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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 151
The 'risk-free rate' assumption in VaR discounting is relevant because:
Select an option first.
Correct answer: B — When present-valuing future cash flows in the P&L calculation, using the risk-free rate affects the mark-to-market value and hence the P&L distribution and VaR
Explanation: B is correct: the present value of future cash flows (bonds, swaps, structured products) depends on the discount rate used. Changes in the risk-free rate affect mark-to-market values and hence the P&L volatility and VaR. For interest rate instruments, this is a primary risk factor. A is wrong — the confidence level is an input, not determined by the rate. C is wrong — VaR is about loss distributions. D is wrong — the relationship is more complex.
Question 152
A risk manager uses 1,000 Monte Carlo simulations for a 99% VaR. The standard error of the quantile estimate is approximately:
Select an option first.
Correct answer: B — Relatively large — only 10 observations fall in the 1% tail, making the 99th percentile estimate imprecise; increasing to 100,000 simulations reduces this to ~√10 fewer standard errors
Explanation: B is correct: with 1,000 simulations at 99% confidence, only 10 observations fall below the VaR threshold. Estimating the quantile from 10 observations is highly imprecise. With 100,000 simulations, 1,000 observations fall in the tail — much better precision. The standard error of the quantile ≈ √(p(1−p)/n)/f(q_p). A is wrong — 1,000 simulations gives substantial sampling error at 99%. C is wrong — all sampling-based methods have estimation error. D is wrong — standard error of quantile ≠ portfolio standard deviation.
Question 153
The 'risk measure' used for capital allocation in RAROC (Risk-Adjusted Return on Capital) is typically:
Select an option first.
Correct answer: B — Economic capital — the internally estimated capital required to support unexpected losses at a high confidence level (e.g. 99.97%) over a one-year horizon
Explanation: B is correct: RAROC = Risk-Adjusted Return / Economic Capital. Economic capital is the firm's own estimate of required capital, typically set at a confidence level corresponding to an AA credit rating (~99.97%). Business units with higher economic capital per unit of return get lower RAROC. A is wrong — market cap is external. C is wrong — regulatory capital is a floor. D is wrong — notional ignores risk profile.
Question 154
The 'autocorrelation of squared returns' (or absolute returns) is used to detect:
Select an option first.
Correct answer: B — Volatility clustering — high absolute returns tend to be followed by high absolute returns, and low by low, indicating ARCH/GARCH effects
Explanation: B is correct: the autocorrelation function of r²_t or |r_t| exhibits significant positive autocorrelation at many lags (often decaying slowly), indicating that large return magnitudes cluster together. This is the empirical signature of volatility clustering motivating GARCH models. A is wrong — price trends show up in return autocorrelation. C is wrong — mean reversion shows negative return autocorrelation. D is wrong — seasonality produces periodic spikes in the ACF.
Question 155
When VaR is calculated using Monte Carlo simulation, increasing the correlation between two risk factors from 0.3 to 0.8 will:
Select an option first.
Correct answer: B — Increase the portfolio VaR — higher correlation means less diversification benefit, so the combined risk is larger
Explanation: B is correct: portfolio variance = σ₁² + σ₂² + 2ρσ₁σ₂. Higher ρ → larger portfolio variance → larger VaR. At ρ=1, VaR = VaR₁ + VaR₂ (no diversification). At ρ=0.8 vs 0.3, the portfolio has significantly less diversification benefit. A is wrong — correlation directly affects portfolio VaR. C is wrong — higher correlation increases VaR. D is wrong — VaR is affected by correlation too.
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Question 156
The 'floor' on the VaR estimate in FRTB stressed ES computation prevents:
Select an option first.
Correct answer: B — The capital from falling below a minimum level even when current volatility is very low — if current ES is below 75% of the stressed ES, the stressed ES is used with a scaling factor to maintain a capital floor
Explanation: B is correct: the FRTB IMA stressed ES computation uses: ES = max(ES_t, 0.75 × ES_stressed) effectively — ensuring capital doesn't fall to negligible levels during unusually calm markets. The stressed ES provides the floor, preventing procyclical capital reduction. A is wrong — the floor prevents ES from being too low. C is wrong — historical data is used for the stressed period. D is wrong — the floor prevents under-capitalisation during calm periods.
Question 157
The 'distribution of VaR estimates' across banks for identical portfolios shows that:
Select an option first.
Correct answer: B — Significant dispersion exists — the same portfolio can have VaR estimates varying by factors of 2-3 across banks using different models, windows, and assumptions
Explanation: B is correct: academic studies (e.g. Mariathasan and Merrouche 2014, Vallascas and Hagendorff 2013) show that identical portfolios produce widely varying VaR estimates across banks, reflecting model diversity and possibly strategic model choice. FRTB's output floor and PLA test aim to reduce this dispersion. A is wrong — significant dispersion exists empirically. C is wrong — no size relationship. D is wrong — standardisation is partial and ongoing.
Question 158
Under a 'fat-tailed' return distribution, compared to the normal distribution:
Select an option first.
Correct answer: B — The 99th percentile is higher than the normal quantile, but the 95th percentile may be close to the normal — fat tails primarily affect extreme quantiles
Explanation: B is correct: fat-tailed distributions (like the t-distribution or stable distributions) have higher probability in the extreme tails than the normal. This makes high-confidence quantiles (99%, 99.9%) substantially higher than the normal estimate. Moderate quantiles (90-95%) may be close to normal since the fat-tail effect is most pronounced in the extreme tail. A is wrong — fat tails increase high-confidence VaR. C is wrong — quantiles differ. D is wrong — ES is higher for fat-tailed distributions.
Question 159
The concept of 'conditional independence' in credit portfolio modelling means that defaults are:
Select an option first.
Correct answer: B — Independent conditional on a common factor (e.g. the state of the economy) — once the economic factor is known, individual default probabilities are determined and defaults are independent of each other
Explanation: B is correct: in factor models for credit portfolios, conditional on the systematic factor (e.g. macro state), individual obligors' asset values are independent. This is used in the Basel IRB formula and CDO modelling. Unconditionally, defaults are correlated through their shared sensitivity to the systematic factor. A is wrong — unconditionally, defaults are correlated. C is wrong — full correlation only at extreme systematic factor values. D is wrong — credit default risk is part of market risk for trading books.
Question 160
EWMA volatility assigns weights to past observations that:
Select an option first.
Correct answer: B — Decrease exponentially with age — σ²_t = λσ²_{t-1} + (1−λ)r²_{t-1}
Explanation: B is correct: EWMA uses the recursion with decay factor λ (typically 0.94 for RiskMetrics). Recent data receives higher weight (1−λ), older data gets (1−λ)λ^k. A is wrong — equal weights describe SMA. C is wrong — increasing weights would overweight old data. D is wrong — option-implied vol is market-derived.
Question 161
The effective number of days in an EWMA model with λ=0.94 is approximately:
Select an option first.
Correct answer: B — 17 days (≈ 1/(1−λ) = 1/0.06)
Explanation: B is correct: effective sample size ≈ 1/(1−λ) = 1/0.06 ≈ 16.7 days. A is wrong — 6 = 1/(1−0.84). C is wrong — 94 is λ×100. D is wrong — 250 is a full year.
Question 162
GARCH(1,1) models conditional variance as:
Select an option first.
Correct answer: B — σ²_t = ω + αr²_{t-1} + βσ²_{t-1} — capturing long-run mean (ω), new shock impact (α), and variance persistence (β)
Explanation: B is correct: GARCH(1,1) (Bollerslev 1986) has three terms: ω (intercept), α (ARCH term for new shocks), β (GARCH persistence). Long-run variance = ω/(1−α−β) requires α+β
Question 163
For GARCH(1,1) with ω=0.000002, α=0.08, β=0.90, the long-run variance is:
Select an option first.
Correct answer: B — 0.0001 (= ω/(1−α−β) = 0.000002/0.02)
Explanation: B is correct: long-run variance = 0.000002/(1−0.08−0.90) = 0.000002/0.02 = 0.0001 (daily vol = 1%). A is wrong — ω alone is not the long-run variance. C is wrong — double the correct answer. D is wrong — not supported.
Question 164
When α+β is close to 1 in GARCH(1,1):
Select an option first.
Correct answer: B — Volatility shocks are very persistent — high/low volatility periods last for a long time
Explanation: B is correct: half-life of a volatility shock ≈ ln(0.5)/ln(α+β). When α+β=0.99, half-life ≈ 69 days. A is wrong — quick reversion requires small (α+β). C is wrong — persistence and level are independent (depends on ω). D is wrong — vol scaling depends on √T.
Question 165
The IGARCH (Integrated GARCH) model with α+β=1:
Select an option first.
Correct answer: B — Is equivalent to the EWMA model (when ω=0) — the unconditional variance is undefined (infinite)
Explanation: B is correct: IGARCH with ω=0, β=λ, α=1−λ is exactly EWMA. With α+β=1, the unconditional variance diverges to infinity (non-stationary). A is wrong — it is related to EWMA, not high-ω GARCH. C is wrong — no mean reversion in IGARCH. D is wrong — unconditional variance is infinite.
Question 166
GJR-GARCH extends GARCH by:
Select an option first.
Correct answer: B — Adding an asymmetric term γI_{t-1}r²_{t-1} where I=1 for negative returns — capturing the leverage effect
Explanation: B is correct: GJR-GARCH: σ²_t = ω + (α+γI_{t-1})r²_{t-1} + βσ²_{t-1}. For equities γ>0 means negative returns increase volatility more than positive ones. A is wrong — no floor. C is wrong — that is ARMA-GARCH. D is wrong — GJR uses daily data.
Question 167
EGARCH models variance in log form: ln(σ²_t) = ω + α(|z_{t-1}|−E|z_{t-1}|) + γz_{t-1} + βln(σ²_{t-1}). The advantage is:
Select an option first.
Correct answer: B — Logarithmic specification ensures positive variance without non-negativity constraints on parameters
Explanation: B is correct: modelling ln(σ²) guarantees σ²>0 regardless of parameter signs. A is wrong — EGARCH is more complex. C is wrong — γ captures leverage effect. D is wrong — EGARCH has similar parameter count.
Question 168
Implied volatility is derived by:
Select an option first.
Correct answer: B — Inverting the Black-Scholes model — finding the volatility that makes model price equal market price
Explanation: B is correct: IV is found numerically (e.g. Newton-Raphson) by solving BS(IV) = market price. A is wrong — GARCH gives statistical forecasts. C is wrong — that is historical vol. D is wrong — that is historical vol calculation.
Question 169
The equity options volatility smile shows:
Select an option first.
Correct answer: B — Higher IV for lower strikes (OTM puts) — the skew reflects negative skewness and fat tails in equity returns
Explanation: B is correct: equity IV skew shows higher IV for lower strikes, reflecting demand for downside protection and the leverage effect. C is wrong — constant IV is the BS assumption, violated in practice. A is wrong — ATM typically has lower IV than OTM puts for equities. D is wrong — that is the term structure, not the smile.
Question 170
For currency options, the IV pattern is typically:
Select an option first.
Correct answer: B — Symmetric U-shaped smile — OTM options on both sides have higher IV than ATM
Explanation: B is correct: currency IV smiles are roughly symmetric because both appreciation and depreciation extremes are possible, unlike equity crash-dominated skew. A is wrong — equity-like skew is not typical. C is wrong — currency IV varies with strike. D is wrong — purely upward-sloping is unusual.
Question 171
The volatility term structure plots IV against:
Select an option first.
Correct answer: B — Time to expiry for a given strike
Explanation: B is correct: term structure = IV(T) for fixed strike/moneyness. During high-vol regimes, short-dated IV > long-dated (inverted). In calm markets, term structure is mildly upward-sloping. A is wrong — that is the smile. C is wrong — underlying price is not the dimension. D is wrong — both maturity and moneyness dimensions are needed.
Question 172
The volatility surface is a 2D plot of IV as a function of:
Select an option first.
Correct answer: B — Both strike (or delta) and time to expiry
Explanation: B is correct: volatility surface IV(K,T) captures smile across strikes and term structure across maturities. It must be arbitrage-free (no butterfly or calendar spreads). A is wrong — only maturity gives just the term structure. C is wrong — delta alone misses maturity. D is wrong — gamma is not a dimension.
Question 173
Linear (Pearson) correlation has a key limitation for risk management because:
Select an option first.
Correct answer: B — It only fully characterises dependence for elliptical distributions — the same correlation can arise from distributions with very different tail behaviour
Explanation: B is correct: for non-elliptical distributions, two bivariate distributions can share identical Pearson ρ but have completely different tail dependence. This motivated the development of copulas. A is wrong — bounded range is a feature. C is wrong — ρ can be negative. D is wrong — marginals can differ.
Question 174
A copula function links the joint distribution to marginals through Sklar's theorem:
Select an option first.
Correct answer: B — F(x₁,...,xₙ) = C(F₁(x₁),...,Fₙ(xₙ)) — separating marginals from dependence structure
Explanation: B is correct: Sklar's theorem: any joint distribution decomposes into marginals and a copula C mapping uniform(0,1) margins to their joint distribution. A is wrong — averaging CDFs ≠ Sklar's theorem. C is wrong — that is the independence copula. D is wrong — multivariate normal is a parametric model.
Question 175
The Gaussian copula's key weakness is:
Select an option first.
Correct answer: B — Zero tail dependence — joint extreme events are not more likely than independence, underestimating crisis co-movements
Explanation: B is correct: Gaussian copula has tail independence (λ_U = λ_L = 0). This means extreme events are less correlated than implied by the general correlation. The 2008 CDO crisis exposed this weakness. A is wrong — negative correlations are possible. C is wrong — any marginals can be used with Sklar's theorem. D is wrong — multivariate Gaussian copulas exist.
Question 176
The Student's t copula has which advantage over Gaussian for tail risk?
Select an option first.
Correct answer: B — Symmetric positive tail dependence — simultaneous extreme events are more likely, better capturing crisis co-movements
Explanation: B is correct: t copula with low degrees of freedom has λ_U = λ_L > 0. As df→∞, t copula→Gaussian copula. A is wrong — t copula has extra df parameter. C is wrong — t copula captures tail events. D is wrong — t copula has positive tail dependence.
Question 177
The Clayton copula exhibits:
Select an option first.
Correct answer: B — Lower tail dependence only — suitable for credit portfolios where joint defaults cluster
Explanation: B is correct: Clayton copula has λ_L > 0 and λ_U = 0. It models situations where joint losses (lower tail) are more correlated than joint gains. Useful for credit portfolios. A is wrong — Clayton is asymmetric. C is wrong — upper tail dependence describes Gumbel. D is wrong — independence copula has zero in both tails.
Question 178
EVT is applied in risk management to:
Select an option first.
Correct answer: B — Estimate probabilities of rare catastrophic losses — using limiting distributions for extremes
Explanation: B is correct: EVT provides rigorous statistical tools for tail estimation using limit theorems for extrema. A is wrong — EVT focuses on tails. C is wrong — GARCH is a time-series model. D is wrong — option pricing uses different models.
Question 179
The Generalised Extreme Value (GEV) distribution arises from:
Select an option first.
Correct answer: B — The extreme value theorem for block maxima — combining Gumbel (ξ=0), Fréchet (ξ>0), and Weibull (ξ
Explanation: B is correct: Fisher-Tippett theorem: standardised block maxima converge to GEV as block size→∞. For financial returns with heavy tails, ξ>0 (Fréchet domain). A is wrong — CLT applies to sums. C is wrong — convolution produces aggregate distributions. D is wrong — normal transformation gives different distributions.
Question 180
The POT (Peaks-Over-Threshold) approach models:
Select an option first.
Correct answer: B — Exceedances above a threshold using the Generalised Pareto Distribution (GPD)
Explanation: B is correct: by Pickands-Balkema-de Haan theorem, threshold exceedances converge to GPD: G(y;ξ,β) = 1−(1+ξy/β)^{−1/ξ}. A is wrong — block maxima use GEV. C is wrong — GPD focuses on the tail only. D is wrong — that is filtered HS.
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