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**Active return** refers to the return on the portfolio above the return on the benchmark. That is,

$$ \text{Active return} = R_P-R_b $$

**Active risk**, also known traditionally as **tracking error **or** tracking risk**, is a risk that a portfolio manager creates in an attempt to outperform benchmark returns against which it is compared. In addition, active risk helps a portfolio manager achieve higher returns for investors. In other words, it is the standard deviation of active returns.

$$ \text{Active risk} = \text{Tracking error (TE)} = s(R_P-R_b) $$

Where:

- S is the sample standard deviation.
- \(R_P\) is the return of the portfolio.
- \(R_b\) is the benchmark return.

Information ratio (IR) is a measure of returns of a portfolio beyond the returns of a benchmark in comparison with the returns’ volatility. Strictly speaking, a benchmark is typically an index representing the market or a particular sector.

Information Ratio (IR) is used as a measure of a portfolio manager’s skills and ability to generate excess returns relative to a benchmark. It also identifies the consistency of performance by incorporating a tracking risk component into the calculation.

Tracking risk identifies the level of consistency with which a portfolio follows the performance of a benchmark. A low tracking risk implies that a portfolio is closely following the benchmark. A high tracking error implies that a portfolio is volatile relative to the benchmark and that returns are drifting from the benchmark. Investors prefer a **low tracking error**.

Although compared funds may be different, the information risk (IR) formula standardizes the returns by dividing the difference in their performances by their tracking risk.

$$ {IR} =\frac{\bar{R_p}-\bar{R_b}}{S(R_P-R_b) } $$

where:

- \(\bar{R_p}\) is the average of the portfolio returns for the chosen periods.
- \(\bar{R_b}\) is the average of the benchmark returns for the chosen periods.

## Question

Using the following data to calculate the manager’s information ratio:

$$ \begin{array}{c|c|c} \textbf{Period} & \bf{\text{Portfolio Returns } ({R}_{p})} & \bf{\text{Benchmark Returns }({R}_{b})} \\ \hline 1 & 0.0211 & 0.0111 \\ \hline 2 & 0.0091 & 0.0112 \\ \hline 3 & 0.0128 & 0.0091 \\ \hline 4 & 0.0083 & 0.0092 \\ \hline 5 & 0.0160 & 0.0111 \\ \hline 6 & 0.0191 & 0.0183 \end{array} $$

Which of the following is

most likelycorrect about the portfolio in question relative to the benchmark?

- It is close to the benchmark.
- It is volatile relative to the benchmark.
- It has a low information ratio.
## Solution

The correct answer is B.$$ \begin{array}{c|c|c|c} \textbf{Period} & \bf{\text{Portfolio}} & \bf{\text{Benchmark}} & \bf{{R}_{p}-{R}_{b}} \\ & \textbf{returns} & \textbf{returns} & \\ & \bf{({R}_{p})} & \bf{({R}_{b})} & \\ \hline 1 & 0.0211 & 0.0111 & 0.0100 \\ \hline 2 & 0.0091 & 0.0112 & (0.0021) \\ \hline 3 & 0.0128 & 0.0091 & 0.0037 \\ \hline 4 & 0.0083 & 0.0092 & (0.0009) \\ \hline 5 & 0.0160 & 0.0111 & 0.0049 \\ \hline 6 & 0.0191 & 0.0183 & 0.0008 \\ \hline \textbf{Average} & \bf{0.0144} & \bf{0.0117} & \bf{0.0027} \\ \hline \textbf{Standard} & & & \bf{0.0041} \\ \textbf{Deviation} & & & \\ \end{array} $$

$$ \begin{align*} {IR} &=\frac{{\bar{R}}_P-{\bar{R}}_b}{S(R_P-R_b) } \\ & =\frac{0.0144-0.0117}{0.0041}=0.6585\approx\ 0.66 \end{align*} $$

The high information ratio implies that the portfolio is volatile relative to the benchmark.

A is incorrect.The high information ratio shows that the portfolio is not closely following the benchmark.

C is incorrect.Investors prefer a portfolio with a low information ratio.

Reading 41: Using Multifactor Models

*LOS 41 (e) Explain sources of active risk and interpret tracking risk and the information ratio.*