You’re on a roll
Create a free account to save your FRM progress, track your accuracy by topic, and pick up right where you left off.
Free Questions › FRM › Market Risk Measurement and Management
Free FRM Market Risk Measurement and Management Practice Questions & Answers
400 exam-style Market Risk Measurement and Management questions. Pick your answer, hit Check answer, and see the worked solution — free to start, no signup.
100% free · No login to startQuestion 31
Parametric interest rate VaR using DV01 is approximately:
Select an option first.
Correct answer: B — z_α × σ_yield × DV01 — where σ_yield is yield volatility
Explanation: B is correct: P&L change = DV01 × Δy. VaR = z_α × σ(Δy) × DV01. A is wrong — 100 bps scaling is in DV01 already. C is wrong — VaR uses σ × z_α. D is wrong — duration × value = DV01 which is then scaled by vol × z_α.
Question 32
Historical VaR typically uses a lookback window of:
Select an option first.
Correct answer: B — 250 trading days — approximately 1 year
Explanation: B is correct: 250 days provides sufficient observations and recent relevance. Basel requires at least 250 days. A is wrong — 10 days is a holding period, not estimation window. C is wrong — too historical. D is wrong — very old data may be irrelevant.
Question 33
Risk factor VaR approach:
Select an option first.
Correct answer: B — Maps portfolio positions to underlying risk factors, then models their joint distribution to compute portfolio VaR
Explanation: B is correct: risk factor decomposition is standard in institutional VaR systems. A is wrong — risk factors aggregate across securities. C is wrong — decomposition is essential. D is wrong — ratings are for credit risk.
Question 34
A 5-year coupon bond in VaR mapping is represented as:
Select an option first.
Correct answer: B — Zero-coupon bonds at each coupon date and principal maturity — preserving interest rate sensitivity at each curve point
Explanation: B is correct: cash flow mapping assigns each coupon and principal to its payment date as a separate zero-coupon bond. A is wrong — one vertex misrepresents the rate risk at intermediate maturities. C is wrong — two vertices lose intermediate coupon exposure. D is wrong — perpetuity has infinite duration.
Question 35
PCA for yield curve risk identifies:
Select an option first.
Correct answer: B — Uncorrelated principal components (level, slope, curvature) explaining most yield curve variance
Explanation: B is correct: the first three PCs typically explain 95%+ of yield curve movement. A is wrong — PCA transforms, not removes. C is wrong — not standard terminology. D is wrong — PCA provides multiple factors.
Get the full FRM question bank — free
Drop your email and we'll send you fresh FRM practice questions, fully worked solutions and exam-deadline reminders. No spam — unsubscribe in one click.
Want to save your score and take a full mock exam? Create a free account →
Question 36
For large equity portfolios, risk factors used for VaR are typically:
Select an option first.
Correct answer: B — Market index and sector factor returns — reducing dimensionality from hundreds of stocks to a few factors
Explanation: B is correct: factor models (CAPM, Fama-French) decompose stock returns into market and sector exposures. A is wrong — individual stocks require modelling all pairwise correlations. C is wrong — book values are accounting. D is wrong — ratings are for debt.
Question 37
Model risk in VaR refers to:
Select an option first.
Correct answer: B — The risk that the VaR model misrepresents the true loss distribution due to wrong assumptions or estimation errors
Explanation: B is correct: model risk includes wrong distributional assumptions, estimation errors, misspecification of risk factors. A is wrong — model risk doesn't imply excess conservatism. C is wrong — that is operational risk. D is wrong — that is compliance risk.
Question 38
The standard error of the historical VaR quantile estimator:
Select an option first.
Correct answer: B — Increases as n decreases: se ≈ √(p(1−p)/n) / f(q_p), where f is the density at the quantile
Explanation: B is correct: smaller n → larger standard error → imprecise VaR. B correctly states the formula. A is wrong — se decreases with √n, not n. C is wrong — sampling creates uncertainty regardless. D is wrong — se depends on both n and p.
Question 39
The ES Gap (ES − VaR) is:
Select an option first.
Correct answer: A — The credit spread above VaR
Explanation: A is correct: ES Gap = ES − VaR ≥ 0. It measures how much worse losses tend to be once they exceed VaR. B describes it correctly but A is the answer. Wait — A says 'credit spread' which is wrong. Let me recheck. A says credit spread. B says additional expected loss beyond VaR. B is correct. A is wrong — not a credit spread. C is wrong — capital shortfall is different. D is wrong — standard error is statistical.
Question 40
Incremental VaR of a new position measures:
Select an option first.
Correct answer: B — The change in portfolio VaR when the position is added — capturing full correlation effects
Explanation: B is correct: incremental VaR = VaR(portfolio+position) − VaR(portfolio). A is wrong — standalone VaR ignores portfolio context. C is wrong — that is component VaR. D is wrong — not standard.
Question 41
For a continuous unimodal loss distribution, ES and VaR satisfy:
Select an option first.
Correct answer: B — ES ≥ VaR — ES is the average of numbers all exceeding VaR
Explanation: B is correct: ES = E[L|L>VaR] ≥ VaR by definition. A is wrong — impossible. C is wrong — ES > VaR for normal (by φ(z_α)/(1−α) − z_α in σ units). D is wrong — symmetry doesn't equate them.
Question 42
Expected exceptions for a 99% VaR backtested over 250 days:
Select an option first.
Correct answer: A — 2.5
Explanation: A is correct: expected exceptions = (1−0.99) × 250 = 0.01 × 250 = 2.5. B is wrong — 5 corresponds to 98% VaR. C is wrong — 10 corresponds to 96%. D is wrong — 0 exceptions suggests overconservative model.
Question 43
A portfolio's 1-day 99% VaR is $1m. The 10-day 99% VaR using √T scaling is:
Select an option first.
Correct answer: B — $3.162m
Explanation: B is correct: 10-day VaR = $1m × √10 ≈ $3.162m. A is wrong — that uses linear scaling ×10. C is wrong — $1.414m = $1m × √2 (2-day VaR). D is wrong — 2.326 is z_{99%}, not a time factor.
Question 44
For a normal distribution, the ratio of 99.9% to 99% VaR is approximately:
Select an option first.
Correct answer: B — 1.33 (z_{99.9%}/z_{99%} ≈ 3.09/2.33 ≈ 1.33)
Explanation: B is correct: 3.09/2.33 ≈ 1.33. A is wrong — ratio of 10 would require extreme non-normality. C is wrong — 2× requires z_{99.9%} = 4.66. D is wrong — ratio clearly varies.
Question 45
Full valuation for fixed income VaR is preferred over duration approximation when:
Select an option first.
Correct answer: B — Portfolio has significant convexity — large yield moves make linear duration approximation inaccurate
Explanation: B is correct: for portfolios with embedded options or callable bonds, convexity is large and the linear approximation fails for large rate moves. A is wrong — short duration means less convexity. C is wrong — stable rates make both approaches similar. D is wrong — even hedged portfolios have residual sensitivities.
Question 46
Relative VaR uses tracking error volatility (TEV) and confidence level z_α to give:
Select an option first.
Correct answer: B — Active VaR = z_α × TEV
Explanation: B is correct: active VaR = z_α × TEV where TEV = σ(R_portfolio − R_benchmark). A is wrong — that is absolute VaR. C is wrong — that is information ratio. D is wrong — not a standard measure.
Question 47
Under FRTB, Basel replaced 99% 10-day VaR with:
Select an option first.
Correct answer: B — 97.5% ES with liquidity-adjusted horizons
Explanation: B is correct: FRTB adopted 97.5% ES (approximately equivalent in tail coverage to 99% VaR under normality) with liquidity horizons ranging from 10 to 120 days by risk factor. A is wrong — ES is 97.5% not 99%. C is wrong — 99.9% is not the FRTB level. D is wrong — 95% is too low.
Question 48
Subadditivity of ES states that:
Select an option first.
Correct answer: B — ES(X+Y) ≤ ES(X) + ES(Y) — the combined ES never exceeds the sum, rewarding diversification
Explanation: B is correct: ES satisfies subadditivity for all distributions. VaR does not. B is the core coherence property. A is wrong — equality is only in degenerate cases. C is wrong — that would be super-additive. D is wrong — ES subadditivity is general.
Question 49
The Cornish-Fisher expansion adjusts normal VaR using:
Select an option first.
Correct answer: B — Skewness γ₁ and excess kurtosis γ₂ to modify the quantile: z_CF ≈ z_α + (z_α²−1)γ₁/6 + (z_α³−3z_α)γ₂/24 − (2z_α³−5z_α)γ₁²/36
Explanation: B is correct: Cornish-Fisher is a moment-matching approximation adjusting the normal z-score for skewness and excess kurtosis. A is wrong — t-distribution is a different approach. C is wrong — CF adjusts a quantile, not the full distribution. D is wrong — stress multipliers are ad-hoc.
Question 50
The risk budgeting framework uses component or marginal VaR to:
Select an option first.
Correct answer: B — Allocate total portfolio VaR across sub-portfolios so each component's risk contribution stays within its budget
Explanation: B is correct: risk budgeting allocates the total VaR budget using component VaRs that sum to total portfolio VaR. A is wrong — absolute limits are stop-losses. C is wrong — return targets are separate. D is wrong — regulatory capital uses risk-weighted assets.
Question 51
The Kupiec POF test for VaR backtesting uses a:
Select an option first.
Correct answer: B — Likelihood ratio test comparing observed exception count against the expected frequency
Explanation: B is correct: LR_POF compares the likelihood of observing x exceptions under the actual model vs the claimed probability p. Under H₀, LR_POF ~ χ²(1). A is wrong — chi-squared is used but specifically as the LR test. C is wrong — Z-test assumes normal distribution of exceptions. D is wrong — bootstrap is different.
Question 52
Christoffersen's conditional coverage test additionally tests:
Select an option first.
Correct answer: B — Whether VaR exceptions are independent over time — exception clustering indicates failure to capture volatility clustering
Explanation: B is correct: Kupiec tests only frequency. Christoffersen adds an independence test: if today's exception increases tomorrow's probability, the model fails to capture volatility clustering. A is wrong — distributional test is different. C is wrong — confidence level is the null. D is wrong — sector concentration is not tested.
Question 53
Under Basel traffic light backtesting, 10 exceptions over 250 days at 99% VaR places the model in:
Select an option first.
Correct answer: C — Red zone (10+ exceptions)
Explanation: C is correct: 10+ exceptions → red zone → capital multiplier = 4 (highest level), requiring model review. A is wrong — green is 0-4. B is wrong — yellow is 5-9. D is wrong — no orange zone in Basel.
Question 54
A major limitation of backtesting for model quality is:
Select an option first.
Correct answer: B — Tests only exception frequency, not the severity of losses in the tail — cannot fully assess whether the loss distribution is accurately modelled
Explanation: B is correct: backtesting is a blunt test of one summary statistic. A model wrong about tail severity but close on frequency can pass. A is wrong — 250 days is manageable. C is wrong — any model can be backtested. D is wrong — exception count depends on confidence level chosen.
Question 55
ES backtesting is more challenging than VaR backtesting because:
Select an option first.
Correct answer: B — ES requires estimating the mean of the tail — with very few observations exceeding VaR, statistical tests have low power
Explanation: B is correct: with 99% VaR, only ~2.5 exceptions per year occur. Estimating the mean of this tiny sample with acceptable statistical power requires years of data. A is wrong — ES > VaR is definitional but not the challenge. C is wrong — same horizon. D is wrong — ES can be computed from data.
Question 56
The 'loss distribution approach' (LDA) for operational risk VaR models:
Select an option first.
Correct answer: B — Frequency (how often losses occur) and severity (how large each loss is) separately, then convolves them to get total loss distribution
Explanation: B is correct: LDA models frequency (Poisson) and severity (e.g. lognormal) independently, then convolves via Monte Carlo to obtain the annual loss distribution. A is wrong — LDA is for operational risk. C is wrong — LDA uses parametric distributions. D is wrong — LDA has its own framework.
Question 57
A 95% confidence 1-day VaR of $1 million means losses will exceed $1 million on approximately how many days per year (250 trading days)?
Select an option first.
Correct answer: B — 12.5 days
Explanation: B is correct: expected exceptions = (1−0.95) × 250 = 0.05 × 250 = 12.5 days per year. A is wrong — 1 day corresponds to 99.6% confidence. C is wrong — 25 = 10% × 250 (90% VaR). D is wrong — 5 = 2% × 250.
Question 58
For a normally distributed portfolio loss, the relationship between 99% ES and 99% VaR is:
Select an option first.
Correct answer: B — ES > VaR — ES = VaR + φ(z_{0.99})/(1−0.99)/σ, where φ is the normal PDF at z_{0.99}
Explanation: B is correct: for the normal distribution, ES = μ + σ × φ(z_α)/(1−α) where φ is the standard normal density. Since φ(z_α)/(1−α) > z_α, ES > VaR. The exact ES = z_α + φ(z_α)/(1−α) in units of σ. A is wrong — ES ≠ VaR. C is wrong — ES is always ≥ VaR. D is wrong — 2× is not the exact ratio.
Question 59
The 'elliptical distribution' class includes:
Select an option first.
Correct answer: B — Multivariate normal and multivariate t distributions — all distributions whose contours of equal density are ellipses
Explanation: B is correct: elliptical distributions (including normal, t, Laplace, and others) have elliptical level sets and are fully characterised by their mean vector, covariance matrix, and a generator function. For elliptical distributions, correlation fully characterises the dependence structure. A is wrong — multivariate t is also elliptical. C is wrong — normal is thin-tailed and elliptical. D is wrong — finite variance is not the criterion.
Question 60
When VaR is computed using a t-distribution with heavy tails, compared to the normal distribution:
Select an option first.
Correct answer: B — VaR is higher at high confidence levels (99%+) because the t-distribution assigns more probability to extreme events
Explanation: B is correct: the t-distribution has heavier tails than normal. At the 99th percentile, the t quantile exceeds the normal quantile. For example, t(5) 99th percentile ≈ 3.365 vs normal 2.326. A is wrong — higher tail weight means higher VaR. C is wrong — same symmetry but different tail weight means different quantiles. D is wrong — VaR depends on both location and shape.
More free FRM topics
Ten questions in
- The ones you miss are saved as a drill you can repeat
- Your place is kept, on this device and any other
- A streak, if that is the thing that gets you back tomorrow
Every question on this page stays free and open either way.