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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 121
The 'translation invariance' property of a coherent risk measure states:
Select an option first.
Correct answer: B — Adding a deterministic cash amount c reduces the risk measure by exactly c: ρ(X + c) = ρ(X) − c
Explanation: B is correct: translation invariance means that adding a risk-free amount c (a certain cash inflow) reduces the risk measure by exactly c — you need c less capital to achieve safety. ρ(X + c) = ρ(X) − c. A is wrong — adding a risk-free asset reduces risk by the amount added. C is wrong — zero investment doesn't imply zero risk. D is wrong — currency translation is a different concept.
Question 122
The '99% confidence, 10-day VaR' regulatory requirement means that a bank could expect losses exceeding VaR approximately how many times per year?
Select an option first.
Correct answer: B — About 0.4 times per year (1% chance × 25 ten-day periods per year)
Explanation: B is correct: the 10-day VaR is assessed over non-overlapping 10-day windows. In a 250-trading-day year, there are approximately 25 non-overlapping 10-day periods. Expected exceedances = 1% × 25 = 0.25 per year — roughly once every 4 years. B is the closest correct answer (~0.25-0.4). A is wrong — 2.5 corresponds to 1-day VaR exceptions per year. C is wrong — 10 times corresponds to 96% 1-day VaR. D is wrong — once per year corresponds to 1-day 99.6% VaR.
Question 123
A 'factor-based' VaR model for an equity portfolio with 500 stocks uses 5 factors instead of 500 individual stock volatilities. The main benefit is:
Select an option first.
Correct answer: B — Dramatically reduced dimensionality: estimating 5 factor volatilities and their correlations (15 parameters) vs 500 stock volatilities and 124,750 pairwise correlations
Explanation: B is correct: a 5-factor model requires estimating 5×6/2 = 15 variance-covariance parameters plus individual idiosyncratic variances, versus 500×501/2 = 125,250 parameters for the full covariance matrix. The factor model is vastly more parsimonious and statistically stable. A is wrong — factor models may miss idiosyncratic risks. C is wrong — factor models are not systematically more conservative. D is wrong — model risk shifts to factor model specification.
Question 124
The 'phantom income effect' in fixed income VaR arises when:
Select an option first.
Correct answer: B — Fixed income positions show positive P&L attributed to theta (time value) while rates are unchanged — this is a feature of accrual accounting, not a risk
Explanation: B is correct: theta in fixed income represents the positive accrual of interest income as time passes. When running daily P&L attribution, the theta term is always positive for a long bond position (assuming no rate or spread changes). This 'phantom income' appears in the P&L attribution but is not a risk-related P&L — it is the carry/accrual component. A is wrong — that is a mapping issue. C is wrong — actual interest accrues. D is wrong — phantom income is about accrual, not double-counting.
Question 125
The relationship between the 'information ratio' (IR) and active VaR is:
Select an option first.
Correct answer: B — IR = active return / tracking error volatility (TEV) — while active VaR = z_α × TEV, the IR uses TEV directly as the risk denominator
Explanation: B is correct: information ratio = active return / TEV where TEV = σ(portfolio return − benchmark return). Active VaR = z_α × TEV is related: if we know IR and TEV, active VaR = z_α × TEV. The two measures use the same risk denominator (TEV) but the IR expresses return relative to risk. A is wrong — dividing by active VaR would rescale by 1/z_α. C is wrong — that is return/volatility = Sharpe ratio. D is wrong — ES/VaR is the ES gap ratio.
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Question 126
When a portfolio has exposure to jump risk (sudden large discrete price changes), standard VaR based on continuous diffusion models will:
Select an option first.
Correct answer: B — Underestimate true VaR because continuous models assign near-zero probability to sudden large moves that have non-negligible probability in reality
Explanation: B is correct: jump-diffusion models (Merton 1976, Kou 2002) add a Poisson jump process to the geometric Brownian motion. Standard diffusion-only VaR assigns essentially zero probability to sudden large gaps (e.g. a stock dropping 30% overnight on news), while empirically such events occur. VaR from jump models is therefore higher. A is wrong — diffusion models miss jump risk. C is wrong — diffusion models underestimate. D is wrong — VaR depends critically on jump assumptions.
Question 127
The 'maximum likelihood' approach to estimating VaR model parameters is preferred over 'method of moments' because:
Select an option first.
Correct answer: B — MLE provides asymptotically efficient estimates (minimum variance) and handles complex distributional assumptions systematically, while MOM is simpler but less efficient
Explanation: B is correct: MLE is asymptotically efficient (achieves the Cramér-Rao lower bound for variance) and naturally handles any specified distribution. Method of moments matches distributional moments to sample moments, which is simpler but less efficient. For GARCH estimation, MLE is standard. A is wrong — MLE is often computationally intensive. C is wrong — method of moments is consistent (not always biased). D is wrong — MLE explicitly requires a distributional assumption (likelihood function).
Question 128
For a multi-period VaR calculation under mean reversion in returns (negative autocorrelation), the T-day VaR would be:
Select an option first.
Correct answer: B — Less than √T × 1-day VaR — negative autocorrelation (mean reversion) means multi-period variance grows slower than T × daily variance
Explanation: B is correct: for a mean-reverting (negatively autocorrelated) return series, shocks partially reverse over time. The multi-period variance is: Var(r_{1:T}) = T × σ² + 2Σ_{k=1}^{T-1}(T−k)Cov(r_t, r_{t+k}). With negative autocorrelations, the covariance terms are negative, making multi-period variance < T × σ². Hence T-day VaR < √T × 1-day VaR. A is wrong — positive autocorrelation amplifies multi-period risk. C is wrong — autocorrelation affects scaling. D is wrong — linear scaling is worse than √T for positive autocorrelation.
Question 129
The 'Gamma P&L' from an options portfolio for a given day equals:
Select an option first.
Correct answer: B — ½ × Gamma × (price change)² — the second-order price sensitivity
Explanation: B is correct: the Taylor expansion P&L ≈ δΔS + ½ΓΔS² + νΔσ + θΔt. The pure gamma contribution is ½ΓΔS², which is always positive for long option positions (gamma > 0 for long calls and puts). A is wrong — that is delta P&L. C is wrong — that is vega P&L. D is wrong — that is theta P&L.
Question 130
The 'P&L explain' report would show an unexplained P&L when:
Select an option first.
Correct answer: B — There are risk factors not included in the risk model — the unexplained portion = actual P&L − sum of all Greek-attributed P&Ls, indicating model incompleteness
Explanation: B is correct: unexplained P&L (or 'P&L explain gap') arises when the risk model misses risk factors or when approximation errors accumulate. For example, if the model omits correlation risk, a correlation-sensitive position will have P&L that doesn't match the delta/gamma/vega attribution. A is wrong — perfect capture gives zero unexplained. C is wrong — profits don't create unexplained P&L. D is wrong — exact BS would mean zero unexplained for vanilla options.
Question 131
For a long straddle position (long call + long put at the same strike), the VaR primarily arises from:
Select an option first.
Correct answer: B — Vega risk — the position loses value if implied volatility falls, regardless of whether prices rise or fall
Explanation: B is correct: a straddle is delta-neutral (approximately) and has positive gamma and vega. The primary VaR driver is vega — if IV falls significantly, the straddle loses value even if prices don't move. At-the-money options have the highest vega. The straddle also pays theta (loses from time decay), but the VaR is primarily from adverse vega moves. A is wrong — the straddle is approximately delta-neutral. C is wrong — positive gamma means large moves are beneficial. D is wrong — theta is a cost, not a VaR source.
Question 132
The 'benchmark VaR exceedance probability' test checks:
Select an option first.
Correct answer: B — Whether the exception rate conditional on previous exceptions is consistent with the model — testing if consecutive exceptions cluster more than expected
Explanation: B is correct: this test (related to Christoffersen's conditional coverage) checks for independence of exceptions. If the probability of an exception tomorrow is higher given an exception today, the model fails to capture volatility clustering. A is wrong — unconditional count testing is Kupiec. C is wrong — benchmark comparison is different. D is wrong — P&L benchmarking is a performance measure.
Question 133
A 'VaR exceedance cluster' in backtesting indicates:
Select an option first.
Correct answer: B — Model failure — consecutive exceptions suggest the model doesn't capture current market regime (high volatility period) properly
Explanation: B is correct: under a correctly specified VaR model, exceptions should be randomly distributed over time (independent). A cluster of consecutive exceptions indicates the model is not updating rapidly enough to reflect current high-volatility conditions — GARCH or filtered historical simulation would help. A is wrong — clustering indicates model failure. C is wrong — seasonal patterns would be spread throughout the year. D is wrong — clustering is systematically related to volatility regimes.
Question 134
When implementing Monte Carlo VaR for a portfolio with 10,000 options, the 'speed-accuracy trade-off' is managed by:
Select an option first.
Correct answer: B — Using importance sampling or variance reduction techniques (antithetic variables, stratified sampling) to improve accuracy for the same number of simulations, or using delta approximations for less important positions
Explanation: B is correct: variance reduction techniques improve MC efficiency: (1) antithetic variables: use −z when z is a random draw; (2) stratified sampling: ensure all regions of the distribution are sampled; (3) importance sampling: oversample the tail. These improve the accuracy of tail quantile estimates for the same simulation count. A is wrong — 100 simulations is insufficient. C is wrong — analytical formulas are not available for all exotic options. D is wrong — ignoring non-linear positions creates large errors.
Question 135
The 'parametric VaR' for a portfolio of bonds assumes which distribution for daily yield changes?
Select an option first.
Correct answer: B — Normal distribution — allowing parametric computation of portfolio P&L VaR from yield change volatility, DV01, and covariance matrix
Explanation: B is correct: the standard parametric approach assumes yield changes are normally distributed. P&L ≈ DV01 × Δy, and with Δy ~ N(0, σ²_y), P&L ~ N(0, DV01² × σ²_y). The parametric bond VaR = z_α × DV01 × σ_yield. A is wrong — lognormal applies to prices, not yield changes (which can be negative). C is wrong — t-distribution gives a non-parametric extension. D is wrong — uniform distribution is not used.
Question 136
The 'Filtered Historical Simulation' (FHS) advantage over plain historical simulation is that it:
Select an option first.
Correct answer: B — Adjusts historical scenarios for the current volatility regime — scaling old scenarios by current GARCH volatility / historical volatility, making them relevant to current market conditions
Explanation: B is correct: FHS scales each historical innovation (return/GARCH vol) by the ratio of current GARCH forecast to the historical GARCH volatility when the observation occurred. This means scenarios from a calm historical period are amplified if current conditions are volatile, and vice versa. A is wrong — same window, better scaling. C is wrong — no random simulation in FHS. D is wrong — backtesting is still required.
Question 137
When a portfolio manager claims their portfolio has zero VaR, this is possible when:
Select an option first.
Correct answer: B — The portfolio consists entirely of cash or risk-free assets, or when the portfolio is perfectly hedged with all risk factor sensitivities equal to zero
Explanation: B is correct: VaR = 0 if and only if the portfolio has no uncertain P&L — either fully in risk-free assets or perfectly hedged across all risk dimensions. A is wrong — zero return ≠ zero risk. C is wrong — low leverage reduces but doesn't eliminate VaR. D is wrong — diversification reduces VaR but doesn't make it zero unless all systematic risk is hedged.
Question 138
The 'economic capital' versus 'regulatory capital' distinction means that a well-managed bank:
Select an option first.
Correct answer: B — May hold more economic capital internally estimated at higher confidence levels than required regulatory capital — especially for risks not fully captured by the regulatory framework
Explanation: B is correct: regulatory capital is a standardised minimum floor. Economic capital is the bank's own assessment of capital needed to cover all risks at a high confidence level (e.g. 99.97% for one-year AA credit rating equivalence). Economic capital is often used for RAROC, desk limits, and strategic planning. A is wrong — economic capital often exceeds regulatory. C is wrong — regulatory capital is a binding constraint. D is wrong — economic capital is more appropriate for internal decisions.
Question 139
In the 'risk-neutral pricing' framework, VaR computation requires the:
Select an option first.
Correct answer: B — Physical (real-world) probability measure P — because VaR measures actual loss probabilities, not option-pricing probabilities
Explanation: B is correct: VaR is a measure of actual (real-world) loss probabilities. It must be computed under the physical measure P. Using the risk-neutral measure Q (used for option pricing) would give wrong probabilities since Q removes the equity risk premium. A is wrong — Q gives risk-neutral probabilities for pricing, not real-world loss distribution. C is wrong — averaging is not appropriate. D is wrong — forward measure is for specific derivative types.
Question 140
A risk manager backtests a 95% 1-day VaR model over 500 days and observes 40 exceptions. The expected number is 25. The model:
Select an option first.
Correct answer: B — Shows significant underestimation — 40/500 = 8% exception rate vs expected 5%, which is (40-25)/√(500×0.05×0.95) ≈ 15/4.87 ≈ 3.1 standard deviations above expected
Explanation: B is correct: standard deviation of exception count = √(n×p×(1−p)) = √(500×0.05×0.95) = √23.75 ≈ 4.87. Observed excess = 40−25 = 15. Z-score = 15/4.87 ≈ 3.08 — highly significant (p < 0.002). The model systematically underestimates risk. A is wrong — 40 is 3 standard deviations above expected. C is wrong — too many exceptions indicates underestimation (VaR too low). D is wrong — 500 observations is sufficient for testing.
Question 141
The 'portfolio effect' in multi-asset VaR means that:
Select an option first.
Correct answer: B — Portfolio VaR < sum of standalone VaRs when assets have imperfect correlations — diversification reduces the combined risk
Explanation: B is correct: the portfolio effect (diversification benefit) = Σ VaR_i − portfolio VaR > 0 when correlations < 1. This is why banks report both standalone VaR (sum of parts) and portfolio VaR (accounting for diversification). A is wrong — sum applies only when ρ=1 for all pairs. C is wrong — assets have different individual VaRs. D is wrong — more assets typically reduces VaR through diversification.
Question 142
When implementing historical simulation for a global portfolio, 'synchronisation' is needed because:
Select an option first.
Correct answer: B — Markets in different time zones close at different local times — to compute portfolio P&L for a given day, the risk manager needs all prices from the same global close (or uses overlapping returns for different time zones)
Explanation: B is correct: a portfolio of US, European, and Asian securities has close times spanning 24 hours. Using today's Tokyo close with today's New York close would mix prices from different points in global market time. Common solutions: use each market's previous close for the New York close, or use globally synchronised end-of-day prices from a data vendor. A is wrong — currency is handled separately. C is wrong — frequency is a data availability issue. D is wrong — volatility doesn't affect synchronisation.
Question 143
The 'expected loss (EL)' differs from 'unexpected loss (UL)' in credit risk management in that:
Select an option first.
Correct answer: B — EL = PD × LGD × EAD is the statistical mean loss (priced into loan spreads), while UL = standard deviation of losses (or quantile minus EL) requires capital allocation to absorb
Explanation: B is correct: expected loss is the statistical average loss, priced into loan rates/spreads and covered by provisioning. Unexpected loss is the deviation above EL that occurs in bad years and requires capital. Economic capital covers unexpected losses at a high confidence level: EC = VaR(confidence) − EL. A is wrong — they are distinct. C is wrong — EL is provisioned (not capital); UL requires capital. D is wrong — by definition EL ≤ 99th percentile loss (economic capital includes EL + UL).
Question 144
The '1% worst-case scenario' in a VaR model for a 250-day backtest corresponds to:
Select an option first.
Correct answer: B — Approximately the 2nd or 3rd worst observation in the sorted P&L series
Explanation: B is correct: 99% VaR over 250 days: 1% tail = 2.5 observations. The VaR is typically the 2nd or 3rd worst loss from the 250-day history (interpolating between the 2nd and 3rd worst). A is wrong — 1 observation would be the 99.6th percentile (1/250). C is wrong — 12th observation is the 95.2th percentile (12/250 ≈ 4.8% tail). D is wrong — average of 5 worst would be ~97.5% ES.
Question 145
The 'first-order condition' for minimising portfolio VaR with respect to portfolio weights requires:
Select an option first.
Correct answer: B — The marginal VaR of each asset to be equal — in the optimal minimum-VaR portfolio, all assets contribute equally at the margin
Explanation: B is correct: at the minimum-variance (minimum-VaR for normal distributions) portfolio, the first-order condition is that the marginal contribution of each asset to portfolio variance is equal for all assets with non-zero weights. This is analogous to the equal marginal utility condition in consumer theory. A is wrong — the minimum-VaR portfolio doesn't necessarily have the highest return. C is wrong — individual VaR can be non-zero. D is wrong — zero correlation would allow further reduction.
Question 146
The 'adverse scenario' for a long equity/short bond portfolio (a risk-on position) would be:
Select an option first.
Correct answer: B — Equity prices fall AND bond yields rise simultaneously — both legs lose: equity falls hurt the long, rate rises hurt the short bond position (bonds fall in price)
Explanation: B is correct: a risk-on long equity/short bond position (common in risk-seeking environments) would be adversely affected by: (1) equity market sell-off (long equity position loses); (2) rate rises (bond prices fall, so short bond position also loses since the short seller must return appreciated bonds). This is a 'double-hit' scenario. A is wrong — this scenario would help the position. C and D are wrong — only one leg losing.
Question 147
The 'Spectral Risk Measure' generalisation allows different risk weights for different parts of the loss distribution. A risk-averse investor would specify weights that:
Select an option first.
Correct answer: B — Increase as the quantile level increases — assigning more weight to extreme losses than small losses, reflecting stronger aversion to catastrophic outcomes
Explanation: B is correct: a risk-averse spectral risk measure uses an increasing (non-decreasing) weighting function φ(p) — assigning progressively more weight to worse outcomes. VaR uses a weight of 1 at p=α and 0 elsewhere. ES uses uniform weight above α. More risk-averse measures use increasing weights in the tail. A is wrong — uniform weights give ES. C is wrong — decreasing weights would reward bad outcomes. D is wrong — zero below threshold gives VaR.
Question 148
A trader with a delta-hedged options book faces which primary risk when the underlying moves sharply?
Select an option first.
Correct answer: B — Gamma risk — the delta hedge becomes imperfect for large moves, and if the trader is short gamma (sold options), large moves cause losses
Explanation: B is correct: a short gamma position (from selling options) has a concave P&L profile with respect to the underlying. When the underlying moves significantly, the P&L loss from the adverse gamma exposure exceeds the premium received. The delta hedge neutralises the linear risk but not the convexity (gamma) risk. A is wrong — delta neutrality hedges only first-order risk. C is wrong — theta is time decay. D is wrong — vega is the vol sensitivity.
Question 149
The 'portfolio insurance' strategy using options involves:
Select an option first.
Correct answer: B — Buying put options or dynamically replicating a put (delta-hedging) to protect a stock portfolio against downside losses while retaining upside potential
Explanation: B is correct: portfolio insurance (Leland-Rubinstein 1976) involves buying put options on the portfolio as a floor protection. If puts are expensive, dynamic replication (CPPI or delta hedging synthetic puts) can achieve the same effect. The strategy caps downside while preserving upside (at the cost of option premiums). A is wrong — selling puts increases downside risk. C is wrong — only downside is limited. D is wrong — futures eliminate upside too.
Question 150
The 'risk management cycle' in a trading firm involves all of the following EXCEPT:
Select an option first.
Correct answer: B — Guaranteeing no losses in any given day
Explanation: B is correct: risk management reduces and controls risk but cannot eliminate it. No risk management system can guarantee daily profitability. The risk management cycle includes: identification, measurement, monitoring, reporting, and mitigation — but not guaranteeing zero losses. A, C, D are all genuine components of risk management. B is not part of risk management.
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