Question 1 of 63
An investor with an investment horizon longer than a bond's Macaulay duration is most concerned about:
id: 4
model: Claude Sonnet
topic: Reinvestment vs Price Risk
Explanation
<h3>First Principles Thinking: Duration Gap and Risk Exposure</h3><p><strong>A is correct.</strong> From the duration gap concept, \( Duration\,Gap = MacDur - Investment\,Horizon \). When the horizon exceeds duration (Duration Gap < 0), the investor will hold the bond beyond the duration point where reinvestment and price risks offset. With a long horizon, many coupon payments must be reinvested over extended periods, making their future value highly sensitive to reinvestment rates. Price risk at sale diminishes because the sale occurs far in the future (or at maturity), while compound interest on reinvested coupons dominates total return. For example, a 30-year buy-and-hold investor in a 10-year duration bond faces 20+ years of reinvestment, making falling rates the primary threat. This relationship is fundamental: long horizons shift risk from price to reinvestment.</p><p>B is incorrect because price risk dominates when the investment horizon is shorter than Macaulay duration (Duration Gap > 0), not longer. Short-horizon investors sell bonds before maturity, exposing them to market price fluctuations from yield changes. They have less time to reinvest coupons, so reinvestment risk is minimal. The question specifies a longer horizon, reversing this relationship.</p><p>C is incorrect because interest rate risk always exists through either reinvestment or price channels (or both). Only at exactly the Macaulay duration point do these risks approximately offset, creating a "duration-matched" position. Longer or shorter horizons create exposure to one dominant risk type, never eliminating risk entirely unless the bond is held to maturity and the investor is indifferent to reinvestment rates.</p>
Question 2 of 63
If a bond is trading at $98 in the market, but its value calculated using spot rates is $99, an arbitrageur would profit by:
id: 10
model: Gemini 3
topic: No-Arbitrage Principle
Explanation
<h3>First Principles Thinking: Buy Low, Sell High</h3><p><strong>A is correct.</strong> The 'Law of One Price' states that a bundle of goods (the bond) must sell for the sum of the prices of its parts (the strips valued at spot rates). Here, the bundle is cheap ($98) compared to the sum of the parts ($99). An arbitrageur buys the undervalued asset (the bond at 98) and sells the overvalued components (strips at 99) to lock in a risk-free profit of $1.</p><p>B is incorrect: This would involve selling the cheap asset and buying the expensive one, guaranteeing a loss.</p><p>C is incorrect: While transaction costs exist, in a theoretical context, a price discrepancy implies an arbitrage opportunity.</p>
Question 3 of 63
A zero-coupon bond matures in 1 year and is currently trading at $90 with a face value of $100. The 1-year spot rate is closest to:
id: 14
model: Gemini 3
topic: Price to Spot Rate (Mental Math)
Explanation
<h3>First Principles Thinking: Yield Calculation</h3><p><strong>A is correct.</strong> The spot rate is the percentage return required to grow the price to the face value. Formula: $r = \frac{FV}{PV} - 1$. Here, $r = \frac{100}{90} - 1 = \frac{10}{90} = \frac{1}{9}$. Since $1/9$ is approximately $0.1111$, the rate is $11.1\%$.</p><p>B is incorrect: This calculates yield as a percentage of Face Value (discount yield), i.e., $10/100$.</p><p>C is incorrect: This assumes the $10 discount is the yield directly.</p>
Question 4 of 63
The primary advantage of using spot rates rather than a single yield-to-maturity for bond pricing is that spot rates:
id: 21
model: Claude Sonnet
topic: Term Structure Application
Explanation
<h3>First Principles Thinking: Spot Rate Advantages</h3><p><strong>A is correct.</strong> When pricing using a single YTM, the implicit assumption is that all cash flows are discounted at the same rate (\( PV = \sum PMT/(1+r)^i \)), regardless of timing. This assumes the term structure is flat and that reinvestment opportunities are constant. Spot rate pricing recognizes that rates vary by maturity (\( PV = \sum PMT/(1+Z_i)^i \)), using the appropriate rate for each cash flow's timing. This eliminates the unrealistic constant-reinvestment assumption embedded in YTM. For example, in an upward-sloping curve, early cash flows should be discounted at lower rates and later flows at higher rates, which spot rates capture but YTM does not. Spot rate pricing thus provides more accurate, no-arbitrage valuations that reflect actual market term structure, especially important when the curve is not flat. YTM remains useful as a summary statistic but is inferior for precise valuation.</p><p>B is incorrect because spot rates do not systematically result in higher prices than YTM. The price relationship depends on curve shape: in upward-sloping curves, spot rate pricing often yields lower prices for premium bonds and higher prices for discount bonds compared to using a single YTM. The advantage is accuracy and no-arbitrage consistency, not directional price bias. Both methods can produce identical prices if YTM happens to equal the par rate.</p><p>C is incorrect because spot rate pricing requires more calculation, not less. You must apply different discount rates (\( Z_1, Z_2, ... Z_N \)) to each cash flow rather than a single YTM (r). The computational burden increases with the number of cash flows. However, this additional complexity is worthwhile because it produces arbitrage-free prices reflecting actual market conditions rather than the simplification of assuming a flat term structure implicit in single-YTM pricing.</p>
Question 5 of 63
A buy-and-hold investor purchases a bond at par. If interest rates immediately rise and remain elevated, the investor's horizon yield will most likely be:
id: 8
model: Claude Sonnet
topic: Interest Rate Change Impact
Explanation
<h3>First Principles Thinking: Buy-and-Hold Return Dynamics</h3><p><strong>C is correct.</strong> For a buy-and-hold investor (holding to maturity), there is no price risk—the bond redeems at par regardless of rate changes. The only effect is reinvestment risk. When rates rise, all coupon payments are reinvested at higher rates, increasing the future value of reinvested coupons: \( FV_{coupons} = \sum c \times (1+r_{new})^{remaining\,periods} \). Since \( r_{new} > r_{original} \), \( FV_{coupons} \) increases, boosting horizon yield above the original YTM. For example, if YTM was 6% and rates rise to 7%, coupons compound at 7%, not 6%, enhancing total return. This fundamental relationship shows buy-and-hold investors benefit from rising rates through superior reinvestment opportunities, a counterintuitive result that reverses the typical "rising rates hurt bonds" narrative.</p><p>B is incorrect because equal YTM requires constant reinvestment rates matching the original yield. The question specifies rates "rise and remain elevated," breaking this assumption. Horizon yield equals original YTM only under three conditions: (1) hold to maturity, (2) no default, and (3) reinvest at original YTM. Condition 3 is violated, so realized yield must differ from original YTM.</p><p>A is incorrect because lower horizon yield occurs when reinvestment rates fall, not rise. Buy-and-hold investors suffer when rates decline because coupons are reinvested at below-original rates, reducing \( FV_{coupons} \). The question specifies rising rates, which benefits reinvestment. This answer confuses the direction of interest rate impact for buy-and-hold versus mark-to-market investors.</p>
Question 6 of 63
If the spot rate curve is upward sloping (normal), which of the following relationships is true?
id: 7
model: Gemini 3
topic: Yield Curve Shape
Explanation
<h3>First Principles Thinking: Marginal Rates</h3><p><strong>A is correct.</strong> When the yield curve slopes upward, the marginal cost of borrowing for an additional year (the forward rate) must be higher than the average cost (the spot rate). Think of a GPA: to raise your average (spot) from 3.0 to 3.5, your next semester grade (forward) must be higher than 3.5. Par rates are weighted averages of spot rates, so they lag behind the spot curve. Thus, Forward > Spot > Par.</p><p>B is incorrect: This describes an inverted curve relationship where future rates are dragging the average down.</p><p>C is incorrect: This is the order for an inverted yield curve.</p>
Question 7 of 63
Positive convexity is most beneficial to bond investors when:
id: 17
model: Claude Sonnet
topic: Convexity Benefit
Explanation
<h3>First Principles Thinking: Convexity Value</h3><p><strong>B is correct.</strong> Convexity's value arises from its asymmetric payoff: prices rise more than duration predicts when yields fall, and fall less than duration predicts when yields rise. This asymmetry is most valuable when yields move significantly and frequently (high volatility). The convexity adjustment \( \frac{1}{2} \times Convexity \times (\Delta y)^2 \) scales with the square of yield changes, so larger moves generate disproportionately larger benefits. With volatile yields, investors repeatedly capture the convexity gain on both upward and downward moves, creating a "gamma" profit similar to options trading. Low-volatility environments minimize \( (\Delta y)^2 \), reducing convexity's contribution. This principle explains why investors pay premiums for high-convexity bonds in uncertain rate environments.</p><p>A is incorrect because stable yields (low volatility) minimize convexity's benefit. When \( \Delta y \approx 0 \), the convexity term \( \frac{1}{2} \times Convexity \times (\Delta y)^2 \approx 0 \) contributes negligibly to returns. Duration provides adequate price estimates for small changes, making convexity's second-order correction irrelevant. Investors shouldn't pay much for convexity in stable environments—it's "insurance" that's rarely needed, making it expensive relative to benefit.</p><p>C is incorrect because credit spread widening relates to credit risk, not interest rate risk or convexity benefits. Convexity measures sensitivity to benchmark yield changes, not credit spread changes. Wider spreads reduce bond prices through higher discount rates but don't interact with convexity's mathematical properties regarding benchmark rate volatility. Credit spread changes are typically modeled separately from duration/convexity frameworks, requiring different risk measures like credit duration or DTS (duration times spread).</p>
Question 8 of 63
A portfolio manager wants to reduce interest rate risk by $500,000 per 1% yield change. This requires adjusting:
id: 18
model: Claude Sonnet
topic: Money Duration Application
Explanation
<h3>First Principles Thinking: Money Duration Application</h3><p><strong>B is correct.</strong> Money duration directly measures dollar price change per 1% yield change, so reducing it by $500,000 directly achieves the goal. If current \( MoneyDur = \$10M \) and target is \( \$9.5M \), the manager reduces money duration by \( \$500k \). This can be accomplished by selling bonds, adding short-duration bonds, or using derivatives. Money duration is the appropriate metric for position-level risk management because it translates percentage sensitivities into dollar terms, enabling direct comparison with risk limits, P&L targets, and hedge requirements. This is essential for practical risk management where absolute dollar exposures matter more than percentage sensitivities.</p><p>A is incorrect because modified duration is a percentage measure, not a dollar measure. Reducing modified duration by 5 (from, say, 6 to 1) doesn't directly specify the dollar risk reduction—the outcome depends on portfolio value. If the portfolio is $10M at MD=6, money duration is $600k (\( 6 \times 10M \times 0.01 \)). Reducing MD to 1 gives \( 1 \times 10M \times 0.01 = 100k \), a $500k reduction. But without knowing portfolio value, adjusting "modified duration by 5" doesn't precisely target a $500k money duration reduction.</p><p>C is incorrect because Macaulay duration measures weighted-average time (in years), not dollar risk. "500,000 years" is a nonsensical unit for duration. Even if this meant reducing Macaulay duration by 5 years (interpreting "500,000" as a typo), it still doesn't directly translate to dollar risk without knowing portfolio value and yield levels. Macaulay duration is primarily a theoretical construct; for risk management, modified or money duration is appropriate.</p>
Question 9 of 63
An implied forward rate is best described as a:
id: 7
model: Claude Sonnet
topic: Forward Rate Definition
Explanation
<h3>First Principles Thinking: Forward Rate Concept</h3><p><strong>A is correct.</strong> Starting from the no-arbitrage principle, an investor must be indifferent between two strategies: (1) investing for B periods at spot rate \( Z_B \), or (2) investing for A periods at \( Z_A \) then reinvesting for (B-A) periods at the forward rate \( IFR_{A,B} \). Equilibrium requires: \( (1+Z_A)^A \times (1+IFR_{A,B})^{B-A} = (1+Z_B)^B \). Solving for \( IFR_{A,B} \) gives the breakeven reinvestment rate. If the realized future rate exceeds this forward rate, the roll-over strategy outperforms; if below, buy-and-hold wins. Forward rates thus represent market expectations embedded in the current spot curve and prevent arbitrage opportunities between different maturity strategies.</p><p>B is incorrect because forward rates are derived from current spot rates and represent future expected rates, not historical realized rates. While ex-post realized rates can be compared to previously implied forward rates to assess forecast accuracy (testing the expectations hypothesis), forward rates themselves are forward-looking constructs calculated from today's term structure, not backward-looking historical data.</p><p>C is incorrect because spot rates already incorporate inflation expectations through the Fisher equation relationship, and forward rates are derived from spot rates, not by adjusting them further. Forward rates represent future real plus inflation expectations for specific future periods, but they are not calculated by applying an inflation adjustment to current spot rates.</p>
Question 10 of 63
Pricing bonds using spot rates establishes no-arbitrage prices because:
id: 19
model: Claude Sonnet
topic: No-Arbitrage Pricing
Explanation
<h3>First Principles Thinking: No-Arbitrage Principle</h3><p><strong>A is correct.</strong> Starting from the law of one price, assets with identical cash flows must have identical prices, or arbitrage opportunities exist. A coupon bond can be replicated by purchasing zero-coupon bonds matching each cash flow. If the coupon bond price differs from the cost of this replicating portfolio, arbitrageurs can buy the cheaper and sell the more expensive, earning riskless profit. Pricing using spot rates ensures \( PV_{bond} = \sum PMT_i/(1+Z_i)^i = \sum PV_{zero,i} \), where each term equals the price of a zero-coupon bond. This equality prevents replication arbitrage. Any deviation would be immediately exploited in efficient markets, driving prices back to the no-arbitrage relationship. This is why spot rates are fundamental to all fixed-income valuation.</p><p>B is incorrect because no-arbitrage pricing does not require bonds to trade at par. Bonds can trade at premium, par, or discount depending on the relationship between coupon rates and spot rates, all while satisfying no-arbitrage conditions. Par pricing is a special case when the coupon equals the par rate, not a general requirement for arbitrage-free markets.</p><p>C is incorrect because forward rates represent breakeven expectations, not guaranteed realized rates. Forward rates are derived from current spot rates and can differ from subsequently realized rates. No-arbitrage applies to current pricing relationships, not future rate realizations. Realized rates may exceed or fall short of forward rates, but this doesn't create arbitrage opportunities at the time of pricing using current spot rates.</p>
Question 11 of 63
An investor buys a 5-year bond at par (100) with 6% annual coupons. If the bond is sold after 2 years at 102, and coupons are reinvested at 6%, the investor's horizon yield is closest to:
id: 7
model: Claude Sonnet
topic: Horizon Yield Calculation
Explanation
<h3>First Principles Thinking: Horizon Yield Computation</h3><p><strong>B is correct.</strong> Horizon yield is the IRR: \( r = [(FV\,of\,coupons + Sale\,Price) / Purchase\,Price]^{1/horizon} - 1 \). Future value of coupons at 6%: \( FV = 6 \times 1.06 + 6 = 6.36 + 6 = 12.36 \). Total proceeds: \( 12.36 + 102 = 114.36 \). Horizon yield: \( r = (114.36/100)^{1/2} - 1 = (1.1436)^{0.5} - 1 \approx 1.069 - 1 = 6.9\% \approx 7\% \). Mental check: 7% for 2 years gives \( 100 \times 1.07^2 = 100 \times 1.1449 = 114.49 \approx 114.36 \). The 2-point capital gain (102 vs 100) boosts return above the 6% coupon/reinvestment rate. This demonstrates how capital gains enhance realized returns beyond yield-to-maturity when bonds are sold above purchase price.</p><p>A is incorrect because 6% would only be achieved if the bond were sold at par (100) with 6% reinvestment, matching the constant-yield trajectory. The actual sale price of 102 creates a 2-point capital gain (2% of initial investment over 2 years, adding approximately 1% annually), boosting the horizon yield above 6%. This answer ignores the capital gain component of total return.</p><p>C is incorrect because 8% significantly overstates the horizon yield. This might result from adding the 2% total capital gain (102-100 = 2) directly to the 6% coupon rate, yielding \( 6\% + 2\% = 8\% \), which fails to account for: (1) the 2% gain is over 2 years (≈1% annually), and (2) proper compounding through the IRR formula is required. Simple addition violates time value of money principles.</p>
Question 12 of 63
The three sources of return from investing in a fixed-rate bond are:
id: 6
model: Claude Sonnet
topic: Sources of Bond Return
Explanation
<h3>First Principles Thinking: Bond Return Components</h3><p><strong>A is correct.</strong> From the fundamental decomposition of bond returns, investors receive: (1) scheduled coupon payments (coupon income), (2) interest earned on reinvested coupons (reinvestment income), and (3) price change if sold before maturity or pull-to-par effect (capital gain/loss). The total return formula is \( Total\,Return = (Coupon\,Income + Reinvestment\,Income + Price\,Change) / Initial\,Price \). These three components fully account for all cash flows and value changes. Coupon income is contractual and certain (absent default). Reinvestment income depends on future rates. Capital gain/loss depends on yield changes and holding period. This tripartite decomposition is essential for analyzing interest rate risk: rising rates boost reinvestment but create capital losses; falling rates reduce reinvestment but create capital gains.</p><p>B is incorrect because "price appreciation" is too narrow—it only captures gains, not losses, and "credit spread" is not a source of return but rather a component of yield that affects pricing. Credit spreads influence returns indirectly through price changes, but they are not a direct return source like coupon payments. This option confuses risk factors (spreads) with return components.</p><p>C is incorrect because bonds do not pay dividends—that's an equity feature. Bonds pay coupons (interest income), not dividends. While "interest income" broadly covers coupons and reinvestment, the phrasing "dividend income" reveals a fundamental confusion between debt and equity securities. This option mixes asset class terminology inappropriately.</p>
Question 13 of 63
If all spot rates in the economy increase by 1%, the price of a standard fixed-coupon bond will:
id: 19
model: Gemini 3
topic: Bond Price & Spot Rates (Inverse)
Explanation
<h3>First Principles Thinking: Denominator Effect</h3><p><strong>A is correct.</strong> Bond pricing formula: $Price = \Sigma [CF_t / (1+S_t)^t]$. Spot rates ($S_t$) appear in the denominator. If the denominator increases (rates go up), the fraction (Present Value) decreases. This is the fundamental inverse relationship between interest rates and bond prices.</p><p>B is incorrect: Prices move inversely to yields.</p><p>C is incorrect: Prices are only unchanged if the cash flows adjust (like a floating rate note), but this is a fixed-coupon bond.</p>
Question 14 of 63
Given 1-year and 2-year spot rates of 4% and 5%, a 2-year bond with 6% annual coupons will trade at approximately:
id: 15
model: Claude Sonnet
topic: Bond Price with Two Spot Rates
Explanation
<h3>First Principles Thinking: Premium/Discount Determination</h3><p><strong>B is correct.</strong> Price equals \( PV = 6/1.04 + 106/(1.05)^2 = 5.77 + 96.15 = 101.92 \), which exceeds 100 (par). The bond trades at a premium because the 6% coupon exceeds both relevant spot rates (4% and 5%). From first principles, when cash flows exceed what the market requires (spot rates), investors pay a premium to obtain those superior flows. The premium of 1.92 represents the excess present value of receiving 6% coupons when the market only requires 4-5% returns. This no-arbitrage pricing ensures investors cannot construct a cheaper synthetic equivalent using zero-coupon bonds, maintaining market efficiency.</p><p>A is incorrect because par pricing (PV = 100) occurs when the coupon rate equals the yield-to-maturity, which would require the weighted average discount rate to equal 6%. Since both spot rates (4%, 5%) are substantially below 6%, the bond must trade above par. The par rate for this spot curve would be approximately 4.98%, well below the 6% coupon, confirming premium pricing.</p><p>C is incorrect because discount pricing (PV < 100) requires the coupon to fall below the discount rates. Here, the 6% coupon exceeds both the 4% and 5% spot rates, creating excess cash flows that command a premium. Discount pricing would require spot rates above 6%, making the bond's coupons deficient relative to market requirements, which contradicts the given rates.</p>
Question 15 of 63
Given a 1-year spot rate of 2% and a 2-year spot rate of 4%, the 1-year forward rate starting in 1 year (1y1y) is closest to:
id: 8
model: Claude Sonnet
topic: Forward Rate Calculation
Explanation
<h3>First Principles Thinking: Forward Rate Derivation</h3><p><strong>C is correct.</strong> From the no-arbitrage relationship \( (1+Z_A)^A \times (1+IFR_{A,B})^{B-A} = (1+Z_B)^B \), we solve for the forward rate. Here, A=1, B=2, \( Z_1 = 2\% \), \( Z_2 = 4\% \). Substituting: \( (1.02)^1 \times (1+IFR_{1,1})^1 = (1.04)^2 \). Calculate: \( (1.04)^2 = 1.0816 \), so \( 1.02 \times (1+IFR_{1,1}) = 1.0816 \). Solving: \( 1+IFR_{1,1} = 1.0816/1.02 = 1.0604 \), thus \( IFR_{1,1} = 6.04\% \). This high forward rate reflects the steep upward slope from 2% to 4%: the implied year-2 rate must be much higher to compensate for the low year-1 rate. This demonstrates how forward rates amplify yield curve slopes and extract marginal rates.</p><p>B is incorrect because 6.00% results from computational rounding errors or approximation. The precise calculation yields 6.04%. More importantly, using exactly 6.00% fails to satisfy the no-arbitrage condition: \( 1.02 \times 1.06 = 1.0812 \neq (1.04)^2 = 1.0816 \). This small difference would create arbitrage opportunities in frictionless markets.</p><p>A is incorrect because 3.00% is the simple arithmetic average of 2% and 4%, which ignores compounding mechanics. Forward rates are not linear averages but geometric relationships derived from compound interest. Using 3% violates no-arbitrage: \( 1.02 \times 1.03 = 1.0506 \) versus \( (1.04)^2 = 1.0816 \), a massive discrepancy that would permit riskless profits through maturity arbitrage strategies.</p>
Question 16 of 63
A 2-year bond with a $100 par value pays an annual coupon of $5. The 1-year spot rate is 5% and the 2-year spot rate is 6%. To calculate the price, you sum:
id: 9
model: Gemini 3
topic: Bond Pricing (Sum of PVs)
Explanation
<h3>First Principles Thinking: No-Arbitrage Valuation</h3><p><strong>A is correct.</strong> Under no-arbitrage valuation, each individual cash flow is viewed as a separate zero-coupon bond and discounted at the spot rate specific to its timing. The Year 1 cash flow ($5) is discounted at $S_1$ (5%). The Year 2 cash flow ($5 coupon + $100 principal = $105) is discounted at $S_2$ (6%).</p><p>B is incorrect: This discounts all flows at $S_1$, ignoring the term structure.</p><p>C is incorrect: This discounts all flows at $S_2$, effectively treating the spot rate as a flat YTM.</p>
Question 17 of 63
In a flat yield curve environment where all spot rates are 6%, the par rate for a 3-year bond is:
id: 13
model: Gemini 3
topic: Flat Yield Curve
Explanation
<h3>First Principles Thinking: Flat Curve Equality</h3><p><strong>A is correct.</strong> If the yield curve is flat, the discount rate for every cash flow is identical (6%). Mathematically, if you discount every coupon and the principal at the same rate $r$, the bond will trade at par only if the coupon rate equals $r$. Thus, Spot Rate = YTM = Par Rate = Forward Rate in a flat environment.</p><p>B is incorrect: Par rates differ from spot rates only when the curve slopes.</p><p>C is incorrect: Same as above; with a constant discount rate, the coupon must match that rate to price at par.</p>
Question 18 of 63
A spot rate is best described as the:
id: 1
model: Claude Sonnet
topic: Spot Rate Definition
Explanation
<h3>First Principles Thinking: Defining Spot Rates</h3><p><strong>A is correct.</strong> Starting from the definition of discount rates, a spot rate is the market discount rate for default-risk-free zero-coupon bonds, also called zero rates or strip rates. These rates represent the pure time value of money for a given maturity with no coupon reinvestment risk. The spot curve is built from yields on zero-coupon bonds or derived from recently issued government bonds. By definition, since zero-coupon bonds have no interim cash flows, their yield-to-maturity equals the spot rate for that maturity. The formula is \( PV = FV/(1+Z_N)^N \), where \( Z_N \) is the spot rate. This fundamental relationship makes spot rates the building blocks for all term structure analysis and ensures no-arbitrage pricing.</p><p>B is incorrect because current yield is simply the annual coupon payment divided by the bond's current price (\( CY = Annual\,Coupon/Price \)), ignoring time value of money and maturity considerations. This is not a spot rate, which specifically applies to zero-coupon bonds and represents the discount rate for a single future cash flow at a specific maturity.</p><p>C is incorrect because forward rates are rates for periods beginning in the future, not today. While a spot rate does start today, it represents immediate investment, whereas forward rates represent future reinvestment rates. The 1-year spot rate could be called the 0y1y forward rate, but this terminology conflates two distinct concepts.</p>
Question 19 of 63
A bond has Macaulay duration of 8 years. An investor with a 5-year investment horizon has a duration gap of:
id: 5
model: Claude Sonnet
topic: Duration Gap Calculation
Explanation
<h3>First Principles Thinking: Duration Gap Formula</h3><p><strong>B is correct.</strong> From the definition \( Duration\,Gap = Macaulay\,Duration - Investment\,Horizon = 8 - 5 = +3\,years \). The positive gap indicates the bond's duration exceeds the investor's horizon, meaning the investor faces price risk from rising rates. The bond will be sold 3 years "early" relative to its duration point, exposing the investor to price fluctuations. If yields rise during the holding period, the sale price will be below the constant-yield trajectory, creating a capital loss. The sign of the gap is critical: positive gaps signal price risk dominance, negative gaps signal reinvestment risk dominance, and zero gap indicates approximate hedging where the two risks offset.</p><p>A is incorrect because -3 years reverses the subtraction order, calculating \( Horizon - Duration = 5 - 8 = -3 \). This inverts the duration gap formula. While mathematically producing a negative result, it misidentifies the risk exposure: the investor actually has a positive gap and faces price risk (rising rates hurt), not the reinvestment risk that a true negative gap would indicate (falling rates hurt).</p><p>C is incorrect because 13 years adds duration and horizon (\( 8 + 5 = 13 \)) rather than subtracting them. Duration gap is specifically the difference between these quantities, not their sum. Adding them has no meaningful interpretation in interest rate risk analysis and would vastly overstate the risk mismatch. The gap measures the directional and magnitude mismatch, requiring subtraction.</p>
Question 20 of 63
The 1-year spot rate is 4% and the 1-year forward rate one year from now (1y1y) is 6%. The approximate 2-year spot rate is closest to:
id: 4
model: Gemini 3
topic: Spot-Forward Relationship (Calculation)
Explanation
<h3>First Principles Thinking: Arithmetic Average Approximation</h3><p><strong>A is correct.</strong> The relationship between spot and forward rates implies that investing for 2 years at the 2-year spot rate should yield roughly the same as investing for 1 year at the 1-year spot rate and rolling it over at the forward rate. The 2-year spot rate is approximately the arithmetic average of the periodic rates: $(4\% + 6\%) / 2 = 5\%$. (The geometric mean would be $\sqrt{1.04 \times 1.06} - 1 \approx 4.995\%$, extremely close to 5%).</p><p>B is incorrect: This sums the rates ($4+6=10$) without averaging them over the 2-year period.</p><p>C is incorrect: This is a subtraction error or random guess unrelated to the averaging principle.</p>
Question 21 of 63
If the spot curve is upward sloping, implied forward rates will most likely be:
id: 9
model: Claude Sonnet
topic: Forward Rates and Curve Slope
Explanation
<h3>First Principles Thinking: Yield Curve Relationships</h3><p><strong>C is correct.</strong> From the forward rate formula \( IFR_{A,B} = [(1+Z_B)^B / (1+Z_A)^A]^{1/(B-A)} - 1 \), when \( Z_B > Z_A \) (upward slope), the numerator compounds a higher rate over more periods than the denominator, creating a ratio > (1+Z_B), thus \( IFR_{A,B} > Z_B > Z_A \). Intuitively, forward rates represent the marginal or incremental rate for extending maturity one more period. In an upward-sloping curve, this marginal rate exceeds the average (spot rate). For example, if \( Z_1=2\% \) and \( Z_2=4\% \), the 1y1y forward is 6.04%, well above both spot rates. This fundamental relationship explains why forward curves lie above upward-sloping spot curves and reflects rising future rate expectations.</p><p>A is incorrect because forward rates below spot rates characterize downward-sloping (inverted) yield curves where \( Z_B < Z_A \). In that scenario, extending maturity lowers expected return, so the marginal rate (forward) falls below the average (spot). For instance, if \( Z_1=4\% \) and \( Z_2=2\% \), the forward rate would be approximately 0%, below both spot rates.</p><p>B is incorrect because forward rates equal spot rates only when the spot curve is perfectly flat (\( Z_A = Z_B \) for all maturities). In this special case, \( IFR_{A,B} = Z_A = Z_B \), meaning no term premium exists and rates are expected to remain constant. This contradicts the premise of an upward-sloping curve.</p>
Question 22 of 63
A bond with modified duration of 5, priced at 98 per 100 face value, and a position size of $10 million face value has money duration closest to:
id: 9
model: Claude Sonnet
topic: Money Duration Definition
Explanation
<h3>First Principles Thinking: Money Duration Formula</h3><p><strong>A is correct.</strong> Money duration measures dollar price change per 1% yield change: \( MoneyDur = ModDur \times Market\,Value \). Market value = \( 10,000,000 \times 0.98 = 9,800,000 \). Thus \( MoneyDur = 5 \times 9,800,000 = 49,000,000 \) per 100% yield change. Per 1% (0.01) yield change: \( 49,000,000 \times 0.01 = 490,000 \). Alternatively, \( MoneyDur = 5 \times 9,800,000 = \$490,000 \) directly (modified duration already incorporates the "per unit yield change" scaling). Mental check: 5% of $9.8M ≈ $490k. Money duration converts percentage sensitivity into dollar terms, enabling position-level risk management and is essential for hedging calculations where absolute dollar exposure matters more than percentage changes.</p><p>B is incorrect because $500,000 uses par value ($10M) instead of market value ($9.8M): \( 5 \times 10,000,000 \times 0.01 = 500,000 \). Bond risk metrics must use current market value, not face value, because price changes occur from current levels, not par. A bond trading at 98 has less dollar value at risk than one at 100, so using par overstates money duration by approximately 2%.</p><p>C is incorrect because $5,000,000 results from forgetting to scale by the yield change magnitude or misunderstanding money duration units. This equals 50% of the market value, which would only occur for a 10% (1,000 bp) yield change: \( 5 \times 0.10 = 0.50 = 50\% \). Money duration should reflect a standard 1% yield change, not 10%, making this answer 10x too large.</p>
Question 23 of 63
If all spot rates equal 5% regardless of maturity, all implied forward rates will be:
id: 13
model: Claude Sonnet
topic: Flat Yield Curve Forward
Explanation
<h3>First Principles Thinking: Flat Curve Properties</h3><p><strong>B is correct.</strong> When the spot curve is flat at rate r (\( Z_A = Z_B = r \) for all maturities), the forward rate formula becomes \( (1+r)^A \times (1+IFR_{A,B})^{B-A} = (1+r)^B \). Simplifying: \( (1+IFR_{A,B})^{B-A} = (1+r)^{B-A} \), therefore \( IFR_{A,B} = r = 5\% \). Intuitively, a flat curve means no term premium and no expectations of rate changes—investors are indifferent between all maturity strategies. The breakeven reinvestment rate must equal the current rate. This is the only curve shape where spot rates, par rates, and forward rates are all identical at every maturity. It represents pure time value of money without risk or expectation premiums, a theoretical benchmark rarely observed in practice.</p><p>A is incorrect because forward rates below the flat spot rate would imply a downward-sloping curve where longer spot rates decrease. But we're told all spot rates equal 5%, which is definitionally flat. If forwards were below 5%, then extending maturity would lower returns, contradicting the flat curve premise where all returns are identical regardless of investment horizon.</p><p>C is incorrect because forward rates above 5% would imply an upward-sloping curve where longer spot rates exceed shorter ones. The forward rate represents the marginal rate for extending maturity; if it exceeded 5%, the spot curve would slope upward, not remain flat at 5%. This would violate the stated condition of constant 5% spot rates across all maturities.</p>
Question 24 of 63
The forward rate notation '2y1y' refers to a rate for a:
id: 3
model: Gemini 3
topic: Forward Rate Notation
Explanation
<h3>First Principles Thinking: Forward Notation</h3><p><strong>A is correct.</strong> In standard CFA notation (AyBy), 'A' represents the start time of the loan measured from today, and 'B' represents the tenor (length) of the loan. Therefore, '2y1y' indicates a loan that starts in 2 years and lasts for 1 year.</p><p>B is incorrect: This would be denoted as '1y2y' (starts in 1 year, lasts 2 years).</p><p>C is incorrect: This would be denoted as '1y1y' (starts in 1 year, lasts 1 year).</p>
Question 25 of 63
Given 1-year and 2-year spot rates of 2% and 4% respectively, the 2-year par rate is closest to:
id: 6
model: Claude Sonnet
topic: Par Rate Calculation
Explanation
<h3>First Principles Thinking: Par Rate Computation</h3><p><strong>B is correct.</strong> From the par rate formula \( 100 = c/1.02 + (c+100)/(1.04)^2 \), we solve for the par coupon c. Calculate: \( 100/(1.04)^2 = 100/1.0816 = 92.46 \). Thus \( 100 - 92.46 = 7.54 = c(1/1.02 + 1/1.0816) = c(0.9804 + 0.9246) = c(1.905) \). Solving: \( c = 7.54/1.905 = 3.96\% \). The par rate is below the 4% 2-year spot rate because the lower 2% 1-year rate reduces the required coupon. Intuitively, receiving one coupon discounted at only 2% instead of 4% means less total coupon is needed to reach par value. This demonstrates how par rates reflect the entire spot curve structure, not just the final maturity.</p><p>A is incorrect because 3.00% is the simple arithmetic average of 2% and 4%, which ignores present value mechanics. Par rates are not simple averages but solutions to the bond pricing equation at par. Using 3% would yield \( PV = 3/1.02 + 103/(1.04)^2 = 2.94 + 95.33 = 98.27 \), well below par, proving this is not the correct par rate.</p><p>C is incorrect because 4.00% would be the par rate only if both spot rates equaled 4%. Since \( Z_1 = 2\% < 4\% \), the par rate must be below 4%. Using 4% yields \( PV = 4/1.02 + 104/(1.04)^2 = 3.92 + 96.38 = 100.30 \), slightly above par, confirming the par rate is slightly below 4%.</p>
Question 26 of 63
Calculating the realized return of a bond to match its calculated YTM requires that all coupons are reinvested at:
id: 21
model: Gemini 3
topic: Reinvestment Assumption
Explanation
<h3>First Principles Thinking: Internal Rate of Return</h3><p><strong>A is correct.</strong> The YTM is mathematically an Internal Rate of Return (IRR). A fundamental property of the IRR calculation is the implicit assumption that intermediate cash flows (coupons) are reinvested at the IRR (the YTM) itself until maturity. If reinvested at a different rate, the actual realized return will differ from the YTM.</p><p>B is incorrect: Reinvesting at spot rates is a more realistic assumption for total return analysis, but strictly speaking, YTM assumes a single constant reinvestment rate.</p><p>C is incorrect: The risk-free rate is usually lower than the YTM of a corporate bond.</p>
Question 27 of 63
For a forward rate denoted as '3y2y', the loan period ends:
id: 15
model: Gemini 3
topic: Forward Rate Period Identification
Explanation
<h3>First Principles Thinking: Timeline Addition</h3><p><strong>A is correct.</strong> The notation 'AyBy' means the loan <em>starts</em> at year A and <em>lasts</em> for B years. Therefore, the maturity date (end of the loan) is $Start + Duration = A + B$. Here, $3 + 2 = 5$ years from today.</p><p>B is incorrect: This is when the loan begins.</p><p>C is incorrect: This is the duration of the loan, not the end date.</p>
Question 28 of 63
In a downward-sloping (inverted) yield curve, forward rates will most likely be:
id: 20
model: Claude Sonnet
topic: Inverted Curve Forward Rates
Explanation
<h3>First Principles Thinking: Inverted Curve Dynamics</h3><p><strong>A is correct.</strong> In a downward-sloping curve, \( Z_A > Z_B \) for \( A < B \). From \( IFR_{A,B} = [(1+Z_B)^B / (1+Z_A)^A]^{1/(B-A)} - 1 \), when \( Z_B < Z_A \), the numerator compounds a lower rate, making the ratio < (1+Z_A), thus \( IFR_{A,B} < Z_A \). For example, if \( Z_1=6\% \) and \( Z_2=4\% \), then \( 1.06 \times (1+IFR_{1,1}) = (1.04)^2 = 1.0816 \), yielding \( IFR_{1,1} = 1.0816/1.06 - 1 = 2.04\% \), well below both spot rates. Intuitively, the forward rate is the marginal rate for extending maturity; in an inverted curve, this marginal rate must be very low (or even negative) to drag down the average from the high short rate to the low long rate. This relationship often signals expectations of economic slowdown or central bank easing.</p><p>B is incorrect because forward rates equal spot rates only when the curve is flat (\( Z_A = Z_B \) at all maturities). An inverted curve, by definition, has different spot rates at different maturities, breaking the equality between spot and forward rates. The marginal (forward) rate must differ from the average (spot) rate when the curve has slope.</p><p>C is incorrect because forward rates higher than spot rates characterize upward-sloping curves, not inverted ones. In an upward curve, the marginal rate for extending maturity exceeds the average, pulling spot rates higher. An inverted curve represents the opposite: declining rates where forwards fall below spots, reflecting expectations of future rate decreases or economic contraction that reverses the normal upward-sloping relationship.</p>
Question 29 of 63
A spot rate is best described as the discount rate applicable to:
id: 1
model: Gemini 3
topic: Spot Rate Definition
Explanation
<h3>First Principles Thinking: Spot Rates</h3><p><strong>A is correct.</strong> A spot rate (or zero-coupon rate) is the yield on a pure discount bond with a specific maturity. It represents the time value of money for a distinct point in time, free from reinvestment risk associated with intermediate coupons. Therefore, it is the appropriate discount rate for a single, distinct cash flow occurring at that specific maturity.</p><p>B is incorrect: A rate applicable to a series of payments is typically a Yield to Maturity (YTM) or a par rate, which is a weighted average of spot rates.</p><p>C is incorrect: The YTM is a single uniform discount rate that equates the present value of all of a bond's diverse cash flows (coupons and principal) to its price, essentially averaging the term structure.</p>
Question 30 of 63
An investor sells a bond before maturity. If interest rates rise after purchase, the investor will most likely experience:
id: 19
model: Claude Sonnet
topic: Rising Rates Impact
Explanation
<h3>First Principles Thinking: Price-Yield Relationship</h3><p><strong>B is correct.</strong> From the fundamental inverse relationship between bond prices and yields, when yields rise, prices fall: \( P = \sum CF_t / (1+y)^t \). Higher y (discount rate) reduces present value of all cash flows. If the investor sells at the new lower price, they realize a capital loss relative to the purchase price. For example, buying at par (100) when yields are 5%, then selling after yields rise to 6%, the bond trades below par—say 95—creating a 5-point capital loss. This is pure price risk: the mark-to-market loss from adverse rate movements. This principle is fundamental to understanding bond investing: holding period shorter than maturity exposes investors to price risk from yield changes.</p><p>A is incorrect because capital gains occur when prices rise, which happens when yields fall, not rise. The question specifies rising rates, which causes falling prices. A capital gain would require selling at a price above purchase price, impossible when rates have risen and pushed prices down. This answer reverses the price-yield relationship.</p><p>C is incorrect because "no capital gain or loss" only occurs if: (1) the bond is held to maturity (redemption at par eliminates price risk), or (2) yields remain unchanged so the sale price equals the constant-yield trajectory. The question specifies rising rates and sale before maturity, guaranteeing a price change. Even if the bond was bought at a discount, rising rates push the price further below the trajectory, creating a mark-to-market loss relative to the expected path.</p>
Question 31 of 63
A bond has Macaulay duration of 5 years and yield-to-maturity of 4%. Its modified duration is closest to:
id: 2
model: Claude Sonnet
topic: Modified Duration Calculation
Explanation
<h3>First Principles Thinking: Modified Duration Formula</h3><p><strong>A is correct.</strong> From the fundamental relationship between Macaulay and modified duration, \( ModDur = MacDur / (1 + r) \), where r is the periodic yield. For annual compounding, \( ModDur = 5 / 1.04 = 4.808 \approx 4.81 \). This can be verified mentally: dividing 5 by 1.04 is slightly less than 5/1.0 = 5, and since 4% of 5 is 0.20, the result should be approximately 5 - 0.19 = 4.81. Modified duration converts Macaulay duration into a direct measure of price sensitivity: a 1% yield change causes approximately a 4.81% price change. The adjustment factor \( 1/(1+r) \) accounts for the present value discounting mechanism and ensures modified duration properly estimates first-order price changes.</p><p>B is incorrect because 5.00 is the Macaulay duration, not the modified duration. While related, these are distinct measures: Macaulay duration is a time measure (weighted average years), while modified duration is a price sensitivity measure (percentage price change per yield change). The conversion formula must be applied to obtain modified duration from Macaulay duration.</p><p>C is incorrect because 5.20 results from multiplying rather than dividing by (1 + r): \( 5 \times 1.04 = 5.20 \). This reverses the correct formula direction. Modified duration must be less than Macaulay duration for positive yields, since we divide by a number greater than 1. Using 5.20 would overestimate price sensitivity and violate the mathematical relationship between these duration measures.</p>
Question 32 of 63
The 1-year spot rate is 3% and the 2-year spot rate is 5%. The implied 1-year forward rate one year from now (1y1y) is closest to:
id: 6
model: Gemini 3
topic: Forward Rate Calculation (Approx)
Explanation
<h3>First Principles Thinking: Bootstrapping Forwards</h3><p><strong>A is correct.</strong> Using the linear approximation rule (which is sufficient for 'closest to' estimates with low rates): $2 \times S_2 \approx S_1 + 1y1y$. Rearranging for the forward rate: $1y1y \approx (2 \times 5\%) - 3\% = 10\% - 3\% = 7\%$. The investor needs a higher 2nd-year return to justify locking in 5% for two years versus just 3% for the first year.</p><p>B is incorrect: This is the simple average of the two spot rates, not the forward rate.</p><p>C is incorrect: This subtracts the rates ($5-3=2$) without accounting for the fact that the 5% spot rate applies to two years, not one.</p>
Question 33 of 63
A 3-year zero-coupon bond is priced at 90 per 100 face value. The 3-year spot rate is closest to:
id: 10
model: Claude Sonnet
topic: Spot Rate from Zero-Coupon Price
Explanation
<h3>First Principles Thinking: Extracting Spot Rates</h3><p><strong>B is correct.</strong> From the zero-coupon bond pricing formula \( PV = FV/(1+Z_N)^N \), we solve for the spot rate: \( Z_N = (FV/PV)^{1/N} - 1 \). Substituting \( FV=100, PV=90, N=3 \): \( Z_3 = (100/90)^{1/3} - 1 = (1.1111)^{0.3333} - 1 \). Mental calculation: \( 1.11^{1/3} \approx 1.0357 \), so \( Z_3 \approx 3.57\% \). Verification: \( 90 \times (1.0357)^3 \approx 90 \times 1.111 \approx 100 \). This demonstrates how spot rates are observable directly from zero-coupon bond prices, making them fundamental building blocks. The discount of 10 over 3 years translates to approximately 3.57% annual compounding, accounting for the geometric nature of compound interest.</p><p>A is incorrect because 3.33% results from using simple interest rather than compound interest: \( (100-90)/(3 \times 90) = 10/270 = 3.70\% \) or \( 10/(3 \times 100) = 3.33\% \). Both approaches ignore compounding over multiple periods. Using 3.33% yields \( 90 \times (1.0333)^3 \approx 99.1 \), below the 100 face value, proving this rate is too low.</p><p>C is incorrect because 10.00% confuses the total percentage discount (10%) with the annualized spot rate. The bond is priced at 90% of par, representing a 10% total discount, but this must be annualized over 3 years with compounding. Using 10% yields \( 90 \times (1.10)^3 = 90 \times 1.331 = 119.8 \), far exceeding par, showing this rate is grossly too high.</p>
Question 34 of 63
For a zero-coupon bond, the Yield to Maturity (YTM) is always equal to:
id: 18
model: Gemini 3
topic: Spot Rate vs YTM (Zero Coupon)
Explanation
<h3>First Principles Thinking: Single Cash Flow Discounting</h3><p><strong>A is correct.</strong> A zero-coupon bond has only one cash flow at maturity. The spot rate is defined as the discount rate for a single cash flow at a specific maturity. The YTM is the single rate that equates price to PV. Since there is only one term in the equation ($P = FV/(1+r)^t$), the YTM ($r$) and the spot rate ($S_t$) are mathematically identical.</p><p>B is incorrect: Par rate applies to coupon bonds trading at par, not zeros (which trade at a discount).</p><p>C is incorrect: A zero-coupon bond has a coupon rate of 0%, which is not its yield (unless interest rates are zero).</p>
Question 35 of 63
A portfolio consists of 50% Bond A (duration 3) and 50% Bond B (duration 7). The portfolio duration is:
id: 13
model: Claude Sonnet
topic: Portfolio Duration
Explanation
<h3>First Principles Thinking: Weighted-Average Duration</h3><p><strong>B is correct.</strong> Portfolio duration is the weighted average of individual bond durations: \( Port\,Dur = w_A \times Dur_A + w_B \times Dur_B = 0.5 \times 3 + 0.5 \times 7 = 1.5 + 3.5 = 5.0 \). Mental calculation: average of 3 and 7 = (3+7)/2 = 5. This assumes equal weighting (50/50). The portfolio duration of 5 means a 1% yield change across both bonds would cause approximately a 5% portfolio value change. This weighted-average approach is practical and widely used, though it assumes parallel yield curve shifts. The simplicity of this calculation makes portfolio duration an essential tool for managing interest rate risk at the portfolio level, allowing managers to quickly assess aggregate exposure.</p><p>A is incorrect because 3.5 would be the weighted average if weights were 75% Bond A and 25% Bond B: \( 0.75 \times 3 + 0.25 \times 7 = 2.25 + 1.75 = 4.0 \), which doesn't match 3.5 either, or if calculated as half of the lower duration: \( 7/2 = 3.5 \). This doesn't properly weight both positions and misapplies the weighted-average formula.</p><p>C is incorrect because 10.0 sums the two durations (\( 3 + 7 = 10 \)) rather than calculating the weighted average. Portfolio duration is not additive—it requires weighting by portfolio proportions. Simply adding durations ignores that each bond represents only part of the portfolio. This error would overstate portfolio interest rate risk by a factor of 2 in this equal-weighted case.</p>
Question 36 of 63
Given spot rates of 2% (1-year), 3% (2-year), and 4% (3-year), the price of a 3-year bond with 5% annual coupons and face value 100 is closest to:
id: 4
model: Claude Sonnet
topic: Coupon Bond Pricing with Spot Rates
Explanation
<h3>First Principles Thinking: Multi-Period Cash Flow Valuation</h3><p><strong>A is correct.</strong> From the fundamental present value principle, each cash flow must be discounted at the spot rate corresponding to its timing: \( PV = \sum PMT_i/(1+Z_i)^i + FV/(1+Z_N)^N \). Applying: Year 1: \( 5/1.02 = 4.90 \); Year 2: \( 5/(1.03)^2 = 5/1.0609 = 4.71 \); Year 3: \( 105/(1.04)^3 = 105/1.1249 = 93.35 \). Summing: \( 4.90 + 4.71 + 93.35 = 102.96 \), closest to 102.78. The bond trades at a premium because the 5% coupon exceeds all spot rates (2%, 3%, 4%), providing excess cash flows. This no-arbitrage pricing prevents profitable arbitrage: you cannot replicate this bond more cheaply using zero-coupon bonds.</p><p>B is incorrect because 100.00 (par) would only occur if the coupon rate equaled the yield-to-maturity, which would require the weighted average discount effect of the spot rates to equal 5%. Since all given spot rates (2%, 3%, 4%) are below the 5% coupon, the bond must trade at a premium above par to compensate for the superior cash flows.</p><p>C is incorrect because 97.45 represents a discount price, which occurs only when discount rates exceed the coupon rate. Since all spot rates (2-4%) are below the 5% coupon, the bond must trade above par. A discount price would require spot rates substantially above 5%, creating deficient coupons relative to market requirements.</p>
Question 37 of 63
Given a 2-year spot rate of 4%, the price of a 2-year zero-coupon bond with face value of 100 is closest to:
id: 2
model: Claude Sonnet
topic: Zero-Coupon Bond Pricing
Explanation
<h3>First Principles Thinking: Zero-Coupon Bond Valuation</h3><p><strong>A is correct.</strong> From the fundamental present value formula, the price of a zero-coupon bond equals the face value discounted at the spot rate for its maturity: \( PV = FV/(1+Z_N)^N \). For a 2-year bond with \( Z_2 = 4\% \) and \( FV = 100 \), we calculate \( PV = 100/(1.04)^2 = 100/1.0816 = 92.46 \). This can be verified mentally: \( (1.04)^2 \) is slightly more than 1.08, so the price should be slightly less than 100/1.08 ≈ 92.6. The calculation uses only the maturity-matching spot rate because there are no interim cash flows. This illustrates the no-arbitrage principle: spot rates ensure consistent pricing across maturities without creating profit opportunities.</p><p>B is incorrect because 96.00 would result from using approximately a 2% discount rate for 2 years (\( 100/(1.02)^2 ≈ 96.12 \)), which is half the correct 4% rate. This reflects a misconception of using simple rather than compound interest, or dividing the annual rate by the maturity period, neither of which is correct for spot rate discounting.</p><p>C is incorrect because 98.00 results from discounting at approximately 1% annually (\( 100/(1.01)^2 ≈ 98.03 \)), which is far too low. This might reflect using a single-period 2% simple discount (\( 100 × 0.98 \)) rather than proper compound discounting over two periods at 4% per period.</p>
Question 38 of 63
Given a 1-year spot rate of 3% and a 2-year spot rate of 5%, the 1-year forward rate one year from now (1y1y) is closest to:
id: 12
model: Claude Sonnet
topic: Simple Forward Rate
Explanation
<h3>First Principles Thinking: Forward Rate Mechanics</h3><p><strong>C is correct.</strong> Using \( (1+Z_1)^1 \times (1+IFR_{1,1})^1 = (1+Z_2)^2 \), we solve: \( 1.03 \times (1+IFR_{1,1}) = (1.05)^2 = 1.1025 \). Thus \( 1+IFR_{1,1} = 1.1025/1.03 = 1.0704 \), yielding \( IFR_{1,1} = 7.04\% \). Mental check: The spot rate rises from 3% to 5% (200 bp increase), so the forward rate must significantly exceed 5% to create this upward slope. The 7.04% forward rate means an investor locking in 5% for 2 years achieves the same result as earning 3% in year 1 and 7.04% in year 2, demonstrating the breakeven concept. This large forward rate amplifies the curve's slope and extracts the marginal year-2 rate.</p><p>B is incorrect because 5.00% is simply the 2-year spot rate, not the forward rate. The forward rate must exceed the longer spot rate when the curve slopes upward. Using 5% would imply \( 1.03 \times 1.05 = 1.0815 \), but \( (1.05)^2 = 1.1025 \), showing 5% is too low and violates the no-arbitrage condition by a significant margin.</p><p>A is incorrect because 4.00% is the simple average of 3% and 5%, ignoring compound interest. Forward rates are geometric, not arithmetic. Using 4% yields \( 1.03 \times 1.04 = 1.0712 \), versus the required \( (1.05)^2 = 1.1025 \). This massive gap (1.0712 vs 1.1025) would create substantial arbitrage profits, demonstrating that averaging fails to capture compounding dynamics.</p>
Question 39 of 63
A zero-coupon bond pays $121 in two years. If the 2-year spot rate is 10%, the current price of the bond is:
id: 5
model: Gemini 3
topic: Zero-Coupon Bond Pricing
Explanation
<h3>First Principles Thinking: Discounting</h3><p><strong>B is correct.</strong> The price of a zero-coupon bond is calculated as $Price = \frac{Face Value}{(1+S_2)^2}$. Substituting the simple numbers: $Price = \frac{121}{(1.10)^2}$. Since $1.10 \times 1.10 = 1.21$, the calculation becomes $121 / 1.21 = 100$.</p><p>A is incorrect: This discounts for only one year ($121/1.1 = 110$).</p><p>C is incorrect: This approximates the discount or assumes simple interest subtraction ($121 - 2 \times 10 = 101$).</p>
Question 40 of 63
Which bond characteristic will increase a bond's duration, all else equal?
id: 14
model: Claude Sonnet
topic: Duration Properties
Explanation
<h3>First Principles Thinking: Duration Determinants</h3><p><strong>C is correct.</strong> Duration is a weighted average of time to cash flows. Longer maturity pushes cash flows further into the future, increasing the weighted average time and thus duration. From the duration formula \( MacDur = \sum (t \times w_t) \), extending maturity adds distant cash flows with large t values, pulling the average upward. For example, extending maturity from 5 to 10 years adds 5 years of coupon payments and delays principal by 5 years, substantially increasing duration. This relationship is monotonic but nonlinear—duration increases with maturity but at a decreasing rate, approaching a limit for perpetual bonds. Longer maturity means greater price sensitivity to yield changes because distant cash flows have larger present value volatility.</p><p>A is incorrect because higher coupon rates decrease duration, not increase it. Higher coupons shift cash flow weighting toward earlier periods (larger near-term coupons) and away from principal (which is relatively smaller), reducing the weighted-average time. For example, a zero-coupon bond has duration equal to maturity, while a high-coupon bond has duration well below maturity. Higher coupons provide more "ballast" in early periods, reducing sensitivity to yield changes.</p><p>B is incorrect because higher yields decrease duration, not increase it. Higher discount rates reduce the present value of distant cash flows more than near-term cash flows (exponential discounting), shifting weight toward earlier periods and lowering the weighted average. Mathematically, \( PV_t = CF_t / (1+y)^t \), so higher y disproportionately reduces weight on large-t cash flows. This makes high-yield bonds less rate-sensitive than low-yield bonds of similar maturity.</p>
Question 41 of 63
A bond has modified duration of 6 and yield-to-maturity of 5%. If yields increase to 6%, the bond's approximate percentage price change is:
id: 3
model: Claude Sonnet
topic: Duration and Price Change
Explanation
<h3>First Principles Thinking: Duration Price Sensitivity</h3><p><strong>B is correct.</strong> From the duration approximation formula \( \Delta P/P \approx -ModDur \times \Delta y \), where \( \Delta y = 0.06 - 0.05 = 0.01 = 1\% \). Substituting: \( \Delta P/P \approx -6 \times 0.01 = -0.06 = -6\% \). The negative sign reflects the inverse relationship between bond prices and yields: when yields rise, prices fall. The magnitude of 6% equals the modified duration multiplied by the 1% (100 bp) yield change, demonstrating duration's role as a linear price sensitivity measure. This can be verified intuitively: duration of 6 means approximately 6% price change per 1% yield change. Since yields increased by 1%, price falls by approximately 6%.</p><p>A is incorrect because it shows a positive price change (+6%), which violates the fundamental inverse relationship between bond prices and yields. When yields increase, bond prices must decrease to maintain market equilibrium—new bonds offer higher yields, making existing lower-yielding bonds less valuable. The sign error likely stems from forgetting the negative sign in the duration formula.</p><p>C is incorrect because -12% would result from a 2% (200 bp) yield change, not a 1% change: \( -6 \times 0.02 = -12\% \). This error doubles the actual yield change, possibly confusing the 6% yield level with a 6% yield change, or misinterpreting the calculation. The yield change is specifically 1% (from 5% to 6%), not 2%.</p>
Question 42 of 63
A bond has modified duration of 4 and convexity of 20. If yields fall by 2%, the estimated percentage price change is closest to:
id: 12
model: Claude Sonnet
topic: Duration-Convexity Price Estimate
Explanation
<h3>First Principles Thinking: Combined Duration-Convexity Formula</h3><p><strong>B is correct.</strong> Using \( \Delta P/P \approx -Duration \times \Delta y + \frac{1}{2} \times Convexity \times (\Delta y)^2 \). With \( \Delta y = -0.02 \) (2% decline): Duration effect = \( -4 \times (-0.02) = +0.08 = +8.0\% \). Convexity effect = \( 0.5 \times 20 \times (-0.02)^2 = 10 \times 0.0004 = 0.004 = +0.4\% \). Total = \( 8.0\% + 0.4\% = 8.4\% \). Mental calculation: duration gives 8%, convexity adds \( 10 \times 4\,bps^2 = 40\,bps = 0.4\% \). The positive convexity boosts the price increase beyond what duration alone predicts, demonstrating the beneficial asymmetry. This example shows why investors value convexity—it enhances gains in favorable rate environments.</p><p>A is incorrect because +8.0% is the duration-only estimate, ignoring convexity. While duration provides a good first-order approximation, for a 2% (200 bp) yield change—a relatively large move—the convexity adjustment becomes material (+0.4%). Omitting convexity underestimates the actual price change for option-free bonds, leading to conservative price estimates and suboptimal risk management.</p><p>C is incorrect because +7.6% would result from subtracting rather than adding the convexity adjustment: \( 8.0\% - 0.4\% = 7.6\% \). This error stems from misunderstanding convexity's always-positive contribution for option-free bonds. Whether yields rise or fall, convexity improves outcomes relative to the duration-only estimate. Subtracting it inverts this beneficial property and would only apply to bonds with negative convexity (e.g., callable bonds under certain conditions).</p>
Question 43 of 63
Convexity measures the:
id: 10
model: Claude Sonnet
topic: Convexity Definition
Explanation
<h3>First Principles Thinking: Convexity Concept</h3><p><strong>B is correct.</strong> From calculus-based bond pricing, the first derivative of price with respect to yield gives duration (linear approximation), while the second derivative gives convexity (curvature). Convexity captures how duration itself changes as yields change—the rate of change of the rate of change. The price-yield relationship is curved (convex), not linear. Duration approximates this curve with a tangent line (first-order), but for large yield changes, the approximation error grows. Convexity (second-order) corrects this error: \( \Delta P/P \approx -Duration \times \Delta y + \frac{1}{2} Convexity \times (\Delta y)^2 \). The convexity term is always positive for option-free bonds, creating beneficial asymmetry: prices rise more when yields fall and fall less when yields rise than duration alone predicts. This mathematical property makes convexity inherently valuable.</p><p>A is incorrect because first-order linear price change is measured by duration, not convexity. Duration provides the slope of the price-yield curve at a point (the tangent line), which is a linear approximation. Convexity addresses the limitation of this linear approximation by measuring how the curve deviates from the tangent line—the curvature or second-order effect.</p><p>C is incorrect because average maturity of cash flows describes Macaulay duration, not convexity. Macaulay duration is a weighted-average time measure (in years), while convexity measures curvature (dimensionless or years-squared depending on formulation). These are distinct concepts: duration is about timing, convexity is about price behavior non-linearity.</p>
Question 44 of 63
A par rate is best defined as the coupon rate that:
id: 5
model: Claude Sonnet
topic: Par Rate Definition
Explanation
<h3>First Principles Thinking: Par Rate Concept</h3><p><strong>A is correct.</strong> Starting from bond pricing fundamentals, a bond is priced at par (PV = 100) when the present value of all cash flows discounted at spot rates equals the face value. The par rate is the coupon rate (and yield-to-maturity) that satisfies: \( 100 = \sum_{i=1}^{N} c/(1+Z_i)^i + 100/(1+Z_N)^N \), where c is the par coupon. Solving for c yields the par rate. This represents the coupon required for a new bond issuance to trade exactly at par given the current spot curve. Par rates are fundamental to constructing benchmark yield curves because they eliminate price distortions from premium/discount bonds. Government bond par curves are key references for pricing corporate and other bonds.</p><p>B is incorrect because the par rate is not simply the longest-maturity spot rate \( Z_N \). While related, the par rate incorporates all spot rates up to maturity through the present value formula, creating a weighted calculation. Only when the spot curve is perfectly flat does the par rate equal the spot rate at all maturities.</p><p>C is incorrect because par rates are not arithmetic averages of spot rates. They are derived from solving the bond pricing equation where PV = par, incorporating present value mechanics with proper compounding and time-weighting. Simple averaging would ignore the exponential discounting and unequal contributions of different maturity cash flows to total price.</p>
Question 45 of 63
The ideal dataset for constructing a government bond spot curve consists of:
id: 16
model: Claude Sonnet
topic: Spot Curve Construction
Explanation
<h3>First Principles Thinking: Spot Curve Data Requirements</h3><p><strong>A is correct.</strong> From the definition of spot rates as yields on zero-coupon bonds, the ideal dataset is yields-to-maturity on default-risk-free zero-coupon government bonds across a full range of maturities. Zero-coupon bonds eliminate coupon reinvestment risk and provide pure single-period rates. The formula \( Z_N = (FV/PV)^{1/N} - 1 \) directly extracts the spot rate from each bond's price without requiring bootstrapping or matrix pricing. These bonds satisfy the "other things equal" requirement: same currency, credit risk, liquidity, and tax status, differing only in maturity. This ensures the spot curve isolates the pure term structure effect without confounding factors. In practice, government zero-coupon bonds (strips) serve this purpose, though their limited availability often necessitates deriving spot curves from coupon bonds using bootstrapping.</p><p>B is incorrect because high-coupon premium bonds introduce coupon reinvestment risk and tax complexities that obscure pure spot rates. Premium pricing also indicates the bonds were issued when rates were lower, making them less liquid and potentially creating tax differentials between capital losses and interest income in some jurisdictions. These factors violate the "other things equal" assumption necessary for term structure isolation.</p><p>C is incorrect because corporate bonds contain credit risk premiums that vary by issuer, maturity, and market conditions, contaminating the term structure signal. The government bond spot curve represents default-risk-free rates, the fundamental benchmark. Corporate bonds require credit spreads added to government spot rates, making them inappropriate for constructing the base spot curve used in all fixed-income valuation.</p>
Question 46 of 63
For coupon-paying bonds, Macaulay duration is always:
id: 20
model: Claude Sonnet
topic: Duration and Maturity
Explanation
<h3>First Principles Thinking: Duration-Maturity Relationship</h3><p><strong>C is correct.</strong> Macaulay duration is the weighted-average time to cash flows: \( MacDur = \sum (t \times w_t) \), where \( w_t = PV_t / Price \). For coupon bonds, cash flows arrive before maturity (periodic coupons), so some weight is placed on times less than maturity. This pulls the weighted average below the final maturity date. For example, a 10-year bond paying annual coupons has cash flows at years 1, 2, ...10. Since weight is distributed across all years (not just year 10), the weighted average must be less than 10. Only zero-coupon bonds have all weight at maturity, making duration equal maturity. This relationship is fundamental: coupons act as "ballast," anchoring duration below maturity and reducing interest rate sensitivity.</p><p>B is incorrect because having duration exceed maturity would require negative cash flows or negative weights, which don't exist for standard bonds. Duration is a weighted average of times when positive cash flows arrive; it cannot exceed the latest cash flow time (maturity). This would violate the mathematical properties of weighted averages, which must fall between the minimum and maximum values being averaged.</p><p>A is incorrect because duration equals maturity only for zero-coupon bonds (no interim cash flows) or perpetual bonds in a special limiting case. For any coupon-paying bond with finite maturity, some cash arrives before the maturity date, making the weighted-average time less than maturity. This answer ignores the fundamental impact of coupon payments on the duration calculation.</p>
Question 47 of 63
For an option-free bond, the convexity adjustment to the duration-based price estimate is always:
id: 11
model: Claude Sonnet
topic: Convexity Effect
Explanation
<h3>First Principles Thinking: Convexity Sign Property</h3><p><strong>A is correct.</strong> From the convexity adjustment formula \( \frac{1}{2} \times Convexity \times (\Delta y)^2 \), the term \( (\Delta y)^2 \) is always positive (squaring eliminates sign), and convexity for option-free bonds is always positive (second derivative \( d^2P/dy^2 > 0 \) due to the exponential discounting function). Therefore, \( \frac{1}{2} \times (+) \times (+) = (+) \) regardless of whether yields rise or fall. This creates the beneficial asymmetry: when yields fall, convexity adds to the price increase beyond duration's estimate; when yields rise, convexity reduces the price decrease below duration's estimate. The bond price is always higher than the linear (duration-only) approximation would suggest. This property makes convexity unambiguously valuable to investors.</p><p>B is incorrect because negative convexity adjustments only occur for bonds with embedded options (like callable bonds) under certain conditions, specifically when the option is near-the-money and likely to be exercised. For option-free bonds, the mathematical structure ensures positive convexity. Negative convexity would imply prices fall more than duration predicts when yields rise and rise less when yields fall—an unfavorable asymmetry not present in option-free bonds.</p><p>C is incorrect because zero convexity would require either zero curvature (impossible for exponential discount functions) or zero yield change. Even for small yield changes, convexity is non-zero; it's just that its contribution to total price change is small (since it scales with \( (\Delta y)^2 \)). Only in the theoretical limit of infinitesimal yield changes does the convexity term become negligible, but it's never exactly zero.</p>
Question 48 of 63
For a zero-coupon bond, Macaulay duration is equal to:
id: 1
model: Claude Sonnet
topic: Macaulay Duration Definition
Explanation
<h3>First Principles Thinking: Zero-Coupon Bond Duration</h3><p><strong>A is correct.</strong> Starting from the definition of Macaulay duration as the weighted average time to receipt of cash flows, a zero-coupon bond has only one cash flow at maturity. Since all weight (100%) is concentrated at the maturity date, the weighted average time equals the time-to-maturity. Mathematically, \( MacDur = \sum (t \times w_t) \), where \( w_t \) is the weight of each cash flow. For a zero-coupon bond, \( w_{maturity} = 1.0 \) and all other weights are zero, so \( MacDur = maturity \times 1.0 = maturity \). This fundamental property makes zero-coupon bonds unique: their duration always equals their maturity unless measured between coupon dates. This relationship serves as a benchmark for understanding that coupon-paying bonds always have duration less than maturity due to interim cash flows.</p><p>B is incorrect because half the time-to-maturity would apply only if cash flows were evenly split between mid-life and maturity, which is not the case for zero-coupon bonds. This might reflect confusion with the average life concept for amortizing securities or a misapplication of the midpoint formula, neither of which applies to duration calculations.</p><p>C is incorrect because zero duration would imply immediate receipt of all cash flows (cash has zero duration). A zero-coupon bond delivers its single payment at maturity, not immediately, so its duration must be positive and equal to the time remaining until that payment. Zero duration would only apply to cash or instruments with no time value component.</p>
Question 49 of 63
The par rate for a specific maturity is best defined as the:
id: 8
model: Gemini 3
topic: Par Rate Definition
Explanation
<h3>First Principles Thinking: Definition of Par</h3><p><strong>A is correct.</strong> By definition, a 'par bond' is one that trades at 100% of its face value. The par rate is the specific coupon rate required to set the present value of the bond's cash flows (discounted at the respective spot rates) equal to its par value.</p><p>B is incorrect: The YTM of a zero-coupon bond is the spot rate, not the par rate (unless the curve is flat).</p><p>C is incorrect: The forward rate is a component of the spot rate, not the coupon rate that equates price to par.</p>
Question 50 of 63
A bond priced at 100 falls to 96 when yields rise by 100 bps. Its approximate modified duration is:
id: 16
model: Claude Sonnet
topic: Simple Duration Calculation
Explanation
<h3>First Principles Thinking: Duration from Price Change</h3><p><strong>B is correct.</strong> Rearranging the duration formula \( \Delta P/P \approx -ModDur \times \Delta y \) to solve for duration: \( ModDur \approx -(\Delta P/P) / \Delta y \). Here \( \Delta P = 96 - 100 = -4 \), so \( \Delta P/P = -4/100 = -0.04 = -4\% \), and \( \Delta y = 0.01 \). Thus \( ModDur \approx -(-0.04) / 0.01 = 0.04 / 0.01 = 4 \). Mental check: 4% price drop for 1% yield rise implies duration of 4. This reverse-engineering of duration from observed price changes is commonly used by traders and risk managers to calibrate models and estimate real-world sensitivities, validating theoretical duration calculations against market behavior.</p><p>A is incorrect because duration of 2 would imply only a 2% price change for a 1% yield change: \( 2 \times 1\% = 2\% \), but the observed change is 4%. This answer halves the actual sensitivity, possibly confusing the magnitude of the price change (4 points) with the percentage change (4%) or miscalculating the ratio. Using duration of 2 would underestimate risk by 50%.</p><p>C is incorrect because duration of 8 would imply an 8% price change for a 1% yield change, double the observed 4% change. This might result from using the absolute price change (4 points) without converting to a percentage, or doubling the calculation somewhere: \( 8 = 2 \times 4 \). This overstates interest rate sensitivity by a factor of 2, leading to excessive hedging or risk aversion.</p>
Question 51 of 63
A risk-free bond has a single payment of $105 due in one year. If the 1-year spot rate is 5%, the price of the bond is:
id: 2
model: Gemini 3
topic: Bond Pricing with Spot Rates
Explanation
<h3>First Principles Thinking: PV Calculation</h3><p><strong>A is correct.</strong> The price of a bond is the present value of its future cash flows. Here, there is one cash flow ($105) at time $t=1$. The discount rate is the 1-year spot rate ($5\%$). The calculation is $PV = \frac{CF}{(1+r)^t} = \frac{105}{1.05} = 100$.</p><p>B is incorrect: This ignores the time value of money entirely, assuming the present value equals the future value.</p><p>C is incorrect: This likely results from a estimation error or applying the discount incorrectly (e.g., $105 \times 0.95$).</p>
Question 52 of 63
Which equation correctly links the 2-year spot rate ($S_2$), the 1-year spot rate ($S_1$), and the 1-year forward rate one year from now ($1y1y$)?
id: 12
model: Gemini 3
topic: Basic Forward Equation
Explanation
<h3>First Principles Thinking: Geometric Compounding</h3><p><strong>A is correct.</strong> The no-arbitrage condition requires that the total return from investing for 2 years (geometric compounding of the 2-year spot rate) must equal the return from investing for 1 year and then reinvesting for a second year (compounding $S_1$ and then the forward rate). Therefore, the compound factors must multiply.</p><p>B is incorrect: Returns compound multiplicatively, not additively (though logs would add).</p><p>C is incorrect: This is a linear approximation, not the exact relationship.</p>
Question 53 of 63
If an investor's horizon equals the bond's Macaulay duration, the investor is most hedged against:
id: 15
model: Claude Sonnet
topic: Price Risk vs Reinvestment Risk
Explanation
<h3>First Principles Thinking: Duration Matching</h3><p><strong>A is correct.</strong> When investment horizon equals Macaulay duration (duration gap = 0), reinvestment risk and price risk approximately offset each other for parallel yield shifts. If rates rise, the loss from selling the bond below its original trajectory is roughly offset by gains from reinvesting coupons at higher rates. If rates fall, the gain from selling at a premium offsets the loss from reinvesting at lower rates. This is the fundamental insight of immunization: matching duration to horizon creates a hedge against interest rate risk. The offsetting occurs because duration represents the holding period where these two countervailing forces balance. This principle is used in liability-driven investing and pension fund management to lock in returns regardless of rate changes.</p><p>B is incorrect because credit risk (default risk) is unrelated to duration matching. Duration hedges against interest rate movements, not issuer creditworthiness deterioration. Credit risk depends on the issuer's financial health, business prospects, and economic conditions affecting default probability. Duration matching provides no protection if the issuer defaults or is downgraded—that requires credit analysis, diversification, or credit derivatives.</p><p>C is incorrect because liquidity risk (inability to sell quickly without price concessions) is independent of duration matching. Liquidity depends on market depth, bid-ask spreads, and trading volume, not the relationship between duration and holding period. Duration matching addresses interest rate risk through the reinvestment/price risk offset mechanism, but it cannot protect against illiquid markets where forced selling creates transaction costs and price impacts unrelated to fundamental value.</p>
Question 54 of 63
If a 1-year zero-coupon bond trades at 98 and a 2-year zero-coupon bond trades at 94, the 2-year spot rate is closest to:
id: 11
model: Claude Sonnet
topic: Two-Period Spot Rates
Explanation
<h3>First Principles Thinking: Multi-Period Spot Rate</h3><p><strong>C is correct.</strong> Using the formula \( Z_N = (FV/PV)^{1/N} - 1 \) with \( PV=94, FV=100, N=2 \): \( Z_2 = (100/94)^{1/2} - 1 = (1.0638)^{0.5} - 1 \). Mental calculation: \( \sqrt{1.064} \approx 1.0319 \), so \( Z_2 \approx 3.19\% \). Verification: \( 94 \times (1.0319)^2 \approx 94 \times 1.0648 \approx 100.09 \approx 100 \). Note the 1-year bond price of 98 is given but not needed for this calculation—each zero-coupon bond's spot rate depends only on its own price and maturity. This illustrates that spot rates at different maturities are independent observations from the term structure, though they collectively form the spot curve.</p><p>B is incorrect because 3.00% is an approximation using simple interest: \( (100-94)/(2 \times 94) = 6/188 \approx 3.19\% \) or \( 6/(2 \times 100) = 3.00\% \). The latter uses the face value as the base, which understates the true yield. Using 3.00% yields \( 94 \times (1.03)^2 = 94 \times 1.0609 \approx 99.7 \), slightly below par, indicating the rate is too low.</p><p>A is incorrect because 2.04% appears to be the 1-year spot rate derived from the 1-year bond (\( (100/98)^{1/1} - 1 \approx 2.04\% \)), not the 2-year rate. This confuses different maturities. Using 2.04% for the 2-year bond yields \( 94 \times (1.0204)^2 \approx 94 \times 1.0412 \approx 97.9 \), well below par, proving this is the wrong maturity rate.</p>
Question 55 of 63
If you can earn 3% for one year or 5% annually for two years, the market is effectively implying that rates next year will be:
id: 17
model: Gemini 3
topic: Implied Forward Rate Logic
Explanation
<h3>First Principles Thinking: Break-Even Rates</h3><p><strong>A is correct.</strong> To be indifferent between investing for 1 year at 3% or locking in 5% for two years, the rate in the second year must compensate for the low starting rate. Since the average return over 2 years (5%) is higher than the first year (3%), the second year must be significantly higher than the average to pull it up. Specifically, roughly $2 \times 5 - 3 = 7\%$. Thus, it must be higher than 5%.</p><p>B is incorrect: If the future rate were 5%, the average of 3% and 5% would be 4%, not the 5% required.</p><p>C is incorrect: Lower rates would pull the 2-year average down below 3%, not up to 5%.</p>
Question 56 of 63
If the 1-year spot rate is 25%, the 1-year discount factor is:
id: 11
model: Gemini 3
topic: Discount Factor Calculation
Explanation
<h3>First Principles Thinking: Discount Factors</h3><p><strong>A is correct.</strong> The discount factor ($DF$) is the present value of $1 to be received in the future. The formula is $DF = \frac{1}{1+r}$. With $r = 0.25$, $DF = \frac{1}{1.25}$. Multiplying numerator and denominator by 4 gives $4/5$, which is $0.80$.</p><p>B is incorrect: This is $(1+r)$, the future value factor, not the discount factor.</p><p>C is incorrect: This calculates $1 - r$ ($1 - 0.25$), which is a linear approximation valid only for very small rates, not 25%.</p>
Question 57 of 63
A 2-year forward rate starting in 1 year (1y2y) of 6% can be interpreted as the rate that makes an investor indifferent between:
id: 17
model: Claude Sonnet
topic: Forward Rate Interpretation
Explanation
<h3>First Principles Thinking: Forward Rate Breakeven</h3><p><strong>A is correct.</strong> The forward rate notation 1y2y means a 2-year rate beginning 1 year from now, ending 3 years from now (1+2=3). From the no-arbitrage relationship \( (1+Z_1)^1 \times (1+IFR_{1,2})^2 = (1+Z_3)^3 \), the forward rate makes the terminal wealth equal whether investing for 3 years at \( Z_3 \) or for 1 year at \( Z_1 \) then reinvesting for 2 years at \( IFR_{1,2} = 6\% \). This is the breakeven reinvestment rate—if the realized 1-year rate in year 2-3 averages 6%, both strategies yield identical returns. If realized rates exceed 6%, the roll-over strategy wins; if below, the buy-and-hold strategy is superior. This relationship prevents arbitrage between maturity strategies.</p><p>B is incorrect because this describes the 1y1y forward rate (1-year rate beginning in 1 year), which ends at year 2, not year 3. The 1y1y forward creates indifference between investing for 2 years versus 1 year + 1 year reinvestment. The question specifies a 1y2y forward, which is a 2-year reinvestment period starting in year 1, totaling 3 years, not 2.</p><p>C is incorrect because this describes a simple spot rate comparison between \( Z_1 \) and \( Z_2 \), not a forward rate. No reinvestment is involved—it's simply choosing between two buy-and-hold strategies of different maturities. Forward rates specifically address the reinvestment decision after an initial investment period, which this option does not capture.</p>
Question 58 of 63
A bond is purchased at par and held to maturity. If interest rates fall immediately after purchase, the investor's realized return will most likely be:
id: 21
model: Claude Sonnet
topic: Falling Rates Scenario
Explanation
<h3>First Principles Thinking: Reinvestment Risk for Buy-and-Hold</h3><p><strong>A is correct.</strong> Buy-and-hold investors (to maturity) face only reinvestment risk, not price risk, since bonds redeem at par. When rates fall, coupons are reinvested at lower rates than the original YTM, reducing the future value of reinvested coupons: \( FV_{coupons} = \sum c \times (1+r_{new})^{remaining} \), where \( r_{new} < r_{original} \). Since total return includes both coupon and reinvestment income, lower reinvestment rates drag realized return below the original YTM. For example, buying at 6% YTM but reinvesting coupons at 4% produces \( Horizon\,Yield < 6\% \). This fundamental relationship shows that YTM equals realized return only if reinvestment occurs at the same YTM—a restrictive assumption rarely met in practice.</p><p>B is incorrect because equal returns require reinvesting coupons at exactly the original YTM. The question specifies falling rates, violating this assumption. The YTM at purchase is a promised yield contingent on specific reinvestment assumptions. When those assumptions fail (rates change), realized yield diverges from promised yield. This answer misunderstands YTM as a guaranteed return rather than a scenario-dependent expectation.</p><p>C is incorrect because higher returns occur when rates rise, not fall. Rising rates boost reinvestment income for buy-and-hold investors, increasing total return above the original YTM. Falling rates have the opposite effect, reducing reinvestment income. This answer confuses the direction of reinvestment risk impact—it correctly identifies that rate changes affect returns but reverses which direction benefits buy-and-hold investors.</p>
Question 59 of 63
In an upward-sloping yield curve environment, the par rate will most likely be:
id: 14
model: Claude Sonnet
topic: Par vs Spot Relationship
Explanation
<h3>First Principles Thinking: Par-Spot Relationship</h3><p><strong>A is correct.</strong> In an upward-sloping curve, earlier spot rates are lower than later ones (\( Z_1 < Z_2 < Z_3 \)). The par rate is calculated such that \( 100 = \sum c/(1+Z_i)^i + 100/(1+Z_N)^N \). Since early coupons are discounted at lower rates than the final spot rate \( Z_N \), they contribute more present value per unit of coupon. Therefore, the par coupon c can be less than \( Z_N \) and still achieve par pricing. For example, with \( Z_1=2\%, Z_2=4\% \), the 2-year par rate is approximately 3.96%, below the 4% spot rate. This relationship is fundamental: in upward-sloping curves, Par < Spot < Forward at each maturity, reflecting the averaging effect of par rates versus marginal (forward) and terminal (spot) rates.</p><p>B is incorrect because par rates equal spot rates only when the curve is flat. In that special case, all discounting occurs at the same rate, so the par coupon exactly equals the constant spot rate. With an upward slope, the differential discounting of early versus late cash flows breaks this equality, making the par rate a weighted average below the terminal spot rate.</p><p>C is incorrect because par rates above spot rates characterize downward-sloping (inverted) curves where \( Z_1 > Z_N \). Early coupons discounted at high rates contribute less PV, requiring higher par coupons to reach par pricing. The upward-sloping premise guarantees the reverse: par rates fall below spot rates as early low-rate discounting boosts coupon PV contributions.</p>
Question 60 of 63
Given $(1+S_3)^3 = 1.331$ and $(1+S_2)^2 = 1.21$, the 1-year forward rate two years from now ($1+2y1y$) is:
id: 20
model: Gemini 3
topic: Simple Forward Calculation
Explanation
<h3>First Principles Thinking: Division of Growth Factors</h3><p><strong>A is correct.</strong> The relationship is $(1+S_3)^3 = (1+S_2)^2 \times (1+2y1y)$. To isolate the forward component, divide the 3-year growth factor by the 2-year growth factor: $\frac{1.331}{1.21}$. Recognizing powers of 11: $1.331 = 1.1^3$ and $1.21 = 1.1^2$. Therefore, $1.1^3 / 1.1^2 = 1.1^1 = 1.10$. The forward rate is 10%.</p><p>B is incorrect: Calculation error.</p><p>C is incorrect: This is the difference ($1.331 - 1.21$), not the ratio.</p>
Question 61 of 63
An upward-sloping spot curve indicates that:
id: 3
model: Claude Sonnet
topic: Spot Curve Terminology
Explanation
<h3>First Principles Thinking: Yield Curve Shape</h3><p><strong>A is correct.</strong> Starting from the definition of a curve's slope in coordinate geometry, an upward-sloping curve has increasing y-values as x-values increase. In the spot curve context, the y-axis represents interest rates (spot rates \( Z_N \)) and the x-axis represents time-to-maturity (N). Therefore, an upward-sloping spot curve means \( Z_2 > Z_1 \), \( Z_3 > Z_2 \), etc. This is the normal or typical yield curve shape under standard economic conditions, reflecting expectations of higher rates in the future, inflation risk premiums, or liquidity premiums for bearing longer-maturity risk. The upward slope compensates investors for tying up funds for extended periods and facing greater uncertainty.</p><p>B is incorrect because this describes a downward-sloping or inverted yield curve, where \( Z_1 > Z_2 > Z_3 \). An inverted curve is less common and often associated with expectations of economic slowdown, central bank tightening followed by easing, or declining future inflation. It represents the opposite slope from an upward-sloping curve.</p><p>C is incorrect because equal spot rates across all maturities (\( Z_1 = Z_2 = Z_3 = ... \)) describe a flat yield curve with zero slope, not an upward-sloping one. A flat curve suggests no term premium and that investors expect stable interest rates, making forward rates equal current spot rates at all horizons.</p>
Question 62 of 63
The 2-year spot rate is 4% and the 3-year spot rate is 5%. The approximate 1-year forward rate two years from now ($2y1y$) is:
id: 16
model: Gemini 3
topic: Approximation of Forward Rate
Explanation
<h3>First Principles Thinking: Weighted Averages</h3><p><strong>A is correct.</strong> Using the approximation formula: $Total Return (3yr) \approx Total Return (2yr) + Forward Rate$. In annual terms: $3 \times S_3 \approx 2 \times S_2 + 1 \times (2y1y)$. Rearranging: $(3 \times 5\%) - (2 \times 4\%) = 15\% - 8\% = 7\%$. The rate must jump significantly to pull the 2-year average of 4% up to a 3-year average of 5%.</p><p>B is incorrect: This is the simple difference ($5-4$), which ignores the duration weighting.</p><p>C is incorrect: This is the simple average, which is conceptually unrelated to bootstrapping.</p>
Question 63 of 63
If a 1-year zero-coupon bond trades at 96, the 1-year spot rate is closest to:
id: 18
model: Claude Sonnet
topic: Spot Rate Bootstrap
Explanation
<h3>First Principles Thinking: Single-Period Spot Rate</h3><p><strong>B is correct.</strong> Using \( Z_N = (FV/PV)^{1/N} - 1 \) with \( PV=96, FV=100, N=1 \): \( Z_1 = (100/96)^1 - 1 = 1.0417 - 1 = 0.0417 = 4.17\% \). Mental calculation: \( 100/96 = 1.0417 \), so the return is 4.17%. Verification: \( 96 \times 1.0417 = 100 \). This demonstrates the fundamental relationship between price and yield for zero-coupon bonds. The bond trades at 96 (a 4-point discount from par), which over 1 year translates to a 4.17% return on the 96 investment. For single-period instruments, the spot rate formula simplifies to the basic return calculation: (FV - PV)/PV = (100-96)/96 = 4/96 = 4.17%, eliminating the need for exponents.</p><p>A is incorrect because 4.00% results from using the face value as the denominator: (100-96)/100 = 4%. This calculates the discount as a percentage of face value rather than the invested amount (price), understating the true yield. Using 4% yields \( 96 \times 1.04 = 99.84 \), below the 100 face value, proving this rate is too low to explain the pricing.</p><p>C is incorrect because 96.00% confuses the price (96) with the spot rate, representing a fundamental misunderstanding. Spot rates are percentage yields, not prices. A 96% spot rate would imply \( PV = 100/1.96 = 51.02 \), meaning the bond would trade around 51, not 96, showing this confuses two entirely different concepts.</p>