Passively managing individual stocks

In "Common Sense on Mutual Funds,"  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."

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

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.

In the case of a US taxpayer living outside the US (but still subject to citizenship-based taxation): 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.
 * Locally-domiciled index funds are treated as Passive Foreign Investment Companies (PFICs) 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).

Demerits
Why might one not want to do this?

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.

Psychology
Owning individual stocks instead of an index fund is kind of like seeing how the proverbial sausage is made. 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.

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

B_{nm} = \left(\sqrt{\frac{\frac{1}{n}+\frac{n-1}{n}\rho}{\frac{1}{m}+\frac{m-1}{m}\rho}}-1\right)EP $$ where $$\rho$$ is the expected correlation between any pair of stocks, and $$EP$$ is the expected equity premium. (In his paper, he uses $$\rho$$ = 0.08, and $$EP$$ = 8.79%/year.

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). EWC has $$m$$ = 97 stocks, and an ER of 0.49%/year. If the investor holds at least $$n$$ individual Canadian stocks such that $$B_{nm}$$ < 0.49%/year, that investor will have a higher expected net return by holding those individual stocks than by holding the ETF. Using the somewhat bold assumption that $$\rho$$ and $$EP$$ 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) for Japanese equities, which has 317 holdings and 0.49%/year expense ratio. That investor would have a higher expected net 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) 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 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 that the minimum number of stocks according to this formula goes down for lower values of $$EP$$, and higher values of $$\rho$$.

Ikenberry, Shockley and Womack
Ikenberry, Shockley and Womack 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&P500 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&P500 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&P500 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 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 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, 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 volatility 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.