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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 271
The 'calibration error' in fitting a GARCH model to data is smallest when:
Select an option first.
Correct answer: B — The model captures the key stylised facts of the data (volatility clustering, fat tails, asymmetry) while being parsimonious — GARCH(1,1) often achieves this with minimal parameters
Explanation: B is correct: model parsimony (few parameters) reduces overfitting while capturing key properties. GARCH(1,1) with 3 parameters (ω, α, β) captures volatility clustering and is typically sufficient. Higher-order GARCH rarely improves out-of-sample performance. A is wrong — more parameters can cause overfitting. C is wrong — 10 days is too little data. D is wrong — IV calibration gives risk-neutral parameters.
Question 272
The 'Heston model correlation' parameter ρ between asset return and volatility processes primarily controls:
Select an option first.
Correct answer: B — The skew of the implied volatility smile — negative ρ (typical for equities) produces a downward-sloping skew (higher IV for lower strikes), while positive ρ produces upward-sloping skew
Explanation: B is correct: in the Heston model, ρ correlates asset return dS and variance dv. Negative ρ means: when asset price falls (bad day), volatility rises → the left tail is fatter → OTM puts are more valuable → higher IV for lower strikes (skew). A is wrong — the level is controlled by the long-run variance θ. C is wrong — curvature is driven by ξ (vol-of-vol). D is wrong — term structure is driven by the mean reversion speed κ.
Question 273
The 'GARCH MIDAS' (Mixed Data Sampling) model addresses the issue of:
Select an option first.
Correct answer: B — Incorporating low-frequency macroeconomic variables (quarterly GDP, annual inflation) into the high-frequency GARCH volatility equation — allowing macroeconomic conditions to influence the long-run component of volatility
Explanation: B is correct: GARCH-MIDAS (Engle et al. 2006) decomposes conditional variance into: (1) a long-run component driven by low-frequency macro variables (sampled quarterly/annually); (2) a short-run GARCH component capturing daily volatility clustering. This allows macroeconomic regime changes to feed into volatility. A is wrong — standard GARCH estimation is by MLE. C is wrong — GARCH option pricing uses Duan (1995). D is wrong — tick data estimation uses realised kernels.
Question 274
Under the 'peaks-over-threshold' method, the choice of threshold u involves a trade-off between:
Select an option first.
Correct answer: B — Bias and variance — a high threshold reduces bias (GPD approximation improves) but increases variance (fewer exceedances, less data); a low threshold reduces variance but increases bias
Explanation: B is correct: the classic statistical bias-variance trade-off: higher u → better GPD approximation (less bias) but fewer exceedances → higher estimation variance. Lower u → more data (lower variance) but GPD may not fit well (higher bias). The optimal threshold balances these. The MEF plot and Hill plot help identify this balance empirically. A is wrong — no direct cost consideration. C is wrong — confidence level is a separate choice. D is wrong — smoothness is not the issue.
Question 275
The 'generalised hyperbolic' (GH) distribution family includes which special cases?
Select an option first.
Correct answer: B — Normal hyperbolic, hyperbolic, variance gamma, normal inverse Gaussian (NIG), and Student's t — making it a flexible family encompassing many financial return distributions
Explanation: B is correct: the generalised hyperbolic distribution (Barndorff-Nielsen 1977) is a 5-parameter family with special cases: VG (λ=1), NIG (λ=−½), hyperbolic (λ=1 with specific parameters), and t (λ=−ν/2). It can model symmetric or skewed, light or heavy tailed distributions. A is wrong — more special cases exist. C is wrong — Cauchy and Laplace are different families. D is wrong — EVT distributions are a separate family.
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Question 276
The 'volatility targeting' approach in portfolio management adjusts position sizes to:
Select an option first.
Correct answer: B — Maintain a constant target volatility level — when GARCH/realised volatility rises above target, positions are reduced; when it falls below target, positions are increased
Explanation: B is correct: volatility-targeted strategies use dynamic leverage: position_size_t = target_vol / σ_t. In high-volatility periods, leverage is reduced (fewer losses in drawdowns); in low-volatility periods, leverage is increased. This tends to improve Sharpe ratios and reduce maximum drawdowns. A is wrong — maximising volatility is not the goal. C is wrong — tracking benchmark creates tracking error. D is wrong — 1% is not necessarily the target.
Question 277
The 'copula density' c(u₁,...,uₙ) is related to the joint density f(x₁,...,xₙ) and marginal densities f_i(x_i) through:
Select an option first.
Correct answer: B — f(x₁,...,xₙ) = c(F₁(x₁),...,Fₙ(xₙ)) × Π f_i(x_i) — the joint density equals the copula density times the product of marginal densities
Explanation: B is correct: differentiating Sklar's theorem F(x₁,...,xₙ) = C(F₁(x₁),...,Fₙ(xₙ)) with respect to each x_i gives: f(x₁,...,xₙ) = c(F₁(x₁),...,Fₙ(xₙ)) × Π f_i(x_i). This decomposition separates the joint density into a copula density (capturing dependence) and marginal densities (capturing individual distributions). A is wrong — product, not quotient of marginals. C is wrong — uses CDFs not densities. D is wrong — holds for all distributions.
Question 278
The 'realised beta' between a stock and the market index from high-frequency data is computed as:
Select an option first.
Correct answer: B — The ratio of realised covariance to realised market variance — both computed from intraday returns, providing a more accurate and timely measure of systematic risk
Explanation: B is correct: realised beta_t = RCov(stock, market)_t / RVar(market)_t, where both are computed from intraday data. This provides daily estimates of beta that respond quickly to changing market conditions. A is wrong — daily beta from 252-day regression is slower to update. C is wrong — annual beta is very slow to update. D is wrong — correlation ≠ beta (beta = ρ × σ_stock/σ_market).
Question 279
The 'Fama-French three-factor model' extends CAPM by adding:
Select an option first.
Correct answer: B — A size factor (SMB: small minus big) and a value factor (HML: high minus low book-to-market) — capturing additional systematic risks beyond the market factor
Explanation: B is correct: Fama-French (1993) adds SMB (return of small stocks minus large stocks) and HML (return of high book-to-market stocks minus low B/M stocks) to the CAPM market factor. These factors explain cross-sectional return differences not captured by market beta alone. A is wrong — momentum (Carhart) is a fourth factor. C is wrong — risk-free rate is already in CAPM. D is wrong — volatility factor is separate (e.g. betting against beta).
Question 280
Backtesting a VaR model involves:
Select an option first.
Correct answer: B — Comparing predicted VaR with actual realised P&L — exceptions occur when actual loss exceeds VaR
Explanation: B is correct: backtesting is an ex-post comparison of VaR threshold with realised P&L. A is wrong — hypothetical scenarios are stress testing. C is wrong — competitor comparison is benchmarking. D is wrong — options validate implied vol models.
Question 281
'Clean P&L' for VaR backtesting excludes:
Select an option first.
Correct answer: B — Fees, intraday trading, and P&L from activity not related to the static end-of-day position — only overnight holding P&L
Explanation: B is correct: clean P&L = P&L of yesterday's positions under today's prices, with no new trades. This isolates the risk being measured by VaR. A is wrong — all P&Ls are included. C is wrong — rate risk is in clean P&L. D is wrong — FX is in clean P&L.
Question 282
The binomial backtesting test tests:
Select an option first.
Correct answer: B — Whether the exception probability equals (1−c) and exceptions are independent
Explanation: B is correct: under H₀, exceptions follow Binomial(n, 1−c). The test checks whether the observed exception count is consistent with the null probability p=1−c. A is wrong — window length is an input. C is wrong — normality is not the null. D is wrong — the test is two-sided.
Question 283
Under Basel backtesting, 10 exceptions in 250 days at 99% VaR:
Select an option first.
Correct answer: B — Places model in the red zone — multiplier = 4, requiring model review
Explanation: B is correct: 10+ exceptions → red zone (multiplier = 4, maximum). A is wrong — 0-4 is acceptable green. B is correct. C is wrong — yellow is 5-9. D is wrong — no trading ban.
Question 284
The main limitation of backtesting is:
Select an option first.
Correct answer: B — Tests only exception frequency — cannot assess whether the entire loss distribution is accurate, especially tail severity
Explanation: B is correct: backtesting is a blunt one-statistic test. Tail severity (how bad losses are given they exceed VaR) is not tested. A is wrong — 250 days is standard. C is wrong — any model can be backtested. D is wrong — exception count reflects confidence level.
Question 285
ES backtesting is harder than VaR backtesting because:
Select an option first.
Correct answer: B — ES requires estimating the mean of the tail — with very few exceedances per year, statistical tests have very low power
Explanation: B is correct: with 99% VaR and 250 days, only ~2.5 exceedances occur per year. Estimating the mean of this tiny sample is unreliable. A is wrong — ES > VaR is definitional. C is wrong — same horizon. D is wrong — ES can be computed.
Question 286
Stress testing is designed to:
Select an option first.
Correct answer: B — Assess losses under extreme but plausible scenarios beyond the VaR confidence interval
Explanation: B is correct: stress testing complements VaR by examining specific extreme scenarios. A is wrong — backtesting validates VaR. C is wrong — stress testing focuses on losses. D is wrong — stress testing supplements VaR.
Question 287
Historical scenario analysis applies:
Select an option first.
Correct answer: B — Actual historical events (1987 crash, 2008 GFC, 2020 COVID) to the current portfolio
Explanation: B is correct: historical scenarios replay actual observed risk factor movements on the current portfolio. A is wrong — those are hypothetical scenarios. C is wrong — Monte Carlo uses random simulation. D is wrong — regulatory scenarios may differ from historical ones.
Question 288
Hypothetical stress scenarios are constructed by:
Select an option first.
Correct answer: B — Defining plausible but not necessarily historical scenarios using expert judgment — e.g. 'rates rise 300bps and equities fall 40%'
Explanation: B is correct: hypothetical scenarios capture specific vulnerabilities using forward-looking expert judgment. A is wrong — random selection misses the point. C is wrong — EVT is statistical extrapolation. D is wrong — traders provide input but risk managers design scenarios.
Question 289
A reverse stress test begins with:
Select an option first.
Correct answer: B — An outcome (e.g. business-threatening loss) and works backwards to identify what market moves would cause it
Explanation: B is correct: reverse stress testing starts from the outcome and identifies the portfolio's key vulnerabilities. A is wrong — standard stress tests start from shocks. C is wrong — VaR is a different measure. D is wrong — historical scenarios work forward.
Question 290
FRTB was introduced primarily to address:
Select an option first.
Correct answer: B — Trading/banking book boundary inconsistencies, VaR inadequacy for tail risk (replaced by ES), and insufficient capital for illiquid positions
Explanation: B is correct: FRTB addressed: (1) clearer trading/banking book boundary; (2) replaced VaR with ES at 97.5%; (3) liquidity-adjusted horizons; (4) new SA; (5) desk-level model approval. A is wrong — VaR inadequacy was one key issue. C is wrong — SVaR was already in Basel 2.5. D is wrong — capital was generally insufficient.
Question 291
FRTB's liquidity horizon improvement:
Select an option first.
Correct answer: B — Assigns varying horizons (10-120 days) to different risk factors — preventing underestimation of liquidation risk for illiquid positions
Explanation: B is correct: LH₁=10 (major FX), LH₂=20 (investment-grade credit), LH₃=40 (EM equities), LH₄=60 (small-cap equity), LH₅=120 (illiquid credit). A is wrong — 10 days was the pre-FRTB uniform horizon. C is wrong — repo is not the main innovation. D is wrong — bid-ask is a different measure.
Question 292
FRTB IMA approval is granted at:
Select an option first.
Correct answer: B — Trading desk level — each desk passes P&L attribution and backtesting independently or reverts to SA
Explanation: B is correct: desk-level approval prevents a good firm-level model from hiding desk-specific weaknesses. A is wrong — firm-level was pre-FRTB. C is wrong — risk factor modellability is a separate concept. D is wrong — regulators supervise but desks are the approval unit.
Question 293
FRTB's P&L attribution test requires:
Select an option first.
Correct answer: B — Comparison of risk-theoretical P&L (from the risk model) with hypothetical P&L (from actual market data) — close agreement validates the model captures all material risk factors
Explanation: B is correct: PLA tests whether the risk model adequately represents all significant risk factors. A is wrong — that is backtesting. C is wrong — trade data reporting is separate. D is wrong — VaR is not used in PLA.
Question 294
A 'non-modellable risk factor' (NMRF) under FRTB:
Select an option first.
Correct answer: B — Lacks sufficient real-price observations (< 24 per year) — receives a standardised stress capital charge instead of ES modelling
Explanation: B is correct: NMRF criterion: < 24 verifiable real price observations per year, or more than 31 days between consecutive observations. NMRFs get a separate stress scenario capital charge (SES). A is wrong — some NMRFs have limited data. C is wrong — banking book uses IRRBB. D is wrong — NMRFs span all asset classes.
Question 295
FRTB SA sensitivity-based method (SBM) aggregates capital from:
Select an option first.
Correct answer: B — Delta, vega, and curvature sensitivities to prescribed risk factors across risk classes, combined using prescribed correlations
Explanation: B is correct: SBM: (1) delta risk from sensitivities to prescribed factors; (2) vega risk from IV sensitivities; (3) curvature risk from second-order price effects. Aggregated within and across risk classes using low/medium/high correlation scenarios. A is wrong — SA does not use VaR. C is wrong — no historical simulation. D is wrong — multiple risk factors per class.
Question 296
FRTB 'jump to default' (JtD) charge captures:
Select an option first.
Correct answer: B — The P&L impact of an issuer suddenly defaulting — instantaneous loss without prior credit deterioration
Explanation: B is correct: JtD captures the discontinuous default event not fully captured by continuous credit spread dynamics in ES models. A is wrong — rate jumps are in rate risk factors. C is wrong — equity earnings are in equity risk. D is wrong — FX intervention risk is in FX risk factors.
Question 297
FRTB default risk charge (DRC) applies to:
Select an option first.
Correct answer: B — Trading book positions with jump-to-default risk — non-securitisation (bonds, equities) and securitisation positions
Explanation: B is correct: DRC covers trading book credit positions: (1) non-securitisation JtD; (2) correlation trading portfolio; (3) securitisation non-correlation trading. It supplements ES-based IMA capital. A is wrong — DRC is separate from VaR/ES. C is wrong — banking book uses IRRBB. D is wrong — corporates are included.
Question 298
Mapping a call option for VaR requires knowing:
Select an option first.
Correct answer: B — Delta, gamma, vega, and the underlying's VaR and volatility — to compute first and second-order price effects and volatility sensitivity
Explanation: B is correct: delta-gamma-vega mapping requires all three Greeks plus risk factor distributions. A is wrong — intrinsic value alone is insufficient. C is wrong — K and T are inputs to compute Greeks. D is wrong — r and q affect option value but insufficient for VaR.
Question 299
Duration mapping to key rate VaR:
Select an option first.
Correct answer: B — Decomposes portfolio rate sensitivity into sensitivities to specific maturity points, then computes VaR using the covariance of key rate changes
Explanation: B is correct: key rate DV01 at each node × joint distribution of key rate changes = key rate VaR. A is wrong — one vertex misses curve risk. C is wrong — parallel shift misses twist and butterfly. D is wrong — on-the-run mapping is a different convention.
Question 300
IRRBB sensitivity analysis measures:
Select an option first.
Correct answer: B — Changes in EVE and NII under prescribed rate shock scenarios (parallel up/down, steepener, flattener, short-rate up/down)
Explanation: B is correct: BCBS 2016 IRRBB standard requires EVE and NII sensitivity under six standardised scenarios. A is wrong — VaR is for the trading book. C is wrong — credit risk uses RWA. D is wrong — that is liquidity risk.
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