Overview
The multi-factor performance attribution is a simple application of the multi-factor risk model. That is, the (excess) return of the portfolio is modeled as
{% r_{excess} = \alpha + \beta_1 r_1 + ... + \beta_n r_n +\epsilon %}
where {% r_i %} is the return of the {% i^{th} %} factor, and {% \epsilon %} is the idiosyncratic risk.
Attribution
The first step is to define a set of indices that represent each factor. (total market risk, value, growth, small cap) Then, run a OLS Regression on every asset in the investable universe. This will give you the value of {% \beta_i %} for each asset. The portfolio factor weightings are then the weighted average of the {% \beta's %} of each asset in the portofolio.
Once the factor betas are calculated, the return due to each factor can be calculated, by multiplying the portfolio beta with the factor returns. The unexplained (idiosyncratic) portion of the return is then the portion of the return that is left over.
{% \epsilon = r_{excess} - \alpha + \beta_1 r_1 + ... + \beta_n r_n %}