Active return

Active return is the difference between the return of an investment portfolio and the return of its benchmark over a specified period. It measures whether a portfolio has outperformed or underperformed its benchmark.

Calculation

Active return is calculated as:

$Active Return = Portfolio Return - Benchmark Return$

or, for period $t$ :

$A_{t} = R_{p, t} - R_{b, t}$

where:

$A_{t}$ = active return;
$R_{p, t}$ = portfolio return;
$R_{b, t}$ = benchmark return.

Example

If a portfolio returns 12% during a year and its benchmark returns 10%:

$Active Return = 12 % - 10 % = 2 %$

The portfolio has therefore generated a positive active return of 2 percentage points.

If the portfolio returns 8% while the benchmark returns 10%:

$Active Return = 8 % - 10 % = - 2 %$

The portfolio has generated a negative active return of 2 percentage points.

Positive and negative active return

A positive active return indicates that the portfolio outperformed its benchmark over the measurement period.

A negative active return indicates that the portfolio underperformed its benchmark.

A zero active return indicates that the portfolio and benchmark produced the same return over the measurement period.

Active management

Active return is particularly relevant to actively managed portfolios.

An active manager may deliberately hold securities or portfolio weights that differ from the benchmark in an attempt to generate returns above the benchmark.

The resulting performance difference is measured by active return.

Passive management

Passive investment strategies generally seek to closely track a benchmark rather than outperform it.

A passive fund may therefore have a small positive or negative active return because of:

management fees;
transaction costs;
taxes;
cash holdings;
portfolio-rebalancing costs;
differences in dividend treatment;
tracking error.

Active return and tracking error

Active return and tracking error measure different aspects of portfolio performance.

Active return measures the difference between portfolio and benchmark returns:

$Active Return = R_{p} - R_{b}$

Tracking error measures the variability of active returns over multiple periods:

$Tracking Error = σ (R_{p} - R_{b})$

Therefore, active return measures the direction and magnitude of relative performance, while tracking error measures the consistency or variability of that relative performance.

Active return and information ratio

The Information ratio relates active return to tracking error.

A simplified expression is:

$Information Ratio = \frac{Average Active Return}{Tracking Error}$

A higher information ratio generally indicates greater active return relative to the variability of that active return.

Cumulative active return

For a single measurement period, active return can be calculated directly by subtracting the benchmark return from the portfolio return.

For multiple periods, cumulative portfolio and benchmark returns should generally be compounded before calculating the cumulative relative performance.

If portfolio returns are $R_{p, 1}, R_{p, 2}, \dots, R_{p, n}$ and benchmark returns are $R_{b, 1}, R_{b, 2}, \dots, R_{b, n}$ , cumulative relative performance can be represented as:

$Cumulative Active Return = \frac{\prod_{t = 1}^{n} (1 + R_{p, t})}{\prod_{t = 1}^{n} (1 + R_{b, t})} - 1$

This differs from simply adding the individual active returns.

Importance

Active return is a fundamental measure of portfolio performance. It allows investors and portfolio managers to determine whether a portfolio has outperformed or underperformed its benchmark and is commonly used alongside tracking error and the information ratio to evaluate active investment strategies.