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Free FRM Valuation and Risk Models Practice Questions & Answers
394 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 1
Value-at-Risk (VaR) at a 99% confidence level over one day answers which question?
Select an option first.
Correct answer: B — What is the maximum loss that will not be exceeded on 99% of trading days?
Explanation: VaR at a given confidence level answers: 'What loss level will not be exceeded X% of the time?' At 99%, losses exceed the VaR figure only 1% of trading days (roughly 2-3 days per year). A describes expected loss; C describes stressed loss; D is not a VaR definition.
Question 2
Expected Shortfall (ES) is always __ than VaR at the same confidence level.
Select an option first.
Correct answer: C — Greater than or equal to
Explanation: ES (CVaR) is the average loss conditional on being in the tail — i.e., given that loss > VaR. Because it is an average of losses beyond VaR, it is always ≥ VaR. ES = VaR only in the degenerate case where all tail losses are exactly equal to VaR, which is practically impossible.
Question 3
Which of the following statements about the normal distribution is correct in the context of financial risk?
Select an option first.
Correct answer: B — The normal distribution is fully described by its mean and standard deviation.
Explanation: The normal (Gaussian) distribution is completely parameterised by mean (μ) and standard deviation (σ). A is false — empirical returns exhibit fat tails and skewness. C is reversed — actual returns have fatter tails than the normal. D is partially true but irrelevant to the description of the normal itself.
Question 4
At a 95% confidence level, the one-tailed z-score used in parametric VaR is approximately:
Select an option first.
Correct answer: B — 1.65
Explanation: The 95% one-tailed z-score is 1.645 (≈1.65). At 99%, z ≈ 2.326. At 90%, z ≈ 1.282. At 97.5% (two-tailed 95%), z ≈ 1.96. Memorising these cut-offs is essential for parametric VaR calculations.
Question 5
If daily returns are i.i.d. and normally distributed, a 1-day VaR can be scaled to a 10-day VaR by:
Select an option first.
Correct answer: B — Multiplying by √10
Explanation: Under i.i.d. (independently and identically distributed) normally distributed returns, variance scales linearly with time, so standard deviation (and VaR) scales with the square root of time. 10-day VaR = 1-day VaR × √10. This is the square-root-of-time rule.
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Question 6
Which of the following is the primary advantage of historical simulation VaR over parametric (normal) VaR?
Select an option first.
Correct answer: B — It captures the actual empirical distribution of returns, including fat tails and skewness, without imposing a specific parametric form.
Explanation: Historical simulation uses actual past returns ranked by magnitude, avoiding any distributional assumption. This allows it to capture empirical fat tails and skewness that the normal distribution misses. A and C are wrong; D is false — it needs a large sample to be meaningful.
Question 7
Historical simulation VaR at the 99% confidence level using 500 days of data corresponds to approximately which observation in the ranked loss distribution?
Select an option first.
Correct answer: A — The 5th-worst loss
Explanation: At 99% confidence, VaR = the loss at the 1st percentile. With 500 observations, 1% × 500 = 5. So the VaR is approximately the 5th-worst daily loss in the sample. The 1st-worst is the maximum loss; the 495th is near the median.
Question 8
Monte Carlo VaR involves:
Select an option first.
Correct answer: C — Simulating thousands of random future scenarios under assumed distributions and reading the VaR from the resulting P&L distribution.
Explanation: Monte Carlo VaR: specify distributions and correlations → generate large numbers of random paths → reprice the portfolio under each → rank losses → read the VaR percentile. A is historical simulation; B is parametric; D is a volatility-weighted historical variation.
Question 9
Which of the following is a key limitation of historical simulation VaR?
Select an option first.
Correct answer: C — Its estimates depend entirely on the historical window chosen and will miss risks absent from that window.
Explanation: Historical simulation is backward-looking. If the chosen window (e.g., 500 trading days) doesn't include a particular crisis type, that risk will not appear in the VaR estimate. A is the parametric approach's characteristic; B is wrong; D is wrong — it can understate risk if the window was calm.
Question 10
Volatility in financial markets is described as 'mean-reverting' because:
Select an option first.
Correct answer: B — After periods of high or low volatility, it tends to return to a long-run average level.
Explanation: Mean reversion in volatility means it oscillates around a long-run average — after a spike (crisis) or trough (calm markets), volatility gradually reverts towards its historical mean. GARCH models explicitly incorporate this property. A, C, and D are all incorrect.
Question 11
Volatility clustering refers to the empirical observation that:
Select an option first.
Correct answer: B — High-volatility periods tend to be followed by more high-volatility periods, and calm periods by more calm periods.
Explanation: Volatility clustering (documented by Mandelbrot, formalised by Engle's ARCH model): volatility is autocorrelated — large moves (positive or negative) tend to be followed by more large moves. This violates the i.i.d. assumption and is a key driver of GARCH model development.
Question 12
The EWMA (Exponentially Weighted Moving Average) volatility model assigns:
Select an option first.
Correct answer: B — Exponentially declining weights to past observations, giving more importance to recent data.
Explanation: EWMA: σ²_t = λσ²_(t-1) + (1-λ)r²_(t-1). The decay factor λ (typically ~0.94 for daily data) means recent squared returns get higher weight, and weights decline exponentially with age. This allows volatility to react quickly to new information while retaining some memory of the past.
Question 13
In the EWMA model, a higher decay factor (λ closer to 1) means:
Select an option first.
Correct answer: C — The model places more weight on older observations relative to recent ones, making it slower to adapt to new conditions.
Explanation: With λ closer to 1, the weight on the previous variance (λ) is large and the weight on the new squared return (1−λ) is small. This makes the model slow to respond to new data — useful for stable long-term volatility estimates but less responsive to sudden market changes.
Question 14
The GARCH(1,1) model for conditional variance is written as σ²_t = ω + α·r²_(t-1) + β·σ²_(t-1). The long-run average variance is:
Select an option first.
Correct answer: A — ω / (1 − α − β)
Explanation: In GARCH(1,1), the unconditional (long-run) variance = ω / (1 − α − β), provided α + β < 1 (stationarity condition). If α + β ≥ 1, the process is integrated (IGARCH) and variance does not mean-revert. This formula is fundamental to GARCH interpretation.
Question 15
For a GARCH(1,1) model to be stationary (mean-reverting), which condition must hold?
Select an option first.
Correct answer: C — α + β < 1
Explanation: Stationarity requires α + β < 1. When α + β = 1 (IGARCH), variance is a random walk with no mean reversion. When α + β > 1, the process is explosive. The closer α + β is to 1 (but still less), the slower the mean reversion.
Question 16
Which of the following best describes the difference between realised volatility and implied volatility?
Select an option first.
Correct answer: B — Realised volatility is computed from historical price movements; implied volatility is extracted from market option prices.
Explanation: Realised (historical) volatility = statistical measure of past price variability. Implied volatility = the volatility that, when plugged into an option pricing model (e.g., BSM), produces the observed market option price. It represents the market's forward-looking expectation of volatility. A, C, D are all wrong.
Question 17
VaR is said to fail the sub-additivity property when:
Select an option first.
Correct answer: B — The combined VaR of two portfolios can exceed the sum of their individual VaRs under certain distributions.
Explanation: A coherent risk measure must be sub-additive: ρ(A+B) ≤ ρ(A) + ρ(B). VaR violates this under certain non-normal distributions — combining two positions can produce a higher VaR than the sum of the individual VaRs. This is a key reason Expected Shortfall (which is sub-additive) is preferred by regulators.
Question 18
Which of the following is a coherent risk measure?
Select an option first.
Correct answer: C — Expected Shortfall (CVaR)
Explanation: A coherent risk measure satisfies four axioms: monotonicity, sub-additivity, positive homogeneity, and translation invariance. Expected Shortfall satisfies all four. VaR fails sub-additivity under certain distributions. Standard deviation and variance lack monotonicity in the risk sense.
Question 19
The mean-variance framework for portfolio selection assumes investors choose portfolios based on:
Select an option first.
Correct answer: C — Both expected return (higher is better) and variance/standard deviation (lower is better).
Explanation: Markowitz's mean-variance framework: rational, risk-averse investors prefer higher expected return and lower variance. This two-dimensional criterion defines the efficient frontier. A ignores risk; B ignores return; D goes beyond the two-moment framework of standard MPT.
Question 20
A one-day 99% VaR of $2 million means that, on any given trading day:
Select an option first.
Correct answer: B — The portfolio will lose more than $2 million approximately 1% of trading days.
Explanation: VaR at 99%: losses will exceed $2m on approximately 1% of days (≈2-3 days/year). A confuses VaR with expected loss; C confuses VaR with maximum loss; D misidentifies the direction (VaR is a loss measure, not profit guarantee).
Question 21
Which of the following is the primary purpose of back-testing a VaR model?
Select an option first.
Correct answer: B — To compare actual daily P&L against the VaR estimate to verify whether the exception rate matches the model's intended confidence level.
Explanation: Back-testing counts VaR exceptions (days when actual loss > VaR). For a 99% VaR, the expected exception rate is 1% (~2.5 per year). Too many exceptions indicate the model understates risk. This is a fundamental model validation technique. A, C, D are separate processes.
Question 22
A VaR model generates 15 exceptions in 250 trading days at the 99% confidence level. This outcome suggests:
Select an option first.
Correct answer: C — The model significantly underestimates risk — far more exceptions than the expected ~2.5 occurred.
Explanation: Expected exceptions at 99% VaR over 250 days = 1% × 250 = 2.5. Actual = 15 — almost 6× the expected rate. Under Basel's traffic-light framework, this falls in the red zone (>10 exceptions), requiring additional capital and model investigation. A and B are wrong; D doesn't follow from the data.
Question 23
The delta-normal VaR approach approximates option positions using their deltas and then applies:
Select an option first.
Correct answer: B — A parametric (normal distribution) VaR calculation to the linearised portfolio.
Explanation: Delta-normal: convert option positions to their delta-equivalent underlying exposures, then compute VaR using the variance-covariance (parametric normal) method on the linearised positions. This is fast but ignores gamma (non-linearity). A uses delta but then simulates, C fully reprices, D is scenario analysis.
Question 24
The primary limitation of the delta-normal VaR approach for options is:
Select an option first.
Correct answer: B — It ignores the non-linear (gamma) component of option price changes, undervaluing risk for large moves.
Explanation: Delta approximates the option price change as linear in the underlying. For large moves, gamma (curvature) creates significant P&L changes that the linear approximation misses. This matters most for options with large gamma (e.g., near-expiry at-the-money options). A, C, D are incorrect.
Question 25
Stressed VaR (SVaR) differs from regular VaR in that it uses:
Select an option first.
Correct answer: C — A historical window calibrated to a period of significant financial market stress.
Explanation: SVaR (introduced under Basel 2.5) is calibrated to a 12-month stressed period (e.g., 2007-2009 financial crisis). This prevents VaR from becoming artificially low during calm market periods (pro-cyclicality problem). A, B, D are all incorrect descriptions.
Question 26
Which of the following explains why VaR estimates can be pro-cyclical?
Select an option first.
Correct answer: B — During calm periods, low historical volatility produces low VaR, encouraging leverage; during crises, high volatility produces high VaR, forcing deleveraging — amplifying the cycle.
Explanation: Pro-cyclicality of VaR: in good times, realised volatility is low → VaR is low → firms lever up. In bad times, volatility spikes → VaR explodes → firms must deleverage exactly when it is most harmful. This is a core critique of VaR-based regulation, addressed by stressed VaR and counter-cyclical capital buffers.
Question 27
Which of the following is an advantage of Monte Carlo VaR over historical simulation?
Select an option first.
Correct answer: B — Monte Carlo can incorporate forward-looking distributional assumptions and generate scenarios not present in history.
Explanation: Monte Carlo's advantage: it can be calibrated to any distribution (including fat-tailed ones), model time-varying volatility, and generate novel stress scenarios not in history. A is wrong — it requires explicit distributional assumptions. C is wrong — it is computationally intensive. D requires explicit fat-tail specification.
Question 28
In a parametric VaR calculation, if daily portfolio volatility is $500,000 and the z-score for 99% confidence is 2.33, the 1-day 99% VaR is:
Select an option first.
Correct answer: C — $1,165,000
Explanation: Parametric VaR = z × σ = 2.326 × $500,000 ≈ $1,163,000 ≈ $1,165,000. This assumes zero mean return (conservative). A is just σ; B doubles σ without the z-score; D multiplies by 4.66.
Question 29
The square-root-of-time (SRT) rule for scaling VaR assumes which critical condition?
Select an option first.
Correct answer: B — Returns are independently and identically distributed (i.i.d.).
Explanation: SRT rule: VaR_T = VaR_1 × √T. This derivation requires returns to be i.i.d. — no autocorrelation and no changing volatility. Violations (volatility clustering, mean-reverting prices, trending markets) make the rule inaccurate, typically understating multi-day VaR during volatile periods.
Question 30
Which of the following correctly describes 'weighted historical simulation' VaR?
Select an option first.
Correct answer: B — Assigning different weights to historical observations — typically higher weights to recent data — before computing the simulated VaR.
Explanation: Weighted (or volatility-weighted) historical simulation modifies standard historical simulation by giving more weight to recent data (similar to EWMA logic), allowing the model to respond more rapidly to changing market conditions while retaining the non-parametric distributional properties of historical simulation.
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