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Free Questions › FRM › Valuation and Risk Models
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 91
The Calmar ratio divides annualised return by:
Select an option first.
Correct answer: B — Maximum drawdown.
Explanation: Calmar ratio = Annualised return / Maximum drawdown. Used extensively in hedge fund evaluation where drawdown risk is a primary investor concern. Sharpe (A) uses SD; Treynor uses beta (C); Information ratio uses tracking error (D).
Question 92
Tracking error is defined as:
Select an option first.
Correct answer: B — Standard deviation of active returns (portfolio return minus benchmark return).
Explanation: Tracking error = σ(Rp − Rb). Measures consistency of active deviation from benchmark. Low TE = index-like; high TE = significant active bets. It is the denominator of the Information ratio.
Question 93
Holding 500 shares at $40 (daily σ = 2%), the 1-day 99% VaR is approximately:
Select an option first.
Correct answer: B — $932
Explanation: Portfolio value = $20,000. Dollar σ = 2% × $20,000 = $400. VaR = 2.326 × $400 = $930 ≈ $932. A is just σ; C doubles; D uses wrong scaling.
Question 94
A 1-day 99% VaR of $500,000 scaled to 10 days using the square-root rule gives:
Select an option first.
Correct answer: B — $1,581,139
Explanation: 10-day VaR = $500,000 × √10 = $500,000 × 3.1623 = $1,581,139. A multiplies by 10 (linear — only valid for variance, not VaR directly); C uses √5; D leaves unchanged.
Question 95
Risk decomposition in portfolio management primarily serves to:
Select an option first.
Correct answer: B — Attribute total portfolio risk to its constituent positions or risk factors for targeted hedging.
Explanation: Risk decomposition (marginal VaR, component VaR, factor contributions) identifies the largest drivers of total portfolio risk, enabling focused hedging where it matters most. A is administrative; C is standalone VaR; D is performance attribution.
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Question 96
Volatility of volatility (vol-of-vol) captures:
Select an option first.
Correct answer: B — How much volatility itself fluctuates over time.
Explanation: Vol-of-vol measures the variability of the volatility estimate itself. High vol-of-vol implies unstable, unpredictable volatility regimes — making single-point GARCH/EWMA estimates less reliable. It drives the Volga Greek and matters for exotic option pricing and regime detection.
Question 97
Two independent assets each with 1-day 99% VaR = $1 million. Under zero correlation and normality, the combined VaR is approximately:
Select an option first.
Correct answer: B — $1.41 million
Explanation: Zero correlation: Portfolio VaR = √(VaR₁² + VaR₂²) = √2 × $1m = $1.414m. A assumes perfect positive correlation (no diversification); C assumes perfect negative correlation; D uses √3. Diversification benefit = $2m − $1.41m = $0.59m.
Question 98
A delta-hedged options portfolio with gamma = 60 experiences an underlying move of $3. Approximate gamma P&L:
Select an option first.
Correct answer: B — $270
Explanation: Gamma P&L = ½ × Γ × (ΔS)² = ½ × 60 × 9 = $270. A uses Γ × ΔS (missing ½ and squaring); C doubles; D uses ½ × Γ × ΔS (not squared). The ½ coefficient comes from the Taylor expansion.
Question 99
EWMA with λ=0.97, current daily variance=0.000225, yesterday's return=−3%. Today's updated variance:
Select an option first.
Correct answer: B — 0.000245
Explanation: σ²_t = 0.97×0.000225 + 0.03×(0.03)² = 0.00021825 + 0.000027 = 0.000245. A omits the new information term; C and D overstate the shock impact. The 3% return squared = 0.0009; weighted by (1−λ)=0.03 gives 0.000027.
Question 100
A portfolio's monthly SD is 3%. The 95% one-month VaR as a percentage of portfolio value:
Select an option first.
Correct answer: B — 4.93%
Explanation: Monthly VaR₉₅ = 1.645 × 3% = 4.935% ≈ 4.93%. A is just σ; C uses z=1.96 (two-tailed 95%); D uses z=2.33 (99%).
Question 101
Risk-neutral probability Q versus real-world probability P: the key distinction is:
Select an option first.
Correct answer: B — Under Q all assets earn the risk-free rate (used for derivative pricing); P reflects actual returns including risk premia (used for VaR and default modelling).
Explanation: Under Q (equivalent martingale measure): discounting at rf gives correct derivative prices. Under P: assets earn their actual expected returns. VaR, physical default probability, and stress testing use P. Option pricing and derivative valuation use Q. A, C, D are incorrect.
Question 102
In VaR methodology, 'mapping' means:
Select an option first.
Correct answer: B — Converting complex or illiquid positions into simpler liquid risk factors for which market data exists.
Explanation: Mapping: instruments without direct price history (mortgages, exotic OTC derivatives) are expressed as equivalent exposures in standard liquid risk factors (yield curve vertices, equity index, FX rates). This enables VaR computation even for illiquid or complex instruments. A and C are unrelated; D is scaling.
Question 103
A position with negative marginal VaR means:
Select an option first.
Correct answer: B — Adding more of this position reduces total portfolio VaR — it acts as a natural hedge.
Explanation: Negative marginal VaR: the position is negatively correlated with the rest of the portfolio — increasing it reduces total risk. This is a hedging signal. Zero marginal VaR implies zero correlation (C). A confuses return direction with risk contribution; D is incorrect — negative marginal VaR is mathematically valid and meaningful.
Question 104
EWMA effective memory at λ=0.94 is approximately:
Select an option first.
Correct answer: B — 17 days
Explanation: Effective observations = 1/(1−λ) = 1/0.06 ≈ 16.7 ≈ 17 days. EWMA reacts much faster than a 250-day rolling window. At λ=0.97, effective days ≈ 33. This short memory is why EWMA responds quickly to market events but also why it is susceptible to the ghost effect.
Question 105
GARCH(1,1) with ω=0.00005, α=0.10, β=0.85, prior variance=0.0002, today's r²=0.01. New variance:
Select an option first.
Correct answer: A — 0.00122
Explanation: σ²_t = 0.00005 + 0.10×0.01 + 0.85×0.0002 = 0.00005 + 0.001 + 0.00017 = 0.00122. The large squared return (r²=0.01 = 1%) dominates the update through the α coefficient. B and C are too high; D is too low.
Question 106
The 'price of risk' in factor models represents:
Select an option first.
Correct answer: B — Expected excess return per unit of factor volatility — compensation for bearing systematic risk.
Explanation: Market price of risk λ = (E[R_factor]−rf)/σ_factor = factor Sharpe ratio. In CAPM: λ=(Rm−rf)/σm. Used in Girsanov's theorem (risk-neutral change of measure) and in quantifying how much investors demand per unit of systematic risk. A, C, D are incorrect.
Question 107
Implied volatility systematically overestimates realised volatility on average because:
Select an option first.
Correct answer: B — Implied vol is risk-neutral and embeds a variance risk premium — investors pay extra for variance insurance, inflating implied vol above expected realised vol.
Explanation: Variance risk premium: option buyers pay a premium above the expected realised variance for variance insurance. On average, implied vol > expected realised vol. Using implied vol in a physical-measure VaR model introduces upward bias. This is distinct from model calibration error (A) or data issues (C, D).
Question 108
Selling deep OTM puts to reduce VaR is dangerous because:
Select an option first.
Correct answer: B — It reduces measured VaR (captures only up to 99th percentile) while dramatically increasing losses beyond the threshold — catastrophic in tail events.
Explanation: VaR gaming: selling OTM puts lowers daily volatility and reduces VaR. But when the rare tail event occurs (put goes deep ITM), losses are catastrophic. Expected Shortfall would rise dramatically even as VaR falls — exactly why regulators moved to ES under the FRTB. ES captures the severity of tail losses that VaR ignores.
Question 109
Under CAPM equilibrium, the portfolio with the highest Sharpe ratio is:
Select an option first.
Correct answer: B — The market portfolio — all assets are correctly priced, so active management cannot improve the Sharpe ratio.
Explanation: Under CAPM assumptions (identical views, no transaction costs, rational agents), the market portfolio is the tangency portfolio with the highest Sharpe ratio. No investor can systematically outperform it on a risk-adjusted basis. This underpins the case for passive index investing. A, C, D are suboptimal.
Question 110
The two-fund separation theorem states:
Select an option first.
Correct answer: B — All investors combine the risk-free asset with the same risky tangency portfolio — only proportions differ by risk tolerance.
Explanation: Two-fund separation: (1) risk-free asset + (2) tangency portfolio. Risk-averse investors allocate more to risk-free; aggressive investors lever the tangency. The composition of the risky portion is identical for all rational investors — only the cash/risky split varies. This is the theoretical basis for passive investing.
Question 111
Doubling all portfolio positions under a coherent risk measure (positive homogeneity) changes ES by:
Select an option first.
Correct answer: B — Doubles ES.
Explanation: Positive homogeneity: ρ(λX) = λ×ρ(X). Doubling all positions doubles ES. This is one of the four coherence axioms (with monotonicity, sub-additivity, and translation invariance). This assumes no market impact from the larger position — valid for small portfolios relative to market size.
Question 112
For portfolios with options, the most accurate VaR method is:
Select an option first.
Correct answer: B — Historical simulation with full repricing of all instruments under each scenario.
Explanation: Full-revaluation historical simulation: each option is repriced under each historical scenario using BSM or binomial, correctly capturing gamma and other non-linearities. A and C miss gamma; D ignores diversification. Full repricing is computationally intensive but captures the true non-linear P&L distribution.
Question 113
97.5% ES under normality is approximately equivalent to which VaR level?
Select an option first.
Correct answer: B — 99% VaR
Explanation: Under normality, ES at 97.5% ≈ VaR at 99%. This motivated FRTB's move from 99% VaR to 97.5% ES — approximately equivalent capital under normality, but ES is coherent (sub-additive) and captures tail severity. This equivalence breaks down for fat-tailed distributions.
Question 114
The Sortino ratio improves on Sharpe for non-normal returns by:
Select an option first.
Correct answer: B — Using only downside deviation (SD of negative returns) rather than total SD.
Explanation: Sortino = (Rp−target)/downside deviation. For negatively skewed returns, Sharpe penalises upside volatility equally — Sortino only penalises harmful downside deviation. This gives a fairer picture of performance for strategies with asymmetric return distributions (e.g., options strategies). A is Treynor; C and D address different issues.
Question 115
The GARCH(1,1) α parameter measures:
Select an option first.
Correct answer: B — How strongly today's variance responds to last period's squared return shock.
Explanation: α (ARCH coefficient) = reaction to shocks. High α → model very reactive to new information. β (GARCH coefficient) = persistence of past variance. Long-run variance = ω/(1−α−β). The larger α, the faster the model responds to market events — useful for capturing fast-moving volatility regimes.
Question 116
Adding a position with negative incremental VaR: the correct action is:
Select an option first.
Correct answer: B — Accept it — it reduces total portfolio VaR by hedging existing exposures.
Explanation: Negative incremental VaR: the new position is negatively correlated with existing exposures and reduces total portfolio risk. This is the definition of a natural hedge. Adding it improves the risk profile. A misinterprets the sign; C is regulatory misrepresentation; D makes no risk management sense.
Question 117
GARCH persistence (α+β) = 0.99 means:
Select an option first.
Correct answer: B — Volatility shocks are very long-lived — half-life ≈ 69 days.
Explanation: Persistence near 1: shock half-life = ln(0.5)/ln(α+β) = ln(0.5)/ln(0.99) ≈ 69 days. At α+β=1 (IGARCH), shocks are permanent. At α+β=0.5, half-life ≈ 1 day. High persistence is typical of equity volatility — vol spikes from a market crash take months to fully dissipate. A describes low persistence; C applies when α+β>1; D describes zero persistence.
Question 118
Normal distribution VaR with portfolio excess kurtosis = 3 (fat tails). The model most likely:
Select an option first.
Correct answer: B — Underestimates the true 99% VaR — fat tails increase the probability of large losses beyond the normal distribution's prediction.
Explanation: Excess kurtosis > 0 (leptokurtosis) means more probability mass in the tails than the normal distribution assigns. The true 99th percentile loss exceeds the normal VaR estimate. Empirically, normal VaR understates 99% losses by 20-50% for equity portfolios, depending on the period and market conditions.
Question 119
FRTB 'non-modellable risk factors' (NMRFs) are treated separately because:
Select an option first.
Correct answer: B — They lack sufficient market data (fewer than 24 observable prices in 12 months) to calibrate a reliable ES model — requiring conservative stressed scenario charges.
Explanation: FRTB NMRFs: risk factors with insufficient market data are capitalised using stressed scenario charges rather than the ES model. This prevents capital arbitrage via exotic or illiquid risk factors. The threshold of 24 observable prices per year ensures only liquid, data-rich factors qualify for IMA treatment.
Question 120
A 130/30 long-short equity portfolio has higher risk than a 100% long portfolio with the same net exposure because:
Select an option first.
Correct answer: B — Higher gross exposure (160% vs 100%) creates more position-level idiosyncratic risk, short-squeeze risk, and factor exposures that net-exposure VaR understates.
Explanation: 130/30: same net market exposure but 60% more gross positions. Full-position VaR (not net-exposure VaR) is required to capture: idiosyncratic short-leg risks, potential short squeezes, factor crowding, and financing risk of the short positions. A and D are wrong; C is incorrect — short positions add, not reduce, idiosyncratic risk.
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