Active Risk Formula (CFA): Tracking Error Explained

Total risk tells you how much a portfolio moves. Active risk tells you how much it moves differently from its benchmark. That distinction is what the CFA curriculum cares about whenever it evaluates an active manager, and it is the source of most exam questions on this topic. The ideas appear in performance evaluation at Level 1 and are developed most fully in the Level 2 reading on active portfolio management and in Level 3 performance measurement.

The three building blocks

Measure	Formula	What it tells you
Active return	RA = RP − RB	How much the portfolio beat (or lagged) its benchmark in a period
Active risk (tracking error)	σA = σ(RP − RB)	How variable those active returns are
Information ratio	IR = R̄A ÷ σA	Active return earned per unit of active risk

Note the order of operations. Active risk is the standard deviation of the difference in returns. It is not the difference between the two standard deviations. That single point is the most common trap on the exam.

Worked example: calculating active risk and the IR

A fund and its benchmark produced these annual returns:

Year	Fund	Benchmark	Active return
1	12%	10%	+2%
2	8%	7%	+1%
3	−3%	−5%	+2%
4	15%	12%	+3%
5	9%	10%	−1%

Step 1: average active return. (2 + 1 + 2 + 3 − 1) ÷ 5 = 1.40%.

Step 2: active risk. Deviations from the mean are 0.6, −0.4, 0.6, 1.6 and −2.4. Squared, they sum to 9.20. Dividing by n − 1 = 4 gives a variance of 2.30, so active risk = √2.30 = 1.52%.

Step 3: information ratio. 1.40 ÷ 1.52 = 0.92.

On the BA II Plus, enter the active returns in the data worksheet and read Sx from the statistics worksheet. That gives the sample standard deviation directly and avoids arithmetic slips.

What high and low active risk mean

An index fund aims for active risk close to zero. An enhanced index fund might run active risk of 1–2%. A concentrated stock picker can run active risk well above 5%. None of these is good or bad on its own. Active risk is a budget: it tells you how far the manager is allowed to stray from the benchmark. The question is whether the manager earns enough active return for the risk they spend, which is exactly what the information ratio measures.

Compare this with the Sharpe ratio, which uses total risk and the risk-free rate. A manager can have a high Sharpe ratio simply because the benchmark did well, while having an information ratio near zero. Our guide to Sharpe vs Treynor vs information ratio shows which one a question is asking for.

Level 2 extensions

Optimal active risk

For an unconstrained manager, the amount of active risk that maximises the portfolio's Sharpe ratio is:

σ_A* = (IR ÷ SR_B) × σ_B

If a manager has an IR of 0.5, the benchmark Sharpe ratio is 0.4 and benchmark volatility is 16%, the optimal active risk is (0.5 ÷ 0.4) × 16% = 20%. The resulting portfolio Sharpe ratio is √(SR_B² + IR²) = √(0.16 + 0.25) = 0.64.

The fundamental law of active management

The expected information ratio can be written as IR ≈ TC × IC × √BR, where TC is the transfer coefficient, IC is the information coefficient and BR is breadth. A manager with an IC of 0.05 making 400 independent bets with a transfer coefficient of 0.6 has an expected IR of about 0.6 × 0.05 × √400 = 0.60. We cover the transfer coefficient in detail in the transfer coefficient explained.

Decomposing active risk

In multifactor models, active risk squared splits into two parts: active factor risk (from deliberate factor tilts) and active specific risk (from individual security bets). If active factor risk is 3% and active specific risk is 4%, total active risk is √(3² + 4²) = 5%. Note that the components add as variances, not as standard deviations.

Five exam traps

1. Subtracting standard deviations. σ_P − σ_B is not active risk. You need the standard deviation of the return differences.

2. Using the risk-free rate. The information ratio uses the benchmark, not the risk-free rate. If you see R_f in your working, you have built a Sharpe ratio by mistake.

3. Adding risk components directly. Active factor risk and active specific risk combine as squares.

4. Assuming leverage changes the IR. Scaling active positions up or down changes active return and active risk in proportion, so the information ratio stays the same. The Sharpe ratio of the overall portfolio does change.

5. Confusing ex ante and ex post. Ex ante tracking error is a forecast from a risk model. Ex post tracking error is measured from realised returns, as in the worked example. Questions sometimes test which one a manager should use for a given purpose.

Frequently asked questions

What is the formula for active risk in the CFA curriculum?

Active risk is the standard deviation of active returns: σ(R_P − R_B), where R_P is the portfolio return and R_B is the benchmark return.

Is active risk the same as tracking error?

Yes. The CFA curriculum uses the two terms interchangeably.

How is the information ratio calculated?

Divide the average active return by active risk. For example, an average active return of 1.4% with active risk of 1.52% gives an information ratio of about 0.92.

What is a good information ratio?

There is no fixed cut-off, but sustained information ratios above about 0.5 are generally considered strong for active managers.

Does leverage change the information ratio?

No. Scaling active positions changes active return and active risk proportionally, so the information ratio is unchanged.

What is the difference between active risk and total risk?

Total risk is the standard deviation of the portfolio's returns. Active risk is the standard deviation of the portfolio's returns relative to its benchmark.

How is optimal active risk calculated?

Optimal active risk equals the information ratio divided by the benchmark Sharpe ratio, multiplied by benchmark volatility.

How do active factor risk and active specific risk combine?

As variances: active risk squared equals active factor risk squared plus active specific risk squared.