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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 181
The GPD shape parameter ξ determines:
Select an option first.
Correct answer: B — Tail heaviness: ξ>0 (Pareto/heavy), ξ=0 (exponential), ξ
Explanation: B is correct: ξ is the key EVT parameter. For financial data ξ typically ranges 0.2–0.5 (heavy tails). A is wrong — threshold u is chosen separately. C is wrong — exceedance count depends on threshold choice. D is wrong — mean excess function involves both ξ and β.
Question 182
The Hill estimator provides:
Select an option first.
Correct answer: B — An estimate of the tail index α = 1/ξ using the k largest observations
Explanation: B is correct: Hill estimator = (1/k)Σln(X_{(n-i+1)}/X_{(n-k)}) for i=1 to k. The Hill plot helps identify stable regions to choose k. A is wrong — mean uses all data. C is wrong — threshold selection uses the mean excess plot. D is wrong — Hill is for heavy-tail estimation.
Question 183
The mean excess function (MEF) plot is used to:
Select an option first.
Correct answer: B — Select the EVT threshold u — the MEF should be linear above a good threshold (positive slope for heavy tails)
Explanation: B is correct: e(u) = (β+ξu)/(1−ξ) for GPD — linear in u. Plotting empirical MEF and finding the region where it is approximately linear guides threshold selection. A is wrong — MEF is not a correlation tool. C is wrong — independence testing uses Ljung-Box. D is wrong — GARCH uses maximum likelihood.
Question 184
The tail dependence coefficient λ measures:
Select an option first.
Correct answer: B — The probability of one variable being extreme given the other is extreme: λ_U = lim_{q→1} P(X>F_X^{-1}(q)|Y>F_Y^{-1}(q))
Explanation: B is correct: λ_U quantifies joint extreme probability beyond what independence predicts. Gaussian copula λ_U=0; t copula λ_U>0. A is wrong — Pearson correlation is not tail-specific. C is wrong — variance is second-moment. D is wrong — that would be a conditional ES.
Question 185
Correlation breakdown during market stress refers to:
Select an option first.
Correct answer: B — Correlations spiking toward 1 as many assets fall simultaneously — diversification disappears precisely when needed most
Explanation: B is correct: empirically, asset correlations increase dramatically during financial crises as risk-off selling affects all markets. Models based on normal-period correlations underestimate crisis risk. A is wrong — correlations increase, not decrease. C is wrong — GARCH captures clustering but not correlation spikes. D is wrong — volatility dynamics are separate.
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Question 186
DCC (Dynamic Conditional Correlation) extends GARCH by:
Select an option first.
Correct answer: B — Allowing the correlation matrix to vary over time using GARCH-like dynamics
Explanation: B is correct: DCC (Engle 2002) models time-varying conditional correlations: Q_t = (1−a−b)Q̄ + aε_{t-1}ε'_{t-1} + bQ_{t-1}, normalised to give valid correlations. A is wrong — that is GJR-GARCH. C is wrong — EVT is separate. D is wrong — copulas model the full dependence.
Question 187
Realised volatility (RV) from high-frequency data is advantageous because:
Select an option first.
Correct answer: B — It is a model-free estimator of daily variance from summed squared intraday returns — converging to true variance as sampling frequency increases
Explanation: B is correct: RV = Σr²_{t,i} converges to quadratic variation as frequency→∞. A is wrong — RV is model-free. C is wrong — microstructure noise biases high-frequency RV; realised kernels address this. D is wrong — RV is backward-looking.
Question 188
The VIX is derived from:
Select an option first.
Correct answer: B — A model-free formula using a wide strip of S&P 500 option prices across strikes — measuring 30-day expected variance
Explanation: B is correct: VIX uses a portfolio replication across all listed strikes to estimate the risk-neutral expected variance over 30 days. A is wrong — VIX is market-derived. C is wrong — historical vol is backward-looking. D is wrong — old VXO used single ATM; modern VIX uses a strip.
Question 189
The leverage effect in equity volatility refers to:
Select an option first.
Correct answer: B — The empirical observation that equity volatility increases more after negative returns than positive returns of equal magnitude
Explanation: B is correct: after negative returns, the D/E ratio rises, increasing financial risk and future volatility. This asymmetry produces the volatility skew and is captured by GJR-GARCH and EGARCH. A is wrong — the direction of causality is stock price → leverage ratio → volatility. C is wrong — interest rate effects are rho. D is wrong — leveraged ETF volatility is a different phenomenon.
Question 190
The SABR model for interest rate options is widely used because:
Select an option first.
Correct answer: B — It models both the forward rate and its volatility as correlated stochastic processes, producing a semi-closed-form IV smile that calibrates well to caps, floors, and swaptions
Explanation: B is correct: SABR (Hagan et al.) generates a calibratable implied vol smile via: dF = σF^β dW₁, dσ = ασ dW₂ with correlation ρ. A is wrong — SABR has stochastic volatility. C is wrong — no EVT in SABR. D is wrong — GARCH is a time-series model.
Question 191
Under GARCH, 'vol of vol' relates to:
Select an option first.
Correct answer: B — The variability of GARCH conditional variance over time — driven by parameter α (reaction speed) and β (persistence)
Explanation: B is correct: the variance of σ²_t over time represents vol of vol in the GARCH context. Higher α → faster, larger variance movements → more vol of vol. A is wrong — that is unconditional volatility. C is wrong — ω is the long-run intercept. D is wrong — decay determines update speed.
Question 192
For block maxima EVT, a GEV with ξ=0.30 implies:
Select an option first.
Correct answer: B — Fréchet domain — power-law tail decay, typical of financial data with heavy tails
Explanation: B is correct: ξ>0 → Fréchet domain → heavy power-law tails. Typical equity/credit loss data has ξ∈[0.2, 0.5]. A is wrong — bounded tail requires ξ0.5 (here ξ=0.30 gives finite variance).
Question 193
Copulas are invariant under monotone transformations because:
Select an option first.
Correct answer: B — Copulas are defined on uniform(0,1) margins — the probability integral transform preserves rank ordering, making dependence structure independent of marginal choice
Explanation: B is correct: since copulas use the PIT (uniform margins), the dependence structure captured is rank-based and invariant to monotone transformations of variables. A is wrong — Sklar's theorem allows any marginals. C is wrong — multivariate copulas exist. D is wrong — copula-based VaR accounts for diversification.
Question 194
The Heston model's Feller condition κθ > ξ²/2 ensures:
Select an option first.
Correct answer: B — The variance process stays strictly positive
Explanation: B is correct: the Feller condition prevents the CIR variance process from reaching zero (or going negative), maintaining strict positivity. A is wrong — mean reversion κ(θ−v) prevents explosion. C is wrong — Heston has two state variables regardless. D is wrong — closed-form exists independent of Feller.
Question 195
The Diebold-Yilmaz connectedness index measures:
Select an option first.
Correct answer: B — System-wide financial connectedness from VAR forecast error variance decompositions — fraction of each variable's forecast error explained by shocks from others
Explanation: B is correct: D-Y connectedness identifies how much of variable i's forecast error variance is attributed to shocks in variable j, measuring risk transmission across the financial system. A is wrong — it's not GARCH-specific. C is wrong — diversification uses correlations. D is wrong — tail dependence uses copulas.
Question 196
Bottom-up VaR aggregation differs from top-down by:
Select an option first.
Correct answer: B — Starting from individual positions and aggregating using correlations up to firm-level — vs top-down which starts at firm level and allocates down
Explanation: B is correct: bottom-up is granular (position→desk→BU→firm); top-down starts with observed firm-level volatility and decomposes. Bottom-up is more precise but computationally intensive. A is wrong — both can use any methodology. C is wrong — bottom-up explicitly uses correlations. D is wrong — enterprise risk management covers all types.
Question 197
Under GARCH(1,1), the h-day conditional VaR forecast uses:
Select an option first.
Correct answer: B — The square root of the sum of h-step variance forecasts — σ²(t+j|t) converging toward long-run variance VL
Explanation: B is correct: GARCH h-step forecast: σ²(t+h|t) = VL + (α+β)^{h-1}(σ²(t+1|t)−VL). H-day VaR = z_α × √Σσ²(t+j|t). A is wrong — linear scaling ignores mean reversion. C is wrong — long-run × h ignores current regime. D is wrong — GARCH provides explicit multi-step forecasts.
Question 198
The variance risk premium (VRP) is:
Select an option first.
Correct answer: B — Risk-neutral expected variance minus physical expected variance — investors pay a premium for variance insurance (typically VRPrealised vol)
Explanation: B is correct: VRP = E^Q[σ²] − E^P[σ²] = IV² − GARCH forecast. VRP is typically negative for equities (IV > realised vol), reflecting non-diversifiable variance risk. A is wrong — VRP is structural. C is wrong — ω is an intercept. D is wrong — ex-ante vs ex-post confusion.
Question 199
The Normal Inverse Gaussian (NIG) distribution is useful because:
Select an option first.
Correct answer: B — It models skewness and excess kurtosis, providing better fit to fat-tailed financial returns
Explanation: B is correct: NIG is a four-parameter distribution fitting asymmetry and heavy tails observed in daily returns. It can serve as the innovation distribution in GARCH models. A is wrong — NIG has heavier tails. C is wrong — NIG has four parameters. D is wrong — NIG is used for daily and higher-frequency data.
Question 200
ARMA-GARCH combines:
Select an option first.
Correct answer: B — ARMA model for conditional mean with GARCH model for conditional variance — handling both return predictability and volatility clustering
Explanation: B is correct: r_t = μ_t (ARMA) + σ_t z_t (GARCH). This handles autocorrelation in returns and volatility clustering simultaneously. A is wrong — ARMA is for the mean. C is wrong — ARCH and GARCH are related models. D is wrong — that is filtered HS.
Question 201
Block maxima EVT applied to annual stock losses gives:
Select an option first.
Correct answer: B — GEV distribution — combining Gumbel, Fréchet, and Weibull families depending on the tail behaviour
Explanation: B is correct: Fisher-Tippett theorem: annual maxima converge to GEV. For heavy-tailed financials, GEV has ξ>0 (Fréchet). A is wrong — maxima are not normal. C is wrong — lognormal is for products. D is wrong — exponential relates to Gumbel only.
Question 202
The POT method's linearity of the MEF above threshold u indicates:
Select an option first.
Correct answer: B — GPD is a good fit for exceedances — GPD has a linear MEF: e(u) = (β+ξu)/(1−ξ)
Explanation: B is correct: linear MEF signals GPD appropriateness. This is the standard graphical diagnostic for threshold selection. A is wrong — normality isn't implied. C is wrong — linearity checks fit quality. D is wrong — linearity suggests the threshold is appropriate.
Question 203
The 'correlation smile' in CDO pricing refers to:
Select an option first.
Correct answer: B — Implied correlation varying across tranches — equity tranches imply low correlation; senior tranches imply high — inconsistent with a single Gaussian copula parameter
Explanation: B is correct: when market prices are used to imply correlation via the Gaussian copula, the implied correlation varies across the capital structure — analogous to the equity volatility smile. This shows the Gaussian copula cannot price all tranches consistently. A is wrong — no standard CDO tranche options. C is wrong — this is within the copula model. D is wrong — it's cross-tranche, not temporal.
Question 204
RiskMetrics monthly volatility from daily EWMA σ_d uses:
Select an option first.
Correct answer: B — Monthly σ = √20 × σ_d (since monthly variance = 20 × daily variance under i.i.d.)
Explanation: B is correct: under i.i.d., monthly variance = 20 × daily variance. Monthly σ = √20 × σ_d ≈ 4.47 × σ_d. A is wrong — multiplying vol (not variance) by 20 would be wrong. C is wrong — different data frequency gives a different estimate. D is wrong — GARCH term structure accounts for mean reversion.
Question 205
The GARCH-in-mean (GARCH-M) model adds the conditional variance to the mean equation because:
Select an option first.
Correct answer: B — It captures the risk-return trade-off — investors demand higher expected returns for higher expected variance: r_t = μ + λσ²_t + ε_t
Explanation: B is correct: GARCH-M (Engle, Lilien, Robins 1987) models the conditional mean as a function of the conditional variance, capturing the CAPM-like idea that expected return increases with expected risk. A is wrong — GARCH-M is more complex. C is wrong — autocorrelation may remain. D is wrong — positivity is ensured by parameter constraints.
Question 206
The 'news impact curve' (NIC) for a GARCH model shows:
Select an option first.
Correct answer: B — The mapping from the magnitude of a shock (lagged return) to the next period's conditional variance — GARCH gives a symmetric parabola; GJR-GARCH gives an asymmetric NIC with steeper response to negative shocks
Explanation: B is correct: the NIC plots σ²_t as a function of r_{t-1} (holding σ²_{t-1} fixed). For GARCH(1,1), NIC is a symmetric parabola (ω + βσ² + αr²). For GJR-GARCH, the NIC has a steeper left branch (negative returns → higher volatility). A is wrong — NIC is a statistical concept. C is wrong — VaR sensitivity is a different measure. D is wrong — earnings-specific volatility is different.
Question 207
The 'variance targeting' approach in GARCH estimation sets:
Select an option first.
Correct answer: A — The long-run variance ω/(1−α−β) to the sample variance of returns, then estimates only α and β — improving estimation efficiency
Explanation: A is correct: variance targeting (Engle and Mezrich 1996) fixes ω = VL × (1−α−β) where VL = sample variance, then estimates only α and β. This reduces the degrees of freedom in estimation and often improves out-of-sample forecasting. B is wrong — the initial variance is typically set to the sample variance. C is wrong — only VL is fixed to historical data. D is wrong — GARCH allows time-varying volatility.
Question 208
The 'realised kernel' estimator for daily variance is developed to:
Select an option first.
Correct answer: B — Reduce the bias from microstructure noise in high-frequency realised variance — using a kernel-weighted function of autocovariances to remove noise while preserving the variance signal
Explanation: B is correct: the realised kernel (Barndorff-Nielsen et al.) is a noise-robust estimator of integrated variance: RK = Σ_{h=−H}^{H} k(h/H) Σ r_{j} r_{j+h}, where k is a kernel function. It removes microstructure noise bias present in the simple realised variance. A is wrong — sampling frequency is a parameter choice. C is wrong — realised kernels estimate variance, not GARCH. D is wrong — no direct option pricing.
Question 209
The 'variance ratio' test for market efficiency tests whether:
Select an option first.
Correct answer: B — The variance of k-period returns equals k times the 1-period variance — if true, returns are uncorrelated (random walk); variance ratio ≠ k indicates autocorrelation
Explanation: B is correct: VR(k) = Var(k-period return) / (k × Var(1-period return)). Under a random walk: VR(k) = 1. VR > 1 indicates positive autocorrelation (momentum); VR < 1 indicates negative autocorrelation (mean reversion). A is wrong — it tests temporal persistence, not cross-period equality. C is wrong — ARCH tests test variance constancy. D is wrong — VaR validation is backtesting.
Question 210
The 'integrated variance' in continuous-time finance is:
Select an option first.
Correct answer: B — ∫₀ᵀ σ²_t dt — the total instantaneous variance accumulated over the period T, which realised variance (from high-frequency data) estimates
Explanation: B is correct: in continuous-time models, the price process follows dS = μS dt + σ_t S dW. The integrated variance over [0,T] is IV = ∫₀ᵀ σ²_t dt. Realised variance converges to IV as the sampling frequency increases (under mild conditions). A is wrong — discrete sum is the estimator, not the definition. C is wrong — integrated variance is path-dependent. D is wrong — GARCH long-run is the unconditional variance.
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