Overview
An active portfolio is a portfolio that is compared to a given benchmark. Weights in the portfolio that differ from the benchmark weights are referred to as "active" weights.
Active Weights
In order to understand how close a portfolio looks to a given benchmark, the portfolio weights are often decomposed as a benchmark weight, plus an active weight as follows.
{% \vec{w} = \vec{b} + \vec{a} %}
where we define:
- {% \vec{b} %} is the benchmark weight
- {% \vec{a} %} is the active weight
Portfolio weights are equal to the benchmark weights plus a set of active weights, where we require the active weights to sum to zero.
If we are given a set of return forecasts {% \vec{f} %}, then the expected active return (the return relative to the benchmark) is
{% \vec{a}^T \vec{f} %}
The active risk is defined to be the variance of the portfolio due to the active weights.
(that is, the variance of the portfolio with the benchmark as numeraire)
{% \omega = \frac{1}{2} \vec{a}^T \Sigma \vec{a} %}
NOTE: these equations are stated using
linear algebra