Portfolio Concepts

CHAPTER 11

PORTFOLIO CONCEPTS

Mean–variance portfolio theory, the oldest and perhaps most accepted part of modern portfolio theory, provides the theoretical foundation for examining the roles of risk and return in portfolio selection.

Mean–variance portfolio theory is based on the idea that the value of investment opportunities can be meaningfully measured in terms of mean return and variance of return.

Markowitz called this approach to portfolio formation mean–variance analysis.

Mean–variance analysis is based on the following assumptions:

1. All investors are risk averse; they prefer less risk to more for the same level of expected return.1

2. Expected returns for all assets are known.

3. The variances and covariances of all asset returns are known.

4. Investors need only know the expected returns, variances, and covariances of returns to determine optimal portfolios. They can ignore skewness, kurtosis, and other attributes of a distribution.2

5. There are no transaction costs or taxes.

An investor’s objective in using a mean–variance approach to portfolio selection is to choose an efficient portfolio. An efficient portfolio is one offering the highest expected return for a given level of risk as measured by variance or standard deviation of return.

The minimum-variance frontier shows the minimum variance that can be achieved for a given level of expected return. The minimum-variance frontier is a more useful concept than the portfolio possibilities curve because it also applies to portfolios with more than two assets.

From Figure 11-1, note that the variance of the global minimum-variance portfolio (the one with the smallest variance) appears to be close to 96.43 (Point A) when the expected return of the portfolio is 6.43. This global minimum-variance portfolio has 14.3 percent of assets in large-cap stocks and 85.7 percent of assets in government bonds. Given these assumed returns, standard deviations, and correlation, a portfolio manager should not choose a portfolio with less than 14.3 percent of assets in large-cap stocks because any such portfolio will have both a higher variance and a lower expected return than the global minimum-variance portfolio.

All of the points on the minimum-variance frontier below Point A are inferior to the global

minimum-variance portfolio, and they should be avoided.

Financial economists often say that portfolios located below the global minimum variance portfolio (Point A in Figure 11-1) are dominated by others that have the same variances but higher expected returns. Because these dominated portfolios use risk inefficiently, they are inefficient portfolios. The portion of the minimum-variance frontier beginning with the global minimum-variance portfolio and continuing above it is called

the efficient frontier. Portfolios lying on the efficient frontier offer the maximum expected return for their level of variance of return.

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.

In summary, when the correlation between two portfolios is less than +1, diversification offers potential benefits. As we lower the correlation coefficient toward −1, holding other values constant, the potential benefits to diversification increase.

2.2. Extension to the Three-Asset Case

A fundamental economic principle states that one is never worse off for having additional choices. At worst, an investor can ignore the additional choices and be no worse off than initially. Often, however, a new asset permits us to move to a superior minimum-variance frontier.

For any portfolio composed of three assets with portfolio weights w1, w2, and w3, the expected return on the portfolio, E(Rp), is

E(Rp) = w1E(R1) + w2E(R2) + w3E(R3)

In this three-asset case, however, determining the optimal combinations of assets is much more difficult than it was in the two-asset example. In the two-asset case, the percentage of assets in large-cap stocks was simply 100 percent minus the percentage of assets in government bonds. But with three assets, we need a method to determine what combination of assets will produce the lowest variance for any particular expected return. At least we know the minimum expected return (the return that would result from putting all assets in government bonds, 5 percent) and the maximum expected return (the return from putting no assets in government bonds, 15 percent). For any level of expected return between the minimum and maximum levels, we must solve for the portfolio weights that will result in the lowest risk for that level of expected return. We use an optimizer (a specialized computer program or a spreadsheet with this capability) to provide these weights.

2.3. Determining the Minimum-Variance Frontier for Many Assets

For a portfolio of n assets, the expected return on the portfolio is

The variance of return on the portfolio is

2.4. Diversification and Portfolio Size

How many different stocks must we hold in order to have a well-diversified portfolio? How does covariance or correlation interact with portfolio size in determining a portfolio’s risk?

As the number of stocks, n, increases, the contribution of the variance of the individual

stocks becomes very small because (1/n)σ2 has a limit of 0 as n becomes large. Also, the contribution of the average covariance across stocks to the portfolio variance stays nonzero because “[(n – 1)/ n ]Cov” has a limit of Cov as n becomes large. Therefore, as the number of assets in the portfolio becomes large, portfolio variance approximately equals average covariance. In large portfolios, average covariance—capturing how assets move together—becomes more important than average individual risk or variance.

With a higher correlation, the investor would need fewer stocks to obtain the same percentage of minimum possible portfolio variance.

One common belief among investors is that almost all of the benefits of diversification can be achieved with a portfolio of only 30 stocks.

The number of stocks needed to achieve a particular diversification gain depends on the average correlation among stock returns: The lower the average correlation, the greater the

...