When should an investor favor picking individual stocks over holding a stock index fund? In their November 2012 paper entitled “On Diversification”, Ben Jacobsen and Frans de Roon derive from Modern Portfolio Theory simple rules to compare concentrated investment in a portfolio of one or a few stocks to a broad, diversified (value-weighted) benchmark portfolio. The essential rule is that a concentrated portfolio is preferable to the benchmark portfolio if the product of its expected Sharpe ratio and the expected correlation of its returns with the benchmark’s returns exceeds the expected Sharpe ratio of the benchmark. They apply derivative thumb rules to real stocks to determine conditions under which stock picking is preferable to buying and holding a diversified benchmark portfolio. Using theoretical derivations and monthly returns and fundamentals for the 500 largest non-financial companies as of the end of the sample period with a history of at least five years during 1926 through 2011,* they find that:*

- From theory, for stocks with returns typically correlated with a diversified value-weighted benchmark:
- A one-stock portfolio is preferable to the benchmark if its expected Sharpe ratio (expected return in excess of the risk-free rate) is at least twice (four times) that of the benchmark.
- An equal-weighted portfolio of four or five stocks is preferable to the benchmark if the expected portfolio Sharpe ratio (expected excess return) is at least 50% higher than (twice) that of the benchmark.
- A n equal-weighted portfolio of eight to ten stocks is preferable to the benchmark if the expected portfolio Sharpe ratio (expected excess return) is at least 25% (70%) higher than that of the benchmark.

- Using empirical data for the 500 selected stocks to construct a value-weighted benchmark:
- The average annual excess return on the benchmark is 8.78%, with standard deviation 18.2% and Sharpe ratio 0.48. The average annual excess return for the individual stocks is 17.2%, with average standard deviation 39.0% and average Sharpe ratio 0.13. The average correlation of individual stock returns with those of benchmark is 0.45. So, empirically, a typically correlated single stock is preferable to the benchmark when its annual expected Sharpe ratio exceeds 1.07.
- About 31% of individual stocks are preferable to the benchmark (expected Sharpe ratio times expected correlation of returns with the benchmark exceeds the expected Sharpe ratio of the benchmark), and another 38% of the stocks are close.
- About 45% (51%) of equal-weighted five-stock (ten-stock) portfolios are preferable to the benchmark, and another 37% (36%) are close.
- Based on NBER-specified economic cycles, stock picking tends to beat benchmark investing during contractions, while the benchmark tends to win during expansions.
- Based on expected performance data from Value Line during 1975 through 2001, about 40% of individual stocks are preferable to the benchmark.
- Individual stocks preferable to the benchmark tend to have high (but not the highest) market capitalizations, high book-to-market ratios, high earnings-to-price ratios, high cash flow-to-earnings ratios and strong price momentum. They generally do not exhibit short-term reversals and have average dividend yields. They prevail in non-durables, energy, shops and healthcare industries and are rare in durables, telecom and other industries.

- However, an equal-weighted benchmark of all 500 stocks dominates stock picking.

In summary,* evidence indicates that concentrated stock portfolios are very often competitive with buying and holding a value-weighted diversified benchmark portfolio.*

Cautions regarding findings include:

- Modern Portfolio Theory assumes normality of asset return distributions. To the extent that actual distributions are not normal, the theory breaks down.
- The rules of thumb that simply relate stock and benchmark expected Sharpe ratios assume the stock is “typical” with respect to the correlation of its returns with those of the benchmark. The rules of thumb that simply relate stock and benchmark expected returns assume the stock is “typical” with respect to both return correlation and return volatility. Investors should prudently verify stock typicality before applying these rules.
- The method of identify economic expansions and contractions is retrospective. Investors would have difficulty discriminating between these two states in real time.