Active Space

Let's consider now an application in active space. To do so, we now need to include benchmark returns for stocks (US) and bonds (FI). We estimate the covariance matrix Q for all four returns series, which we will refer to as the combined covariance matrix V_c:

Again, the top left quadrant is the 2 × 2 covariance matrix (V) for the program returns while the bottom right 2 × 2 matrix are benchmark covariances. The top right and bottom left 2 × 2 matrices estimate covariances across programs and benchmarks. The covariances in active space can be estimated by transforming V_c as in equation (8) generating V_a = m'V_cm:

Thus, the active risks on stocks and bonds are the square roots of the diagonal (2.70, 1.83).

In the previous example, equilibrium returns were the forward-looking expectations = (7.1, 5.3). Since these returns are what managers expect to prevail over the long run, then they are, in essence, managers’ expectations of equilibrium. Views, on the other hand, denote short-run deviations from equilibrium. From the previous example, the return view vector = (9, 6) suggests active returns equal to (2.9, 0.7).

Now, suppose management targets an alpha on the portfolio equal to 2 percent. Then the minimum variance active portfolio that achieves the target return (but with active weights constrained to sum to zero) is the solution to the system:

where λ and μ are Lagrange multipliers. Notice the presence of V_a in the left-most matrix. On the right-hand side, the third element equal to 2 is the targeted alpha. We wish to solve this system and extract the solution for the active weights (w_{a 1}, w_{a 2}). Inverting the matrix and multiplying by the vector on the right-hand side yields (0.87, –0.87) = w_a. The active portfolio, in the presence of the views, overweights stocks relative to its benchmark (likewise, it underweights bonds). This makes sense in light of managers expecting a 300-basis point spread over equilibrium between stock and bond returns.

What is the active risk on this portfolio? It is (w_aV_aw_a′)^1/2 = 3.10. (Had we instead targeted a 1 percent active return on the portfolio, w_a would have been (0.43, –0.43) with active risk equal to 1.55.)