Assumes 3.5% risk-free return and 4.0% market risk premium.
This approach is straightforward to implement in practice, even with more nuanced assumptions. The current risk-free return is readily observable. There are a variety of methods for estimating the expected return of a diversified market portfolio. For example, we might adjust the historical risk premium to accord with current risk levels and add this adjusted risk premium to the current risk-free return. We may expect some asset classes to produce returns that differ from those that would occur if markets were in equilibrium and perfectly integrated, especially if they are not typically arbitraged against other asset classes. Suppose we expect commodities to return 7.0% and we assign as much confidence to this view as we do to the equilibrium return. We can blend the equilibrium estimate with our view to derive expected return. The final column of Table 2.1 shows the expected returns for each of the asset classes in our analysis.
Estimating Standard Deviations and Correlations
We also need to estimate the standard deviations of the asset classes as well as the correlations between each pair of asset classes. We estimate these values, shown in Table 2.2, from the monthly returns for the period beginning in January 1976 and ending in December 2015.
TABLE 2.2 Standard Deviations and Correlations
| Asset Classes | Standard Deviations (%) | Correlations | ||||||
|---|---|---|---|---|---|---|---|---|
| a | b | c | d | e | f | |||
| a | US Equities | 16.6 | ||||||
| b | Foreign Developed Market Equities | 18.6 | 0.66 | |||||
| c | Emerging Market Equities | 26.6 | 0.63 | 0.68 | ||||
| d | Treasury Bonds | 5.7 | 0.10 | 0.03 | −0.02 | |||
| e | US Corporate Bonds | 7.3 | 0.31 | 0.24 | 0.22 | 0.86 | ||
| f | Commodities | 20.6 | 0.16 | 0.29 | 0.27 | −0.07 | 0.02 | |
| g | Cash Equivalents | 1.1 | 0.02 | 0.02 | 0.03 | 0.18 | 0.09 | 0.06 |
Efficient Portfolios
With this information, we use optimization to combine asset classes efficiently, so that for a given level of expected return the efficiently combined asset classes offer the lowest level of risk, measured as the standard deviation. A continuum of these portfolios plotted in dimensions of expected return and standard deviation is called the efficient frontier, as we discussed earlier.
There are a variety of methods for identifying portfolios that reside along the efficient frontier. We next describe two methods and illustrate them with a hypothetical portfolio that consists of just two asset classes: stocks and bonds.
Matrix Inversion
To begin, we define a portfolio's expected return and risk.
As noted earlier, the expected return of a portfolio is simply the weighted average of the assets' expected returns. Equation 2.1 shows expected return for a portfolio consisting of only stocks and bonds:
In Equation 2.1,
As noted earlier, portfolio risk is a little trickier. It is defined as volatility, and it is measured by the standard deviation or variance (the standard deviation squared) around the portfolio's expected return. To compute a portfolio's variance, we must consider not only the variance of the asset class returns, but also the extent to which they covary. The variance of a portfolio of stocks and bonds is computed as follows:
Here