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Free Questions › FRM › Foundations of Risk Management
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 211
Which of the following is a key property that makes Expected Shortfall (ES) superior to VaR as a coherent risk measure?
Select an option first.
Correct answer: B — ES satisfies the sub-additivity property — the combined portfolio ES is always ≤ the sum of individual ESs, ensuring diversification is properly rewarded
Explanation: A coherent risk measure must be sub-additive: ρ(A + B) ≤ ρ(A) + ρ(B). VaR fails sub-additivity under certain distributions (it can suggest that combining two portfolios increases risk). ES satisfies all four coherence axioms. C and D are incorrect; A is also incorrect.
Question 212
Stress VaR (SVaR), introduced under Basel 2.5, differs from regular VaR in that it:
Select an option first.
Correct answer: B — Is calibrated to a period of significant financial stress rather than the most recent data window
Explanation: SVaR uses a historical window that includes a period of significant financial stress (e.g., 2007-09 crisis) to ensure capital requirements are not artificially low during calm periods. This addresses the pro-cyclicality problem of standard VaR.
Question 213
Which of the following best explains why the square-root-of-time rule for scaling VaR may underestimate risk over longer horizons?
Select an option first.
Correct answer: B — The assumption of i.i.d. returns breaks down in practice due to volatility clustering, fat tails, and mean reversion in prices
Explanation: The sqrt(T) rule requires i.i.d. normally distributed returns. In reality: volatility clusters (violations of i.i.d.), returns exhibit fat tails (non-normal), and some price series mean-revert. These all make the simple scaling rule inaccurate.
Question 214
A portfolio has the following characteristics: Expected return = 10%, Standard deviation = 20%, Risk-free rate = 3%. The Sharpe ratio is:
Select an option first.
Correct answer: A — 0.35
Explanation: Sharpe = (10% − 3%) / 20% = 7% / 20% = 0.35. B uses (10/20); C uses (8/20); D inverts the ratio. The Sharpe ratio of 0.35 means the portfolio earns 0.35 units of excess return per unit of total risk.
Question 215
In the CAPM framework, the Security Market Line (SML) shows:
Select an option first.
Correct answer: B — The relationship between systematic risk (beta) and expected return — any security plotted above the SML is undervalued
Explanation: The SML plots expected return vs. beta for individual securities. Securities above the SML have positive alpha (undervalued; expected return exceeds CAPM prediction). A describes the CML (total risk); C is the efficient frontier; D is fixed income.
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Question 216
A bond with a face value of $1,000, a 6% annual coupon, 3 years to maturity, and a required yield of 8% has a price of approximately:
Select an option first.
Correct answer: A — $948
Explanation: PV = 60/(1.08) + 60/(1.08)² + 1060/(1.08)³ = 55.56 + 51.44 + 841.40 = $948.40. B is par (yield = coupon rate); C is wrong direction; D is close but not the precise answer.
Question 217
A portfolio manager runs a back-test and finds 12 exceptions out of 250 trading days using a 99% VaR model. Which conclusion is most justified?
Select an option first.
Correct answer: B — The model appears to be underestimating risk — expected exceptions are ~2.5, so 12 suggests the model is too optimistic
Explanation: Expected exceptions at 99% VaR = 1% × 250 = 2.5 days. Actual = 12, far more than expected. Under Basel rules, this falls in the 'red zone' (>10 exceptions), indicating the model significantly underestimates risk and requires recalibration.
Question 218
The Efficient Market Hypothesis (EMH) in its strong form states that:
Select an option first.
Correct answer: C — All information — public and private (insider) — is fully reflected in market prices
Explanation: The three forms: weak (historical prices only), semi-strong (all public information), and strong (all information including insider). Strong-form EMH implies no investor, not even insiders, can consistently earn abnormal returns.
Question 219
A swap where one party pays a fixed interest rate and receives a floating rate is used to:
Select an option first.
Correct answer: B — Convert a fixed-rate liability into an effective floating-rate liability
Explanation: Paying fixed and receiving floating on a swap synthetically converts fixed-rate funding into floating-rate funding. This is used by borrowers who want to reduce fixed-rate exposure. A incorrectly describes a no-change transaction; C is too absolute; D is wrong — this reduces duration.
Question 220
Which of the following correctly states the relationship between EBITDA, EBIT, and net income (top-down)?
Select an option first.
Correct answer: B — EBITDA → deduct D&A → EBIT → deduct interest and taxes → net income
Explanation: The correct waterfall: EBITDA minus D&A = EBIT; minus interest = EBT; minus taxes = net income. A works bottom-up but states it incorrectly; C and D have the wrong sequence/direction.
Question 221
A firm's free cash flow (FCF) is best described as:
Select an option first.
Correct answer: B — Operating cash flow minus capital expenditure
Explanation: Free Cash Flow = Operating Cash Flow − Capital Expenditure. It represents cash available to all capital providers after reinvestment needs. A is retained earnings; C omits taxes, D&A, and working capital changes; D is gross profit.
Question 222
A long position in a forward contract on an asset worth $100 today with a delivery price of $105 in one year. If the risk-free rate is 4%, what is the fair forward price and is this contract fairly priced?
Select an option first.
Correct answer: A — Fair price = $104, contract is overpriced by $1
Explanation: Fair forward price = S × (1+r)^T = $100 × 1.04 = $104. The delivery price of $105 exceeds the fair value — the long must pay too much, so the contract is overpriced for the long buyer by $1.
Question 223
Which of the following is a property of a 'coherent' risk measure?
Select an option first.
Correct answer: B — It must satisfy monotonicity, sub-additivity, positive homogeneity, and translation invariance
Explanation: The four axioms of coherent risk measures (Artzner et al., 1999): (1) Monotonicity, (2) Sub-additivity, (3) Positive homogeneity, (4) Translation invariance. VaR fails sub-additivity under certain conditions; Expected Shortfall satisfies all four. A, C, D are all incorrect.
Question 224
Delta-normal VaR for a portfolio of options calculates VaR by:
Select an option first.
Correct answer: B — Approximating option positions using their deltas and applying the parametric VaR methodology to the linearised exposures
Explanation: Delta-normal VaR maps option positions to equivalent underlying delta positions and then applies parametric (normal distribution) VaR. It is fast but inaccurate for large moves because it ignores gamma (curvature). A is full Monte Carlo revaluation; C is historical simulation; D is not a standard approach.
Question 225
A trader has a gamma of 0.05 on an option position. This means:
Select an option first.
Correct answer: B — For a $1 change in the underlying, the delta of the option changes by 0.05
Explanation: Gamma = ∂Δ/∂S is the second derivative of option price with respect to the underlying. It measures the rate of change of delta. A is delta; C is approximately delta for at-the-money options but not the definition of gamma; D is theta.
Question 226
In extreme value theory (EVT), the key advantage over standard VaR approaches is that EVT:
Select an option first.
Correct answer: B — Directly models the tails of the loss distribution without imposing a specific distribution on the entire data set
Explanation: EVT focuses on modelling the tail of the distribution (extreme losses) using distributions specifically designed for extremes (Generalised Pareto Distribution for exceedances). This avoids forcing the entire distribution into a normal or log-normal framework. A, C, D are all incorrect.
Question 227
Which of the following best describes the concept of 'value at risk contributions' (VaRC) in portfolio risk decomposition?
Select an option first.
Correct answer: B — The marginal contribution of each position to the overall portfolio VaR, allowing identification of the largest risk drivers
Explanation: VaR contributions (marginal VaR × position size) decompose total portfolio VaR across individual positions, revealing which positions drive the most risk. This enables targeted hedging. A ignores risk contribution; C ignores diversification (standalone VaR); D is capital allocation.
Question 228
The formula for Net Present Value (NPV) of a project is:
Select an option first.
Correct answer: B — Sum of all discounted future cash flows minus the initial investment
Explanation: NPV = Σ [CFt / (1+r)^t] − Initial Investment. If NPV > 0, the project creates value. A ignores the time value of money; C confuses IRR and NPV; D is accounting profit not NPV.
Question 229
If a bond trades at a price below its face value (par), it is said to be trading:
Select an option first.
Correct answer: C — At a discount
Explanation: A bond trading below par is at a discount — its yield exceeds its coupon rate. A bond at a premium trades above par (yield < coupon). At par means price = face value (yield = coupon).
Question 230
Which of the following correctly describes the relationship between a bond's coupon rate and its yield when the bond is priced at par?
Select an option first.
Correct answer: C — Coupon rate = yield
Explanation: When a bond trades at par (price = face value), the coupon rate exactly equals the yield. This is because the PV of cash flows discounted at the coupon rate equals the face value. Above par: coupon > yield; below par: coupon < yield.
Question 231
Which of the following best describes 'price discovery' in financial markets?
Select an option first.
Correct answer: B — The process by which market prices are determined through the interaction of buyers and sellers, incorporating all available information
Explanation: Price discovery is a fundamental market function — through trading activity, market prices continuously aggregate and reflect available information. Efficient markets are good at price discovery; illiquid or opaque markets suffer from poor price discovery.
Question 232
The weighted average cost of capital (WACC) represents:
Select an option first.
Correct answer: B — The blended cost of all capital sources (debt and equity), weighted by their proportion in the firm's capital structure
Explanation: WACC = (E/V)×Re + (D/V)×Rd×(1−T). It is the hurdle rate for investment decisions — projects must earn at least WACC to create value. A is only debt cost; C is the cost of equity (Ke); D is the yield on the firm's bonds.
Question 233
The 'yield curve' in fixed income markets typically refers to:
Select an option first.
Correct answer: B — The relationship between bond yields and their maturities for securities of the same credit quality
Explanation: The yield curve plots yields (y-axis) against maturities (x-axis) for same-quality bonds (typically government bonds). A normal curve slopes upward; inverted curves (short rates > long rates) have historically preceded recessions.
Question 234
Which of the following describes a 'normal' (upward sloping) yield curve?
Select an option first.
Correct answer: C — Long-term yields are higher than short-term yields
Explanation: A normal yield curve slopes upward — investors require a higher yield (term premium) for lending money over longer periods due to greater uncertainty and opportunity cost. An inverted curve (A) is when short rates exceed long rates. B is a flat curve.
Question 235
Which of the following correctly defines 'accrued interest' on a bond?
Select an option first.
Correct answer: B — The interest that has accumulated since the last coupon payment date but has not yet been paid
Explanation: Accrued interest is the portion of the next coupon that has accrued since the last coupon date. The 'dirty' (full) price of a bond = clean price + accrued interest. Buyers compensate sellers for accrued interest at settlement.
Question 236
The 'quick ratio' is also known as:
Select an option first.
Correct answer: B — The acid-test ratio
Explanation: The quick (acid-test) ratio = (Cash + Short-term investments + Receivables) / Current liabilities. It is a more stringent liquidity test than the current ratio because it excludes inventory. LCR (D) is a banking regulatory metric — not the same as the quick ratio.
Question 237
If a firm has a debt-to-equity ratio of 2.0, it means:
Select an option first.
Correct answer: B — For every $2 of debt, there is $1 of equity
Explanation: D/E = 2.0 means the firm has $2 of debt for every $1 of equity. Total debt is twice total equity. A inverts the ratio. C describes D/E = 1.0. D describes an equity multiplier of 2.0 (Total assets / Total equity = 2.0, which implies D/E = 1.0 not 2.0).
Question 238
An asset has a beta of 0.5. If the market rises by 10%, what return would CAPM predict for this asset (risk-free rate = 2%, market return = 10%)?
Select an option first.
Correct answer: C — 6%
Explanation: E(Ri) = Rf + β × (Rm − Rf) = 2% + 0.5 × (10% − 2%) = 2% + 0.5 × 8% = 2% + 4% = 6%. A simply halves the market return; B ignores beta; D adds instead of adjusting.
Question 239
Which of the following financial ratios measures how efficiently a firm is using its assets to generate sales?
Select an option first.
Correct answer: C — Asset turnover ratio (Revenue / Total assets)
Explanation: Asset turnover = Revenue / Total assets. Higher turnover means the firm generates more sales per dollar of assets — a measure of operational efficiency. NIM (A) is profitability; ROE (B) is equity efficiency; D is leverage.
Question 240
In discounted cash flow (DCF) analysis, the discount rate used is typically:
Select an option first.
Correct answer: B — The weighted average cost of capital (WACC) for firm valuation, or the required return on equity for equity valuation
Explanation: In DCF, the discount rate reflects the risk of the cash flows being valued. For firm-level free cash flows → WACC; for equity free cash flows → cost of equity (Ke). Always using the risk-free rate (C) would overvalue risky projects.
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