Overview
One of the most common ways to evaulate the performance of a manager is to compare them to a benchmark. In its simplest form, you can just compare the total returns of the manager vs those of the benchmark.
As simple as this is, it ignores risk. That is, the manager could outperform the benchmark on average simply by taking more risk. For example, a manager trying to beat the S&P could just leverage up the benchmark by borrowing money and buying the benchmark. This is clearly not an example of manager skill.
Most measures of manager performance attempt to account for the risk that the manager takes. Typically this involves modeling the managers portfolio within the context of a single index model or more broadly, multi factor model
Portable Alpha
Portable alpha utilizes the ideas of APT and Factor Investing to try to measure the excess return that a manager has earned, over and above the returns generated from the risks that the manager took. The fundamental equation of return in APT is given as
{% r = \alpha + \sum \beta_{j}f_j + \epsilon_i %}
where {% f_j %} are factor risks, {% \beta_j %} are the factor exposures and {% \alpha %} is the manager skill over and above
the factor risks.
When there is a Single Factor Risk, usually a benchmark for the fund, such as the S&P500 or other index, the equation becomes
{% r = \alpha + \sum \beta f + \epsilon %}
which is the
CAPM model.
Information Ratio
The information ratio is a number that is computed from comparing the returns of the portofolio in question to a pre-defined benchmark. It utilizes a single index model to do the comparison. Starting with the basic single index equation,
{% r(t) = \alpha + \beta r_m(t) + \epsilon %}
- {% r(t) %} is the excess return of the managers portfolio over the risk free rate at time t.
- {% r_m(t) %} is the excess return of the market over the risk free rate at time t.
The residual return (also called the active return) at time t is defined to be
{% \theta (t) = r(t) - \beta \times r_m (t) %}
The information ratio is defined to be
{% IR = \alpha / \sigma (\theta) %}
where {% \sigma(X) %} is the standard deviation of {% X %}.
The information ratio can be estimated using a standard OLS regression to calculate the {% \alpha %} (intercept) of the regression as well as its standard error.