are expected to generate return, or unproductive when they are too large or unintended. Thus, knowing the level of risk in a portfolio is not enough. The investor needs to measure where the risk is coming from. The best way to understand the sources of risk in a portfolio is simply to measure the impact on the overall portfolio risk of separate small changes in each component in the portfolio. This marginal measurement forms the basis for a decomposition of portfolio risk. It identifies the hot spots in the portfolio, the components to which portfolio risk is most sensitive. The decomposition of risk is similar to but different from the marginal contribution to portfolio risk, which was described in Chapter 2. In forming optimal portfolios we try to equalize across all assets the ratio of the contribution to expected return from each asset with its contribution to portfolio risk. In that case we measure the change in portfolio risk that is caused by a unit addition of the asset to the portfolio. We might, for example, consider adding a unit of a new asset that is not currently in the portfolio. Such an addition will generally impact portfolio risk, either increasing or decreasing it. In measuring the decomposition of risk, though, we focus not on a unit change, but rather on what happens to portfolio risk when there is a percentage change in the portfolio weight. This difference in measuring marginal risk in the context of risk decomposition should be intuitive. In the first context we want to be very cautious about adding an asset to a portfolio. If the asset creates significant risk at the margin, we need to get paid an expected return premium for taking that marginal risk. In contrast, when measuring where risk is in a current portfolio we want to know how important are existing positions; if we don't already own an asset then it cannot be a source of risk in the current portfolio. For well-behaved measures of risk, the total portfolio risk is equal to the sum of the marginal percentage changes in all the portfolio components. Thus, the percentage contribution to risk of each component of the portfolio is simply the mar-