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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 1
Value at Risk (VaR) at confidence level α and horizon T is best defined as:
Select an option first.
Correct answer: B — The loss level that will not be exceeded with probability α over the horizon T
Explanation: B is correct: VaR(α,T) is the quantile such that P(Loss > VaR) = 1−α. A is wrong — expected loss is the mean, not the quantile. C is wrong — that is Expected Shortfall. D is wrong — that is a volatility measure.
Question 2
A portfolio has a 1-day 99% VaR of $500,000. This means:
Select an option first.
Correct answer: B — On 99% of days, losses will not exceed $500,000
Explanation: B is correct: the 99% confidence level states losses stay below $500,000 on 99% of days. A is wrong — losses can exceed $500,000 by any amount on those 1% of days. C is wrong — that is ES. D is wrong — no relationship to total portfolio loss.
Question 3
The parametric (variance-covariance) VaR approach assumes:
Select an option first.
Correct answer: B — Returns are independently and identically normally distributed
Explanation: B is correct: parametric VaR uses the normal distribution for returns, computing VaR = z_α × σ. A is wrong — t-distribution is an extension. C is wrong — that is historical simulation. D is wrong — GPD is used in EVT.
Question 4
For a portfolio with mean zero and daily standard deviation $100,000, the 1-day 99% VaR is approximately:
Select an option first.
Correct answer: C — $233,000
Explanation: C is correct: 99% VaR = z_{0.99} × σ = 2.326 × $100,000 ≈ $233,000. A is wrong — that is just σ. B is wrong — 1.65 × σ gives 95% VaR. D is wrong — 3.28σ corresponds to a higher confidence level.
Question 5
The square root of time rule for scaling VaR assumes:
Select an option first.
Correct answer: B — Daily returns are i.i.d. — losses scale with √T for a T-day horizon
Explanation: B is correct: under i.i.d. returns, T-day variance = T × daily variance, so T-day VaR = 1-day VaR × √T. A is wrong — fat tails and autocorrelation violate i.i.d. C is wrong — volatility scales with √T, not linearly. D is wrong — that would be linear scaling.
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Question 6
Expected Shortfall (ES) is defined as:
Select an option first.
Correct answer: B — The expected value of losses conditional on losses exceeding VaR
Explanation: B is correct: ES_α = E[L | L > VaR_α]. A is wrong — ES is an average, not a maximum. C is wrong — ES uses mean, not median. D is wrong — that would be expected loss (mean over entire distribution).
Question 7
ES is preferred over VaR as a risk measure because:
Select an option first.
Correct answer: B — ES is a coherent risk measure satisfying subadditivity, meaning diversification is properly rewarded
Explanation: B is correct: VaR violates subadditivity for non-normal distributions; ES always satisfies it. A is wrong — ES ≥ VaR by definition. C is wrong — ES is more complex. D is wrong — both can use parametric or non-parametric methods.
Question 8
A coherent risk measure satisfies:
Select an option first.
Correct answer: A — Monotonicity, subadditivity, positive homogeneity, and translation invariance
Explanation: A is correct: Artzner et al. (1999) defined coherent risk measures with these four axioms. VaR satisfies all except subadditivity; ES satisfies all four. B, C, D describe properties of other mathematical objects.
Question 9
Historical simulation VaR is computed by:
Select an option first.
Correct answer: B — Ordering past portfolio P&Ls from worst to best and reading the appropriate percentile
Explanation: B is correct: HS applies historical return scenarios to the current portfolio, orders the resulting P&Ls, and reads off the VaR quantile. A is wrong — that is parametric VaR. C is wrong — that is Monte Carlo. D is wrong — that is EVT.
Question 10
A key limitation of historical simulation VaR is:
Select an option first.
Correct answer: B — It depends entirely on the historical period used — if the lookback window lacks stress events, VaR is underestimated
Explanation: B is correct: HS is bounded by its historical window. A is wrong — HS is distribution-free (a strength). C is wrong — HS can handle options by applying historical factor changes. D is wrong — HS can either over- or underestimate.
Question 11
Monte Carlo VaR:
Select an option first.
Correct answer: B — Generates thousands of random risk factor scenarios and revalues the portfolio under each
Explanation: B is correct: MC draws from specified joint distributions across risk factors, revalues the portfolio, and reads the VaR from the resulting P&L distribution. A is wrong — that is HS. C is wrong — that is parametric VaR. D is wrong — that is stress testing.
Question 12
Full revaluation in Monte Carlo differs from the delta approximation because:
Select an option first.
Correct answer: B — Full revaluation reprices each instrument using its pricing model for each scenario, capturing all non-linearities
Explanation: B is correct: full revaluation captures gamma, convexity, and all higher-order effects. A is wrong — delta approximation uses constant delta. C is wrong — full revaluation is more accurate. D is wrong — it can handle any instrument.
Question 13
VaR mapping decomposes a complex portfolio into:
Select an option first.
Correct answer: B — Standardised risk factors whose VaR can be calculated and aggregated
Explanation: B is correct: VaR mapping expresses complex instruments in terms of standardised risk primitives (zero-coupon bonds, spot FX, etc.). A is wrong — mapping reduces to primitives, not all underlyings. C is wrong — that is credit risk. D is wrong — that is historical simulation.
Question 14
For a two-asset portfolio with individual VaRs of $200 and $300 and correlation 0.5, the parametric portfolio VaR is approximately:
Select an option first.
Correct answer: A — $436
Explanation: A is correct: VaR² = 200² + 300² + 2×0.5×200×300 = 40,000+90,000+60,000 = 190,000. VaR = √190,000 ≈ $436. B is wrong — that is the sum. C and D are not supported.
Question 15
Marginal VaR of an asset measures:
Select an option first.
Correct answer: B — The change in portfolio VaR per unit increase in the position — reflecting each unit's contribution to total risk
Explanation: B is correct: marginal VaR = ∂(portfolio VaR)/∂(weight_i). A is wrong — that is standalone VaR. C is wrong — that is incremental VaR. D is wrong — that is closer to component VaR.
Question 16
Component VaR is additive in that:
Select an option first.
Correct answer: B — Sum of all component VaRs equals total portfolio VaR — component VaR_i = marginal VaR_i × weight_i
Explanation: B is correct: Σ component VaR_i = total portfolio VaR by definition. A is wrong — component ≠ marginal. C is wrong — incremental VaR is the discrete change from removing a position. D is wrong — component VaR can be negative for diversifying assets.
Question 17
The delta-normal approach for options VaR:
Select an option first.
Correct answer: B — Uses delta to linearise the option, treating it as an equivalent underlying position — ignoring gamma
Explanation: B is correct: delta-normal replaces each option with its delta-equivalent in the underlying, linearising the P&L. Gamma and other higher-order effects are ignored. A is wrong — no full revaluation. C is wrong — lognormal is the underlying assumption. D is wrong — delta-normal avoids Monte Carlo.
Question 18
The delta-gamma approximation for option P&L adds:
Select an option first.
Correct answer: B — ½ × Gamma × (ΔS)² — capturing the curvature of the option price relationship
Explanation: B is correct: the Taylor expansion ΔV ≈ δΔS + ½ΓΔS². Adding gamma improves accuracy for non-linear options. A is wrong — that would be delta-gamma-vega. C is wrong — theta is time decay. D is wrong — rho is rate sensitivity.
Question 19
A bond portfolio with DV01 = $50,000 faces a 20 bps rate move. The approximate P&L is:
Select an option first.
Correct answer: B — $1,000,000 loss if rates rise
Explanation: B is correct: P&L ≈ DV01 × Δy(bps) = $50,000 × 20 = $1,000,000. For a long bond position, rising rates cause losses. A is wrong — DV01 × 1bp, not 20bps. C is wrong — would require DV01 × 50. D is wrong — rates rising causes losses for long bonds.
Question 20
P&L attribution decomposes portfolio value changes into:
Select an option first.
Correct answer: B — Contributions from risk factors: delta P&L, gamma P&L, vega P&L, theta, and other Greeks
Explanation: B is correct: attribution sums delta×ΔS + ½γΔS² + vega×Δσ + theta×Δt + rho×Δr etc. This validates the risk model. A is wrong — not a standard decomposition. C is wrong — P&L attribution is a market risk tool. D is wrong — time decomposition is different.
Question 21
Relative VaR (active VaR) measures:
Select an option first.
Correct answer: B — The VaR of active returns (portfolio return minus benchmark) — tracking error VaR
Explanation: B is correct: relative VaR = z_α × TEV (tracking error volatility). It measures underperformance risk relative to a benchmark. A is wrong — that is absolute VaR. C is wrong — notional-based risk. D is wrong — no industry comparison.
Question 22
Pre-FRTB Basel regulations required VaR at:
Select an option first.
Correct answer: B — 99% confidence, 10-day horizon with at least 1 year of historical data
Explanation: B is correct: Basel II/2.5 required 99%, 10-day VaR using at least 250 trading days. A is wrong — 95% is common internally but not the Basel regulatory level. C is wrong — 99.9%/1-day is not the Basel standard. D is wrong — 97.5% ES is the FRTB standard.
Question 23
VaR's key limitation is that it:
Select an option first.
Correct answer: B — Does not convey the magnitude of losses beyond the threshold — only that losses will exceed it with probability 1−α
Explanation: B is correct: VaR says nothing about how bad losses can be once they exceed the threshold. A, C, D are wrong — VaR provides no tail severity info, is not coherent, and higher confidence still doesn't address severity.
Question 24
A loss distribution has 99% VaR of $10m and 99% ES of $15m. This means:
Select an option first.
Correct answer: B — When losses exceed $10m, the average loss is $15m
Explanation: B is correct: ES = E[L|L > $10m] = $15m. A is wrong — average daily loss is the unconditional mean. C is wrong — losses can exceed $15m. D is wrong — there is no finite maximum.
Question 25
Spectral risk measures generalise ES by:
Select an option first.
Correct answer: B — Applying a weighting function φ(p) across all quantile levels — ES is a special case using uniform weight above the VaR threshold
Explanation: B is correct: spectral risk measures ρ(X) = ∫φ(p)q_p(X)dp with non-decreasing φ. ES uses φ(p)=1/(1−α) for p>α. A is wrong — single quantile gives VaR. C is wrong — no spectral distribution. D is wrong — market prices don't determine weights.
Question 26
For normal returns with σ, the ratio of 99% VaR to 95% VaR is approximately:
Select an option first.
Correct answer: B — 1.414 (= 2.326/1.645 ≈ 1.41)
Explanation: B is correct: z_{99%}/z_{95%} = 2.326/1.645 ≈ 1.414. A is wrong — 10× would imply extreme non-normality. C is wrong — 2× would require z_{99%} ≈ 3.29. D is wrong — ratio differs for different confidence levels.
Question 27
Weighted historical simulation (WHS) improves on simple HS by:
Select an option first.
Correct answer: B — Assigning higher weights to more recent observations through an exponential decay factor
Explanation: B is correct: WHS (e.g. BRW) assigns weight (1−λ)λ^(k-1) to observation k days old. Recent data receives higher weight. A is wrong — same window, different weights. C is wrong — that is filtered HS. D is wrong — WHS is purely empirical.
Question 28
Age-weighted HS's main disadvantage is:
Select an option first.
Correct answer: B — If weights decay rapidly, the effective sample size shrinks, reducing statistical reliability of the tail estimate
Explanation: B is correct: fast decay → fewer effective observations → higher estimation error. A is wrong — multi-asset portfolios can use WHS. C is wrong — no distributional assumption. D is wrong — WHS can handle options.
Question 29
Filtered historical simulation (FHS) combines:
Select an option first.
Correct answer: B — Historical simulation with GARCH volatility models — standardising past returns by current volatility estimates
Explanation: B is correct: FHS standardises historical innovations by GARCH volatility, then rescales by current conditional volatility. A is wrong — not a 50/50 mix. C is wrong — no stress testing. D is wrong — EVT is a separate extension.
Question 30
A QQ-plot in risk management is used to:
Select an option first.
Correct answer: B — Visually compare whether an empirical distribution matches a theoretical one by plotting corresponding quantiles
Explanation: B is correct: QQ-plot graphs empirical vs theoretical quantiles. Deviations from the 45° line indicate misfit — heavy tails show points above the line in extremes. A is wrong — that is a CDF plot. C is wrong — that is a scatter plot. D is wrong — ACF/PACF plots detect autocorrelation.
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