Performance Attribution
The preceding first two cases present us with two distinct optimal portfolios. The portfolio returns are simply the optimally weighted average of the expected returns on the individual securities, that is:
where the wi are the optimal weights. For portfolio 1, the mean return is 3.4 percent, while for portfolio 2, it is the stipulated 3.6 percent. The expected variance of the portfolio returns is given by 
, and the portfolio risk (volatility) is the square root of this number. For portfolio 1, it is 29.3 percent (this is the left-most point on the efficient frontier) and a bit higher for portfolio 2 (31 percent) because this portfolio required a higher rate of return. See the spreadsheet for computational details.
Portfolio managers are interested in measuring the contribution to portfolio risk as the weight allocated to a specific security change. This is called the marginal contribution to risk (MCR), denoted by:
This is easily derived—recall that σ is the portfolio risk, which is equal to
. The derivative of this with respect to w′ is MCR. The MCR is a therefore a vector; it shows the contribution to risk on the margin for a given change in the portfolio weights. Let's put this in perspective, using portfolio 2. Again, the computational details are provided on the spreadsheet. Since V is a 2 × 2 square matrix and w is a 2 × 1 column vector with σ a scalar, then MCR is a 2 × 1 vector. For portfolio 2, it is 
′. The interpretation is that if the allocation to asset 2 were to rise one unit, then portfolio risk rises 39 percent. Since our weights are in percents, then a 1 percent change produces 0.39 percent change in portfolio risk. More generally, however, we focus on the relative differences in MCR across assets—here, for example, the marginal contribution to portfolio risk is roughly twice as high for asset 2 relative to asset 1, and this has important implications for the portfolio manager contemplating tilting his portfolio in the direction of asset 2. We discuss the concept of overall portfolio risk in the chapter on risk budgeting.
Let me introduce you to one more risk measure—Allocation Attribution. This is a decomposition of total portfolio risk and is given by 
. The notation “.∗” denotes element wise multiplication (see the spreadsheet for details). This is the MCR that is weighted by the individual optimal allocations. For portfolio 2, it is (0.25, 0.75), respectively, indicating that our portfolio risk is three times as concentrated in asset 2 relative to asset 1.
The impact that portfolio restrictions have on risk attribution can be significant, which I show in Table 6.1.
Table 6.1 Risk Attribution—Portfolios 3 and 4.
Allowing asset 3 to be shorted in portfolio 3 means that its allocation attribution—contribution to total risk on the portfolio (which was 26.2 percent)— is negative. On the other hand, asset 1 accounts for roughly three times the risk as asset 2 because of its overweighting. On the margin, however, asset 2 has the highest contribution to risk. When short sales are restricted (portfolio 3), this picture changes; clearly asset 3 can contribute nothing to risk since it has zero weight. If we were to allocate to asset 3 on the margin, then risk would increase in the portfolio by 34 basis points for each 1 percent risk in asset 3 (and asset 2) allocation. Finally, when we restrict both short sales and cap the weight to asset 2, then the risk contribution from asset 2 must decline, which it does, while the contributions to risk on the margin from assets 1 and 3 rise. The allocation attribution is heavily concentrated on asset 1 because there is a zero allocation to asset 3 and a capped allocation to asset 2.