Skip to content

FRM Exam Part II · Portfolio Risk: Analytical Methods

Tracking Error and Benchmark-Relative Risk Explained

Updated 11 October 2026 · Fact-checked

Tracking error is the standard deviation of active return, which is portfolio return minus benchmark return. Compute it from active weights (portfolio weight minus benchmark weight) as √(wₐ′Σwₐ). Relative VaR scales tracking error by a z-score and the portfolio value. Information ratio divides active return by tracking error.

Understand Tracking Error and Benchmark-Relative Risk

A benchmark-relative investor does not care about total loss alone. A fund manager is judged against an index. So the risk that matters is how far the portfolio can drift from that index.

Active return = portfolio return − benchmark return. Tracking error (TE), also called active risk, is the standard deviation of active return. A TE of 2% means that, roughly, active return falls within ± 2% of its mean about two-thirds of the time, if returns are approximately normal.

To compute it from holdings, use active weights: wₐ = w_P − w_B for each asset. Active weights sum to zero when both portfolio and benchmark are fully invested. The active portfolio is a long-short portfolio, so its variance is wₐ′Σwₐ, where Σ is the covariance matrix of asset returns. Tracking error is the square root. Note that the benchmark's own risk drops out. Only the bets away from the benchmark count.

Relative VaR (also called tracking error VaR) is VaR of the active return. Under normality with mean active return assumed zero, Relative VaR = z × TE × portfolio value. It answers: how much can I underperform the benchmark, at a given confidence level, over the horizon? Absolute VaR uses total portfolio volatility instead.

Do not confuse TE with the information ratio (IR). TE measures risk. IR measures reward per unit of that risk: IR = expected active return ÷ TE. A manager can have a low TE and a high IR, or the reverse.

Key formulas to remember

Active return
Rₐ = R_P − R_B
Portfolio return minus benchmark return for the same period.
Active weights
wₐ,i = w_P,i − w_B,i
Sum to zero if both portfolio and benchmark are fully invested.
Tracking error (two-step)
TE = σ(Rₐ) = √(σ_P² + σ_B² − 2ρσ_Pσ_B)
ρ is the correlation between portfolio and benchmark returns. Equals √(wₐ′Σwₐ).
Tracking error (active weights)
TE = √(wₐ′ Σ wₐ) = √(ΣᵢΣⱼ wₐ,i wₐ,j σᵢⱼ)
Use the covariance matrix of asset returns. Result is in the same time unit as the covariances.
Relative VaR (normal)
Relative VaR = z × TE × V
Assumes zero mean active return. z = 1.645 at 95%, 2.326 at 99% (one-tailed). V is portfolio value.
Information ratio
IR = E(Rₐ) ÷ TE
Both measured over the same period, usually annualised.
Time scaling
TE(annual) = TE(monthly) × √12
Valid when active returns are independent over time.

How to solve Tracking Error and Benchmark-Relative Risk questions

Use this order for any question on tracking error, active risk or relative VaR.

  1. 1Identify what is asked: TE, relative VaR, or IR. Note the horizon and confidence level.
  2. 2Check whether the data are portfolio and benchmark volatilities with a correlation, or weights and a covariance matrix.
  3. 3If weights are given, compute active weights as portfolio minus benchmark for each asset.
  4. 4Compute active variance. For two assets: wₐ₁²σ₁² + wₐ₂²σ₂² + 2wₐ₁wₐ₂σ₁₂. If given volatilities and correlation, convert to covariance first.
  5. 5Take the square root to get TE. Convert units (monthly to annual) using √time.
  6. 6For relative VaR, multiply TE by z and by portfolio value. Use the mean only if the question gives one.
  7. 7For IR, divide expected active return by TE over the same period.
  8. 8State the interpretation in one line, for example 'at 95% confidence, underperformance should not exceed X in a year'.

Quickest way: Fast active-variance check

When to use it: Two- or three-asset portfolios with a short covariance input, or when options differ by clear orders of magnitude.

  1. Write active weights first. Spot if they are zero: zero active weight means zero TE contribution.
  2. With two assets and fully invested portfolios, active weights are +a and −a. Then TE² = a²(σ₁² + σ₂² − 2σ₁₂).
  3. Take the square root. Estimate with perfect squares before computing.
  4. Multiply by z last, and scale time at the end.
  5. Sanity check: TE must be non-negative and cannot exceed |a| × (σ₁ + σ₂).

Common mistakes in Tracking Error and Benchmark-Relative Risk

  • Using portfolio weights instead of active weights in the covariance formula.

    Students recall the standard portfolio variance formula and apply it directly.

    Fix: Subtract benchmark weights first. Then use wₐ′Σwₐ.

  • Treating tracking error as the same thing as information ratio.

    Both are called benchmark-relative measures and appear together.

    Fix: TE is a standard deviation (risk). IR is active return divided by TE (reward per unit risk).

  • Dropping the covariance term, or using the wrong sign for negative active weights.

    With +a and −a weights, the cross term is negative, and students forget to include it.

    Fix: Keep signs on active weights. The cross term is 2wₐ₁wₐ₂σ₁₂, negative when weights have opposite signs and covariance is positive.

  • Forgetting to scale the horizon, or scaling variance by √t.

    Mixing up variance (scales with t) and standard deviation (scales with √t).

    Fix: Scale TE by √t. Scale variance by t.

  • Computing relative VaR with total portfolio volatility.

    Relative VaR looks like ordinary VaR.

    Fix: Use TE, not σ_P. Absolute VaR uses σ_P. Relative VaR uses TE.

  • Using a two-tailed z-value for a one-tailed VaR.

    Confusion between confidence interval and VaR quantile.

    Fix: For 95% VaR use 1.645. For 99% use 2.326.

Worked examples

Example 1

A portfolio holds 60% in Equity A and 40% in Equity B. The benchmark holds 50% in each. Annual volatilities are 20% for A and 10% for B, and the correlation is 0.5. Calculate the annual tracking error.

Show the solution
  1. Active weights: A = 60% − 50% = +10%; B = 40% − 50% = −10%.
  2. Covariance σAB = 0.5 × 0.20 × 0.10 = 0.01.
  3. Active variance = (0.10)²(0.20)² + (−0.10)²(0.10)² + 2(0.10)(−0.10)(0.01).
  4. = 0.01 × 0.04 + 0.01 × 0.01 − 0.0002 = 0.0004 + 0.0001 − 0.0002 = 0.0003.
  5. TE = √0.0003 = 0.01732, about 1.73%.

Answer: Annual tracking error ≈ 1.73%.

Example 2

A USD 200 million fund has an annual tracking error of 3% versus its benchmark. Assume normal active returns with zero mean. Calculate the 95% one-year relative VaR, and the information ratio if expected active return is 1.2% a year.

Show the solution
  1. Relative VaR = z × TE × V = 1.645 × 0.03 × 200 million.
  2. 1.645 × 0.03 = 0.04935.
  3. 0.04935 × 200 million = USD 9.87 million.
  4. IR = E(Rₐ) ÷ TE = 1.2% ÷ 3% = 0.4.

Answer: 95% one-year relative VaR ≈ USD 9.87 million; information ratio = 0.4.

Exam tips

  • Look for the word 'active' or 'relative'. It signals active weights, not portfolio weights.
  • Check the unit of the volatilities (monthly or annual) before computing, and scale at the end.
  • Questions often ask you to interpret: a higher TE means more room to deviate from the benchmark, not worse performance.
  • Know that IR compares managers on risk-adjusted skill, while TE alone only measures risk. Expect options that swap the two.
  • Always check whether the question gives a mean active return. If not, assume zero in relative VaR.

Practice questions from Portfolio Risk: Analytical Methods

Tracking Error and Benchmark-Relative Risk in other exams

The same ground in other exams, if you are preparing for more than one or want another angle on it.

Tracking Error and Benchmark-Relative Risk: frequently asked questions

What is the tracking error formula in FRM Part II?

Tracking error is the standard deviation of active return: TE = √(wₐ′Σwₐ), where wₐ is the vector of active weights and Σ is the covariance matrix. With volatilities and correlation, TE = √(σ_P² + σ_B² − 2ρσ_Pσ_B).

What is the difference between tracking error and information ratio?

Tracking error measures the volatility of active return, so it is a risk measure. Information ratio is expected active return divided by tracking error, so it measures return earned per unit of active risk.

How is relative VaR different from ordinary VaR?

Ordinary VaR measures potential loss in portfolio value. Relative VaR measures potential underperformance against a benchmark, using tracking error in place of portfolio volatility.

Do active weights always sum to zero?

Yes, when both the portfolio and the benchmark are fully invested with weights summing to 100%. If the portfolio holds cash or leverage, they will not sum to zero, so check the data.