A diversification benefit is a reduction in portfolio standard deviation of return through diversification without an accompanying decrease in expected return. Portfolio diversification is affected by the number of assets in the portfolio and the correlation between these assets. Correlation The trade-off between risk and return for a portfolio depends not only on the expected asset returns and variances but also on the correlation of asset returns. The correlation between two assets represents the degree to which assets are related. The correlation is the engine that drives the whole theory of portfolio diversification. The following figure illustrates minimum-variance frontier of a two-asset portfolio for four different correlations. The endpoints (X and Y) for all the frontiers are the same, since at each endpoint the expected return and standard deviation are simply the expected return and standard deviation of either asset.

• A: correlation = 1. This indicates a perfect linear relationship between the two assets. Diversification has no potential benefits.
• B: correlation = 0.5. Portfolio diversification can be achieved. The lower the correlation, the greater the diversification benefits.
• C: correlation = 0. This indicates there is no linear relationship between the two assets. More diversification can be achieved then B.
• D: correlation = -1. This indicates a perfect inverse linear relationship. Notice the minimum-variance frontier has two linear segments: XZ and ZY. XZ (line D) is the efficient frontier. The risk of the portfolio can be reduced to zero if desired.

The conclusion: As the correlation between two assets decreases, the diversification benefits increase. Effect of Number of Assets on Portfolio Diversification For an equally-weighted portfolio, its variance is σ2-bar is the average variance of return across all stocks, and Cov-bar is the average covariance of all pairs of two stocks. Note that if n gets large enough:

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• The first component becomes very small.
• The second component gets close to Cov-bar.

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Therefore, the variance of an equally-weighted portfolio approximately equals the average covariance as the number of assets becomes large. Example Assume portfolio A has 2 assets and portfolio B has 30 assets. They are both equally weighted. The average asset variance is 0.5 and the average covariance is 0.3. The variance of A is 1/2 0.5 + 1/2 0.3 = 0.4. The variance of B is 1/30 0.5 + 29/30 0.3 = 0.31. Portfolio B which has more assets has a lower variance. In general, as the number of stocks increases, the variance of the portfolio will decrease.

• It takes less than 30 stocks to achieve 90% of the diversification benefit.
• The higher the average correlation, the greater the number of stocks needed to achieve a specified risk reduction.
• If you add more stocks to the portfolio, the standard deviation of the portfolio will eventually reach the level of the market portfolio.

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