# Passively managing individual stocks

It is widely accepted among Bogleheads that the best portfolios incorporate passive management, low costs, wide diversification and tax efficiency. The simplest way to achieve such attributes is by using a low-cost passive index fund such as those offered by Vanguard. However, some may find that constructing your own portfolio of individual stocks may offer some advantages on cost and/or taxes, such that the benefits may be sufficient to account for the potentially inferior diversification on the portfolio.

In Common Sense on Mutual Funds, [1] Jack Bogle suggests that a reasonable alternative to an index fund for some investors would be to hold a well-diversified portfolio of individual stocks, as long as they are held long-term, with a minimum of trading costs incurred. This article outlines some suggestions for how to build a portfolio of individual stocks to cover at least part of one's overall stock allocation. It will also attempt to summarize the advantages and possible pitfalls of doing so.

Note that the discussion here assumes that one is not trying to beat the market, but rather, by passively managing individual stocks create a "DIY index fund."

Holding a well-diversified portfolio of passively managed individual stocks may offer cost advantages, and more opportunities for tax-loss harvesting or for donating appreciated assets than do index funds. Special factors affecting US citizens living abroad may also make holding individual stocks a suitable option.

### Costs

An obvious advantage that individual stocks have over an index fund is an expense ratio (ER) of zero. Depending on what class of stocks one is trying to cover, this may or may not be a significant advantage in itself. In the case of single-country or -industry stocks, for example, where index funds often have ERs of 0.5% or more, the savings solely due to ER of using individual stocks would be more substantial than in the case of the broader S&P 500 index.

A hidden cost in mutual funds is the impact of turnover, since those costs are not reflected in the expense ratio. To the extent that an individual can realize lower turnover than the relevant index fund, they will likely have lower costs. An index fund may be forced to have turnover to track the companies being deleted and added to the index, while an individual who has a limited portfolio isn't forced to make as many changes.

### Taxes

Individual stocks present many more opportunities for tax-loss harvesting than do index funds. The potential advantage has been estimated to be equivalent to about 0.5%/year, depending on tax bracket. [2] [3]

An investor with individual stocks can gain additional tax management benefits through donations, gifting and dividend yield management. These are discussed in more depth below in the tax management section.

In the case of a US taxpayer living outside the US (but still subject to citizenship-based taxation):

• Locally-domiciled index funds are treated as a passive foreign investment company (PFIC) by the IRS, which results in extremely unfavorable taxation; and
• US-domiciled funds may not be available, may face unfavorable or double taxation in the country of residence, and/or may have high expense ratios (for example, single-country funds covering the country of residence).

Given these constraints, such a person may find a portfolio of individual stocks the best way to cover, e.g., the domestic stock market of the country of residence.

While creating a portfolio of individual stocks has some advantages, keep the following caveats in mind. An individual stock portfolio will be less diversified than the market portfolio, and will be exposed to the problem of market skewness. In addition, a portfolio of individual stocks is likely to require heavier bookkeeping burdens than investing in a total market index fund. Finally, empirical studies have shown that individual investors exhibit multiple behavioral errors that result in poor performance.

### Diversification

Any subset of stocks is by definition less diversified than the whole market, and creates exposure to greater unsystematic risk.

### Skewness

Individual stocks are generally observed to exhibit positive skewness, meaning that most stocks will have a somewhat lower return than the market average, with a few having much higher return. As noted by Bernstein[4] and many others, the result is that any subset of stocks is likely have a lower return than the whole market, due to failing to include some of the highest performers; i.e., the median return is lower than the mean return.

This skewness impact is more pronounced in smaller stocks, so a strategy of investing only in Large Capital individual stocks (and using mutual funds for diversification in other areas) tends to be less exposed to this issue.

### Documentation

There may be more cost bases (see below) to keep track for a collection of individual stocks than for a single index fund, depending on how the purchases were carried out (single lump-sum into the index instead of monthly purchases, for example). In addition, individual stocks sometimes create taxable or other paperwork-generating events such as going private, getting bought out, going bankrupt and splitting. This is particularly the case with smaller cap stocks.

### Behavioral effects

Owning individual stocks instead of an index fund is kind of like seeing how the proverbial sausage is made.[note 1] Some of your stocks will rocket up, some will go out of business, and, due to the skewness mentioned above, the majority will underperform the market as a whole. The same things happen inside an index fund, of course, but hidden from view. While these gyrations can provide tax-savings opportunities, they also take some getting used to. Not everyone is of a mindset to watch the sausage production process in their portfolio with equanimity. Know thyself.

These difficulties are evident in empirical studies of the stock trading behavior of individual stock market investors. Odean and Barber have studied individual performance (beginning in 1991) [5] and found that overconfident individual investors [6] "(1) underperform standard benchmarks (e.g., a low cost index fund), (2) sell winning investments while holding losing investments (the “disposition effect”), [7] 3) are heavily influenced by limited attention and past return performance in their purchase decisions, (4) engage in naïve reinforcement learning by repeating past behaviors that coincided with pleasure while avoiding past behaviors that generated pain, and (5) tend to hold undiversified stock portfolios. These behaviors deleteriously affect the financial well being of individual investors." [8]

## How many stocks?

There are several different ways to estimate how many stocks one should aim to hold at minimum. A few approaches found in the academic literature are discussed here. While none of the papers discussed below provide a definitive answer to that question, they are instructive in illuminating different angles on it.

### Statman

Statman[9] provides the following formula for the benefit ${\displaystyle{\displaystyle B_{nm}}}$ in terms of risk-adjusted return gained by moving from ${\displaystyle{\displaystyle n}}$ stocks to ${\displaystyle{\displaystyle m}}$ stocks:

${\displaystyle{\displaystyle B_{nm}=\left({\sqrt{\frac{{\frac{1}{n}}+{\frac{n-% 1}{n}}\rho}{{\frac{1}{m}}+{\frac{m-1}{m}}\rho}}}-1\right)EP}}$

where ${\displaystyle{\displaystyle\rho}}$ is the expected correlation between any pair of stocks, and ${\displaystyle{\displaystyle EP}}$ is the expected equity premium. (In his paper, he uses ${\displaystyle{\displaystyle\rho}}$ = 0.08 [note 2], and ${\displaystyle{\displaystyle EP}}$ = 8.79%/year. [note 3]

This formula assumes simple Gaussian return distributions with zero skewness (which is known not to be correct, as noted above), and requires the use of some assumptions about correlation and equity premium, but with those caveats in mind, it is useful to get a ballpark idea of the minimum number of stocks to aim for.

Let's take as an example an investor in Canada who is a US taxpayer, for whom the best non-PFIC alternative for covering Canadian stocks might be the iShares MSCI Canada ETF (EWC). [10] EWC has ${\displaystyle{\displaystyle m}}$ = 97 stocks, and an ER of 0.49%/year. If the investor holds at least ${\displaystyle{\displaystyle n}}$ individual Canadian stocks such that ${\displaystyle{\displaystyle B_{nm}}}$ < 0.49%/year, that investor will have a higher expected risk-adjusted return by holding those individual stocks than by holding the ETF.[note 4] Using the somewhat bold assumption that ${\displaystyle{\displaystyle\rho}}$ and ${\displaystyle{\displaystyle EP}}$ for the Canadian market are similar to the values Statman uses above for the US market, the investor expectantly comes out ahead over EWC after about 50 stocks.

For another example, a US taxpayer in Japan might consider iShares MSCI Japan ETF (EWJ) [11] for Japanese equities, which has 317 holdings and 0.49%/year expense ratio. That investor would have a higher expected risk-adjusted return than EWJ after about 75 individual Japanese stocks.

For the above two investors, the minimum required number of stocks would go down if possible savings due to tax-loss harvesting are taken into account. However, since they both have to pay taxes to two different governments, based on gains and losses calculated in two different currencies that float relative to each other, the tax-loss harvesting opportunities would likely be considerably reduced as compared to a taxpayer subject only to one country's tax code.

A US-based investor, on the other hand, might instead expect for tax-loss harvesting to provide a bigger benefit than ER savings. In comparison to an S&P 500 index fund such as the Vanguard S&P 500 ETF (VOO) [12] with ER of 0.05%/year, if the expected tax savings from tax loss harvesting are 0.5%/year, that investor would expect to come out ahead of VOO on a risk-adjusted basis after about 75 stocks. If the expected tax savings are 1%/year, that number falls to about 40 stocks. Similar considerations would apply to any investor based outside the US who is not also a US taxpayer.[note 5]

Note that the minimum number of stocks according to this formula goes down for lower values of ${\displaystyle{\displaystyle EP}}$, and higher values of ${\displaystyle{\displaystyle\rho}}$.

It might be noted that Statman's formula implicitly assumes portfolios of equally-weighted stocks. For a capitalization-weighted index, the effective number of stocks would be less than the actual number, because the fluctuations of the largest capitalization stocks contribute much more to the overall portfolio volatility than to those of the smallest capitalization stocks (though this would be partially counterbalanced by the higher individual volatilities of smaller-capitalization stocks). If one is comparing a roughly equal-weighted portfolio to a capitalization-weighted index fund, substituting the effective number of stocks in the index for the actual number would produce a somewhat lower suggested minimum number of individual stocks to hold. On the other hand, accounting for skewness should lead to an increase in the suggested minimum number of stocks to hold.

### Surz and Price

Surz and Price (2000) [13] examine diversification measures such as R-squared and tracking error for US individual stock selections over the January 1986 - June 1999 period.

Over this period the overall market experienced a standard deviation of 14.5%. A single stock portfolio provided an average standard deviation of 45.0%; a fifteen stock portfolio averaged 16.5% (range 14.3% - 19.2%); a thirty stock portfolio averaged 15.4% (range 13.9% - 17.5%); a sixty stock portfolio averaged 15.2% (range 13.6% - 17.2%).

While one individual stock has a standard deviation of 45% and the entire market has 14.5%, portfolios of 30 stocks are shown to eliminate 97% of the excess (diversifiable) standard deviation. A 60 stock portfolio eliminates 98% of excess standard deviation.

Surz and Price found that increasing the number of stocks in a portfolio increased the portfolio's R-squared and reduced portfolio tracking error. [note 6]. The authors suggest two methods for reducing tracking error over random stock selection:

1. Professional money managers can use portfolio optimization.
2. A less sophisticated approach can invest in the largest capitalization stocks.

### Ikenberry, Shockley and Womack

Ikenberry, Shockley and Womack[14] use historical backtesting over the 34-year period from 1962 through 1995 to examine the impact of skewness, which they measure as the difference between the mean and median yearly average return for an n-stock portfolio, for several values of n from 15 to 150. They found that for 35-stock portfolios, where the stocks are randomly chosen from the S&P 500 index and equally weighted, the average yearly median portfolio return lags the mean by 0.22%. This number goes down to 0.14% for 50-stock portfolios, 0.09% for 75-stock portfolios, 0.06% for 100-stock portfolios, and 0.03% for 150-stock portfolios. The skewness cost is somewhat lower for capitalization-weighted portfolios, and for equal-weighted portfolios where the probability of selecting a stock is proportional to its market capitalization. For more details, see Table V and Figure 2 of their paper.

They do not find evidence of any systematic return penalty for sub-sampling the S&P 500 index. (However, they are not comparing risk-adjusted returns, just straight returns, so their results cannot be directly compared with, e.g., Statman's formula.) They find that for the portfolios where the holdings are capitalization weighted, or equally weighted but selected with a probability proportional to capitalization, the yearly mean and median returns slightly exceed the return of the S&P 500 index itself, though not by statistically significant amounts (about 1-sigma in standard deviation of the mean, if calculated using the yearly numbers in their Table II). They also find that equal-probability, equal-weighted portfolios show about a 2.5% boost in mean and median returns compared to the capitalization-weighted portfolios, and slightly more compared to the S&P 500 index itself. This is only a little over a 2-sigma effect (again, calculating from their yearly numbers in Table II), but suggests the presence of a measurable small-cap premium even within the confines of the S&P 500 universe.

### Domian, Louton and Racine

Domian, Louton and Racine[15] use historical backtesting over the 20-year period from Jan 1985 to Dec 2004 to study the effect of the number of stocks in a portfolio on shortfall risk. They created equal-weighted portfolios from a universe of 1000 stocks, consisting of the 100 largest companies in each of 10 industries, covering 82% of the capitalization of the total market. They calculate ending wealth distributions for random buy-and-hold portfolios of varying numbers of stocks. Using a 1% chance of underperforming US Treasuries over that time period as criterion, they state that at least 164 stocks are needed.

In the context of the present discussion, a more apt comparison might be with the return of a cap-weighted index fund covering that universe. For nearest comparison purposes, the 20-year cumulative return of the MSCI US Large and Mid cap index[16] turned one dollar invested over that same period into $11.96, for an annualized return of 13.2%. Looking at Figure 3 of their paper (Wealth per dollar of initial investment), it can be seen that a 30-stock portfolio had just over a 50% chance of beating the index, a 50-stock portfolio over 55%, and a 100-stock portfolio about a 65% chance. These results presumably reflect the effect of a small-cap premium within the 1000-stock universe. These results do not consider costs. If a cost advantage of 0.5%/year is expected from individual stocks over an index fund tracking the index, the ending value of one dollar in the index with an annualized return of 12.7% (= 13.2%-0.5%) becomes$10.94, at which point a 20-stock portfolio has a greater than 50% chance of beating the index fund. A 30-stock portfolio then has close to a 60% chance of coming out ahead, a 50-stock portfolio better than 65%, and a 100-stock portfolio has better than 75% chance of outperforming the index fund.

Of course, the magnitude of possible shortfalls should also be taken into consideration. Figure 3 shows how that magnitude grows with decreasing numbers of stocks.

### Summary

In general, the answer to "how many stocks are recommended?" is, of course, as many as practically possible. Holding fewer stocks increases unsystematic risk and the effect of skewness on final portfolio value. Holding stocks in equal weights can pick up some small-cap premium, but it should be borne in mind that this also picks up extra volatility. There is no free lunch.

Reducing the number of stocks held is in principle always a negative factor in terms of risk/reward ratio. The only guaranteed benefit can come from lower costs, due to expense ratio and/or savings due to tax management opportunities. The trick is to figure out where the benefit balances out the increased risk. This is, necessarily, a somewhat individualized decision.

## Portfolio construction

### How to choose stocks?

Some potentially useful factors to consider in choosing stocks are:

• Size (capitalization)
• Value (book-to-market ratio)
• Industry (at least early on)
• Liquidity (for smaller stocks)
• Dividend rate (for tax management purposes)

Size and value loadings and industry classification are of interest if one is trying to match or tilt away from market weights. Size and value are established Fama-French risk factors, which some investors like to tilt towards or away from. Domian, Louton and Racine (see above) found that diversification across industries helped reduce dispersion of returns for small portfolio sizes, though it becomes less important for larger numbers of stocks held. Industry classification may also be of interest to an investor who already has too much exposure to the risk of a certain industry through work, and wishes to reduce portfolio exposure to that industry. Low liquidity can be a problem in very small-cap stocks, leading to larger bid-ask spreads. Dividend yield may be of interest to an investor looking to minimize current taxable income.

Some other factors which have support in the academic literature are momentum, profitability, and low-beta. Investors interested in pursuing these tilts may find individual stocks a way to do so in the absence of easily-available index funds that incorporate those tilts.

Note: if the Efficient Market Hypothesis holds, then reading balance sheets, shareholder reports and analyst recommendations is unnecessary and a complete waste of time. The strategy used for deciding which exact stocks to hold should not matter -- one can neither help nor hurt oneself through choice of strategy. Random selection is a perfectly valid strategy, although it isn't much trouble to select stocks from different industries such that one achieves a reasonable diversification across sectors and industries.

### Weighting

The easiest way for an individual investor to buy stocks is in roughly equal weights -- for example, by buying equal-sized amounts of new stocks every month with new money. This will typically lead to a significant small-cap tilt in the portfolio if the stocks are selected more or less at random. If exposure to the size factor is not desired, then weighting one's stocks by relative market capitalization was found by Ikenberry, Shockley and Womack (see above) to yield median returns very close to the overall (capitalization-weighted) index. As a practical matter, this leads to huge disparities in stock holding sizes for an individual investor, and is difficult to maintain as new money is added to the portfolio. An alternative is to buy equal dollar amounts of each stock, but weight the probability of buying a stock in proportion to its market capitalization; Ikenberry, Shockley and Womack found that this generates results that are statistically indistinguishable from weighting individual holdings in proportion to market cap.

### Tracking error

With smaller portfolios, one will experience "tracking error". This is the effect in which your returns in any one period differ from the entire market. While your long-term returns should be similar to the market, in any individual period you will see differences. The best example of this is when comparing US stock returns to international stock returns. Both may have the same long-term returns, but in any one year the returns will certainly differ. If one compares international returns to a US benchmark index, they will see a difference each year and that would be tracking error. If one constructs a portfolio of US large cap stocks and compares the yearly results to the S&P 500, one will see a difference each year. Over the long-term the cumulative results should be similar, but in any given year there will be a difference. Surz and Price address tracking error and they indicate that random portfolios of 15 stocks should expect a tracking error of 8.1%, which reduces to 6.2% for a 30-stock portfolio and 5.3% for a 60 stock portfolio. Surz and Price indicate that by constructing the portfolio with a focus on balance among industries will reduce the tracking error to 5.4% (15 stocks), 4.2% (30 stocks) and 3.5% (60 stocks).

Note that tracking error is harmless to the disciplined long-term investor. Over time, you will generate the same cumulative returns as your benchmark index since your portfolio has the same risk characteristics. [17] However, some investors may be alarmed to see tracking error, especially in years when their small portfolio underperforms the benchmark index.

### Transaction costs

In the interest of minimizing transaction costs (commissions and bid-ask spreads), it is best to avoid turnover as much as possible. If one is considering tax-loss harvesting, for example, then it should only be done if the tax benefit significantly outweighs the transaction costs.

One of the results that Domian, Louton and Racine (above) found was that rebalanced portfolios have lower terminal wealth dispersions. However, such rebalancing would, besides incurring extra transaction costs, systematically result in creating immediate taxable gains, so it is probably best avoided outside of a commission- and tax-free situation. An exception might be made if a single stock grows to become a significant fraction of the size of the portfolio all by itself; in such a case, selling off some losing positions along with paring down the excess position can help minimize the tax impact.

## References

1. John Bogle, Common Sense on Mutual Funds, pp.373 -401. ISBN 0470138130
2. Arnott, Robert D., Berkin, Andrew L.,Ph.D., and Ye, Jia Ph.D, Tax Management and Mismanagement of Taxable Assets, (2000) No. 2., First Quadrant
3. Berkin, Andrew L. and Ye, Jia, Tax Management, Loss Harvesting, and HIFO Accounting, (2003) No. 2, First Quadrant.
4. Bernstein, William, The 15-Stock Diversification Myth, (Fall 2000). Efficient Frontier
5. Barber, Brad M. and Odean, Terrance,The Common Stock Investment Performance of Individual Investors (May 1998). Available at SSRN.
6. Barber, Brad M. and Odean,Terrance, The Courage of Misguided Convictions: The Trading Behavior of Individual Investors Available at SSRN.
7. Odean, Terrance, Are Investors Reluctant to Realize Their Losses? (December 1997). Available at SSRN.
8. Barber, Brad M. and Odean, Terrance,The Behavior of Individual Investors (September 7, 2011). Available at SSRN.
9. Statman, Meir, How Much Diversification is Enough? (October 2002). Available at SSRN.
10. Vanguard S&P 500 ETF (VOO)
11. Surz, Ronald J. and Price, Mitchell, The Truth About Diversification By Numbers, (Winter 2000). Journal of Investing
12. Ikenberry, David L.,Shockley, Richard L., and Womack, Kent L,Why active fund managers often underperform the S&P 500 index: The impact of size and skewness
13. Domian, Dale L. and Louton, David A. and Racine, Marie D., Diversification in Portfolios of Individual Stocks: 100 Stocks are Not Enough (April 2006). Available at SSRN.
14. MSCI, MSCI Index Returns
15. Looking at how different indexes affect performance., Craig Israelsen, JOI, September /October 2012.