Required Conditions
For a given time horizon or assuming returns are expressed in continuous units, it is a remarkably robust portfolio formation process, assuming that at least one of two conditions prevails: either investor preferences toward return and risk can be well described by just mean and variance, or returns are approximately elliptically distributed.
The objective function for mean-variance analysis is a quadratic function, which many investors find problematic because it implies that at a particular level of wealth investors would prefer less wealth to more wealth. Of course, such a preference is not plausible, but as shown by Levy and Markowitz (1979),2 mean and variance can be used to approximate a variety of plausible utility functions across a wide range of returns reasonably well. If this condition is satisfied, it does not matter how returns are distributed, because investors care only about mean and variance.
If this condition is not satisfied, however, mean-variance analysis requires returns to be approximately elliptically distributed. The normal distribution is a special case of an elliptical distribution, which is itself a special case of a symmetric distribution. A normal distribution has skewness equal to zero, and its tails conform to a kurtosis level of three.3 An elliptical distribution, in two dimensions (two asset classes), describes a scatter plot of returns in which the return pairs are evenly distributed along the boundaries of ellipses that are centered on the mean observation of the scatter plot.4 It therefore has skewness of zero just like a normal distribution, but it may have non-normal kurtosis. The same is true for symmetric distributions more generally, though they also allow for return pairs in a two-dimensional scatter plot to be unevenly distributed along the boundaries of ellipses that are centered on the mean observation of the scatter plot, as long as they are distributed symmetrically. A symmetric distribution that comprises subsamples with substantially different correlations would not be elliptical, for example. The practical meaning of these distinctions is that mean-variance analysis, irrespective of investor preferences, is well suited to return distributions that are not skewed, have correlations that are reasonably stable across subsamples, and have relatively uniform kurtosis across asset classes, but may include a higher number of extreme observations than a normal distribution.
Asset Classes
In Chapter 1, we introduced seven characteristics that define an asset class:5
1 The composition of an asset class should be stable.
2 The components of an asset class should be directly investable.
3 The components of an asset class should be similar to each other.
4 An asset class should be dissimilar from other asset classes in the portfolio as well as combinations of the other asset classes.
5 The addition of an asset class to a portfolio should raise its expected utility.
6 An asset class should not require selection skill to identify managers within the asset class.
7 An asset class should have capacity to absorb a meaningful fraction of a portfolio cost-effectively.
For illustrative purposes we begin by considering the following seven asset classes in our asset allocation analysis: domestic equities, foreign developed market equities, emerging market equities, Treasury bonds, US corporate bonds, commodities, and cash equivalents.6
Estimating Expected Returns
Before we estimate expected returns, we must decide which definition of expected return we have in mind. If we base our estimate of expected return on historical results, we might assume that the geometric average best represents the expected return. After all, it measures the rate of growth that occurred historically or what should happen prospectively with even odds of a better or worse result. However, it does not measure what we should expect to happen on average over many repetitions; the arithmetic average gives this value. But there is a more practical reason for choosing the arithmetic average instead of the geometric average as our estimate of expected return. The average of the geometric returns of the asset classes within a portfolio does not equal the geometric return of the portfolio, but the average of the arithmetic returns does indeed equal the portfolio's arithmetic return. Because we wish to express the portfolio's return as the weighted average of the returns of the component asset classes, we are forced to define expected return as the arithmetic average.7 Of course, we are not interested in the arithmetic average of past returns unless we believe that history will repeat itself precisely. We are interested in the arithmetic average of prospective returns.
To estimate expected returns, we start by assuming that markets are fairly priced; therefore, expected returns represent fair compensation for the degree of risk each asset class contributes to a broadly diversified market portfolio. These returns are called equilibrium returns, and we estimate them by first calculating the beta of each asset class with respect to a broad market portfolio. This calculation implicitly reflects the historical standard deviations and correlations of the assets. Next, we estimate the expected return for the market portfolio and the risk-free return. We calculate the equilibrium return of each asset class as the risk-free return plus the product of its beta and the excess return of the market portfolio. Moreover, we can easily adjust the expected return of each asset class to accord with our views about departures from fair value. Suppose we estimate the market's expected return to equal 7.5% and the risk-free return to equal 3.5%. Given these estimates, together with estimates of beta based on monthly returns from January 1976 through December 2015, we derive the equilibrium returns shown in Table 2.1.
TABLE 2.1 Expected Returns
| Asset Classes | Equilibrium Returns (%) | Views (%) | Confidence (%) | Expected Returns (%) |
|---|---|---|---|---|
| US Equities | 8.8 | 8.8 | ||
| Foreign Developed Market Equities | 9.5 | 9.5 | ||
| Emerging Market Equities | 11.4 | 11.4 | ||
| Treasury Bonds | 4.1 | 4.1 | ||
| US Corporate Bonds | 4.9 | 4.9 | ||
| Commodities | 5.4 | 7.0 | 50.0 |
6.2
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