# Active Risk, Tracking Risk and Information Ratio

## Active Risk

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.

### Measuring Active Risk

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

Where:

• S is the sample standard deviation.
• $$R_P$$ is the return of the portfolio.
• $$R_b$$ is the benchmark return.

## Tracking Risk and Information Ratio (IR)

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.

### Information Risk (IR) Formula

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 likely correct about the portfolio in question relative to the benchmark?

1. It is close to the benchmark.
2. It is volatile relative to the benchmark.
3. It has a low information ratio.

#### Solution

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

\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.

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

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