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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 241
The 'normal mixture' distribution for returns:
Select an option first.
Correct answer: A — Blends multiple normal distributions with different means and variances — producing a distribution with heavier tails and possibly bimodal shape, appropriate for modelling regime changes
Explanation: A is correct: a normal mixture: p × N(μ₁, σ₁²) + (1−p) × N(μ₂, σ₂²). Even with symmetric components (μ₁=−μ₂), the mixture has excess kurtosis. With asymmetric means, it can also capture skewness. A correctly describes this. B is wrong — mixture uses multiple normals. C is wrong — t-distribution is a different fat-tailed model. D is wrong — mixtures can have non-zero higher moments.
Question 242
The 'market microstructure noise' in high-frequency realised variance arises from:
Select an option first.
Correct answer: B — Bid-ask spreads, trade reporting errors, and non-synchronous trading — causing observed high-frequency returns to fluctuate around the true price, biasing the realised variance upward
Explanation: B is correct: at very high frequencies, observed prices bounce between bid and ask, creating spurious return variation unrelated to the true price process. This inflates realised variance above the true integrated variance. Solutions include: subsampling, the realised kernel, and TSRV. A is wrong — genuine movements are what we want to measure. C is wrong — GARCH is a model, not noise. D is wrong — seasonal patterns are a different issue.
Question 243
For a portfolio with assets following a multivariate t distribution with ν degrees of freedom, the tail dependence coefficient between any two assets is:
Select an option first.
Correct answer: B — Positive: λ = 2t_{ν+1}(−√((ν+1)(1−ρ)/(1+ρ))) > 0, increasing as ν decreases — showing that joint extreme events become more likely as tails become heavier
Explanation: B is correct: the t copula's tail dependence coefficient depends on both ν and ρ. As ν → ∞, λ → 0 (Gaussian). For finite ν, λ > 0 regardless of correlation magnitude, meaning there is always some probability of joint extreme events. A is wrong — that is the Gaussian case. C is wrong — tail dependence and Pearson correlation are different concepts. D is wrong — tail dependence is defined for all ν > 0.
Question 244
The 'Engle and Manganelli' CAViaR model estimates VaR directly using:
Select an option first.
Correct answer: B — A quantile regression model: VaR_t(α) = β₀ + β₁VaR_{t-1}(α) + β₂g(r_{t-1}), where g is a function of past returns — without specifying the full distribution
Explanation: B is correct: CAViaR (Conditional Autoregressive Value at Risk) models the VaR quantile directly as an autoregressive process in the VaR itself plus past returns, without distributional assumptions. It is estimated via quantile regression. A is wrong — no full distribution is needed. C is wrong — GARCH models variance. D is wrong — historical simulation is empirical.
Question 245
Under the 'minimum variance copula' estimation, the copula is chosen to:
Select an option first.
Correct answer: B — Minimise the variance of a portfolio using copula-implied tail risks — selecting the copula that best represents tail dependence for the specific portfolio composition
Explanation: B is correct: in some approaches, copula selection for risk management purposes considers which copula specification most accurately captures the joint tail behavior relevant to the specific portfolio. Minimum variance copula is one approach where copula parameters minimise portfolio VaR variance. A is wrong — maximising correlation is not the objective. C is wrong — MLE selects the best-fitting copula, not minimum variance. D is wrong — normal is a specific (and often inadequate) choice.
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Question 246
The 'Pickands estimator' for the GPD tail index is based on:
Select an option first.
Correct answer: B — Three order statistics — it uses X_{(n-2k+1)}, X_{(n-k+1)}, and X_{(n)} to estimate ξ: ξ = (1/ln2) × ln((X_{(n-k+1)} − X_{(n-2k+1)}) / (X_{(n)} − X_{(n-k+1)}))
Explanation: B is correct: the Pickands estimator uses three order statistics at different quantile levels to estimate the shape parameter ξ of the GPD. While less efficient than MLE, it is useful for graphical diagnostics (Pickands plot). A is wrong — it uses only the extreme observations. C is wrong — only the largest would give no information about the tail shape. D is wrong — the MEF is used for threshold selection.
Question 247
The 'volatility skew' in equity options versus the 'volatility smile' in currency options reflects:
Select an option first.
Correct answer: B — The asymmetric crash risk in equities (investors fear large market drops) versus the more symmetric currency risks (either currency can appreciate or depreciate strongly)
Explanation: B is correct: equity IV skew reflects the 'crash risk premium' — demand for OTM puts is elevated due to crash fears, while call prices are lower (investors own equities). Currency options show a more symmetric smile because either currency can move adversely. A is wrong — the risk structures differ. C is wrong — currency options do have tail risk. D is wrong — price level is about ATM IV, not the shape.
Question 248
The 'covariance matrix regularisation' technique (e.g. Ledoit-Wolf) is needed because:
Select an option first.
Correct answer: B — Sample covariance matrices from insufficient data are not well-conditioned (ill-posed) — regularisation shrinks them towards a target (e.g. factor model) to improve out-of-sample stability
Explanation: B is correct: for n assets with T observations, if T < n, the sample covariance matrix is singular (not invertible). Even with T > n, sample matrices can be ill-conditioned. Ledoit-Wolf shrinks the sample covariance towards a structured target (identity or factor model) to improve estimation. A is wrong — sample matrices are often too noisy, not conservative. C is wrong — regularisation is needed for stability. D is wrong — correlations can be over- or underestimated.
Question 249
The 'high-frequency data' advantage for VaR models comes from:
Select an option first.
Correct answer: B — More frequent observations within a day enabling model-free volatility estimation (realised variance) that is more accurate and timely than GARCH-based estimates
Explanation: B is correct: intraday data (tick-by-tick or 5-minute returns) enables computation of realised variance daily, which is more informative than the single daily return. This makes daily volatility estimates more accurate and responsive than GARCH models calibrated on daily returns. A is wrong — longer daily time series helps with extreme values; HFD is about intraday. C is wrong — distributional assumptions are still needed for risk measures. D is wrong — option pricing is separate.
Question 250
The 'long memory' property in volatility (Hurst exponent H > 0.5) means:
Select an option first.
Correct answer: B — Volatility shocks decay very slowly — correlations between squared returns at different lags are positive and decay as a power law rather than exponentially, a property captured by FIGARCH models
Explanation: B is correct: empirical evidence shows long-range dependence in volatility (Ding et al. 1993). The autocorrelation of |r_t|^d decays as a power law c × k^{2d-1} for lag k, rather than the exponential decay of GARCH. FIGARCH (Fractionally Integrated GARCH) models this property with a fractional differencing parameter d ∈ (0,0.5). A is wrong — slow decay defines long memory. C is wrong — volatility is time-varying. D is wrong — long memory means high persistence.
Question 251
The 'GARCH volatility forecast' converges to the long-run variance as the horizon increases. The speed of convergence is:
Select an option first.
Correct answer: B — Governed by (α + β) — higher α+β means slower convergence to the long-run mean (longer persistence of current volatility regime)
Explanation: B is correct: the h-step ahead GARCH variance forecast: σ²(t+h|t) = VL + (α+β)^{h-1} × (σ²(t+1|t) − VL). The convergence speed is (α+β)^h — higher α+β → slower convergence. If α+β = 0.98, the half-life of a volatility shock ≈ 34 days. A is wrong — convergence speed depends on α+β. C is wrong — both α and β contribute. D is wrong — same reason.
Question 252
The 'option-adjusted volatility' (OAV) for a callable bond differs from a standard bond's yield volatility because:
Select an option first.
Correct answer: B — The embedded call option means the bond's price sensitivity to rates changes with market levels — OAV must capture this optionality, using option-adjusted spread and OAS duration
Explanation: B is correct: for callable bonds, as rates fall the call option becomes in-the-money, reducing the bond's effective duration (negative convexity). Computing VaR for callable bonds requires modeling the optionality — either through OAS-based duration or full revaluation under rate scenarios. A is wrong — they differ due to optionality. C is wrong — callable bonds have significant interest rate risk. D is wrong — the call option affects interest rate risk (prepayment/refinancing).
Question 253
The 'cross-sectional volatility' in equity markets refers to:
Select an option first.
Correct answer: B — The dispersion of individual stock returns around the market return on a given day — high cross-sectional volatility means stocks are diverging in performance, benefiting long-short strategies
Explanation: B is correct: cross-sectional volatility = standard deviation of individual stock returns on a given day. It measures how spread out stock performances are relative to each other. A is wrong — time-series volatility measures one series over time. C is wrong — that is correlation. D is wrong — equity risk premium variation is time-series volatility.
Question 254
The 'Black-Scholes vega' for a European call option is maximised when the option is:
Select an option first.
Correct answer: B — At-the-money and has longest time to expiry — ATM options have the greatest sensitivity to changes in implied volatility
Explanation: B is correct: vega = S × √T × φ(d₁) where φ is the standard normal PDF. The PDF φ(d₁) is maximised when d₁ = 0 (ATM options). Combined with higher T giving more time for vol changes to matter, ATM long-dated options have the highest vega. A is wrong — deep ITM options have low vega (near-certain exercise). C is wrong — deep OTM options have low vega (exercise is unlikely). D is wrong — very short time to expiry reduces vega.
Question 255
The 'basis risk' in hedging with VIX futures arises because:
Select an option first.
Correct answer: B — The VIX index and VIX futures prices can diverge — VIX futures trade at a premium to spot VIX when the term structure of variance is upward-sloping (contango), meaning holding long VIX futures incurs negative roll yield
Explanation: B is correct: VIX futures are typically in contango (futures price > spot VIX) during calm markets because the market expects volatility to eventually rise from low levels. A long VIX futures position incurs negative roll yield as the futures rolls down toward spot. This basis risk makes VIX futures imperfect hedges for volatility exposure. A is wrong — exchange trading reduces, not creates, basis risk. C is wrong — VIX futures are very liquid. D is wrong — VIX measures implied volatility.
Question 256
The 'term structure of implied volatility' for equity indices is typically:
Select an option first.
Correct answer: B — Downward-sloping when near-term volatility is elevated (stress) and upward-sloping (or flat) in calm periods — reflecting mean reversion in the expected volatility
Explanation: B is correct: IV term structure reflects market expectations of future volatility. In low-vol environments, long-term options have slightly higher IV (some mean reversion premium). During market stress, short-term IV spikes above long-term (inverted term structure) as near-term uncertainty dominates. A is wrong — term structure varies. C is wrong — inverted term structure occurs in stress. D is wrong — term structure applies to all options markets.
Question 257
The 'double exponential' (Laplace) distribution as an innovation distribution for GARCH models provides:
Select an option first.
Correct answer: B — Heavier tails than normal but analytical tractability — the Laplace has kurtosis 6 (vs 3 for normal) and is sometimes used as a compromise between normal and t
Explanation: B is correct: the Laplace (double exponential) distribution: f(x) = (1/b)exp(−|x−μ|/b). It has excess kurtosis = 3 (total kurtosis = 6), heavier tails than normal. It is symmetric and can be used as an innovation distribution in GARCH. A is wrong — Laplace has heavier tails. C is wrong — Laplace has heavier, not lighter, tails. D is wrong — Laplace is symmetric.
Question 258
The 'realised skewness' of daily returns calculated from high-frequency intraday data:
Select an option first.
Correct answer: B — Can be estimated as the sum of cubed standardised intraday returns, providing a model-free measure of intraday return asymmetry
Explanation: B is correct: analogous to realised variance (sum of squared intraday returns), realised skewness = sum of cubed standardised intraday returns. This provides an intraday, model-free estimate of daily return asymmetry. A is wrong — HFD enables richer estimates. C is wrong — even symmetric Brownian motion can exhibit intraday asymmetry. D is wrong — realised skewness approximates, not exactly equals, daily skewness.
Question 259
The 'local volatility' model (Dupire 1994) specifies that implied volatility:
Select an option first.
Correct answer: B — Is a function of current stock price and time: σ_local(S, t) — the unique diffusion that exactly matches all observed option prices simultaneously, derived from the volatility surface
Explanation: B is correct: the Dupire formula derives a local volatility function σ_local(S,t) from the observed implied volatility surface such that the resulting diffusion dS = σ_local(S,t)S dW prices all vanilla options exactly. A is wrong — constant vol is B-S assumption. C is wrong — mean reversion is in stochastic vol models. D is wrong — local vol is calibrated from current option prices, not history.
Question 260
In the 'Nelson-Siegel' model for yield curve fitting, the three factors represent:
Select an option first.
Correct answer: B — Level (parallel shift), slope (short-long spread), and curvature (hump/butterfly shape) of the yield curve — providing a parsimonious three-factor model
Explanation: B is correct: Nelson-Siegel: y(m) = β₀ + β₁(1−e^{−m/τ})/(m/τ) + β₂[(1−e^{−m/τ})/(m/τ) − e^{−m/τ}]. β₀ = long-run level, β₁ = slope (negative for normal curve), β₂ = curvature. Together they capture the three main PCA factors. A is wrong — the factors are not duration buckets. C is wrong — no credit/liquidity factor. D is wrong — it models one yield curve.
Question 261
The 'fat-tail correction' for a parametric VaR model can be implemented by:
Select an option first.
Correct answer: B — Using a Student's t distribution with calibrated degrees of freedom, or applying a Cornish-Fisher adjustment for skewness and excess kurtosis
Explanation: B is correct: fat-tail corrections include: (1) t-distribution (lower df → heavier tails); (2) Cornish-Fisher expansion (adjusts normal quantile using skewness and kurtosis); (3) EVT (fits GPD to the tail directly). A is wrong — lower confidence would give smaller VaR. C is wrong — fewer observations increases estimation noise. D is wrong — excluding extremes would reduce the VaR underestimation.
Question 262
The 'volatility forecast evaluation' using the 'Mincer-Zarnowitz regression' regresses:
Select an option first.
Correct answer: B — Realised variance (or volatility) on the forecasted variance (or volatility): σ²_realised,t = a + b × σ²_forecast,t + ε_t — an unbiased efficient forecast has a=0 and b=1
Explanation: B is correct: the Mincer-Zarnowitz regression tests forecast efficiency: if a≠0 or b≠1, the forecast is biased or inefficient. A GARCH forecast with b=1, a=0, R² close to 1 is a good forecast. A is wrong — autocorrelation regression tests persistence. C is wrong — implied vs historical is the VRP comparison. D is wrong — Kupiec/Christoffersen test exceptions.
Question 263
The 'exponential distribution' is related to the GPD because:
Select an option first.
Correct answer: B — The GPD with ξ=0 reduces to the exponential distribution — the Gumbel extreme value domain has exponential tails, and the corresponding POT threshold exceedances follow the exponential distribution
Explanation: B is correct: GPD(ξ=0, β) = 1 − exp(−y/β) (exponential distribution with mean β). When ξ→0, the GPD converges to the exponential. This corresponds to the Gumbel domain of attraction (distributions with exponential tails). A is wrong — they are directly related. C is wrong — exponential has a thin tail (light tail). D is wrong — the relationship is reversed.
Question 264
The 'correlation between implied and historical volatility' in practice is:
Select an option first.
Correct answer: B — High but not perfect — implied volatility incorporates a risk premium and forward-looking information not in historical data, so IV > RV on average (variance risk premium), and they co-move but diverge
Explanation: B is correct: empirically, IV and RV are positively correlated (both high or low in similar regimes) but IV typically exceeds RV (variance risk premium). The correlation is around 0.7-0.9 in practice. A is wrong — perfect correlation would require no risk premium or information difference. C is wrong — strong positive co-movement exists. D is wrong — both tend to be high in volatile periods.
Question 265
The 'higher-order cumulants' (skewness and kurtosis) of financial returns affect VaR by:
Select an option first.
Correct answer: B — Making the VaR larger than a normal approximation for negative skewness and/or positive excess kurtosis — the Cornish-Fisher correction and EVT both account for these moments
Explanation: B is correct: negative skewness (more probability in the left tail) and positive excess kurtosis (fatter tails) both increase the VaR compared to a normal model. The Cornish-Fisher approximation quantifies this adjustment. EVT directly models the tail. A is wrong — higher moments materially affect quantiles. C is wrong — diversification affects variance. D is wrong — no factor-of-2 relationship.
Question 266
The 'long-term volatility equilibrium' in a GARCH model can be estimated from:
Select an option first.
Correct answer: B — The unconditional variance of the return series: VL = ω/(1−α−β), which represents the level to which GARCH conditional variance reverts in the long run
Explanation: B is correct: VL = ω/(1−α−β) is the long-run (unconditional) variance of the GARCH process. All GARCH conditional variance forecasts converge to VL as the horizon increases. A is wrong — VL uses all parameters. C is wrong — implied vol is market-derived. D is wrong — historical maximum is an extreme observation, not the equilibrium.
Question 267
The 'implied correlation' from FX options in a currency triangle (USD/EUR/GBP) is extracted because:
Select an option first.
Correct answer: B — Cross-currency options contain implicit information about the correlation between the two exchange rates, derived from the triangle no-arbitrage condition: σ²_{EUR/GBP} = σ²_{USD/EUR} + σ²_{USD/GBP} − 2ρσ_{USD/EUR}σ_{USD/GBP}
Explanation: B is correct: for the currency triangle USD/EUR/GBP, the EUR/GBP implied vol is related to the other two vols through the correlation formula. Given observed market IVs for all three pairs, the implied correlation ρ can be extracted. This is widely used in FX risk management. A is wrong — correlation products do exist. C is wrong — correlation is time-varying. D is wrong — regulatory requirement is not the reason.
Question 268
The 'correlation matrix' estimated from historical data must be:
Select an option first.
Correct answer: B — Symmetric and positive semi-definite — correlation between any two variables is between −1 and +1, and all eigenvalues are non-negative
Explanation: B is correct: a valid correlation matrix R satisfies: (1) R_ij ∈ [−1,1]; (2) R_ii = 1; (3) symmetric (R_ij = R_ji); (4) positive semi-definite (all eigenvalues ≥ 0). Historical estimation can produce non-PSD matrices, requiring regularisation (e.g. eigenvalue clipping). A is wrong — diagonal elements are always 1. C is wrong — off-diagonal elements represent correlations. D is wrong — uncorrelated assets have R = identity matrix (which is PSD).
Question 269
The 'multivariate normal' distribution's elliptical contours mean that:
Select an option first.
Correct answer: B — The joint distribution is fully characterised by the mean vector and covariance matrix — and tail dependence is zero for the Gaussian copula
Explanation: B is correct: the multivariate normal distribution has elliptical probability density contours. Key properties: (1) fully parameterised by μ and Σ; (2) zero tail dependence (Gaussian copula); (3) linear correlation captures all dependence. This is a key limitation for risk management: crises show positive tail dependence not captured by the Gaussian. A is wrong — they are correlated, not independent. C is wrong — covariance matrix allows different variances. D is wrong — normal returns can be negative.
Question 270
The 'subordinated process' interpretation of the Variance Gamma model means:
Select an option first.
Correct answer: B — The stock price process S_t = exp(X_{Γ(t)}) where X is Brownian motion and Γ(t) is a gamma-distributed random clock — the gamma 'clock' randomises the business time, creating fat tails
Explanation: B is correct: time-changed Brownian motion: replace calendar time t with random business time Γ(t) ~ Gamma(t/ν, ν). This creates a pure jump process with fat tails and skewness. The gamma clock represents varying 'business activity' — more trading occurs in volatile periods. A is wrong — subordination is a mathematical concept. C is wrong — VG is an alternative to BS. D is wrong — gamma governs time, not price.
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