Chapter 10

Active Portfolio Management

To whom men commit much, of him, they will demand the more.

—Luke 12:48, The New Testament

Portfolios can be managed passively or actively. Passive allocations are typically investments in an index fund. This management style is attractive because it minimizes trading costs and eliminates most management fees while earning the return on the index. The popularity of index funds has increased geometrically as evidenced by the staggering array of choices offered as retirement investment vehicles in various 401(k) and defined contribution plans. Active allocations, on the other hand, are made by managers who believe they can beat the index, presumably because they have superior information or skills, or both, relative to other market participants. Clients hire active managers and pay fees with the anticipation that the active manager will produce risk-adjusted returns above the index (the opportunity cost of funds) net of fees. These returns, referred to as alpha, are the returns to skill—stock selection, superior information, models, and so on—and are uncorrelated with the index return (and therefore beta). Active managers are alpha seekers who measure their performance relative to an agreed-upon benchmark.

Active management can be thought of as shorting the benchmark and going long the active portfolio. The active allocation therefore generates a return differential, relative to the benchmark, as well as a risk differential, referred to as the active portfolio's tracking error. A very simple, yet intuitively clear, model is the single-index model we developed from the CAPM, again, where r_i indexes the return to portfolio i and r_m the return on the market (benchmark):

where ε denotes pricing error that we assume to be zero on average (that is, we assume the single-index model is correctly specified). Subtracting r_m from both sides and taking expectations (E(ε) = 0) creates the portfolio's expected return premium over the benchmark:

This is what I was alluding to earlier when I suggested that the observed premium may be misleading as an alpha measure. Only in the case for which the portfolio's beta is one is the premium pure alpha. As you can see, if the portfolio's beta exceeds one, then the observed return premium is due in part to taking on incremental systematic risk in the amount (β – 1)r_m.

To continue the example, consider the case of a pension fund whose assets are managed actively across two classes—stock and bonds. Suppose the trustees of the fund establish a long-term 70/30 mix of stocks and bonds as the fund's benchmark. A passive investment would therefore consist of investing 70 percent of the fund in an equity index and 30 percent in a bond index. On the other hand, if the trustees have faith in a manager's professed ability to beat the benchmark, net of fees, then they will fund the manager to deviate from the benchmark weights (obviously, it would make no sense to pay fees if the manager were to maintain the benchmark weights). In making that decision, the trustees will consider the foregone opportunity loss by not investing in the benchmark, that is,

where r₁ and r₂ index the two asset classes with respective benchmark weights w_{b 1} and w_{b 2} and compare this to the return on the manager's investment in the active portfolio:

The return premium, now called the active return, is the difference:

The active risk, or tracking error, can be thought of as the standard deviation of the active return:

It is more compact to express these terms in matrix form as we first did in Chapter 5. The active return r_a is therefore the scalar:

where w_p, w_b, and r are vectors (in this case 2 × 1 vectors). This return is what the client is paying for. The active risk, on the other hand, is the risk the client must bear to receive the active return. In matrix form, it is expressed as:

where V_a is the active returns covariance matrix.

Construction of V_a is a bit tricky and it pays to understand how to do it correctly. Recall my claim from before that the active portfolio can be thought of as a short position in the benchmark and a long position in the active portfolio. We can therefore think of the covariance matrix as covariances across both the benchmark returns and the active portfolio returns. In our two-asset case, we have covariances between returns on stocks and bonds in the benchmark, covariances between returns on stocks and bonds in the actively invested portfolio, and covariances between stocks and bonds across the benchmark and the active portfolio (denoted with the subscripts b and p, respectively). Thus, for the two-asset case:

Generalizing to k-assets implies that V_a will have dimension 2k × 2k. The elements of V_a are estimated using the benchmark and the portfolio returns. Recognizing again the short benchmark position, we construct the weight vector w_a such that:

We then estimate the tracking error (active risk) using the following:

So, the client takes risk σ_a in return for expected return r_a. What exactly does this mean? Clearly, any return premium is due to tilts away from the benchmark, that is, (w_p – w_b) ≠ 0, and this is reflected in the active risk estimate. Of interest, therefore, is the price in terms of active risk of deviating from the benchmark:

This is the marginal contribution to active risk (MCAR). Suppose, for example, the two assets were stocks and bonds and the benchmark was a 70/30 mix of these two assets. The client has the choice of investing in this passively or investing with the active manager. Suppose the active manager invests in a portfolio having weights 75/25, indicating a tilt away from the benchmark with an overweight to stocks and an underweight to bonds. Suppose, furthermore, that the manager contemplates taking even more exposure to stocks, that is, by increasing the allocation to stocks beyond 75 percent. MCAR will estimate the contribution to active risk. Let's see now how this works in practice.