Equity Risk Factor Models 385 Note that the term bak(t - 1) is a weighted average of exposures (for a particular factor) of all

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assets in the active portfolio. So, when we refer to taking a derivative with respect to the &th exposure we are not specifying whether that derivative is with respect to the active portfolio weights or the asset exposures. Similarly, we can compute the relative specific contribution to tracking error- (RSCTE(t)). That is, for the ?zth asset, its relative specific contribution to tracking MCIE.(t)="g<t-1>A<tlt-1'"'<t-1> ,20.87, oa(t)2 Note that the sum over all assets of the RSCTEn(?)'s is equal to the proportion of specific risk in tracking error. That is, jRSCTEn(t) = ^- (20-88) Therefore, the sum of RSCTE's over all assets plus the sum of RFCTE's over all factors is equal to one (or 100 percent). When determining a portfolio's sources of risk, it is important that we can drill down to the most detailed level. Sometimes its not enough to know how much a factor (e.g., price momentum) contributes to a portfolio's tracking error. Instead, we may need to know what assets are most responsible for a particular factor's risk contribution. Alternatively stated, suppose our goal is to lower our portfolio's risk that is coming from the price momentum factor. In order to do so we would need to reduce exposure to assets that contribute highly to price momentum's contribution to tracking error.19 This is not the same as simply reducing the weights in assets that have high exposure to price momentum. Rather, we need to reduce the weight in the risky assets that contribute to the price momentum factor's contribution to risk. In order to determine which assets contribute to a particular factor's risk, we need to measure the Kth asset's relative contribution to the &th factor. This measure is given by R =<ft-l)xi^-^xA (2089) This expression tells us how much risk the wth asset contributes to the &th factor. Note that when we sum RFCTE , over all stocks, the result is the factor's con-tribution to tracking error. ^RFCTE^ =RFCTEi (20.90) "=1 'Here, we assume that overweight positions have high price momentum exposure.

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