1t - l)wf{t - 1) (20.62) Note that risks, as defined in terms of standard deviations, are not additive. That is, the factor risk and specific risk do not sum to the managed total risk. Were we to measure risk using variances-see equation (20.60)-in place of standard deviations, then the risks would be additive. In practice, the standard deviation is used as a measure of risk since its units are in returns and not returns squared. Similarly, the forecast variance of the return on the active portfolio is o2Jt) = b*{t-l)l{t I t-l)b"{t-l)T + w'(t- 1)TA(£ \t-l)w"(t-l) (20.63) Equation (20.63) provides a measure of an active portfolio's total risk (squared). In practice, this number is usually reported in standard deviation terms, that is, 0B(f), and is known as tracking error. The active portfolio's factor and specific risk components are given by qLj') = b*(t-imt \t-l)b"(t-l) (20.64) O.W) = w'(t-l)TA(t I t-l}w'(t-l) (20.65) A Risk Budget and Hot Spots One way to evaluate a portfolio's positions is in terms of their contributions to risk. In order to understand the meaning of these contributions, it is useful to think of a portfolio's risk defined in terms of a risk budget. Simply put, a risk budget is the amount of risk that a portfolio manager can allocate to different factors or securities. A portfolio manager managing her portfolio against a benchmark would consider the portfolio's tracking error as representing 100 percent of its overall risk. With a risk budget, we decide how much risk should come from different factors and/or assets. The sum of the contributions to risk from each of the factors and assets is equal to 100 percent. It is important to note that a portfolio's risk budget is separate from the absolute level of risk that the portfolio incurs. For example, a portfolio might have a target tracking error of 5 percent, but currently its realized (and predicted) tracking error is running about 4 percent. In this example, the portfolio has 100 basis points of unused risk that it could employ in order to improve its chances of increasing returns. Contributions to risk are defined by assets (stocks), investment style factors, industry factors, countries, and currencies. (In fact, any factor falls into the framework we discuss in this section.) Contributions to a portfolio's risk (e.g., tracking error) measure a position's marginal impact on that portfolio's risk. They answer questions such as, if we change a position's size by 2 percent, how much does the portfolio's tracking error change? What proportion of my portfolio's overall risk budget comes from a bet geared to the U.S. momentum factor? And how is the risk in my portfolio allocated across different securities and sectors? As contributions to risk measure the marginal effect on risk, they are typically defined in terms of (mathematical) derivatives. This is not to say, however, that