Benchmark Based Manager Performance
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
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 %}.
