Return attribution is based on the cross-sectional model of returns:

Kl(t) = Bi(t-l)Fi(t) + ui(t) (19.4)

where R (?) is an N-vector of local excess returns (over the local risk-free rate) from time t - 1 to t; B?{t - 1) is an N X K matrix of exposures to factors that are available as of t - 1. These factors include investment styles such as growth or momentum and industry classifications. In the case where we may want to attribute return to sources that are contemporaneous (unlike a risk model), the information contained in the exposures matrix will be as of time t. F it) is a K X 1 vector of returns to factors, and u'{t) is an N-vector of mean-zero-specific returns from t-\lot.

There are three steps involved in the return attribution computation based on a factor model. (In the following discussion, we focus on the managed portfolio. However, our results generalize to any portfolio type.)

Step 1: Define a set of exposures to factors and estimate the cross-sectional return model specified by (19.4). This gives estimates of one-period returns to factors, that is, factor returns from period t - 1 to t.

Step 2: Compute the local return on the managed portfolio.

Letting wp{t - 1) represent an N-vector of managed portfolio weights at time t -1, the return on the managed portfolio is given by rl{t) = w*(t-l)TRUt) = b*(t-l)TFUt) + ul(t) (19-5)

where rp (t) b"(t-l) F\t) uUt)

Managed local excess portfolio return from period t-ltot K-vector of managed portfolio exposures K-vector of factor returns Specific local portfolio return

Step 3: Quantify the sources of local return. For example, a managed portfolio with N assets has K + N sources of return-K sources from factor returns and N sources from specific returns (one for each asset).

The source of return from the &th factor is given by the component

Si(t) = b*(t-l)%(t) for* = l, ...,1C (19.6)

The specific return contribution from the nxh asset is simply the return on that asset's specific return times its portfolio weight.

Si{t) = w*{t-\)un{t) "=1, . . . ,N (19.7)

Hence, the portfolio return is the sum of K + N sources of return and can be written as