FRM Exam Part II · VaR and Risk Budgeting in Investment Management

Absolute, Relative and Tracking Error VaR Explained

Updated 11 October 2026 · Fact-checked

Absolute VaR measures the loss in a portfolio's total value. Relative VaR (tracking error VaR) measures the shortfall versus a benchmark, using active return. Tracking error is the standard deviation of active return, from active weights and the covariance matrix. Relative VaR = z × tracking error × portfolio value.

Understand Absolute, Relative and Tracking Error VaR

Absolute VaR asks: how much money could I lose in total? It uses the volatility of the portfolio's own return. A portfolio worth $100 million with 10% annual volatility has a very different absolute VaR from one with 5% volatility.

Relative VaR asks: how far could I fall behind my benchmark? Most asset managers are judged against an index, not against zero. A manager who loses 8% when the index loses 10% has done well. So the risk that matters to the mandate is the risk of the difference.

That difference is the active return: portfolio return minus benchmark return. Its standard deviation is the tracking error (TE), also called active risk. Relative VaR, often called tracking error VaR, is the VaR of active return. Under normality it is a z-score times TE times portfolio value.

To get TE from holdings, work with active weights: portfolio weight minus benchmark weight for each asset. Active weights sum to zero when both are fully invested. TE = √(a′Σa), where a is the vector of active weights and Σ is the covariance matrix of asset returns. A portfolio identical to the benchmark has zero TE, even if its absolute VaR is large.

The two measures answer different questions. A fund can have high absolute VaR and low relative VaR (a closely tracked index fund). It can also have low absolute VaR and high relative VaR (a cash-heavy fund against an equity benchmark).

Key formulas to remember

Active return: R_A = R_P − R_B; Portfolio return minus benchmark return.
Active weights: a_i = w_i(P) − w_i(B); Sum to zero if portfolio and benchmark are both fully invested.
Tracking error (two assets): TE = √(a₁²σ₁² + a₂²σ₂² + 2·a₁·a₂·ρ·σ₁·σ₂); Include the covariance term. Active weights can be negative.
Tracking error (general): TE = √(a′Σa); Σ is the covariance matrix of asset returns.
TE from portfolio and benchmark: TE² = σ_P² + σ_B² − 2·ρ_PB·σ_P·σ_B; Useful when you are given volatilities and their correlation.
Absolute VaR (normal): VaR = z × σ_P × V (less any expected-return term); z = 1.645 at 95%, 2.326 at 99% one-tailed.
Relative VaR (tracking error VaR): Relative VaR = z × TE × V; Same z, but TE replaces σ_P.
Time scaling: σ_T = σ_annual × √(T in years); For example, monthly = annual ÷ √12. Assumes independent returns.
Information ratio link: IR = expected active return ÷ TE; TE is the denominator of the information ratio.

How to solve Absolute, Relative and Tracking Error VaR questions

Use this order for any absolute, relative or tracking error VaR question.

1Decide what is asked: total loss (absolute) or shortfall versus benchmark (relative).
2For relative VaR, compute active weights as portfolio weight minus benchmark weight.
3Get TE: √(a′Σa), or √(σ_P² + σ_B² − 2ρσ_Pσ_B) if portfolio and benchmark volatilities and correlation are given.
4Convert TE to the required horizon by multiplying by √(time) if it is quoted annually.
5Pick z for the confidence level (1.645 for 95%, 2.326 for 99%).
6Multiply: VaR = z × volatility × portfolio value. Use TE for relative, σ_P for absolute.
7State the interpretation: the loss (or lag versus benchmark) not expected to be exceeded at that confidence over that horizon.

Quickest way: Active-weight shortcut

When to use it: Use when the question gives active weights or volatilities and correlation, and options differ clearly in size.

Write the active weights first. Check they sum to zero.
Square each active weight times its volatility and add the cross term.
Take the square root to get TE, then multiply by z and value.
Sense-check: TE should be below the larger of the two volatilities unless correlation is low. Eliminate options that are far off.
Rescale time last, using √T.

Common mistakes in Absolute, Relative and Tracking Error VaR

Using the portfolio's own volatility for relative VaR.
Students treat every VaR as absolute VaR.
Fix: If the question mentions a benchmark or active risk, use TE, not σ_P.
Using portfolio weights instead of active weights in the TE formula.
The formula looks like the portfolio variance formula.
Fix: Subtract benchmark weights first, then apply the same formula to the active weights.
Dropping the covariance term, or ignoring that negative active weights change its sign.
Rushing and treating assets as uncorrelated.
Fix: Always include 2·a₁·a₂·ρ·σ₁·σ₂ with the signs of the active weights.
Adding portfolio and benchmark volatilities to get TE.
Confusing risk of a difference with a sum.
Fix: Use σ_P² + σ_B² − 2ρσ_Pσ_B. High correlation lowers TE.
Forgetting to scale TE to the horizon.
TE is quoted annually but the question asks for a month or ten days.
Fix: Multiply by √(T in years) before applying z.
Assuming zero absolute risk means zero relative risk, or the reverse.
The two measures are blended in memory.
Fix: A benchmark clone has zero TE but full market risk. Judge each measure on its own question.

Worked examples

Example 1

A $200 million equity portfolio has annual volatility of 18%. Its benchmark has annual volatility of 16%. Correlation between portfolio and benchmark returns is 0.95. Using a normal model, find the tracking error and the 95% one-year relative VaR.

Show the solution

TE² = 0.18² + 0.16² − 2 × 0.95 × 0.18 × 0.16.
0.18² = 0.0324. 0.16² = 0.0256. Sum = 0.0580.
2 × 0.95 × 0.18 × 0.16 = 0.05472.
TE² = 0.0580 − 0.05472 = 0.00328.
TE = √0.00328 ≈ 0.05727, about 5.73%.
Relative VaR = 1.645 × 0.05727 × $200 million = $18.84 million.

Answer: TE ≈ 5.73%. The 95% one-year relative VaR is about $18.8 million.

Example 2

A portfolio is 60% Asset X and 40% Asset Y. The benchmark is 50% X and 50% Y. Annual volatilities are 20% for X and 10% for Y, and correlation is 0.30. For a $50 million portfolio, find the 99% one-month relative VaR (normal, 12 months in a year).

Show the solution

Active weights: X = 60% − 50% = +0.10. Y = 40% − 50% = −0.10.
Variance terms: 0.10² × 0.20² = 0.01 × 0.04 = 0.0004. 0.10² × 0.10² = 0.01 × 0.01 = 0.0001.
Cross term: 2 × (0.10) × (−0.10) × 0.30 × 0.20 × 0.10 = −0.02 × 0.006 = −0.00012.
Annual TE² = 0.0004 + 0.0001 − 0.00012 = 0.00038.
Annual TE = √0.00038 ≈ 0.019494, about 1.949%.
Monthly TE = 0.019494 ÷ √12 = 0.019494 ÷ 3.4641 ≈ 0.005627.
Relative VaR = 2.326 × 0.005627 × $50 million ≈ $0.654 million.

Answer: Annual TE ≈ 1.95%. The 99% one-month relative VaR is about $0.65 million.

Exam tips

Read for the word benchmark, active or relative. It tells you to use TE, not portfolio volatility.
Check the horizon and confidence level in the final line of the question before computing.
Compute TE² first and take the square root once. Rounding early often moves you to a wrong option.
Expect interpretation items: low TE means close to benchmark, not low risk. High TE means large deviation from the benchmark in either direction.
Remember that TE is symmetric. It measures deviation above and below the benchmark, so relative VaR focuses on the downside tail only.

Practice questions from VaR and Risk Budgeting in Investment Management

Absolute, Relative and Tracking Error VaR in other exams

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

Absolute, Relative and Tracking Error VaR: frequently asked questions

What is the difference between absolute VaR and relative VaR?

Absolute VaR measures the potential loss in the portfolio's total value, based on its own volatility. Relative VaR measures the potential shortfall against a benchmark, based on the volatility of active return. Use relative VaR when performance is judged against an index.

How do I calculate tracking error from active weights?

Subtract benchmark weights from portfolio weights to get active weights. Then compute TE = √(a′Σa), which for two assets means squaring each active weight times its volatility and adding twice the active weight product times covariance. Take the square root of the total.

Is tracking error the same as active risk?

Yes, in this context. Both mean the standard deviation of the portfolio's return minus the benchmark's return. It is usually quoted on an annual basis.

Can relative VaR be larger than absolute VaR?

Yes, it can. If the portfolio and benchmark are negatively correlated or hold very different exposures, TE can exceed the portfolio's own volatility. For a typical long-only fund against its own market benchmark, TE is much smaller.