to risk using numerical simulation rather than derivatives. Both approaches have their advantages and disadvantages. The primary advantage of the mathematical approach is that the calculations are extremely fast because you have closed-form results for the contributions to risk. The mathematical derivatives that we employ measure the percentage change in risk for a given percentage change in position value. These derivatives are based on the factor model expression for tracking error (squared). o2Jt) = ba(t - l)l(t 11 - l)ba(t-l)T + wa(t -l)TMt 11- l)wa{t - 1) (20.66) Using this expression, we can answer the following three questions: Question 1: How much does tracking error change when there is a change in the number of shares held in the Kth stock position? A related question is, how much does the nth position contribute to the overall tracking error? Question 2: How much does tracking error change when there is a change in the exposure to the &th factor? How much does the &th factor contribute to the total tracking error? Question 3: What is the breakdown of total tracking error to factor and specific risk? We address each question separately. Contributions to Risk by Asset To answer the first question we need to find an expression for the change in tracking error, Oa(t), for a given change in the nxh element (asset) of wa{t - 1), which we represent by ws(t - 1). The N -vector of absolute marginal contributions to tracking error (ACTE) is given by the derivative of the portfolio's tracking error with respect to the position vector wa{t - 1), ACre(*)= * =W-W-V (2o.67) dlVa(t-l) <5a(t) where the ?zth element of ACTE(£), denoted ACTE (?), is the ?zth asset's absolute marginal contribution to tracking error. Since oa(t)2 = ivs{t- l)TV{t 11 - l)ws{t - 1), if we premultiply (20.67) by w"(t - 1)T, we get W^-l)TACra(f)="''(f-1)TV(flf-1)"''(f-1)=o^) (20-68) oa(t) Or we can write equation (20.68) as £<(£-l)ACTE(£) = o-a(£) That is, the tracking error is equal to the weighted average of the absolute marginal contributions to tracking error, where the weights are defined as the active portfolio weights. Dividing both sides of (20.68) by tracking error yields