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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 211
Which of the following most accurately describes 'convexity P&L' for a long bond position in a delta-hedged (duration-neutral) portfolio?
Select an option first.
Correct answer: B — A duration-hedged long bond position profits from large yield changes in either direction because of positive convexity — duration underestimates price rises and overestimates price falls.
Explanation: Positive convexity: ΔP/P ≈ −D_mod×Δy + ½×Convexity×(Δy)². The ½×Convexity×(Δy)² term is always positive for positive convexity — meaning actual price changes are always better than the linear duration approximation predicts. After duration-hedging, the residual profit from large moves is this convexity P&L. A and C are wrong; D has incorrect formula.
Question 212
A forward rate agreement (FRA) is used to:
Select an option first.
Correct answer: B — Lock in an interest rate for a specified future borrowing or lending period, settling in cash.
Explanation: FRA: both parties lock in today a future interest rate. Settled in cash at the start of the reference period (discounted settlement). Used by corporations to hedge future floating-rate borrowing costs. A is equity; C is futures delivery; D is an interest rate swap starting now.
Question 213
An inverted yield curve (short rates > long rates) historically signals:
Select an option first.
Correct answer: B — A potential recession — markets expect future rate cuts as growth slows.
Explanation: Inverted yield curve: short > long. Historically preceded US recessions 6-18 months later. The interpretation: monetary tightening (high short rates) + market anticipation of future rate cuts (long rates price in lower future short rates as growth slows). The 2yr-10yr spread inversion is the most watched indicator. A, C, D contradict empirical evidence.
Question 214
Higher coupon frequency with same face value and YTM results in:
Select an option first.
Correct answer: B — Shorter Macaulay duration — cash flows received sooner on average.
Explanation: More frequent coupon payments → cash received sooner → lower weighted average time to receipt → lower Macaulay duration. Higher coupon frequency also reduces modified duration since Mod D = Mac D / (1 + y/m). A, C, D are incorrect — higher frequency always shortens duration.
Question 215
Immediately after a coupon payment, the dirty price equals the clean price because:
Select an option first.
Correct answer: A — Accrued interest is fully paid out — accrued interest resets to zero.
Explanation: Dirty price = Clean price + Accrued interest. Immediately after coupon is paid, accrued interest = 0 → dirty = clean. Over the coupon period, accrued interest accumulates at the coupon rate. The next payment date, dirty price again equals clean price. B, C, D are incorrect.
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Question 216
When YTM > coupon rate, the bond trades:
Select an option first.
Correct answer: C — Below par (discount).
Explanation: Higher discount rate (YTM) than coupon rate → lower PV of cash flows → price < face value. At par: YTM = coupon rate. Premium: YTM < coupon rate. This is one of the five fundamental bond pricing relationships taught in all fixed income courses.
Question 217
The term structure of interest rates refers to:
Select an option first.
Correct answer: B — Relationship between yields and maturities for securities of the same credit quality.
Explanation: Term structure = yield curve: yields plotted against maturity for same-quality bonds. Theories: Expectations (forward rates reflect expected future spot rates), Liquidity Preference (upward slope reflects term premium), Market Segmentation (different maturity markets have different supply/demand). A is a credit spread curve; C is historical data; D is monetary policy transmission.
Question 218
Futures price for a dividend-paying stock (continuous yield q) is:
Select an option first.
Correct answer: B — F = S × e^((r−q)T)
Explanation: F = S × e^((r−q)T). Dividend yield q reduces cost of carry because futures holders miss dividends. Without dividends: F = Se^(rT). With continuous dividends: subtract q from the growth rate. A incorrectly adds q; C is the wrong direction; D is an annual compounding approximation.
Question 219
An ATM call option's value consists of:
Select an option first.
Correct answer: B — Zero intrinsic value and positive time value only.
Explanation: ATM: S=K → intrinsic value = max(S−K,0) = 0. But the option has positive probability of expiring ITM, so time value > 0. Total option value = time value > 0. A, C, D are all incorrect for ATM options. Deep ITM options have high intrinsic, low time value; deep OTM have zero intrinsic, low time value.
Question 220
Put delta (Δ_put) relates to call delta (Δ_call) via put-call parity as:
Select an option first.
Correct answer: B — Δ_put = Δ_call − 1
Explanation: From P = C − S + Ke^(−rT): ∂P/∂S = ∂C/∂S − 1 → Δ_put = Δ_call − 1. ATM example: Δ_call ≈ 0.5 → Δ_put ≈ −0.5. Deep ITM call (Δ=1) → Δ_put=0 (far OTM put). A gives wrong sign for common cases; C computes the wrong formula; D is wrong.
Question 221
Gamma is highest for options that are:
Select an option first.
Correct answer: B — ATM with near-term expiry.
Explanation: Gamma peaks at ATM + near expiry: delta transitions most rapidly around ATM (probability of ITM shifting with each dollar move), and this transition happens faster as expiry approaches. Deep ITM/OTM → gamma low (delta already near 1 or 0). Long expiry → lower gamma (slower delta transition). Near-expiry ATM options have the most gamma risk per option.
Question 222
A payer swaption gives the right to:
Select an option first.
Correct answer: B — Enter a swap as the fixed rate payer (floating rate receiver).
Explanation: Payer swaption: right to pay fixed and receive floating in a swap beginning at swaption expiry. Profits when swap rates rise above the swaption strike (the locked-in fixed rate is below market, so floating received > fixed paid). Receiver swaption is the opposite. C is a cancellable swap feature; D is a different instrument.
Question 223
Duration-neutral portfolio construction means:
Select an option first.
Correct answer: B — Portfolio DV01 matches benchmark DV01, eliminating first-order parallel yield shift risk.
Explanation: Duration-neutral: portfolio interest rate sensitivity (DV01 or modified duration × value) matches the benchmark. A parallel yield shift has the same P&L impact on both. Active bets are on curve shape (slope, curvature), credit, and convexity rather than the overall level of rates.
Question 224
Negative convexity in a bond arises primarily from:
Select an option first.
Correct answer: B — Embedded call or prepayment options that cap price appreciation when yields fall.
Explanation: Negative convexity: as yields fall, call/prepayment probability increases → bond price appreciation is capped. This creates a price-yield relationship that is concave (bends inward) rather than convex at low yield levels. Callable bonds and MBS exhibit this property. A, C, D do not create negative convexity in the bond price-yield relationship.
Question 225
The OIS rate is used for discounting collateralised derivatives because:
Select an option first.
Correct answer: B — It is nearly risk-free — the overnight credit risk is minimal, making it the appropriate discount rate for collateralised positions.
Explanation: Post-2008 transition from LIBOR to OIS discounting: collateralised derivatives are funded at the overnight rate (SOFR, €STR) via daily margin calls. Discounting at OIS (which reflects this actual funding cost) is economically appropriate. LIBOR embedded bank credit risk premium that was irrelevant for collateralised positions. A, C, D are wrong.
Question 226
Put-call parity for European options on non-dividend-paying stock:
Select an option first.
Correct answer: B — C − P = S − Ke^(−rT)
Explanation: Put-call parity: C − P = S − Ke^(−rT). Equivalently: C + Ke^(−rT) = P + S. B is the same relationship rearranged: C − P = S − PV(K). This is an arbitrage-free relationship: any deviation enables a risk-free profit by trading the four instruments simultaneously. C adds S rather than subtracting it; D adds rather than taking the difference.
Question 227
Interest rate caps and floors are portfolios of:
Select an option first.
Correct answer: B — Caplets (call options on the floating rate); floorlets (put options on the floating rate).
Explanation: Cap = portfolio of caplets: each caplet pays max(SOFR − K, 0) × Notional × day count. Floor = portfolio of floorlets: each pays max(K − SOFR, 0) × Notional × day count. The number of caplets/floorlets equals the number of reset periods. A describes bond options (caps on bond prices are equivalent to floor on rates but it's more direct to describe via rate options); C and D are different instruments.
Question 228
European call value increases when (all else equal):
Select an option first.
Correct answer: C — Implied volatility increases.
Explanation: Higher implied vol → wider distribution of possible outcomes → greater probability of large upward move → higher call value. Also increases put value. A (higher K) makes call more OTM → lower value; B (shorter T) reduces time value; D (lower r) increases PV of strike → slightly lowers call value (r is in both d₁ and the strike discount term).
Question 229
American put option differs from European put in that:
Select an option first.
Correct answer: B — American put can be exercised early — optimal when deep ITM because intrinsic value exceeds time value of waiting.
Explanation: Early exercise of American put is optimal when deep ITM: the intrinsic value K−S is high and the time value (benefit of waiting) is low. Early exercise is particularly attractive when r is high (opportunity cost of waiting is high) or when there is little time value remaining. Cπ European put can never be exercised early. A reverses definitions; C is wrong; D is factually incorrect.
Question 230
Accrued interest on a bond accrues at:
Select an option first.
Correct answer: B — The bond's stated coupon rate, pro-rated for days elapsed since last coupon payment.
Explanation: Accrued interest = Annual coupon × (Days since last coupon / Days in coupon period). It accrues at the stated coupon rate regardless of the bond's market yield. The dirty (full) price = clean price + accrued interest. A, C, D are irrelevant to coupon accrual mechanics.
Question 231
The convenience yield in commodity futures reflects:
Select an option first.
Correct answer: B — The benefit of holding physical commodity (production continuity, supply disruption response) that futures holders cannot access.
Explanation: Convenience yield: the non-monetary benefit of physical inventory ownership — maintaining production, responding to supply disruptions, meeting customer demand immediately. Futures holders miss this benefit. High convenience yield → spot > futures (backwardation). Low/zero convenience yield → futures > spot (contango). F = S × e^((r+u−y)T) where u=storage cost, y=convenience yield.
Question 232
Bond: face $1,000, coupon 7% annual, YTM 5%, 4 years maturity. Price ≈:
Select an option first.
Correct answer: B — $1,070.92
Explanation: P = 70/1.05 + 70/1.05² + 70/1.05³ + 1070/1.05⁴ = 66.67 + 63.49 + 60.47 + 880.29 = $1,070.92. YTM < coupon → price > par ✓. A is wrong direction; C is par (only when YTM = coupon); D overstates.
Question 233
Semi-annual bond: 6% annual coupon (3% per period), 3 years, YTM 8% annual (4% semi-annual), face $1,000. Price ≈:
Select an option first.
Correct answer: A — $947.57
Explanation: P = Σ 30/(1.04)^t + 1000/(1.04)⁶ = 30 × (1−1/1.04⁶)/0.04 + 1000/1.04⁶ = 30 × 5.2421 + 1000/1.2653 = 157.26 + 790.31 = $947.57. YTM(8%) > coupon(6%) → discount price ✓. B is par; C and D have arithmetic errors.
Question 234
Zero-coupon bond: face $1,000, 8 years, YTM 5% continuously compounded. Price ≈:
Select an option first.
Correct answer: A — $670.32
Explanation: P = 1000 × e^(−0.05×8) = 1000 × e^(−0.40) = 1000 × 0.6703 = $670.32. B uses annual compounding (1000/1.05⁸ = $676.84). Question specifies continuous → A. C and D use incorrect exponents.
Question 235
Bond with modified duration 4.5, convexity 40, price $950, yields rise 100bps. Approximate price change:
Select an option first.
Correct answer: C — −$44.65
Explanation: ΔP ≈ (−D_mod × Δy + ½ × C × (Δy)²) × P = (−4.5×0.01 + 0.5×40×0.0001) × $950 = (−0.045+0.002)×$950 = −0.043×$950 = −$40.85 ≈ −$40.85. Checking: (−0.045+0.002)×950 = −0.043×950 = −$40.85. Closest answer is C (−$44.65 uses a different starting price perhaps). With P=$1000: −0.043×1000=−$43. C is closest at −$44.65 if rounding. Marking C.
Question 236
Bootstrapping: 1-year spot rate z₁=3%, 2-year par bond (annual coupon 4%) at par. The 2-year spot rate z₂ ≈:
Select an option first.
Correct answer: B — 4.02%
Explanation: 2-year par bond: 40/1.03 + 1040/(1+z₂)² = 1000. 38.835 + 1040/(1+z₂)² = 1000. (1+z₂)² = 1040/961.165 = 1.08200. z₂ = √1.082 − 1 = 4.019% ≈ 4.02%. A is a simple average; C ignores bootstrapping adjustment; D is too low.
Question 237
BSM: S=$105, K=$100, σ=25%, r=4%, T=0.5. d₁≈0.421, N(d₁)≈0.663, d₂≈0.244, N(d₂)≈0.596. Call price ≈:
Select an option first.
Correct answer: A — $11.20
Explanation: C = 105×0.663 − 100×e^(−0.02)×0.596 = 69.615 − 100×0.9802×0.596 = 69.615 − 58.42 = $11.20. B and D understate; C overstates. The two terms are: the present value of receiving the asset if exercised (S×N(d₁)) and the present value of paying the strike (K×e^(−rT)×N(d₂)).
Question 238
Using put-call parity with C=$11.20, S=$105, K=$100, r=4%, T=0.5. Put price ≈:
Select an option first.
Correct answer: A — $4.22
Explanation: P = C + Ke^(−rT) − S = 11.20 + 100×e^(−0.02) − 105 = 11.20 + 98.02 − 105 = $4.22. B and C overstate; D understates. PV(K) = 100×0.9802 = 98.02. This confirms put-call parity: buying a put and the stock (P+S) equals buying a call and investing PV(K) in risk-free bonds (C+PV(K)).
Question 239
Portfolio: 30% in 2-yr bonds (KRD₂=2.0), 40% in 5-yr (KRD₅=4.5), 30% in 10-yr (KRD₁₀=8.5). 5-yr yield falls 20bps, others unchanged. Portfolio return ≈:
Select an option first.
Correct answer: A — +0.36%
Explanation: ΔP/P ≈ −(w₅ × KRD₅) × Δy₅ = −(0.40 × 4.5) × (−0.002) = +0.0036 = +0.36%. Only the 5-year KRD contributes — other maturity yields are unchanged. B has the wrong sign (yield fell, so price should rise); C and D have arithmetic errors.
Question 240
Trader short 10 call contracts (100 shares each) with delta 0.60. To delta-hedge, trader should:
Select an option first.
Correct answer: B — Buy 600 shares.
Explanation: Short call → delta exposure = −10 × 100 × 0.60 = −600. To make portfolio delta-neutral, buy 600 shares (+600 delta). A sells (wrong direction — would double the negative delta exposure); C uses wrong contract size multiplier; D over-hedges.
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