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Free FRM Valuation and Risk Models Practice Questions & Answers
400 exam-style Valuation and Risk Models 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 primary benefit of full repricing (as in Monte Carlo) versus delta approximation in VaR is:
Select an option first.
Correct answer: B — Full repricing correctly captures non-linear P&L for instruments like options, where delta approximation becomes inaccurate for large moves.
Explanation: Full repricing: for each simulation path, the instrument is valued using its full pricing function (e.g., BSM for options). This correctly captures gamma (convexity) and other non-linearities. Delta approximation is fast but misses curvature — increasingly inaccurate for large underlying moves or near-expiry options.
Question 62
The term 'volatility regime' refers to:
Select an option first.
Correct answer: B — A persistent period of relatively high or low volatility, as opposed to the transient day-to-day fluctuations captured by GARCH models.
Explanation: Volatility regimes are sustained periods of structurally different volatility levels (e.g., the low-volatility regime pre-2008 vs the high-volatility crisis regime). They differ from GARCH volatility clustering, which captures short-term autocorrelation. Regime-switching models (e.g., Markov-switching GARCH) explicitly model regime changes.
Question 63
A bank uses a 99% 1-day VaR under the Basel IMA. It has 12 back-testing exceptions over 250 trading days. What is the most likely regulatory consequence?
Select an option first.
Correct answer: B — A capital add-on of up to 1 additional multiplier on VaR, and a requirement to investigate the model.
Explanation: Basel back-testing traffic lights: green (0-4 exceptions) = no action; yellow (5-9) = investigation; red (10+) = capital add-on of up to 1 on the multiplier (from 3 to 4) and likely supervisory challenge to the model. 12 exceptions = red zone. C is possible but not automatic; D doesn't follow.
Question 64
Why does the square-root-of-time rule underestimate multi-day VaR during periods of volatility clustering?
Select an option first.
Correct answer: B — Volatility clustering means large losses on day 1 are followed by higher volatility on subsequent days, making the actual multi-day loss distribution fatter-tailed than the scaled normal.
Explanation: When volatility clusters, a large loss on day 1 tends to be followed by high volatility on days 2-10 — the multi-day loss variance is greater than T × σ²_daily. The SRT assumes each day's variance is the same constant σ², but GARCH dynamics increase conditional variance after a shock, making the actual 10-day loss distribution more dispersed than predicted by SRT.
Question 65
A risk manager observes that during the 2008 financial crisis, the correlation between previously uncorrelated assets rose sharply. The VaR model, calibrated on pre-crisis data, significantly understated losses. This failure is best attributed to:
Select an option first.
Correct answer: B — The assumption of stable correlations — a key violation during tail events where 'correlation breakdown' occurred in the wrong direction (correlations rose, not diversifying).
Explanation: Crisis-period correlation breakdown: correlations jump toward +1 during systemic events (everyone selling simultaneously, liquidity collapse). Models calibrated on normal-period correlations massively understate portfolio losses. This is why copulas with tail dependence and stressed correlation matrices are needed for crisis-period risk modelling.
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Question 66
The conditional coverage test in VaR back-testing assesses:
Select an option first.
Correct answer: B — Both whether the exception frequency is correct AND whether exceptions are serially independent (clustered exceptions suggest volatility clustering is not captured).
Explanation: The conditional coverage test (Christoffersen, 1998) jointly tests: (1) unconditional coverage — correct exception frequency, AND (2) independence — exceptions should not be clustered (serial independence). A clustered exception pattern suggests the model is not capturing volatility persistence (GARCH effects), even if the total count is correct.
Question 67
In the context of Model Risk, which of the following poses the greatest danger when using VaR models?
Select an option first.
Correct answer: B — Over-reliance on VaR as the sole risk measure, combined with failure to back-test and stress-test the model, creating a false sense of security.
Explanation: Model risk from VaR: the greatest danger is treating VaR as the definitive risk measure without complementary tools (ES, stress tests, scenario analysis) and without ongoing validation (back-testing). Firms that relied solely on VaR in 2008 were blindsided by losses many multiples of their reported VaR. A, C, D are methodological choices, not the core model risk issue.
Question 68
Under the FRTB (Fundamental Review of the Trading Book), which confidence level is used for Expected Shortfall as the primary market risk measure?
Select an option first.
Correct answer: B — 97.5%
Explanation: The FRTB (Basel IV market risk framework) replaced 99% VaR with 97.5% ES as the primary market risk measure. The 97.5% ES is designed to capture roughly the same capital as 99% VaR under normal distributions, while being coherent (sub-additive) and capturing tail severity. This change also introduced liquidity horizons of 10-120 days depending on asset class.
Question 69
Which of the following correctly describes the 'correlation matrix' requirement for multi-asset parametric VaR?
Select an option first.
Correct answer: B — The matrix must be positive semi-definite to ensure portfolio variance is always non-negative.
Explanation: A valid correlation (or covariance) matrix must be positive semi-definite (PSD): all eigenvalues ≥ 0. This guarantees portfolio variance ≥ 0 for all portfolio weights. Empirically estimated matrices from limited data may fail PSD — they need regularisation (e.g., shrinkage, eigenvalue clipping) before use. A, C, D are incorrect requirements.
Question 70
An analyst finds that a portfolio's 99% ES is $5 million and its 99% VaR is $3.5 million. The ratio ES/VaR ≈ 1.43. What does a high ES/VaR ratio indicate?
Select an option first.
Correct answer: C — The tail is heavy — losses beyond VaR are large on average relative to the VaR boundary, suggesting a fat-tailed distribution.
Explanation: ES/VaR ratio: for a normal distribution, 99% ES/99% VaR ≈ 1.13. A ratio of 1.43 indicates fatter tails than the normal — losses in the worst 1% scenarios average 43% more than the VaR boundary. This is typical of equity or credit portfolios with fat-tailed return distributions. A and B incorrectly describe a thin tail; D is a model issue not implied by the ratio.
Question 71
Which of the following best explains why 'extreme value theory' (EVT) is used for tail risk estimation?
Select an option first.
Correct answer: B — EVT directly models the tails of the loss distribution using distributions specifically suited to extreme events (Generalised Pareto Distribution), without imposing a specific form on the entire distribution.
Explanation: EVT (peaks-over-threshold approach using GPD): focuses specifically on observations in the extreme tail, fitting a distribution that is asymptotically appropriate for extremes rather than the centre of the distribution. This overcomes the limitation of normal/historical methods that poorly characterise rare, large losses. A, C, D are incorrect.
Question 72
The term 'liquidity-adjusted VaR' (L-VaR) extends standard VaR by incorporating:
Select an option first.
Correct answer: B — The cost of unwinding a position over its liquidation horizon, adding bid-ask spread and market impact costs to the standard VaR estimate.
Explanation: L-VaR adds the illiquidity premium to standard VaR: L-VaR = VaR + Liquidity cost = VaR + ½ × Spread × Position × √(Liquidation days). This acknowledges that large positions in illiquid markets cannot be exited at mid-price, and the cost of exiting amplifies risk. A, C, D describe different liquidity concepts.
Question 73
A portfolio manager finds that the 1-day VaR scaled to 10 days using the SRT rule is $10 million, but the actual 10-day VaR computed by Monte Carlo is $14 million. This discrepancy most likely reflects:
Select an option first.
Correct answer: C — Volatility clustering — the conditional volatility over 10 days is higher than assumed by the SRT, because a high-vol day 1 is followed by high-vol subsequent days.
Explanation: SRT assumes flat (constant) conditional volatility each day. Monte Carlo with GARCH dynamics: after a large return (high day-1 vol), subsequent conditional variances are also elevated — so the 10-day cumulative variance is greater than 10 × σ²_1day. The SRT underestimates because it cannot capture this GARCH-induced volatility persistence over the holding period.
Question 74
Which of the following best describes the relationship between VaR and the probability distribution of portfolio returns?
Select an option first.
Correct answer: B — VaR is the quantile of the loss distribution at the specified confidence level.
Explanation: VaR_α = the α-quantile of the loss distribution: the loss level L such that P(Loss > L) = 1−α. At 99%, VaR is the 99th percentile of the loss distribution. A is the mean (expected) loss; C relates VaR to σ only under normality with zero mean; D has no standard mathematical basis.
Question 75
Which of the following factors most directly causes the 'volatility smile' (U-shaped pattern of implied vol vs strike) observed in FX options?
Select an option first.
Correct answer: B — Fat tails and distributional uncertainty — OTM options on both sides carry higher implied vol because large moves in either direction are more likely than BSM assumes.
Explanation: FX smile: OTM calls and OTM puts both have higher implied vol than ATM options. This reflects that actual FX return distributions have fat tails on both sides (large moves in either direction are more frequent than log-normal). The smile in FX differs from the skew in equities (where the left tail concern dominates).
Question 76
In the mean-variance framework, when the risk-free asset is introduced, the optimal risky portfolio for all investors is:
Select an option first.
Correct answer: B — The tangency portfolio — the risky portfolio where the Capital Market Line is tangent to the efficient frontier.
Explanation: Two-fund separation theorem: all investors combine the risk-free asset with the same risky portfolio — the tangency (market) portfolio. The proportion invested in the tangency portfolio vs the risk-free asset depends on risk aversion, but the composition of the risky portion is identical for all investors. This is a cornerstone of CAPM.
Question 77
Which of the following best explains why 'diversification benefits' disappear in severe market downturns?
Select an option first.
Correct answer: B — In crises, all assets become driven by a single dominant factor (systemic risk/risk-off sentiment), making correlations converge toward +1 and eliminating the low-correlation benefits modelled in normal times.
Explanation: Tail dependence: in crises, common risk factors (leverage unwinding, liquidity hoarding, sentiment) dominate idiosyncratic factors. Assets that appear uncorrelated in normal times become highly correlated as systemic risk overwhelms diversification. Standard correlation matrices from normal periods dramatically understate crisis-period co-movement.
Question 78
Compared to a 95% confidence VaR, a 99% confidence VaR using the same model and data:
Select an option first.
Correct answer: C — Is higher because it represents a more extreme (rarer) loss threshold.
Explanation: Higher confidence level → further into the tail → higher loss threshold. At 99%, z = 2.326; at 95%, z = 1.645. Under normality: 99% VaR / 95% VaR = 2.326/1.645 ≈ 1.41. So the 99% VaR is about 41% higher than the 95% VaR (not double as in A). B reverses the logic; D is wrong.
Question 79
A risk manager says the 10-day 99% VaR is $15 million. Under the Basel square-root scaling, the corresponding 1-day 99% VaR used as the basis is approximately:
Select an option first.
Correct answer: A — $4.74 million
Explanation: 10-day VaR = 1-day VaR × √10. So 1-day VaR = 10-day VaR / √10 = $15m / 3.162 ≈ $4.74 million. B divides by 10; C multiplies by 10; D divides by 5.
Question 80
The coherence property 'translation invariance' for a risk measure ρ means:
Select an option first.
Correct answer: A — Adding a risk-free cash amount c to the portfolio reduces ρ by exactly c.
Explanation: Translation invariance: ρ(X + c) = ρ(X) − c (in loss terms). Adding certain cash c to a portfolio reduces its risk measure by exactly c. This ensures that holding more cash always reduces risk in a consistent, predictable way. B is positive homogeneity; C is not a coherence axiom; D is sub-additivity.
Question 81
Which of the following is the most significant practical challenge in implementing GARCH-based VaR?
Select an option first.
Correct answer: B — Estimating GARCH parameters (ω, α, β) requires maximum likelihood optimisation, and the estimates can be unstable with limited data or during regime changes.
Explanation: GARCH parameter estimation via MLE can be numerically unstable, especially when the data span includes structural breaks or regime changes. Overfitting is also a risk. Additionally, multivariate GARCH (for portfolios) is very high-dimensional. A, C, D are wrong — GARCH is sophisticated, not simple.
Question 82
Which of the following best describes mark-to-market (MTM) valuation?
Select an option first.
Correct answer: B — Valuing assets at current market prices reflecting today's exit value.
Explanation: Mark-to-market requires positions to be valued at current market prices — the price achievable in a transaction today. This is essential for accurate daily risk measurement (VaR, P&L). A is historical cost accounting; C is DCF valuation; D uses stale prices.
Question 83
P&L attribution in risk management refers to:
Select an option first.
Correct answer: B — Decomposing daily P&L into contributions from each risk factor (delta, gamma, theta, vega).
Explanation: P&L attribution: ΔP ≈ Δ×ΔS + ½Γ×(ΔS)² + Θ×Δt + ν×Δσ. Unexplained P&L reveals model errors or missing risk factors. This is a core model validation tool. A is revenue allocation; C and D are accounting/finance functions.
Question 84
The correct formula for two-asset portfolio variance is:
Select an option first.
Correct answer: B — w₁²σ₁² + w₂²σ₂² + 2w₁w₂Cov(r₁,r₂)
Explanation: Portfolio variance = w₁²σ₁² + w₂²σ₂² + 2w₁w₂ρσ₁σ₂. A adds standard deviations without squaring; C omits diversification benefit embedded in the cross-term; D omits the covariance term entirely. The covariance term captures the diversification effect.
Question 85
The volatility surface in options markets plots:
Select an option first.
Correct answer: B — Implied volatility against both strike and time to expiry.
Explanation: The vol surface is 3D: implied volatility (z) vs strike (x) vs maturity (y). It captures both the smile/skew across strikes and the term structure across maturities. Models like SVI and SABR parameterise and interpolate the surface for consistent derivative pricing.
Question 86
The Sharpe ratio measures:
Select an option first.
Correct answer: B — Excess return per unit of total risk (standard deviation).
Explanation: Sharpe = (Rp − Rf) / σp. Higher Sharpe = more excess return per unit of volatility. A is the Information ratio concept; C is the Treynor ratio; D is the profit factor.
Question 87
As portfolio size grows, which risk is eliminated through diversification?
Select an option first.
Correct answer: B — Idiosyncratic (firm-specific) risk
Explanation: Adding uncorrelated assets reduces idiosyncratic risk toward zero. Systematic (market-wide) risk cannot be diversified away regardless of portfolio size — it is the irreducible component of all equity portfolios. A, C, D all describe non-diversifiable market exposures.
Question 88
Standard deviation is preferred over variance as a risk measure because:
Select an option first.
Correct answer: B — It is expressed in the same units as returns, making it directly interpretable.
Explanation: Variance is in squared units (%²), hard to interpret practically. Standard deviation restores the original units — a 2% daily SD is directly meaningful as a typical daily deviation. A is false when variance < 1; C and D are incorrect.
Question 89
A portfolio with daily mean zero and daily SD $1 million has a 1-day 95% VaR of approximately:
Select an option first.
Correct answer: B — $1.65 million
Explanation: VaR₉₅ = z × σ = 1.645 × $1m = $1.645m ≈ $1.65m. A uses z=1; C is two-tailed 95% (z=1.96); D is 99% z-score (2.326).
Question 90
Maximum drawdown measures:
Select an option first.
Correct answer: B — The largest peak-to-trough decline in portfolio value over a specified period.
Explanation: Maximum drawdown = (Peak − Trough) / Peak. It captures path-dependent, sustained losses — critical for investors and hedge fund allocators who care about the worst cumulative decline experienced, not just daily tail risk.
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