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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 181
Which of the following statements about the normal distribution is correct?
Select an option first.
Correct answer: A — It is characterised by a mean and variance, and is symmetric about its mean
Explanation: The normal (Gaussian) distribution is symmetric, bell-shaped, and fully described by two parameters: mean (μ) and variance (σ²). B, C, and D are all incorrect; skewness of normal = 0, it has well-defined moments, and it takes values over the entire real line.
Question 182
In a standard normal distribution, approximately what percentage of observations fall within two standard deviations of the mean?
Select an option first.
Correct answer: C — 95%
Explanation: The empirical rule: ~68% within ±1σ, ~95% within ±2σ, ~99.7% within ±3σ. This is fundamental to VaR calculations using the normal distribution.
Question 183
Kurtosis measures:
Select an option first.
Correct answer: B — The fatness of the tails relative to a normal distribution
Explanation: Kurtosis measures tail heaviness. Excess kurtosis > 0 (leptokurtosis) indicates fatter tails than the normal distribution. Skewness (A) measures symmetry; mean (C) is central tendency; mode (D) is the most frequent value.
Question 184
Skewness in a return distribution refers to:
Select an option first.
Correct answer: B — The asymmetry of the distribution around its mean
Explanation: Skewness captures asymmetry: positive skew = longer right tail; negative skew = longer left tail. Financial returns often exhibit negative skew (large downside losses are more common than large upside gains). A is mean; C is variance/SD; D is probability-related.
Question 185
Which of the following best describes the concept of 'basis points' used in finance?
Select an option first.
Correct answer: B — One basis point equals 0.01% (or 1/100th of 1%)
Explanation: 1 basis point (bp) = 0.01% = 0.0001. So 100 bp = 1%. This is the standard unit for expressing changes in interest rates, credit spreads, and yields.
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Question 186
Historical simulation VaR is computed by:
Select an option first.
Correct answer: B — Using actual historical returns ranked from worst to best and reading the return at the desired confidence percentile
Explanation: Historical simulation uses actual past returns without distributional assumptions. The 1-day 99% VaR is the 1st percentile of the empirical return distribution. A is parametric/analytical VaR; C is Monte Carlo VaR; D is the sum of individual VaRs (not a standard method).
Question 187
Parametric (analytical) VaR assumes:
Select an option first.
Correct answer: C — Portfolio returns follow a specific distribution (typically normal) characterised by mean and standard deviation
Explanation: Parametric VaR (also called variance-covariance) uses statistical parameters (mean, standard deviation, correlations) and assumes a specific distribution (usually normal). A is historical simulation; B is Monte Carlo; D is non-parametric.
Question 188
Which of the following is a key limitation of historical simulation VaR?
Select an option first.
Correct answer: B — It relies entirely on past data and may not capture risks from events not present in the historical window
Explanation: Historical simulation is backward-looking — if the sample period doesn't include a specific type of crisis, that risk will not appear in the VaR estimate. A is wrong (it makes NO distributional assumptions); C is partially true but not the key limitation; D is incorrect.
Question 189
Monte Carlo simulation for VaR involves:
Select an option first.
Correct answer: B — Generating a large number of random scenarios from specified statistical models to build a distribution of portfolio outcomes
Explanation: Monte Carlo VaR simulates thousands or millions of random paths based on assumed distributions and correlations, producing a full distribution of outcomes. A is historical simulation; C is parametric VaR; D is scenario analysis/stress testing.
Question 190
The primary advantage of Expected Shortfall (ES) over VaR for risk management purposes is:
Select an option first.
Correct answer: B — ES captures the average severity of losses in the tail, not just the probability of exceeding a threshold
Explanation: ES (CVaR) provides information about the magnitude of losses beyond the VaR threshold, addressing VaR's primary blind spot. A is incorrect (ES is more complex); C is incorrect (ES is always ≥ VaR, not lower); D is incorrect (regulators use multiple measures).
Question 191
Back-testing a VaR model involves:
Select an option first.
Correct answer: B — Comparing actual portfolio losses to the VaR estimate to check whether the exception rate matches the expected confidence level
Explanation: Back-testing is the empirical validation of VaR: count the days when actual losses exceeded VaR. For a 99% VaR, exceptions should occur ~1% of the time. Too many exceptions = model underestimates risk. A, C, and D are different processes.
Question 192
Which of the following is the formula for RAROC?
Select an option first.
Correct answer: B — Risk-adjusted net income / Economic capital
Explanation: RAROC = Risk-adjusted net income / Economic capital. It standardises returns across business units by accounting for their economic capital consumption. A is a banking efficiency metric; C is ROE variant; D is a loss ratio, not RAROC.
Question 193
Which of the following describes a positively skewed distribution?
Select an option first.
Correct answer: C — The distribution has a longer right tail; extremely large gains are more common than extremely large losses
Explanation: Positive skewness = longer right tail. Most observations cluster to the left, with occasional large positive values pulling the tail right. A is negative skewness (relevant for most financial returns which are negatively skewed). B is symmetric; D is kurtosis.
Question 194
Which of the following explains why financial return distributions often exhibit 'negative skewness' and 'excess kurtosis'?
Select an option first.
Correct answer: B — Market crashes and liquidity crises produce large, infrequent negative returns and volatility clustering, creating fat left tails and occasional extreme events
Explanation: Financial returns exhibit non-normality because: (1) crashes cause large sudden losses (fat left tails / negative skew), (2) volatility clusters over time, and (3) liquidity evaporation amplifies moves. A and D assume normality incorrectly; C is empirically false.
Question 195
The correlation between the returns of a portfolio and the market is 0.8, and the standard deviations are 15% (portfolio) and 12% (market). What is the portfolio's beta?
Select an option first.
Correct answer: B — 1.0
Explanation: β = ρ × (σp / σm) = 0.8 × (15/12) = 0.8 × 1.25 = 1.0. The portfolio has a beta of 1.0, meaning it moves exactly with the market on average. Candidates must not confuse correlation with beta.
Question 196
A zero-coupon bond with face value $1,000 matures in 5 years. If the required yield is 4% annually, its price is approximately:
Select an option first.
Correct answer: C — $822
Explanation: PV = 1000 / (1.04)^5 = 1000 / 1.2167 ≈ $822. A ($780) uses too high a rate; B ($820) is close but slightly off; D ($962) incorrectly applies a 1-year discount.
Question 197
The relationship between bond price and yield is:
Select an option first.
Correct answer: B — Inverse — when yields rise, bond prices fall
Explanation: Bond price is the present value of future cash flows discounted at the yield. When the discount rate (yield) rises, the PV of those cash flows falls — hence the inverse relationship. This is one of the most fundamental fixed-income concepts.
Question 198
If a firm has total assets of $500 million and total liabilities of $350 million, its equity is:
Select an option first.
Correct answer: B — $150 million
Explanation: Equity = Total assets − Total liabilities = $500m − $350m = $150m. This is the basic accounting identity: Assets = Liabilities + Equity. A adds rather than subtracts; C and D are simply re-stating input values.
Question 199
The internal rate of return (IRR) of an investment project is:
Select an option first.
Correct answer: A — The discount rate at which the NPV equals zero
Explanation: IRR is the rate that equates PV of cash inflows to PV of cash outflows (making NPV = 0). It is used as the hurdle rate comparison: if IRR > cost of capital, the project creates value. B is simple payback; C is CAPM-style required return; D is ROA.
Question 200
The gross profit margin is calculated as:
Select an option first.
Correct answer: B — ((Revenue − Cost of Goods Sold) / Revenue) × 100
Explanation: Gross profit margin = (Revenue − COGS) / Revenue × 100. It measures core production profitability before operating expenses. A is net profit margin; C is ROA variant; D is EBITDA margin.
Question 201
The net profit margin is:
Select an option first.
Correct answer: C — Net income / Revenue
Explanation: Net profit margin = Net income / Revenue. It captures profitability after all costs including interest and tax. A is gross margin; B is operating margin; D is ROA variant.
Question 202
An asset has an expected return of 8% and a standard deviation of 12%. Using the normal distribution, the 95th percentile loss (approximate one-year VaR at 95%) is approximately:
Select an option first.
Correct answer: B — −11.7%
Explanation: At 95% confidence (5% left tail), z = −1.645. VaR ≈ −(μ − z×σ) = −(8% − 1.645×12%) = −(8% − 19.74%) = −11.74% ≈ −11.7%. A just uses 1 standard deviation; C has wrong sign; D is above the mean.
Question 203
Which of the following describes 'volatility clustering' in financial return series?
Select an option first.
Correct answer: B — Periods of high volatility tend to be followed by high volatility, and low volatility by low volatility
Explanation: Volatility clustering — documented by Mandelbrot and formalised in GARCH models — shows that volatility is time-varying and autocorrelated. This violates the constant-variance assumption of simple VaR models and is a key feature of financial time series.
Question 204
GARCH models are primarily used in risk management to:
Select an option first.
Correct answer: B — Forecast time-varying conditional volatility for use in VaR and option pricing models
Explanation: GARCH (Generalized Autoregressive Conditional Heteroskedasticity) models capture volatility clustering by allowing today's variance to depend on yesterday's variance and squared return. They improve VaR estimates by using dynamic rather than rolling-window volatility.
Question 205
The Sortino ratio differs from the Sharpe ratio by using:
Select an option first.
Correct answer: B — Downside deviation (the standard deviation of negative returns only) in the denominator
Explanation: Sortino ratio = (Rp − target rate) / downside deviation. It penalises only downside volatility, recognising that upside variability is not harmful. This makes it preferable for asymmetric return distributions. A is Treynor; C and D are incorrect.
Question 206
An options delta of 0.6 means that:
Select an option first.
Correct answer: B — For a $1 rise in the underlying asset price, the option price is expected to rise by approximately $0.60
Explanation: Delta (Δ) is the first derivative of option price with respect to the underlying price. Δ = 0.6 means the option moves $0.60 per $1 move in the underlying. A approximates the risk-neutral probability of exercise (also close to 0.6 but not the definition); C and D are unrelated.
Question 207
A firm has ROE of 15% and a total equity of $200 million. What is the net income?
Select an option first.
Correct answer: B — $30 million
Explanation: Net income = ROE × Equity = 15% × $200m = $30m. A uses wrong denominator; C inverts the ratio; D applies 1.5%.
Question 208
The DuPont decomposition breaks ROE into three components. Which of the following correctly states these components?
Select an option first.
Correct answer: B — Net profit margin, asset turnover, and equity multiplier (leverage)
Explanation: DuPont: ROE = (Net income/Revenue) × (Revenue/Total assets) × (Total assets/Equity) = Net profit margin × Asset turnover × Equity multiplier. This separates profitability, efficiency, and leverage contributions to ROE.
Question 209
A risk manager calculates a 1-day 99% VaR of $5 million. To scale this to a 10-day VaR under the assumption of i.i.d. normally distributed returns, the calculation would be:
Select an option first.
Correct answer: B — $5m × √10 ≈ $15.8m
Explanation: Under i.i.d. returns, VaR scales with the square root of time: 10-day VaR = 1-day VaR × √10 ≈ $5m × 3.162 ≈ $15.8m. A uses a linear (wrong) scalar; C squares time; D divides (wrong direction).
Question 210
What does a VaR 'exception' (or 'exceedance') indicate?
Select an option first.
Correct answer: B — A trading day where actual losses exceed the VaR threshold
Explanation: An exception is when actual portfolio loss > VaR on a given day. For a 99% VaR, we expect ~1% exceptions (about 2-3 days per year for daily VaR). Too many exceptions suggest the model underestimates risk.
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