3  Expected portfolio return

Asset classes are often combined into portfolios and there is a need to calculate information about expected long-term portfolio returns. In order to illustrate the issues that this raises, suppose that a portfolio comprises two asset classes, equities and bonds. The annual return for the portfolio () is a weighted average of the annual returns on the two asset classes ( and for equities and bonds, respectively):[11]

(2)    

The portfolio can be thought of as a single asset with expected value and variance of annual returns being functions of the expected values, variances and covariance of the component asset classes:

(3)    

(4)    

The same relationships between portfolio annual returns and portfolio geometric returns apply as described above for single assets. In particular, the expected geometric return over time is less than the expected annual return, the difference becomes greater as longer time periods are considered, and the expected portfolio annual return is a more meaningful measure of the expected growth in portfolio wealth than the expected geometric return.

Equation (2) illustrates that the annual portfolio return is a weighted average of the component asset returns. Similarly, Equation (3) illustrates that the expected annual arithmetic portfolio return is a weighted average of the component assets’ expected annual arithmetic returns. However, it is a surprise to many practitioners that the geometric portfolio return is not equal to a weighted average of the component assets’ geometric returns. It is greater than the weighted average. That is:

(5)    

This is a common mistake made when computing portfolio returns. We can see this by decomposing each side of this equation. The result is:[12]

(6)    

This also applies to the calculation of the portfolio expected geometric return. Therefore, if we want to calculate the expected geometric return for a portfolio from the expected geometric returns of the individual asset classes, it is necessary to start with the component asset classes’ expected annual arithmetic returns, and take their weighted average to get the expected portfolio annual arithmetic return, then use that result (along with the portfolio volatility and time horizon) to derive the expected portfolio geometric return for the required time horizon.

The incorrect calculation of the expected geometric return (using the weighted average of the component asset classes’ geometric returns) understates the true portfolio expected geometric return. If this incorrect result is then used when the expected annual return is more appropriate, the bias is even worse than if the correct geometric calculation had been used. Figure 2 illustrates how the correct and incorrect calculations of portfolio returns can vary over time.

Figure 2 – Alternative Calculations of Expected Portfolio Returns

Another counterintuitive result of the nonlinear relationship between expected geometric asset class returns and expected portfolio geometric return is that it is possible for the expected portfolio geometric return to be greater than any of the individual asset class expected geometric returns. To illustrate, Figure 3 shows how the portfolio geometric return of a two-asset portfolio, comprising bonds and equities, changes as the portfolio allocation moves from 0% equities to 100%. In this example, there is a region of portfolio composition, from 55% equities to 100%, in which the portfolio geometric return becomes higher than that of either of the individual asset classes. Of course, this does not always happen – it depends on the structure of the return covariance matrix. Nonetheless, it is usual to find that the expected portfolio geometric return is at the upper end of the spread of the individual asset class expected geometric returns.

Figure 3 – Expected Portfolio Geometric Return