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Free FRM Foundations of Risk Management Practice Questions & Answers
400 exam-style Foundations of Risk Management questions. Pick your answer, hit Check answer, and see the worked solution — free to start, no signup.
100% free · No login to startQuestion 241
A lognormal distribution is often used to model asset prices because:
Select an option first.
Correct answer: B — It ensures asset prices are always positive and that returns (log price ratios) are normally distributed
Explanation: If ln(St/S0) is normally distributed (i.e., log returns are normal), then prices follow a lognormal distribution — always positive (prices cannot go below zero for limited-liability assets). A allows negatives (wrong); C is incorrect (it is more complex than normal); D is wrong (lognormal underestimates fat tails).
Question 242
Which of the following is the correct interpretation of a p-value in hypothesis testing?
Select an option first.
Correct answer: B — The probability of observing a result at least as extreme as the one obtained, assuming the null hypothesis is true
Explanation: The p-value is the probability of observing results as extreme or more extreme than those obtained, IF the null hypothesis were true. It is NOT the probability that H0 is true (A) or that H1 is true (C). The significance level (D) is α, the threshold against which p is compared.
Question 243
Under a parametric VaR approach with normally distributed returns, a 1-day 99% VaR for a position with daily mean return of 0% and daily standard deviation of $100,000 is approximately:
Select an option first.
Correct answer: C — $233,000
Explanation: At 99% (z = 2.326): VaR = z × σ = 2.326 × $100,000 = $232,600 ≈ $233,000. At 95% (z = 1.645): VaR ≈ $165,000 (B). A uses z=1; D uses z=3.28.
Question 244
Which of the following is the key property that distinguishes a stationary time series from a non-stationary one?
Select an option first.
Correct answer: B — A stationary series has constant mean, variance, and autocovariance structure over time, making statistical inference reliable
Explanation: Stationarity is required for most time series statistical techniques to be valid. Non-stationary series (with trends, structural breaks) violate these assumptions. A is unnecessarily restrictive; C and D describe non-stationary characteristics.
Question 245
An Augmented Dickey-Fuller (ADF) test is primarily used to:
Select an option first.
Correct answer: B — Test whether a time series contains a unit root (is non-stationary)
Explanation: The ADF test (H0: unit root / non-stationary; H1: stationary) is the standard test for stationarity. Rejecting H0 means the series is stationary. A is the Engle-Granger/Johansen test; C is the Jarque-Bera test; D is the Durbin-Watson or Ljung-Box test.
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Question 246
Which of the following correctly describes 'autocorrelation' (or serial correlation) in a return series?
Select an option first.
Correct answer: B — The correlation between a return series and its own past values
Explanation: Autocorrelation measures the relationship between a series and its lagged values. Significant autocorrelation in returns suggests predictability (violating weak-form EMH) and is important for risk model calibration. A is cross-asset correlation; C is kurtosis-related; D is tracking.
Question 247
The EWMA (Exponentially Weighted Moving Average) volatility model differs from a simple rolling window model by:
Select an option first.
Correct answer: B — Assigning exponentially declining weights to observations, giving more weight to recent data and less to older data
Explanation: EWMA uses a decay factor (λ) to weight recent observations more heavily: σ²t = λσ²t-1 + (1-λ)r²t-1. This allows volatility to respond quickly to recent market changes. A is the simple rolling window; C is too extreme; D is the constant volatility assumption EWMA is designed to avoid.
Question 248
A portfolio has a 1-day 99% VaR of $2 million. The portfolio manager estimates that on the 1% of days when losses exceed VaR, the average loss is $3.5 million. The Expected Shortfall (ES) at 99% confidence is:
Select an option first.
Correct answer: C — $3.5 million
Explanation: ES at 99% = the average loss given that loss > VaR = $3.5 million. ES is always ≥ VaR (here $3.5m > $2m). A is VaR; B adds VaR and ES incorrectly; D subtracts.
Question 249
Which of the following correctly describes the put-call parity relationship for European options?
Select an option first.
Correct answer: B — Call price − Put price = Underlying price − PV(Strike price)
Explanation: Put-call parity: C − P = S − PV(K), where C = call price, P = put price, S = spot price, PV(K) = present value of strike. This is an arbitrage relationship for European options on non-dividend paying stocks. A rearranges incorrectly; C multiplies; D is only approximately true.
Question 250
For a European call option, which of the following factors increases the option's value (all else equal)?
Select an option first.
Correct answer: C — An increase in the volatility of the underlying asset
Explanation: Higher volatility increases option value (both calls and puts) because it increases the probability of the option expiring deep in the money. A (higher strike) reduces call value; B (less time) reduces value; D is partially correct but stated incorrectly — higher rates increase call value for European options.
Question 251
Which of the following describes 'theta' in options pricing?
Select an option first.
Correct answer: C — The rate at which the option loses value purely due to the passage of time (time decay)
Explanation: Theta (Θ) = ∂V/∂t — time decay. Options lose value as they approach expiry because the probability of a favourable move decreases. A is delta; B is vega; D is gamma.
Question 252
Vega measures:
Select an option first.
Correct answer: A — The sensitivity of an option's price to a 1% change in implied volatility
Explanation: Vega (ν) = ∂V/∂σ — the change in option price per 1% change in implied volatility. Long options have positive vega (benefit from rising vol); short options have negative vega. B is theta; C is gamma; D is rho.
Question 253
The Black-Scholes-Merton model for option pricing assumes which of the following?
Select an option first.
Correct answer: B — Asset prices follow geometric Brownian motion (log returns are normally distributed with constant volatility)
Explanation: BSM assumes: GBM price process, constant volatility, continuous trading, no dividends (base case), constant risk-free rate, no arbitrage. These assumptions are simplifications — fat tails and volatility smiles indicate real-world deviations from BSM. A, C, and D all violate BSM assumptions.
Question 254
Which of the following correctly describes 'implied volatility'?
Select an option first.
Correct answer: C — The volatility that, when input into an option pricing model, produces the observed market price of the option
Explanation: Implied volatility is backed out from the observed option market price using an options model (e.g., BSM). It reflects the market's consensus expectation of future volatility. A is historical (realised) volatility; B is GARCH forecast; D is option price volatility.
Question 255
The 'volatility smile' observed in options markets refers to:
Select an option first.
Correct answer: C — A U-shaped pattern where implied volatility is higher for deep in-the-money and deep out-of-the-money options than for at-the-money options
Explanation: The volatility smile contradicts BSM's constant volatility assumption. In equity markets, a volatility 'skew' (higher implied vol for low strikes/OTM puts) is more common — reflecting demand for downside protection. C is the general smile shape observed in some markets (FX, especially).
Question 256
A firm has EBIT of $50 million, annual interest expense of $10 million, and a tax rate of 25%. What is the firm's net income?
Select an option first.
Correct answer: A — $30 million
Explanation: EBT = EBIT − Interest = $50m − $10m = $40m. Net income = EBT × (1−tax rate) = $40m × 0.75 = $30m. B is EBT; C applies tax to EBIT; D deducts tax incorrectly.
Question 257
Which of the following correctly describes the difference between 'systematic' and 'unsystematic' risk in portfolio theory?
Select an option first.
Correct answer: B — Systematic risk is market-wide and non-diversifiable; unsystematic risk is firm-specific and can be reduced through diversification
Explanation: Systematic (beta) risk = market risk, non-diversifiable. Unsystematic (alpha/idiosyncratic) risk = firm-specific, diversifiable in a large portfolio. A reverses the definitions. C is incorrect — systematic risk cannot be diversified away. D is incorrect.
Question 258
Which of the following is the primary use of the Capital Market Line (CML) in investment theory?
Select an option first.
Correct answer: B — To determine the optimal combination of the risk-free asset and the market portfolio for a specific investor, based on their risk tolerance
Explanation: The CML determines the best portfolio for each investor by combining the risk-free asset and the market (tangency) portfolio in proportions reflecting the investor's risk aversion. A and C describe the SML (for individual assets, not portfolios); D is a single point on the efficient frontier.
Question 259
A risk manager observes that the 10-day VaR calculated using the square-root-of-time rule is lower than the 10-day VaR from a Monte Carlo simulation. Which of the following best explains this discrepancy?
Select an option first.
Correct answer: B — The SRT rule assumes i.i.d. returns; Monte Carlo captures volatility clustering and fat tails, which increase multi-day VaR for realistic return distributions
Explanation: SRT = sqrt(10) × 1-day VaR assumes i.i.d. normal returns. Monte Carlo, when calibrated to real data (with GARCH volatility, fat tails), produces a higher multi-day VaR because: (1) volatility clustering means high-vol days cluster together, and (2) tail events are more frequent than normal. B is the correct explanation.
Question 260
Which of the following is the correct interpretation of the 'DV01' or 'PVBP' of a bond?
Select an option first.
Correct answer: B — The change in the bond's price for a 1 basis point (0.01%) increase in yield
Explanation: DV01 (Dollar Value of a Basis Point) = Modified Duration × Price × 0.0001. It is the price sensitivity to a 1bp yield change — a key practical tool for hedging. A is the yield; C is a different duration concept; D is a coupon calculation.
Question 261
A fixed income portfolio manager wants to hedge interest rate risk using Treasury futures. To calculate the number of futures contracts needed, the primary inputs are:
Select an option first.
Correct answer: B — The portfolio's DV01 and the futures contract's DV01
Explanation: Hedge ratio = Portfolio DV01 / Futures contract DV01. This calculates how many futures contracts, each with a given DV01, are needed to offset the portfolio's interest rate sensitivity. A, C, and D are irrelevant inputs for a duration hedge.
Question 262
Which of the following best describes the 'variance-covariance approach' to calculating portfolio VaR for a multi-asset portfolio?
Select an option first.
Correct answer: C — It expresses portfolio variance as a function of individual asset variances and pairwise covariances (captured in a covariance matrix), then derives VaR analytically
Explanation: Variance-covariance (parametric) VaR: σ²portfolio = w'Σw (where Σ is the covariance matrix, w is the weight vector). VaR = z × σp × Portfolio value. A is Monte Carlo; B is historical simulation; D ignores correlation (overestimates if positive, underestimates if negative).
Question 263
Which of the following best describes 'principal component analysis' (PCA) as applied to yield curve risk?
Select an option first.
Correct answer: B — A technique that identifies the main independent factors driving yield curve movements — typically: level shift, slope change, and curvature change — and expresses portfolio sensitivity to each
Explanation: PCA on the yield curve typically reveals ~3 main components explaining >95% of yield curve movements: (1) parallel shift (level), (2) steepening/flattening (slope), (3) butterfly/curvature. This enables parsimonious hedging and risk decomposition. A, C, D describe different tools.
Question 264
When calculating credit VaR for a loan portfolio, the key challenge compared to market VaR is:
Select an option first.
Correct answer: B — Credit return distributions are highly asymmetric (rare, large losses) and data is sparse, making standard normal-distribution approaches inappropriate
Explanation: Credit loss distributions are: left-skewed (many small gains from interest income, rare large losses from defaults), fat-tailed, and asymmetric. Normal distribution assumptions (valid for market risk) dramatically understate credit tail risk. A, C, and D are all incorrect.
Question 265
The 'convexity adjustment' in fixed income is necessary because:
Select an option first.
Correct answer: B — Duration is only a first-order (linear) approximation; for large yield changes, the actual price change includes a significant second-order (convexity) component that duration alone misses
Explanation: ΔP/P ≈ −D×Δy + ½×Convexity×(Δy)². For small Δy, the convexity term is negligible; for large moves (e.g., 200bp), convexity can be significant. A is incorrect (linear approximation); C is wrong; D is partially true for options-adjusted convexity but standard bonds also have convexity.
Question 266
Which of the following best describes 'copula functions' in the context of credit risk modelling?
Select an option first.
Correct answer: B — A statistical technique for modelling the joint distribution of defaults by separating the marginal distributions from the dependence structure (correlation)
Explanation: Copulas (e.g., Gaussian copula) separate the marginal default probability distributions from the correlation structure, enabling flexible joint default modelling. The Gaussian copula was widely used (and criticised) in CDO pricing before 2008 because it underestimated tail dependence in correlated defaults.
Question 267
What is the key insight of the Modigliani-Miller theorem (in a world without taxes and frictions) regarding capital structure?
Select an option first.
Correct answer: B — A firm's total value is independent of its capital structure — the mix of debt and equity does not affect firm value
Explanation: In a frictionless world (no taxes, no bankruptcy costs, symmetric information), the way a firm is financed does not affect its total value — investors can replicate any leverage ratio themselves (homemade leverage). In reality, taxes and bankruptcy costs create optimal capital structure tradeoffs.
Question 268
Which of the following best describes 'duration drift' and its implications for a fixed income portfolio?
Select an option first.
Correct answer: B — The change in portfolio duration over time as bonds age and market yields change, requiring periodic rebalancing to maintain a target duration hedge
Explanation: Duration drift: as time passes and yields change, a bond's (and portfolio's) duration changes — even without trading. For duration-matched hedges (e.g., pension LDI), this drift means the hedge gradually becomes misaligned, requiring rebalancing. A and D are different concepts; C relates to yield accrual.
Question 269
Which of the following is the correct formula for calculating compound annual growth rate (CAGR)?
Select an option first.
Correct answer: B — (Ending Value / Beginning Value)^(1/n) − 1
Explanation: CAGR = (EV/BV)^(1/n) − 1, where n = number of years. This converts total growth into an annualised equivalent rate. A doesn't annualise properly; C is an average annual absolute change; D has an operator precedence error and is not CAGR.
Question 270
The 'information coefficient' (IC) in active portfolio management measures:
Select an option first.
Correct answer: A — The correlation between a manager's forecasts and actual subsequent returns
Explanation: IC measures forecast skill — the correlation between ex-ante return predictions and ex-post realised returns. IC = 1 is perfect foresight; IC = 0 is no skill; IC = −1 means systematically wrong. B is the information ratio; C is alpha per trade; D is breadth (N in the fundamental law).
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