of contributions across all groups is equal to the portfolio's total return. In practice, we can compute group returns and group contributions for the managed, benchmark, and active portfolios. Examples of groups include: assets, industries, sectors, and percentiles of the distribution of a particular investment style. An "asset group" simply means that each asset is treated as a separate group. In this way the return to an asset group is that asset's total return, and the asset's contribution is the contribution of the individual asset to the entire portfolio return. In the asset grouping approach, one-period active returns are defined in terms of stock selection, allocation effect (also known as group weight), and a so-called interaction effect. Mathematically, the asset grouping model for an active portfolio can be written as: rs(t) = S(t) + A(t) + I(t) (19.13) where S(t) represents the one-period total stock selection component at time t. For a given group of stocks, stock selection is defined as follows. First, compute the difference between the group's return as defined by stocks in the managed portfolio and the (same) group's return as defined by stocks in the benchmark. An industry or sector is an example of a group. Second, multiply this difference by the group's benchmark weight. Mathematically, the stock selection component for the z'th group of stocks at time t is St(t) = w\(t-l)[rup{t)-%b(t)\ (19.14) where rt b(t) = Return on stocks in the benchmark portfolio that belong to the z'th group. For example, r b(t) might represent the return to all telecom stocks in the benchmark portfolio. tit) = Return on stocks in the managed portfolio that belong to the ith group wh{t - 1) = Weight of the z'th group in the benchmark portfolio Summing over all i (i = 1, . . . , I) groups gives us the total stock selection component I S(t) = ^tvUt-l)[rhP(t)-r^b(t)} (19.15) i=i A{t) is the allocation effect (also known as group weight) and measures the impact of over- or underweighting a particular group of stocks. The allocation effect for the 2th group of stocks is defined as I A(t) = ^As(t) (19.16)