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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 91
The 'Greeks ladder' approach to VaR mapping for options books:
Select an option first.
Correct answer: B — Computes P&L attribution at each portfolio level for each Greek (delta, gamma, vega, theta, rho) to understand risk contribution from each sensitivity
Explanation: B is correct: a Greeks ladder breaks down option book risk systematically by each Greek: delta-VaR (price moves), gamma-VaR (convexity), vega-VaR (vol moves), theta (time decay), rho (rate moves). This gives a complete picture of option book risk. A is wrong — delta alone ignores other risks. C is wrong — Greeks are not walk parameters. D is wrong — alphabetical is not the criterion.
Question 92
A delta-neutral options portfolio has which VaR property?
Select an option first.
Correct answer: B — VaR is primarily from gamma (second-order price moves) and vega (volatility changes) — first-order price risk is hedged but higher-order risks remain
Explanation: B is correct: delta-neutrality eliminates linear (first-order) price risk. But gamma risk (from ΔS² terms) and vega risk (from Δσ terms) remain significant. A delta-neutral straddle, for example, still has large VaR from large price moves (gamma) or volatility changes (vega). A is wrong — VaR is non-zero. C is wrong — delta risk is hedged. D is wrong — theta is not the dominant VaR component.
Question 93
In the parametric approach, portfolio VaR for a single asset position is proportional to:
Select an option first.
Correct answer: B — The position size times the asset's VaR per unit — VaR scales linearly with position size for a single asset
Explanation: B is correct: for a single asset, VaR = z_α × σ × position_size. Doubling the position doubles the VaR. For multi-asset portfolios, the relationship is non-linear due to correlation effects. A is wrong — square root applies to number of positions. C is wrong — log scaling is not used. D is wrong — squared scaling is not used.
Question 94
The 'benchmark VaR' compares portfolio VaR to the VaR of a reference benchmark. If portfolio VaR < benchmark VaR:
Select an option first.
Correct answer: B — The portfolio has less absolute market risk than the benchmark — though relative risk (tracking error) may still be significant
Explanation: B is correct: portfolio VaR < benchmark VaR means the portfolio has lower expected tail losses. However, if the portfolio's composition differs from the benchmark, relative (active) VaR (tracking error) may be high. A is wrong — VaR < benchmark VaR doesn't mean riskless. C is wrong — VaR is not a performance measure. D is wrong — idiosyncratic risk is not determined by absolute VaR level.
Question 95
Parametric VaR for a multi-currency portfolio requires:
Select an option first.
Correct answer: B — A joint covariance matrix of returns in the base currency — capturing both individual asset volatilities and cross-asset/cross-currency correlations
Explanation: B is correct: multi-currency portfolio VaR uses the joint covariance matrix of all risk factors in the base currency. This includes: equity volatilities, FX volatilities, interest rate volatilities, and all pairwise correlations. A is wrong — separate currency VaRs miss cross-currency correlations. C is wrong — largest alone ignores diversification. D is wrong — same issue as A.
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Question 96
The 'conditional value at risk' (CVaR) is identical to:
Select an option first.
Correct answer: B — Expected Shortfall — both measure the expected loss given the loss exceeds the VaR threshold
Explanation: B is correct: CVaR = ES = ETL = TailVaR — these are all names for the same measure: E[L | L > VaR_α]. A is wrong — VaR is the threshold, not the conditional expectation. C is wrong — maximum loss is the worst case. D is wrong — VaR minus EL is not a standard measure.
Question 97
For a portfolio consisting only of long positions in a stock index, the 1-day 95% VaR at $1 million portfolio value with 1.5% daily volatility is:
Select an option first.
Correct answer: B — $24,675
Explanation: B is correct: VaR = z_{0.95} × σ × portfolio value = 1.645 × 0.015 × $1,000,000 = $24,675. A is wrong — $14,800 would require z × σ ≈ 1.48%. C is wrong — $30,000 requires 3% = 2×σ. D is wrong — $10,000 requires σ alone without z.
Question 98
The 'estimation window' for VaR determines:
Select an option first.
Correct answer: B — How much historical data is used to estimate volatilities, correlations, and the distribution — longer windows capture more regimes but may include irrelevant historical periods
Explanation: B is correct: the estimation window (lookback period) balances: stability (longer window → more stable estimates) versus responsiveness (shorter window → more responsive to current conditions). Basel minimum is 250 trading days. A is wrong — confidence level is chosen separately. C is wrong — holding period is 1 or 10 days. D is wrong — simulation count is a separate parameter.
Question 99
The 'delta-gamma-vega' P&L approximation for an option is:
Select an option first.
Correct answer: B — ΔV ≈ δΔS + ½ΓΔS² + νΔσ — first-order price, second-order price, and first-order volatility effects
Explanation: B is correct: the Taylor expansion of option P&L includes delta (first-order price), gamma (second-order price), and vega (first-order vol) terms. Higher-order terms (theta×Δt, rho×Δr) can also be added. A is wrong — delta only gives linear approximation. C is wrong — omits gamma term. D is wrong — omits delta term.
Question 100
When VaR assumptions are violated in practice, the most dangerous violations are those that:
Select an option first.
Correct answer: B — Cause VaR to systematically underestimate tail risk — heavy tails, skewness, and crisis-era correlation spikes all lead to larger losses than models predict
Explanation: B is correct: systematic underestimation of tail risk (fat tails, correlation breakdown, model mis-specification) means banks hold insufficient capital precisely when losses are largest. This procyclicality was a key lesson of the 2008 financial crisis. A is wrong — overestimation leads to excess capital. C is wrong — model outperformance is not a dangerous violation. D is wrong — computational load is an operational concern.
Question 101
The 'risk neutral' measure (Q) versus 'physical' measure (P) distinction matters for VaR because:
Select an option first.
Correct answer: B — Physical measure (P) reflects actual historical probabilities and is used for VaR/ES; risk-neutral measure (Q) reflects option-implied probabilities and is used for derivatives pricing — mixing them gives incorrect results
Explanation: B is correct: VaR and ES are statistical measures of actual loss distributions, using the physical (real-world) probability measure P. Option pricing uses the risk-neutral measure Q (no risk premium). Using implied volatility (Q-measure) directly as the P-measure volatility in VaR would conflate the two measures. A is wrong — they give different results. C is wrong — Q-measure VaR is not necessarily higher. D is wrong — P-measure applies to all asset classes.
Question 102
A credit portfolio's loss distribution is typically:
Select an option first.
Correct answer: B — Highly skewed with a heavy right tail — most years have small losses but occasionally extreme losses occur when multiple defaults happen simultaneously
Explanation: B is correct: credit loss distributions are characterised by: (1) most outcomes near zero (defaults are rare); (2) a few catastrophic outcomes (multiple simultaneous defaults in recessions); (3) strong right skewness; (4) heavy tail beyond the 99th percentile. This differs fundamentally from market risk P&L distributions which are more symmetric. A is wrong — credit losses are not symmetric. C is wrong — not uniform. D is wrong — lognormal is for asset prices.
Question 103
In the context of VaR, the 'homogeneity' property of a coherent risk measure means:
Select an option first.
Correct answer: B — Scaling a position by factor λ scales the risk measure by the same factor λ — consistent with proportional risk
Explanation: B is correct: positive homogeneity: ρ(λX) = λρ(X) for λ > 0. This means doubling a position doubles the risk measure. Both VaR and ES satisfy this property. A is wrong — that is a weaker statement (zero position → zero risk). C is wrong — the opposite: larger positions have more risk. D is wrong — additivity is subadditivity's stronger form (only satisfied when correlation = 1).
Question 104
The 'monotonicity' property of a coherent risk measure states:
Select an option first.
Correct answer: A — If asset X always performs better than asset Y in all scenarios, then ρ(X) ≤ ρ(Y) — higher-quality assets have lower or equal risk
Explanation: A is correct: monotonicity: if X ≥ Y in all scenarios (X always provides at least as much as Y), then ρ(X) ≤ ρ(Y). This captures the intuition that a stochastically superior asset should have lower risk. B is wrong — that is positive homogeneity. C is wrong — VaR increases with confidence level but that is not the monotonicity axiom. D is wrong — VaR and ES can rank portfolios differently.
Question 105
For a normally distributed portfolio, the ratio ES/VaR at 99% confidence is approximately:
Select an option first.
Correct answer: B — 1.14 — ES = φ(z_{0.99})/(1−0.99) × σ and VaR = z_{0.99} × σ, giving ratio = φ(2.326)/(0.01 × 2.326) ≈ 0.0267/(0.0233) ≈ 1.14
Explanation: B is correct: ES/VaR = [φ(z_α)/((1−α))] / z_α. For α=0.99: φ(2.326)=0.02668, so ES/VaR = 0.02668/(0.01 × 2.326) = 0.02668/0.02326 ≈ 1.147. A is wrong — ES > VaR. C is wrong — ratio of 2 requires very different parameters. D is wrong — ES cannot be less than VaR.
Question 106
The 'daily earnings at risk' (DEaR) measure used by some banks is essentially:
Select an option first.
Correct answer: B — A 1-day 95% VaR — measuring the potential loss from the daily trading position at 95% confidence
Explanation: B is correct: DEaR (or DEAR) = 1-day 95% VaR. It was popularised by JPMorgan as a daily market risk measure for trading books. A is wrong — DEaR is a statistical VaR measure, not stress testing. C is wrong — it is a market risk measure. D is wrong — not a liquidity measure.
Question 107
When comparing VaR models across banks, a key challenge is:
Select an option first.
Correct answer: B — Banks use different methodologies, lookback windows, correlations, and distributional assumptions — making direct comparison difficult without standardisation
Explanation: B is correct: model diversity across banks means that two banks with identical trading books might report very different VaRs. This 'model multiplicity' reduces the comparability of VaR disclosures. Regulators have partially addressed this through the output floor in FRTB. A is wrong — significant model diversity exists. C is wrong — only certain aspects are standardised. D is wrong — bank size and VaR are not directly related.
Question 108
The 'effective number of bets' in a portfolio (Meucci) provides information about:
Select an option first.
Correct answer: B — The true diversification of risk sources — accounting for correlation, it measures how many independent risk contributions exist in the portfolio
Explanation: B is correct: the effective number of independent bets = 1/Σp²_i where p_i is the proportion of risk from each principal component. A concentrated portfolio (one risk dominates) has ENB ≈ 1; a well-diversified portfolio has ENB close to the number of assets. A is wrong — it measures risk concentration, not position count. C is wrong — it uses portfolio risk attribution. D is wrong — simulation count is separate.
Question 109
Under the variance-covariance approach, the VaR of a two-asset portfolio equals the VaR derived from the full covariance matrix. This formulation:
Select an option first.
Correct answer: A — Is exact only for normal distributions and linear portfolios
Explanation: A is correct: the delta-normal (parametric) VaR is exact only when: (1) returns are jointly normally distributed; (2) the portfolio P&L is a linear function of the underlying returns (no options). For non-normal distributions or non-linear portfolios, the normal VaR is an approximation. B is wrong — not exact for all distributions. C is wrong — applies to any linear portfolio. D is wrong — it can be exact under stated conditions.
Question 110
The concept of 'risk parity' portfolio construction targets:
Select an option first.
Correct answer: B — Equal risk contribution (equal component VaR or equal volatility contribution) from each asset — typically resulting in higher allocation to less volatile assets like bonds
Explanation: B is correct: risk parity equalises each asset's risk contribution to the total portfolio risk. Since bonds are less volatile than equities, risk parity requires more capital in bonds. This contrasts with equal dollar weighting (which concentrates risk in equities). A is wrong — equal dollars gives unequal risk. C is wrong — that maximises Sharpe ratio. D is wrong — minimum VaR is the minimum variance portfolio direction.
Question 111
The 'scenario VaR' approach uses predefined market scenarios to compute P&L, differing from statistical VaR in that:
Select an option first.
Correct answer: B — It does not rely on historical return distributions but instead evaluates specific, deterministic market events — providing insight into particular vulnerabilities even if they are not well-represented in historical data
Explanation: B is correct: scenario VaR evaluates specific predefined shocks (e.g. 'equity market falls 20%, credit spreads widen 100bps, USD strengthens 5%') deterministically. Statistical VaR relies on the estimated distribution of random outcomes. A is wrong — scenario VaR can give lower or higher results. C is wrong — scenarios typically specify correlations through their joint definition. D is wrong — the scenario specification itself embeds assumptions.
Question 112
An analyst computes a 5-day 99% VaR using daily VaR × √5. The daily VaR is $1 million. The 5-day VaR is approximately:
Select an option first.
Correct answer: B — $2.236 million
Explanation: B is correct: 5-day VaR = 1-day VaR × √5 = $1m × 2.236 = $2.236 million. A is wrong — $5m is linear scaling (×5). C is wrong — $1.414m = $1m × √2 (2-day VaR). D is wrong — $3.162m = $1m × √10 (10-day VaR).
Question 113
For parametric VaR, the covariance matrix Σ must be:
Select an option first.
Correct answer: B — Positive semi-definite — allowing computation of portfolio variance as w'Σw ≥ 0
Explanation: B is correct: a valid covariance matrix must be positive semi-definite (PSD) — ensuring w'Σw ≥ 0 for all portfolio weights w (variance cannot be negative). In practice, estimated covariance matrices from insufficient data may violate PSD, requiring regularisation. A is wrong — off-diagonal correlations are essential. C is wrong — PSD (not invertible/positive definite) is required. D is wrong — symmetry alone is insufficient.
Question 114
The VaR 'violation ratio' in backtesting is computed as:
Select an option first.
Correct answer: A — Observed exceptions / Expected exceptions — a ratio significantly above 1 indicates model underestimation of risk
Explanation: A is correct: violation ratio = (observed exceptions) / (expected exceptions) = x / (n × (1−c)). A ratio > 1 means more violations than expected (model too loose); < 1 means fewer (model too conservative). Under the null, the ratio should be around 1. B is wrong — that is a P&L attribution metric. C is wrong — standard deviation tests model variance. D is wrong — averaged exceptions is the frequency, not the ratio.
Question 115
A bond immunisation strategy sets the portfolio's modified duration to match the liabilities' duration. This is related to VaR because:
Select an option first.
Correct answer: B — It reduces the portfolio's interest rate VaR — matching durations ensures that parallel yield curve shifts affect assets and liabilities proportionally, hedging the key interest rate risk
Explanation: B is correct: duration immunisation matches asset and liability interest rate sensitivities, reducing the portfolio's net interest rate exposure. This is directly equivalent to reducing the interest rate VaR to near zero for parallel shifts. However, non-parallel shifts (key rate risk) remain. A is wrong — non-parallel shifts still create VaR. C is wrong — immunisation reduces, not maximises, VaR. D is wrong — immunisation is a fixed-income technique.
Question 116
For a linear portfolio (only linear positions, no options), the parametric VaR using the delta method is:
Select an option first.
Correct answer: B — An exact result when returns are jointly normally distributed — the portfolio P&L is a linear combination of normally distributed factors, which is also normal
Explanation: B is correct: a linear combination of jointly normal random variables is itself normally distributed. For a linear portfolio with normal factor returns, the parametric (delta-normal) VaR is exact — not an approximation. A is wrong — exactness doesn't depend on the number of positions. C is wrong — the parametric VaR is exact (not an overestimate) for normal linear portfolios. D is wrong — it applies to multi-asset linear portfolios.
Question 117
When interest rates are near zero and an increase of 1% seems unlikely, extreme VaR scenarios for interest rate risk might use:
Select an option first.
Correct answer: B — Both upward and downward scenarios — rate increases are possible even from low levels, and upward moves can be large relative to starting rates; the VaR should be computed symmetrically
Explanation: B is correct: VaR models should consider all possible adverse scenarios — even from low absolute rate levels, rates can rise significantly (as seen in 2022 when rates rose 400+ bps from near zero). Restricting VaR to only rate decreases would systematically underestimate risk. A is wrong — upward moves are also a risk factor. C is wrong — most likely moves may not be the most adverse. D is wrong — current regime may differ from historical average.
Question 118
The 'economic capital' for market risk:
Select an option first.
Correct answer: B — Is the internally-estimated capital buffer sufficient to cover unexpected market losses at a specified confidence level (e.g. 99.97%) over a 1-year horizon — typically larger than regulatory minimum VaR-based capital
Explanation: B is correct: economic capital is the bank's own estimate of required capital for unexpected losses, typically at higher confidence (99.9%+) and longer horizon (1 year) than regulatory VaR requirements. It encompasses all risk types. A is wrong — economic capital often exceeds regulatory minimum. C is wrong — economic capital is typically larger. D is wrong — economic capital can be derived from VaR-type calculations at appropriate confidence levels.
Question 119
The 'normal power' (NP) approximation for VaR uses:
Select an option first.
Correct answer: B — The first three moments (mean, variance, skewness) to improve the VaR estimate beyond the normal distribution approximation
Explanation: B is correct: the Normal Power approximation uses: if X has mean μ, variance σ², and skewness γ, then z_NP ≈ z_N + (z_N² − 1)γ/6 − z_N γ²/36 + ..., providing a better approximation than the normal for skewed distributions. A is wrong — using only mean and variance gives the normal approximation. C is wrong — kurtosis alone is insufficient. D is wrong — NP uses through skewness.
Question 120
The 'risk-neutral density' extracted from option prices:
Select an option first.
Correct answer: B — Provides the risk-neutral (Q-measure) distribution of asset prices — it is flatter and has heavier left tails than the physical (P-measure) distribution for equities due to the risk premium and crash fears
Explanation: B is correct: the risk-neutral density (RND) implied from option prices has a pronounced left tail (reflected in the volatility skew) and differs from the physical distribution by the equity risk premium. The RND reflects investor risk aversion and crash fears. A is wrong — RND and physical distribution differ. C is wrong — RND typically has heavier left tails than historical. D is wrong — VaR uses the physical measure.
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