.5% 
40% BM, small 40% EW  0.62  11.6%  16.3% 
40% BM EW  0.47  9.2%  17.0% 
20% BM, small 20% VW  0.63  11.4%  15.4% 
20% BM VW  0.40  8.1%  16.9% 
40% BM, small 40% VW  0.48  9.0%  15.5% 
40% BM VW  0.39  7.7%  16.3% 
Appendix C: Factor regression on selected portfolios
This table provides a monthly time series factor regression on several portfolios, using factors as defined in the Ken French data library.
 Beta: market beta factor.
 SMB: size factor.
 HML: value factor.
 Alpha: excess return not explained by these three factors.
 Pvalues are reported for the null hypothesis that alpha = 0.
 I use 30% as the diversified portfolio rather than 50% because I cannot fully accurately construct a 50% valueweighted portfolio using the Ken French data. For the data on historical returns, I wasn’t concerned about this because it doesn’t make much difference; but for a factor regression, I want to be as accurate as possible.
Beta  SMB  HML  Alpha  pval  

30% BM VW  1.07  0.22  0.79  0.05  0.07 
10% BM VW  1.18  0.55  1.08  0.23  0.002 
10% BM, small 10% VW  0.97  1.57  1.14  0.05  0.74 
30% BM EW  1.01  1.06  0.85  0.24  2e5 
10% BM EW  1.03  1.33  1.11  0.31  0.004 
10% BM, small 10% EW  0.97  1.71  1.21  0.59  0.0008 
Based on this limited sample, it appears that more concentrated portfolios do not have higher market beta, but do tend to have greater exposure to the size and value factors, as well as higher alpha. Most notably, the equalweighted portfolios have more alpha than the valueweighted ones.
If we add leverage to the 30% VW portfolio in an attempt to replicate a more concentrated portfolio like 10% EW, the former will have less alpha and more market beta. This may be undesirable in the context of a broader portfolio that includes other allocations to equities.
I found this result surprising. I would have predicted that concentrated portfolios’ higher return would primarily come from higher factor exposure, especially exposure to the value factor (HML). While my prediction is at least partially true, even the most concentrated portfolio only had 1.21x loading on HML, and a significant portion of the outperformance could not be explained by any factor.
Conventionally, HML is defined as the return of the top 30% of value stocks minus the bottom 30%, valueweighted. If we redefine HML as the top 30% minus the bottom 30% equalweighted (keeping the other factor definitions the same), we get an interesting result:
Beta  SMB  HML EW  Alpha  pval  

30% BM VW  1.13  0.03  0.53  0.21  8e5 
10% BM VW  1.26  0.27  0.78  0.47  9e8 
10% BM, small 10% VW  1.03  1.15  1.08  0.37  0.004 
30% BM EW  1.06  0.79  0.72  0.02  0.7 
10% BM EW  1.09  0.96  0.99  0.06  0.6 
10% BM, small 10% EW  1.02  1.24  1.23  0.09  0.6 
We can see that higher factor loadings on SMB and HML EW fully explain the return of the more concentrated portfolios. Meanwhile, the valueweighted portfolios have statistically significant negative alpha.
One possible explanation is that switching from valueweighting to equalweighting “unlocks” more of the value premium by allowing investors to assign relatively higher weight to the most undervalued stocks, and this component of the value premium is invisible to HML VW.
The other value metrics, E/P and CF/P, exhibit a similar effect: when we regress on a valueweighted HML factor using E/P or CF/P (instead of B/M), the equalweighted portfolios have statistically significant alpha at p<0.001, with 10% EW having more alpha than 30% EW. On an equalweighted HML factor, valueweighted portfolios have negative alphas, but none are statistically significant at p<0.001 (or even p<0.01).
The following table gives factor regressions for selected momentum portfolios. “Mom” is the momentum factor.
Beta  SMB  Mom  Alpha  pval  

30% Mom VW  1.07  0.08  0.37  0.03  0.35 
20% Mom, small 20% VW  1.09  1.32  0.23  0.44  1e4 
30% Mom EW  1.03  0.74  0.27  0.31  1e11 
10% Mom EW  1.11  0.94  0.45  0.28  1e5 
20% Mom, small 20% EW  1.05  1.38  0.17  0.65  1e6 
Here we see qualitatively similar results to the first table, where more concentrated portfolios tend to have stronger exposure to the size and momentum factors, as well as significantly positive alpha.
When we use an equalweighted momentum factor instead of valueweighted, concentrated portfolios still have statistically significant alpha. However, when we replace the momentum factor with an equalweighted longonly factor (that is, 30% Mom EW minus the riskfree rate), none of the momentum portfolios have alpha (either positive or negative).^{20} Relatedly, while equalweighted momentum portfolios did earn higher riskadjusted returns than comparable valueweighted strategies, the difference was not as large as for B/M. These two observations suggest that switching from valueweighting to equalweighting does not matter as much for momentum as it does for value. This makes intuitive sense: equalweighting implicitly tilts toward smallcap value, so it unlocks more of the value premium, but there’s no reason to expect it to do the same for momentum.
Notes

Ang, Hodrick, Xing & Zhang (2006). The CrossSection of Volatility and Expected Returns. ↩

Frazzini & Pedersen (2013). Betting Against Beta. ↩

Fama and French (1992). The CrossSection of Expected Stock Returns. ↩

Jegadeesh and Titman (1993). Returns to Buying Winners and Selling Losers: Implications for Stock Market Efficiency. ↩

Asness, Moskowitz, and Pedersen (2013). Value and Momentum Everywhere. ↩

Baltussen, Swinkels, and van Vliet (2019). Global Factor Premiums. ↩

All returns reported in this essay are geometric returns, not arithmetic. ↩

B/M and Mom historical returns were calculated over the period 1927–2020. E/P and CF/P were calculated over 1951–2020. ↩

I can only show this chart for B/M because that’s the only metric for which the Ken French Data Library includes the necessary data. ↩

It makes sense that the highest returns would show up in stocks that where it’s difficult to actually take advantage of their high return. ↩

The paper Trading Costs by AQR Capital Management found that their own trading costs were significant enough that they probably could not run an equalweighted small cap value strategy like I described above, but they can still implement value and momentum in mid to large caps. AQR invests billions to tens of billions of dollars in each of its funds, so this information doesn’t tell us much about what smaller investors can do, but it at least indicates that investors with billions of dollars probably can’t increase risk and return via concentration in the manner I describe in this essay. Such investors who want to increase return probably need to use leverage. ↩

In Joel Greenblatt’s book, The Little Book that Beats the Market, in which he proposes following a concentrated value investing strategy, he offers a trick to increase tax efficiency:
For individual stocks in which we are showing a loss from our initial purchase price, we will want to sell a few days before our oneyear holding period is up. For those stocks with a gain, we will want to sell a day or two after the oneyear period is up. In that way, all of our gains will receive the advantages of the lower tax rate afforded to longterm capital gains […], and all of our losses will receive shortterm tax treatment […].
This works for US investors; I can’t comment on how taxes work in other countries. ↩

Perhaps the best argument that momentum will persist: There’s good reason to believe that the value premium will persist, and if value can persist, why not momentum? ↩

If 30 stocks are chosen uniformly at random, they would have outperformed historically, because equal weighting has outperformed market cap weighting. But if the 30 stocks were selected at random in proportion to market cap, then they’d have the same expected return as the market, but with higher volatility. ↩

All pvalues are rounded up. For any pvalues smaller than 1e5, I rounded up to the nearest power of 10. ↩

We have discussed concentration along three dimensions: factor concentration, weighting, and market cap. This table compares the most concentrated to the least concentrated portfolios along the first two dimensions, but does not use market cap because not all factors had data on portfolios sorted by market cap. ↩

Technically, the probability (assuming a normal distribution) is higher than that, because I cherrypicked the worst rolling 10month period rather than looking at nonoverlapping periods. ↩

This surprised me—I thought some of the relationships would be statistically insignificant because the data sample only covers 30 years. For comparison, the long/short value factor (HML) was statistically insignificant over this sample (p=0.34), as was market beta (p=0.09). ↩

I did not establish a significance threshold in advance. p<0.001 was the smallest power of 10 at which all three tests were significant. ↩

All alphas fall between 0.05 and 0.07, with the lowest pvalue being 0.28. ↩