Overview
The portfolio return is represented by
{% r_{port} = \sum w_i r_i %}
where {% r_i %} is the return of the {% i^{th} %} asset, and {% w_i %} is the weight of the {% i^{th} %} asset
in the portfolio.
The benchmarket return is
{% r_{bench} = \sum W_i r_i %}
where {% W_i %} is the benchmark weight of the {% i^{th} %} asset.
Asset Allocation
In order to allocate the return to a risk factor, or asset class, the analyst defines a set of market indices (funds) that represent each factor, or class. The return of each index will be labeled {% b_j %}.
Then, each asset is assigned to an index. (Alternatively, they can be given index weights, indicating a percentage to which each asset belongs to each index). The portfolios index weights are computed. That is, if an asset is judged to be a small cap asset, then all that assets weight is assigned to the small cap index. The portfolios asset allocation return is then
{% r_{alloc} = \sum w_j' b_j %}
The asset allocation return represents the return that the portfolio would have achieved if each asset were simply replaced by the
index in which it was a member. (For example, all small cap stocks are assumed to be invested in the small cap index instead)
The active allocation return is simply the portfolios allocation return minus the benchmark return.
Asset Selection
The asset selection return is just the return that is left over after the benchmark and allocation returns have been subtracted from the portfolio return.
{% r_{selection} = r_{port} - r_{bench} - r{alloc} %}