Figure 4: Two risky assets

For a slightly expanded version, let’s consider that we have two assets, \(A\) and \(B\) where the expected return on asset \(A\) is (\(E[r_A]\)) is 0.03 and the expected return on asset \(B\) (\(E[r_B]\)) is 0.07. The standard deviation of asset \(A\) (\(\sigma_A\)) is 0.15 and the standard deviation of asset \(B\) (\(\sigma_B\)) is 0.3. For the plot, let’s consider correlations (\(\rho_{AB}\)) ranging from -1 to 1.

Dominated portfolios: Two risky assets

Now let’s focus on the case where the correlation \(\rho_{AB}\) is 0.2 and identify the dominated portfolios. Here dominated means portfolios that offer a return for a given level of risk that is less that that offered by another portfolio with the same risk. In the plot below, the dominated portfolios are indicated. For each of these points, there is a portfolio offering a higher return for the same level of risk.

Dominated portfolios: Two risky assets and one risk-free

Now suppose we add a risk-free asset with a return (\(r_f\)) of 0.01. Again we will focus on the case where \(\rho_{AB}\) is 0.2.

Now it will turn out there is one mix of the two risky assets that, when combined with the risk-free asset, dominates all other possible portfolios. With a risk-free asset, we can focus on combinations of \(A\) and \(B\) that have (for the risky portion of the portfolio) 0.519 in asset \(A\) and 0.481 in asset \(B\). In effect we can consider this to be a risky asset \(M\) with a mean return (\(E[r_M]\)) of 0.04924 and a standard deviation (\(\sigma_M\)) of 0.1771345 (see the black point on the plot below). We can combine with the risk-free asset to produce portfolios depicted on the plot below in red.

These dominate any combination on the blue curve. For example, a portfolio that invests the proportion 0.18 in the risk-free asset and 0.82 in the risky combination \(M\) (so 0.42558 of the total portfolio in asset \(A\) and 0.39442 in asset \(B\)) will have an expected return of 0.0421768 and a standard deviation of 0.1452503 (see the green point on the plot below)