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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 121
In Basel trading book framework, 'specific risk' differs from 'general market risk' in that:
Select an option first.
Correct answer: B — General market risk = broad market movements (index, rate level); specific risk = idiosyncratic issuer risk (credit spread, individual stock).
Explanation: Specific risk (issuer-specific): single-name credit spread changes, individual equity price moves, default and migration risk. General market risk: systematic moves in interest rate levels, equity indices, FX rates. Both require separate capital treatment under Basel IMA (pre-FRTB) and continue under FRTB's SA and IMA frameworks.
Question 122
Portfolio VaR and ES have both declined over a quarter despite unchanged positions. The most likely explanation:
Select an option first.
Correct answer: B — Market volatility and correlations declined over the estimation window — common in benign periods, creating pro-cyclical risk underestimation.
Explanation: Pro-cyclicality: in calm markets, estimated volatilities and correlations decline → VaR/ES fall even with unchanged positions. This incentivises leverage in good times and forces deleveraging in bad times. Stressed VaR/ES (calibrated to crisis periods) addresses this by anchoring to high-volatility historical windows regardless of current conditions.
Question 123
The correlation matrix in parametric VaR must be positive semi-definite (PSD) because:
Select an option first.
Correct answer: B — PSD guarantees portfolio variance ≥ 0 for all possible portfolio weight vectors.
Explanation: Portfolio variance = w'Σw must be ≥ 0 for all w. This requires Σ to be PSD (all eigenvalues ≥ 0). Empirically estimated matrices from limited data often fail PSD due to sampling noise — regularisation techniques (shrinkage, eigenvalue clipping) are needed before use in VaR computations.
Question 124
'Importance sampling' in Monte Carlo VaR simulation:
Select an option first.
Correct answer: B — Concentrates simulations in the tail of the distribution — generating more tail scenarios per run to reduce estimation error for extreme quantiles.
Explanation: Importance sampling: change the sampling distribution to oversample the tail region, then reweight results to correct for the change. This dramatically reduces the number of simulations needed to accurately estimate extreme percentiles (99.9% VaR, ES). A is equal-weighted historical simulation; C is parametric; D is bootstrap sampling.
Question 125
Long gamma / short theta in a low realised volatility environment leads to:
Select an option first.
Correct answer: B — Daily theta decay exceeding gamma P&L from small moves — net loss if realised vol < implied vol.
Explanation: Long gamma costs theta daily. Breakeven: realised vol must exceed implied vol for gamma P&L (½Γ(ΔS)²) to exceed theta decay (|Θ|×Δt). In low-vol environments where realised vol < implied vol, the position bleeds theta. This is why buying options (long gamma) in calm markets is typically a losing strategy.
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Question 126
ES cannot be 'gamed' as easily as VaR because:
Select an option first.
Correct answer: B — ES averages losses beyond the threshold — strategies concentrating losses just beyond VaR will increase ES even if they reduce VaR.
Explanation: VaR gaming: selling OTM puts moves losses beyond the 99% threshold, reducing VaR while increasing tail severity. ES explicitly captures this average tail loss — any strategy that increases severity beyond the threshold increases ES. This makes ES more robust to the specific arbitrage that plagued VaR-based capital rules.
Question 127
FRTB liquidity horizons (10-120 days) for different risk factor categories reflect:
Select an option first.
Correct answer: B — Different risk factors take different time to hedge or exit under stress — illiquid factors require longer horizons than liquid ones.
Explanation: FRTB horizons: 10 days for G10 FX and large-cap investment-grade credit; up to 120 days for small-cap equity, high-yield, and exotic underlyings. The rationale: VaR/ES horizon should reflect realistic liquidation time under stress for each factor type. A uniform 10-day Basel 2.5 horizon understated risk for illiquid instruments.
Question 128
Back-testing differs from model validation in that:
Select an option first.
Correct answer: B — Back-testing compares VaR to actual P&L for exception counting; model validation is a broader independent review of assumptions, data, methodology, and scope.
Explanation: Back-testing = empirical exception counting (P-value test on exception rate). Model validation = comprehensive review: conceptual soundness, parameter estimation, sensitivity analysis, benchmarking to alternative models, documentation of limitations. Validation subsumes back-testing but is broader in scope. A, C, D are incorrect.
Question 129
High vol-of-vol implies for VaR model accuracy:
Select an option first.
Correct answer: B — VaR estimates become less reliable — single-point volatility estimates are unstable, and confidence intervals around VaR widen significantly.
Explanation: High vol-of-vol: volatility itself is highly uncertain and unpredictable. A GARCH/EWMA model produces a single volatility estimate, but when that estimate changes substantially from day to day, the VaR output is highly sensitive to recent data. Regime-switching models or stochastic volatility models are more appropriate under high vol-of-vol regimes.
Question 130
Clustered VaR exceptions (e.g., 3 consecutive days of exception) are more concerning than scattered exceptions because:
Select an option first.
Correct answer: B — Clustered exceptions indicate the model fails to capture volatility persistence — violations should be independent if GARCH dynamics are properly modelled.
Explanation: Christoffersen conditional coverage test: tests both (1) exception frequency and (2) serial independence. Clustered exceptions mean volatility is higher than the model estimates for extended periods — GARCH persistence is not captured. Even if the total exception count is correct, clustering reveals a model deficiency in handling volatility autocorrelation.
Question 131
PCA on a portfolio covariance matrix primarily enables:
Select an option first.
Correct answer: B — Identifying a small number of independent risk factors explaining most portfolio variance for dimensionality reduction and stable risk attribution.
Explanation: PCA transforms correlated asset returns into uncorrelated principal components ranked by variance explained. A few PCs (e.g., 5-10 for a large equity portfolio) often explain 80-95% of total variance. Benefits: (1) dimensionality reduction, (2) stable risk attribution, (3) identification of key risk drivers for hedging. A, C, D are incorrect.
Question 132
An Information ratio of 0.8 means:
Select an option first.
Correct answer: B — The manager generates 0.8 units of active return per unit of tracking error.
Explanation: IR = Active return / Tracking error = (Rp−Rb) / σ(Rp−Rb). IR = 0.8 means 0.8% alpha per 1% tracking error. Strong IR ≥ 0.5; exceptional IR ≥ 1.0. A is Sharpe ratio; C is R-squared; D is correlation.
Question 133
The most severe challenge for a delta-gamma neutral options portfolio is:
Select an option first.
Correct answer: B — Overnight price gaps — delta-gamma hedging cannot be adjusted before the jump occurs, leaving higher-order unhedged Greeks to create large P&L.
Explanation: Gap risk: delta-gamma hedging assumes continuous price movement. Overnight or intraday gaps (earnings surprise, geopolitical event) cannot be rebalanced. The position's P&L from the jump depends on speed, colour, and other higher-order Greeks that are not managed in standard delta-gamma frameworks.
Question 134
Shrinkage estimation of the covariance matrix is used in portfolio optimisation because:
Select an option first.
Correct answer: B — Sample covariance matrices from limited data have extreme eigenvalues; shrinkage toward a structured target reduces estimation error and produces more stable portfolio weights.
Explanation: Ledoit-Wolf shrinkage: blends the noisy sample covariance matrix with a structured target (e.g., constant correlation, diagonal). The optimal blend minimises out-of-sample portfolio variance. Without shrinkage, mean-variance optimisation amplifies estimation errors, producing extreme and unstable portfolio weights that perform poorly out-of-sample.
Question 135
The joint probability of multiple VaR exceptions being uncorrelated (independent) serves as the null hypothesis in the:
Select an option first.
Correct answer: B — Christoffersen conditional coverage test.
Explanation: Christoffersen (1998): jointly tests (1) unconditional coverage — correct exception frequency, AND (2) independence — exceptions are not serially correlated. The null hypothesis under independence is that the conditional probability of an exception on day t does not depend on whether day t-1 was also an exception. Failing this test suggests GARCH effects are not captured.
Question 136
Which of the following best describes the 'variance reduction' benefit of antithetic variates in Monte Carlo simulation?
Select an option first.
Correct answer: B — For each random scenario Z, also simulate −Z; averaging paired outcomes cancels odd-moment noise and reduces variance of the estimator by up to 50%.
Explanation: Antithetic variates: if Z generates scenario 1 (e.g., stock rises), −Z generates the mirror scenario (stock falls). The pair's average is a better estimator than either alone because positive and negative estimation errors cancel. This can reduce Monte Carlo standard error by up to 50% at no additional computational cost beyond pairing.
Question 137
The 'no-arbitrage' principle in finance states that:
Select an option first.
Correct answer: B — If two portfolios have identical future cash flows, they must have the same current price — otherwise, a risk-free profit opportunity exists.
Explanation: No-arbitrage (law of one price): identical cash flow patterns must have identical prices. If not, an investor can sell the expensive one and buy the cheap one, earning a risk-free profit. This principle underlies derivative pricing, bond pricing, and futures pricing. A is too strong; C and D are incorrect.
Question 138
The 'law of one price' is violated when:
Select an option first.
Correct answer: B — The same asset trades at two different prices in two markets simultaneously, allowing a risk-free profit.
Explanation: Law of one price violation = arbitrage opportunity. If IBM shares trade at $150 in NYSE and $152 in Tokyo, buying in NYC and selling in Tokyo earns a risk-free $2 profit per share. In efficient markets, such gaps close immediately. A is normal (credit spreads differ); C is model difference (no violation); D is expected (different cash flow patterns → different YTMs).
Question 139
The yield to maturity (YTM) of a bond is best described as:
Select an option first.
Correct answer: B — The single discount rate that makes the present value of all bond cash flows equal to the current bond price.
Explanation: YTM is the internal rate of return (IRR) of the bond: solve P = Σ[CF_t / (1+y)^t] for y. It assumes all coupons are reinvested at the YTM (which rarely holds in practice). A is the coupon rate; C is the price return; D is the amortisation rate.
Question 140
A bond with a face value of $1,000, annual coupon of 6%, and 5 years to maturity is priced at $1,000 (par). Its YTM is:
Select an option first.
Correct answer: C — Exactly 6%
Explanation: When a bond trades at par, its YTM exactly equals its coupon rate. This is because discounting the coupon and principal cash flows at the coupon rate produces a PV exactly equal to face value. Below par → YTM > coupon; above par → YTM < coupon.
Question 141
The 'current yield' of a bond is calculated as:
Select an option first.
Correct answer: A — Annual coupon payment divided by the bond's current market price.
Explanation: Current yield = Annual coupon / Current market price. E.g., $60 coupon / $950 price = 6.32%. It differs from YTM because it ignores the capital gain or loss to maturity. C is the coupon rate (coupon/face). D and B are not standard definitions.
Question 142
Duration measures a bond's:
Select an option first.
Correct answer: B — Sensitivity of price to interest rate changes, expressed in years.
Explanation: Duration (modified duration) measures price sensitivity: ΔP/P ≈ −D_mod × Δy. Macaulay duration measures the weighted average time to receive cash flows. Both are expressed in years and both capture interest rate risk. A is total cash flows; C is credit assessment; D is coupon rate.
Question 143
Modified duration is related to Macaulay duration by:
Select an option first.
Correct answer: B — Modified Duration = Macaulay Duration / (1 + y/m)
Explanation: Modified Duration = Macaulay Duration / (1 + y/m), where y is YTM and m is compounding frequency. For annual compounding (m=1): Mod D = Mac D / (1+y). This converts the time-weighted average (Macaulay) into a price sensitivity measure. A multiplies rather than divides; C and D have no mathematical basis.
Question 144
If a bond has a modified duration of 5 years and yields rise by 50 basis points, the approximate percentage price change is:
Select an option first.
Correct answer: B — -2.5%
Explanation: ΔP/P ≈ −Modified Duration × Δy = −5 × 0.005 = −0.025 = −2.5%. Rising yields → falling bond prices (negative sign). A has the wrong sign; C and D apply a 100bp change. Understanding the negative relationship between price and yield is fundamental.
Question 145
Convexity in bond pricing captures:
Select an option first.
Correct answer: B — The curvature of the price-yield relationship — the second-order effect that improves accuracy for large yield changes.
Explanation: Full price change: ΔP/P ≈ −D_mod × Δy + ½ × Convexity × (Δy)². Convexity (the ½C(Δy)² term) adds a positive curvature adjustment — for large yield moves, the linear duration approximation understates actual price gains and overstates price losses. Positive convexity is always beneficial to bondholders. A, C, D are incorrect.
Question 146
DV01 (Dollar Value of a Basis Point) is calculated as:
Select an option first.
Correct answer: B — Modified Duration × Current price × 0.0001
Explanation: DV01 = Modified Duration × Price × 0.0001. It is the dollar price change for a 1 basis point (0.01%) yield change. A uses 1% (100bp); C uses coupon (not duration or price); D is incorrect. DV01 is the practical tool for hedging bond duration — how many futures contracts are needed to offset a DV01.
Question 147
A bond with a DV01 of $500 and a futures contract with a DV01 of $100: how many futures contracts are needed to hedge the bond position?
Select an option first.
Correct answer: C — 5
Explanation: Hedge ratio = Bond DV01 / Futures DV01 = $500 / $100 = 5 futures contracts. Selling 5 futures contracts offsets the bond's DV01 exposure. A uses $100×1; B uses $500/10; D uses $500×1. This is the standard duration hedging calculation.
Question 148
The yield curve 'level' shift refers to:
Select an option first.
Correct answer: B — A parallel shift where all yields across all maturities move by the same amount.
Explanation: Yield curve shifts are decomposed into: (1) Level (parallel) — all maturities move equally; (2) Slope (twist) — short vs long yields change differently; (3) Curvature (butterfly) — the middle of the curve moves relative to the ends. Level shift is the largest component of yield curve variance (~80%).
Question 149
A 'butterfly' shift in the yield curve refers to:
Select an option first.
Correct answer: B — The short and long ends of the curve moving in opposite direction to the middle (or vice versa).
Explanation: Butterfly shift: the middle of the curve (2-5 year) rises (or falls) while both wings (short-term and long-term) move in the opposite direction. This changes the curvature of the yield curve. Portfolios with exposure to the curve's curvature need butterfly hedges (key rate durations or PCA-based hedges).
Question 150
In a binomial tree for option pricing, the underlying asset can move to:
Select an option first.
Correct answer: B — Only two possible prices each period: an 'up' move (u) or a 'down' move (d).
Explanation: The binomial tree discretises the continuous BSM framework into discrete time steps. At each node, the underlying moves to exactly two outcomes: S×u (up) or S×d (down). By increasing the number of steps, the binomial model converges to the BSM. A and D describe continuous models; C is a trinomial tree.
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