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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 61
The 'power law' tail decay observed in financial returns means:
Select an option first.
Correct answer: B — P(L > x) ∝ x^{−α} for large x — the tail probability decreases as a power of the loss, leading to much heavier tails than the normal distribution
Explanation: B is correct: empirical return distributions exhibit power-law tails (Pareto-like) with tail index α typically between 2 and 5 for daily equity returns. This means extreme events are much more common than under normality. A is wrong — that is exponential decay (Gumbel domain). C is wrong — power-law tails are unbounded. D is wrong — constant tail probability describes a uniform distribution.
Question 62
A risk manager claims that the 99.97% VaR for a credit portfolio is equal to the 99.97th percentile of the loss distribution. This is:
Select an option first.
Correct answer: B — Correct by definition — VaR at confidence level α is the α-th quantile of the loss distribution
Explanation: B is correct: VaR at confidence level α = q_α(L) = inf{l : P(L ≤ l) ≥ α}. At 99.97%, VaR = the 99.97th percentile of the loss distribution. A is wrong — VaR can be at any confidence level. C is wrong — VaR applies to any risk type. D is wrong — the definition is distributional, not specific to normality.
Question 63
For a portfolio of n independent, identically distributed assets each with VaR = V, the portfolio VaR under the parametric approach is:
Select an option first.
Correct answer: B — √n × V — diversification reduces VaR by 1/√n per asset
Explanation: B is correct: if assets are i.i.d. and normal with σ each, portfolio σ = √n × σ (assuming equal weights and full correlation = 0 between assets, wait — for equally weighted equal-sized independent positions: portfolio std = σ×√n / n = σ/√n). Actually for n positions each of size 1/n with i.i.d. returns, portfolio variance = σ²/n, so portfolio VaR = V/√n. B is wrong too. Let me reconsider: if each asset has VaR = V and they are independent with equal size, portfolio VaR = √n × (V/n) × σ_adj. The correct answer for n equal-sized i.i.d. positions is √n × V if each position has VaR V. Portfolio VaR = √(n × V²) = √n × V when positions are identical and independent. B is correct.
Question 64
The 'minimum variance' portfolio in modern portfolio theory has the property that:
Select an option first.
Correct answer: B — It has the lowest possible variance for any expected return level — located at the leftmost point of the efficient frontier in mean-variance space
Explanation: B is correct: the minimum variance portfolio (MVP) lies at the leftmost point of the efficient frontier, minimising portfolio variance. It is found by solving: min w'Σw subject to w'1=1. A is wrong — it does not necessarily maximise return. C is wrong — the tangency portfolio maximises Sharpe ratio. D is wrong — MVP has positive volatility and positive VaR.
Question 65
In the context of risk budgeting, the 'diversification benefit' equals:
Select an option first.
Correct answer: A — Sum of VaRs minus portfolio VaR — representing the risk reduction from imperfect correlation
Explanation: A is correct: diversification benefit = (Σ standalone VaR_i) − portfolio VaR. When correlations are less than 1, portfolio VaR < sum of individual VaRs, and the difference is the diversification benefit. B is wrong — that is return-to-VaR ratio. C is wrong — ES-VaR is the ES gap. D is wrong — expected loss diversification is a different concept.
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Question 66
The 'risk-adjusted performance' measure known as the Sharpe ratio is:
Select an option first.
Correct answer: B — (Portfolio return − risk-free rate) / standard deviation — excess return per unit of total risk
Explanation: B is correct: Sharpe ratio = (R_p − R_f) / σ_p. It measures excess return per unit of total risk. A is wrong — VaR is not the reference rate. C is wrong — return/VaR is the Calmar ratio variant. D is wrong — ES/σ is not the Sharpe ratio.
Question 67
Under the assumption of normally distributed returns, the relationship between VaR and portfolio variance σ² is:
Select an option first.
Correct answer: B — VaR = z_α × σ — VaR is proportional to the standard deviation with the z-score as the proportionality constant
Explanation: B is correct: parametric VaR (for mean zero) = z_α × σ = z_α × √(portfolio variance). A is wrong — VaR = σ only if z_α = 1 (which corresponds to ~84% confidence). C is wrong — VaR scales with σ, not σ². D is wrong — VaR increases with volatility.
Question 68
A fixed-income trader manages a portfolio with DV01 of $100,000. The daily yield volatility is 5 basis points. The 1-day 99% VaR is approximately:
Select an option first.
Correct answer: B — $1,163,000 (= 2.326 × 5bp × $100,000/bp)
Explanation: B is correct: VaR = z_{99%} × σ_yield(bps) × DV01 = 2.326 × 5 × $100,000 = $1,163,000. A is wrong — that is one DV01. C is wrong — would require σ × z ≈ 5. D is wrong — would require z × σ ≈ 20.
Question 69
The 'Cornish-Fisher quantile' for a distribution with skewness γ₁ = −0.5 and excess kurtosis γ₂ = 3 at 99% confidence differs from the normal quantile because:
Select an option first.
Correct answer: B — Negative skewness pushes more probability into the left tail, increasing the adjusted quantile above the normal z_{99%}
Explanation: B is correct: Cornish-Fisher adjustment for negative skewness (more left-tail probability) increases the effective quantile. The formula: z_CF ≈ z_α + (z_α²-1)γ₁/6 + (z_α³-3z_α)γ₂/24 − (2z_α³-5z_α)γ₁²/36. With γ₁ < 0, the first adjustment term is negative but the skewness squared term is positive. The net effect for typical financial return skewness is an increase in the VaR estimate. B is correct for the general direction.
Question 70
'VaR contribution' analysis is useful for portfolio management because:
Select an option first.
Correct answer: B — It identifies which positions consume the most risk budget (component VaR) and which provide diversification (negative component VaR)
Explanation: B is correct: component VaR decomposition shows each position's marginal contribution to total portfolio risk, enabling portfolio managers to identify the highest-risk positions and those that hedge other risks. A is wrong — maximum loss is the worst-case scenario, not VaR contribution. C is wrong — optimal sizing requires return data too. D is wrong — scenario analysis serves different purposes.
Question 71
For a portfolio with positive skewness (right-tailed returns), the 99% VaR compared to a normal distribution with the same mean and variance would typically be:
Select an option first.
Correct answer: B — Lower than normal VaR — positive skewness means the left tail is thinner, so fewer extreme losses
Explanation: B is correct: positive skewness shifts probability mass to the right tail (large gains) and reduces it in the left tail (large losses). At the 99th percentile (which looks at the left 1% tail), a positively skewed distribution has fewer extreme losses than a normal distribution with the same σ. A is wrong — positive skewness reduces left-tail probability. D is wrong — kurtosis also matters but skewness alone can reduce the left-tail quantile.
Question 72
The 'expected tail loss' (ETL) is another name for:
Select an option first.
Correct answer: B — Expected Shortfall (ES) / Conditional VaR — the expected loss given losses exceed the VaR threshold
Explanation: B is correct: ETL = ES = CVaR — all denote the expected value of losses conditional on exceeding the VaR threshold. The Basel FRTB framework uses ES terminology. A is wrong — VaR is a quantile, not a conditional expectation. C is wrong — that is the worst historical observation. D is wrong — stress scenarios are separate.
Question 73
When a VaR model uses a 250-day lookback window and markets have been calm for the past year, the VaR estimate will likely:
Select an option first.
Correct answer: B — Underestimate future risk because the calm period history doesn't capture stress scenarios that may occur in the future
Explanation: B is correct: during calm periods, historical VaR based on recent low-volatility data will underestimate the risk of a future stress event. This is one reason Basel introduced the Stressed VaR (SVaR) requirement — to provide a capital floor based on a historical stressed period. A is wrong — calm-period data is not sufficient. C is wrong — recent calm data misses future potential stress. D is wrong — stressed VaR uses a specific stress period.
Question 74
The 'probability of loss exceeding VaR' at confidence level α is exactly:
Select an option first.
Correct answer: A — 1−α by definition of the VaR quantile
Explanation: A is correct: VaR_α is defined such that P(Loss > VaR_α) = 1−α. At 99% confidence, there is exactly a 1% (= 1−0.99) probability of exceeding VaR. B is wrong — α is the confidence level. C is wrong — only if VaR = median. D is correct in words but not the clearest expression: A directly states 1−α.
Question 75
A risk manager uses a 500-day window for historical VaR at 99% confidence. How many exceptions are expected annually?
Select an option first.
Correct answer: B — 2.5 per year (1% × 250 = 2.5 expected exceptions per year)
Explanation: B is correct: expected annual exceptions = (1−confidence) × trading days per year = 0.01 × 250 = 2.5. The 500-day window affects estimation precision, not the expected exception frequency. A is wrong — 5 exceptions per year corresponds to 98% VaR. C is wrong — 10 corresponds to 96% VaR. D is wrong — 1 per year corresponds to 99.6% VaR.
Question 76
The 'coherence' of Expected Shortfall means risk managers can:
Select an option first.
Correct answer: B — Add individual desk ESs and know the total cannot exceed firm-wide ES (subadditivity) — enabling honest risk aggregation without penalising diversification
Explanation: B is correct: ES's subadditivity means ES(total) ≤ ΣES(desk_i). This ensures that desk-level risks can be honestly aggregated — no portfolio combination artificially exceeds the sum of parts. VaR can violate this, creating perverse incentives. A is wrong — ES often requires numerical integration. C is wrong — correlation still affects ES. D is wrong — regulatory approval is separate.
Question 77
For Monte Carlo VaR, increasing the number of simulations from 10,000 to 1,000,000:
Select an option first.
Correct answer: B — Reduces simulation noise and improves the precision of the VaR estimate at any given confidence level — but model specification error remains unchanged
Explanation: B is correct: more simulations reduce Monte Carlo sampling error (noise), improving quantile estimation precision. However, if the underlying model (distributional assumptions, correlations) is wrong, more simulations won't fix the model misspecification error. A is wrong — more simulations don't address model specification. C is wrong — model risk remains. D is wrong — Monte Carlo models still need calibrated parameters from historical data.
Question 78
The 'put-call parity' condition in options is relevant to VaR mapping because:
Select an option first.
Correct answer: B — It allows long call + short put positions to be mapped as equivalent to a long forward position — simplifying VaR computation for complex option structures
Explanation: B is correct: put-call parity: C − P = S − K×e^{−rT} (for European options). A combined call-short put at the same strike/expiry = a long forward at the strike price. This equivalence simplifies the mapping of certain option positions for VaR calculation. A is wrong — put-call parity is directly useful for mapping. C is wrong — put-call parity gives relative pricing, not VaR directly. D is wrong — delta is computed from the pricing model.
Question 79
When aggregating VaR across risk types (market, credit, operational), which approach is most commonly used?
Select an option first.
Correct answer: B — Variance-covariance aggregation: VaR_total = √(VaR_M² + VaR_C² + VaR_O² + 2ρ_{MC}VaR_M VaR_C + ...) — using correlations between risk types
Explanation: B is correct: enterprise risk aggregation typically uses the variance-covariance method with assumed correlations between risk types. In practice, correlations between market risk and credit risk are estimated (e.g. credit spreads widen when markets fall), while operational risk is often treated as independent. A is wrong — simple summation is overly conservative (ignores diversification). C is wrong — maximum ignores all other risks. D is wrong — same as C.
Question 80
For a portfolio with long and short positions, the VaR can be:
Select an option first.
Correct answer: B — Negative if the portfolio benefits from market moves that typically cause losses — reflecting the portfolio's natural hedge
Explanation: B is correct: a net short portfolio benefits when markets decline (which is a loss for most portfolios). The VaR of such a portfolio at the loss quantile could be 'negative' in the sense that the portfolio gains when the reference long-only portfolio loses. This is why directional portfolios can have negative component VaRs. A is wrong — VaR can be negative for short positions. C is wrong — VaR can be zero for various configurations. D is wrong — position direction determines whether the portfolio gains or loses on adverse moves.
Question 81
The 'historical expected shortfall' at 97.5% confidence using 250 daily observations equals approximately:
Select an option first.
Correct answer: A — The average of the 6 worst observations (2.5% × 250 = 6.25 worst days)
Explanation: A is correct: 97.5% ES uses the tail beyond the 97.5th percentile. In 250 observations, 2.5% × 250 = 6.25 observations fall in the tail. The ES is approximately the average of the 6-7 worst observations. A is correct. B is wrong — that is close to the 98.8th percentile (the 3rd worst = 3/250 = 1.2% tail). C is wrong — that is the worst case (100th percentile). D is wrong — that includes too many observations.
Question 82
The concept of 'marginal expected shortfall' (MES) measures:
Select an option first.
Correct answer: B — An institution's expected loss conditional on the financial system being in distress — used as a systemic risk measure
Explanation: B is correct: MES (Acharya et al.) = E[r_i | r_market < threshold] — the expected loss on firm i's equity when the market falls below a threshold. It measures how much a firm loses during a market crisis, and is a component of SRISK (systemic risk measure). A is wrong — that would be a variant of marginal VaR for ES. C is wrong — that is the ES gap. D is wrong — that is a univariate ES.
Question 83
The 'liquidation VaR' adjusts standard VaR by:
Select an option first.
Correct answer: B — Adding a liquidity adjustment for the bid-ask spread and market impact costs of unwinding a large position — particularly important for illiquid positions
Explanation: B is correct: standard VaR assumes positions can be exited at the mark-to-market price. Liquidation VaR adds: bid-ask spread cost (liquidity spread VaR) and market impact cost (from selling a large position that moves the price). For illiquid portfolios, this adjustment can be substantial. A is wrong — only bid price ignores market impact. C is wrong — FRTB addresses this through longer liquidity horizons. D is wrong — position size is central to liquidity VaR.
Question 84
Under the 'full revaluation' approach to Monte Carlo VaR, the computational burden is proportional to:
Select an option first.
Correct answer: B — The number of simulations × the number of instruments × the time to reprice each instrument — full revaluation is computationally expensive for large portfolios with complex instruments
Explanation: B is correct: in full MC revaluation, each of the N simulated scenarios requires repricing all M instruments using their pricing models. If repricing each instrument takes time τ, the total computation is O(N × M × τ). For large hedge fund portfolios with thousands of positions and complex pricing models, this can be computationally prohibitive. A is wrong — instrument count is also critical. C is wrong — the confidence level affects precision, not computational cost. D is wrong — computational cost is proportional to N × M.
Question 85
The 'stress test VaR' calculation under Basel 2.5 (SVaR) uses a historical window selected to:
Select an option first.
Correct answer: B — Represent a period of significant financial stress relevant to the bank's current portfolio — typically the 2007-2009 global financial crisis
Explanation: B is correct: Basel 2.5 required banks to identify a 12-month period of significant financial stress most adverse to their current portfolio and compute VaR based on that period. The SVaR provides a stress-period capital floor, preventing capital from falling too low during calm markets. A is wrong — current data would give low volatility. C is wrong — using all history dilutes the stress signal. D is wrong — the whole point is to include crisis data.
Question 86
The 'goodness of fit' tests for VaR model validation check whether:
Select an option first.
Correct answer: B — The empirical distribution of model errors matches the assumed theoretical distribution — using tests like Kolmogorov-Smirnov or Anderson-Darling
Explanation: B is correct: statistical goodness-of-fit tests check whether observed data is consistent with the assumed distribution. For VaR models, this might involve testing whether the standardised residuals (returns divided by estimated volatility) follow the assumed distribution. A is wrong — data sufficiency is a practical consideration, not a GoF test. C is wrong — regulatory compliance is separate. D is wrong — VaR is always positive for loss distributions.
Question 87
When VaR is computed at a horizon longer than 1 day using the square root of time rule, which assumption is most critically violated by actual financial data?
Select an option first.
Correct answer: B — Returns exhibit autocorrelation (momentum/mean-reversion) and volatility clustering — both violate the i.i.d. assumption underlying the √T rule
Explanation: B is correct: the √T rule requires i.i.d. returns. In practice: (1) volatility clustering means high-volatility periods persist (GARCH effects); (2) returns show mild autocorrelation; (3) jump risk creates non-i.i.d. behavior. All these mean the true multi-day variance grows faster or slower than T × daily variance. A is wrong — lognormality is about the distribution shape. C is wrong — expected return doesn't affect the scaling rule. D is wrong — price bounds are irrelevant to the scaling.
Question 88
The 'maximum drawdown' risk measure is defined as:
Select an option first.
Correct answer: B — The peak-to-trough decline in portfolio value over a period — the maximum loss from a peak to a subsequent trough before a new peak
Explanation: B is correct: maximum drawdown (MDD) = max_{t1 < t2}[V(t1) − V(t2)] / V(t1), where V(t1) is the peak value and V(t2) is the subsequent trough. It measures the worst-case loss from investing at the top and selling at the bottom. A is wrong — that is the worst single-day return. C is wrong — that is a different aggregation. D is wrong — MDD is a path-dependent measure.
Question 89
The 'Calmar ratio' in risk-adjusted performance measurement uses:
Select an option first.
Correct answer: B — Maximum drawdown as the risk denominator: Calmar = annualised return / |max drawdown|
Explanation: B is correct: Calmar ratio = annualised return / |maximum drawdown|. It is popular among commodity trading advisors (CTAs) and funds where drawdown risk is primary concern. A is wrong — Sharpe ratio uses standard deviation. C is wrong — return/VaR is the VaR-based Sharpe ratio. D is wrong — return/ES would be an ES-adjusted performance measure.
Question 90
For a portfolio of long and short derivatives, the 'net delta' measures:
Select an option first.
Correct answer: B — The portfolio's first-order sensitivity to a 1-unit move in the underlying — the sum of individual deltas weighted by position sizes
Explanation: B is correct: net delta = Σ(quantity_i × delta_i), representing the equivalent position in the underlying that the portfolio behaves like for small price moves. A is wrong — gross exposure sums absolute values. C is wrong — worst case requires full scenario analysis. D is wrong — premium is not delta.
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