# Active Return and Active Risk

Active return is the component of a portfolio’s return that results from active management i.e. the decision to overweight or underweight assets in the portfolio. It equals the difference between the portfolio return and the benchmark return. Active risk is the volatility i.e. standard deviation of active return.

There are two approaches to investment management: passive management and active management. In passive management, the investment manager doesn’t attempt to beat the market by underweighting or overweighting different investments but tracks an index that is representative of the investment strategy. In active management, the investment manager tries to beat the market by overweighting investments that are undervalued and underweighting investments that are overvalued. Active return and active risk are relevant to the active investment management approach.

## Active return

Active return is the excess return that results from an investment manager’s attempt to pick good investments. It equals the difference between the portfolio return and the benchmark return. Benchmark return is the return that could be earned by adopting the passive investment approach. It is the return on an index which represents the investment universe that corresponds to the selected investment strategy i.e. broad market index such as S&P 500 or an index representing the investment style (value vs growth) and size classification (small-cap vs large-cap) etc.

$$\text{r} _ \text{A}=\text{r} _ \text{P}-\text{r} _ \text{B}$$

Where rA is the active return, rP is the portfolio return and rB is the benchmark return. Portfolio return equals the weighted average return of the investment portfolio based on the actual portfolio weights and benchmark return is the weighted average return of the benchmark based on index weights.

### Active return vs alpha

Active return is different from alpha, the risk-adjusted excess return in that the active return is the difference between actual return and benchmark return while alpha is the difference between actual return and required return keeping in view the associated systematic risk.

## Active risk

Active risk is a measure of how sustained active return is. It is the volatility i.e. standard deviation of the active return. The higher the variability of active return, the higher the standard deviation (i.e. volatility) and higher the active risk and vice versa.

Active risk is also referred to as tracking error or tracking risk.

$$\text{Active Risk}=\sigma _ \text{A}=\sqrt{\frac{\sum _ {\text{i}}^{\text{n}}{(\text{r} _ \text{i}-\text{r} _ \text{A})}^\text{2}}{\text{n}}}$$

Where σA is the active risk, ri is the individual active return, rA is the average active return and n is the number of observations.

Active risk is measured using the following steps:

• Step 1: calculate the average active return.
• Step 2: subtract the average active return from each value of the active return and square it.
• Step 3: sum up the squared deviations calculated in Step 2 and divide by n.
• Step 4: take the square root of the value arrived at in Step 3.

## Information ratio

Information ratio is a measure of success of an active management strategy. It calculates active return per unit of active risk. A high information ratio highlights that the investment manager has been generating consistent excess returns and that the probability of those excess returns creeping in just by chance is low.

$$\text{Information Ratio}=\frac{\text{r} _ \text{A}}{\sigma _ \text{A}}$$

## Example

Over the last year, you managed a portfolio of five stocks. The following table shows their portfolio weights and actual return.

Stock Portfolio Weight Return
A 20% 15%
B 25% 10%
C 15% -8%
D 25% -2%
E 15% 4%

If the stocks are equally weighted in the benchmark, calculate the active return. If the active return in the previous 5 years was 2%, -1%, 0.5%, 0.75% and -3%, calculate the active risk and information ratio.

We first need to find out the portfolio return which is calculated by multiplying the return for each stock with its portfolio weight and sum all the products. This works out to 4.40%.

Next, we need to find the benchmark return. Since the benchmark has equal weighting of all stocks, its return is the average of the stock returns which is 3.8%.

$$\text{Active Return}\ =\text{r} _ \text{A}=\text{r} _ \text{P}-\text{r} _ \text{B}=\text{4.4%}-\text{3.8%}=\text{0.6%}$$

Given the last five year returns above and the last year return of 0.6%, standard deviation of active returns over the last six years works out to 1.56%.

Information ratio is the ratio of active return to active risk. Assuming the active risk in the last year is 1.8%, the information ratio is 0.33.

$$\text{Information Ratio}=\frac{\text{r} _ \text{A}}{\sigma _ \text{A}}=\frac{\text{0.60%}}{\text{1.8%}}=\text{0.33}$$