Practice: The Term Structure of Interest Rates: Spot, Par, and Forward Curves
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Fixed IncomeThe Term Structure of Interest Rates: Spot, Par, and Forward Curves
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Question 1Exam level
The 1-year spot rate is 3.00% and the 2-year spot rate is 4.00%. The 1-year forward rate one year from now, f(1,1) is closest to:
How sure are you?
Correct: A. The correct answer is 5.01%. Using (1+S2)^2 = (1+S1) x (1+1f1): (1.04)^2 = (1.03) x (1+1f1). (1.0816)/(1.03) = 1.0501. Forward rate = 5.01%. The exam tests whether candidates correctly apply the no-arbitrage compounding relationship rather than simply averaging (3%+4%)/2=3.5%, which is wrong..
B. Simple average of 3% and 4%. Intuitive shortcut that ignores compounding. Forward rates are compounded, not averaged. Averaging is only valid if rates are continuously compounded and the difference is infinitesimal.
C. You might double S2 and subtracts S1: 2x4% - 3% = 5%, then rounds down to 4.5% as 'reasonable'. The arithmetic subtraction approach ignores the geometric nature of compounding. Must use (1+S2)^n / (1+S1)^m - 1.
The 1-year spot rate is 2.50%, the 2-year spot rate is 3.00%, and the 3-year spot rate is 3.50%. The 1-year forward rate two years from now, f(2,1) is closest to:
How sure are you?
Correct: A. The correct answer is 4.51%. Formula: (1+S3)^3 = (1+S2)^2 x (1+2f1). (1.035)^3 = (1.03)^2 x (1+2f1). 1.108718 / 1.0609 = 1.04510. Forward rate = 4.51%. The first subscript is when the period starts, the second is the length..
B. You might read 3-year spot rate and assumes that is also the forward rate for the third period. Spot rates and forward rates are different. The 3-year spot rate is a blended rate covering all three periods; the forward rate is the marginal rate for just the third period.
C. You might linearly extrapolates: if spot rates rise by 0.5% per year, forward rate = 3.5% + 0.5% = 4%. Forward rates are derived geometrically from spot rates. Linear extrapolation ignores compounding and will always give the wrong answer.
An analyst observes the following spot rates: S1 = 2%, S2 = 3%, S3 = 4%. She calculates that a 3-year bond's arbitrage-free price is $980. The bond currently trades at $960. What arbitrage trade should she execute, most likely?
How sure are you?
Correct: A. The correct answer is Buy the bond at $960 and sell (short) the replicating portfolio of zero-coupon bonds at $980, capturing a $20 risk-free profit. The arbitrage-free price ($980) represents what it costs to replicate the bond's cash flows using individual zero-coupon bonds (STRIPS). If the bond is cheaper than the replicating portfolio, buy the cheap asset and sell the expensive one..
B. You might reverse the trade direction. Confuses which asset is cheap vs expensive. The bond at $960 is CHEAP relative to the replicating portfolio at $980. Buy cheap, sell expensive.
C. You might incorrectly focuses on the par value comparison rather than the relative pricing between bond and replicating portfolio. Arbitrage is about relative mispricing between the bond and its replicating portfolio, not about pricing relative to par.
The 2-year spot rate is 4.00%. The 1-year forward rate one year from now is 5.01%. The 1-year spot rate is closest to:
How sure are you?
Correct: A. The correct answer is 3.00%. Rearrange: (1+S2)^2 = (1+S1)(1+1f1). (1.04)^2 = (1+S1)(1.0501). 1.0816 / 1.0501 = 1.03. S1 = 3.00%. This reversal of the standard formula is a common exam variant. Most candidates can solve forward-from-spot but struggle to solve spot-from-forward..
B. You might subtract: 4.00% - (5.01% - 4.00%) = 2.99%, rounds to 2%. Arithmetic subtraction ignores compounding. Must use the geometric rearrangement of the formula.
C. You might average 4.00% and 5.01% = 4.5%. Averaging never works for compounded rates in a term structure calculation.
Which statement best describes the 'arbitrage-free' condition in bond valuation?
How sure are you?
Correct: A. The correct answer is A bond's market price must equal the sum of the present values of each cash flow discounted at the spot rate for that cash flow's maturity. If not, a trader can construct a riskless profit by stripping or reconstituting the bond relative to a portfolio of zero-coupon bonds..
B. YTM and risk-free rate seem related; candidates confuse the definition of no-arbitrage with the definition of fair yield. Arbitrage-free pricing is not about the level of YTM relative to risk-free rates; it is about internal consistency with the spot rate curve.
C. Par bonds trade at face value. You might confuse 'fairly priced' with 'arbitrage-free'. An arbitrage-free bond can trade at a premium or discount to par. The condition is consistency with spot rates, not equality to par.
The 1-year spot rate is 3%, the 2-year spot rate is 4%, and the 3-year spot rate is 5%. The 2-year forward rate starting one year from now, f(1,2) is closest to:
How sure are you?
Correct: A. The correct answer is 6.01%. Formula: (1+S3)^3 = (1+S1)^1 x (1+1f2)^2. (1.05)^3 = (1.03) x (1+1f2)^2. 1.157625 / 1.03 = 1.12390. (1.12390)^(0.5) - 1 = 0.0601 = 6.01% per year. Key: take the nth root where n = length of the forward period (here, 2 years)..
B. You might read off the 3-year spot rate, confusing spot and forward. The 3-year spot rate and the 2-year forward starting in 1 year are entirely different quantities.
C. You might forget the square root step: computes (1.12390) - 1 = 12.39%, then halves it to 6.20% or misestimates. Forgetting the nth root for multi-period forward rates is the most common error. For f(1,2), the right side is (1+1f2)^2, requiring a square root at the end.
A bond has an arbitrage-free value of $1,050 but trades in the market at $1,020. Which of the following is most likely true?
How sure are you?
Correct: A. The correct answer is The bond is underpriced. Arbitrageurs will buy the bond at $1,020 and sell (short) the replicating portfolio of zero-coupon strips at $1,050, earning $30 per bond. This activity will push the bond price up toward $1,050, restoring equilibrium. The arbitrage-free value represents the fair value derived from the spot curve; any deviation is an arbitrage opportunity..
B. You might compare bond price to par ($1,000) rather than to the arbitrage-free value ($1,050). The benchmark for arbitrage is the arbitrage-free value ($1,050), not par. At $1,020, the bond is BELOW its fair value.
C. You might focuse on the absolute level (both above $1,000) rather than the relative mispricing. Arbitrage is about relative mispricing between the bond and replicating portfolio. If market price ≠ arbitrage-free value, an arbitrage exists regardless of the relationship to par.
An investor can either invest for 2 years at the 2-year spot rate of 4% per year, or invest for 1 year at the 1-year spot rate of 3% and then reinvest at whatever rate prevails in year 2. What forward rate would most likely make the investor indifferent between the two strategies?
How sure are you?
Correct: A. The correct answer is 5.01%. This is the break-even forward rate, which is identical to f(1,1). At 5.01%, both strategies yield the same accumulated value: Strategy 1: (1.04)^2 = 1.0816. Strategy 2: (1.03)(1.0501) = 1.0816. The forward rate is the rate that makes you indifferent. It is the break-even reinvestment rate embedded in the spot curve..
B. You might assume the forward rate must equal the current long-term spot rate. The 2-year spot rate is a blended rate covering both periods. The forward rate is the marginal rate for just the second period, which must be higher than 4% to compensate for the lower first-year rate of 3%.
C. Simple arithmetic average of the two spot rates. The break-even forward rate requires the geometric calculation, not arithmetic averaging.
The 1-year spot rate is 5.00%, the 2-year spot rate is 5.50%, and the 3-year spot rate is 6.00%. The price of a 3-year zero-coupon bond with a face value of $1,000 using the arbitrage-free framework is closest to:
How sure are you?
Correct: A. The correct answer is $839.62. A zero-coupon bond has only one cash flow: $1,000 at maturity. Using the 3-year spot rate: Price = $1,000 / (1.06)^3 = $1,000 / 1.191016 = $839.62. For zero-coupon bonds, the arbitrage-free price and the YTM-based price are identical, because there is only one cash flow and the spot rate for that maturity IS the YTM. Spot rates are zero-coupon yields..
B. You might use the 2-year spot rate (5.50%) instead of the 3-year spot rate for a 3-year bond. The spot rate used must match the maturity of the cash flow. A 3-year cash flow uses the 3-year spot rate.
C. You might use the average spot rate: (5%+5.5%+6%)/3 = 5.5% per year, giving $1,000/(1.055)^3. The relevant rate is the 3-year spot rate (6%), not the average of all three spot rates.
A portfolio manager values a 2-year, 6% annual coupon bond (face $1,000) using both YTM-based pricing and arbitrage-free pricing. The 1-year spot rate is 5% and the 2-year spot rate is 7%. Under which condition would the two methods give the same bond price, most likely?
How sure are you?
Correct: A. The correct answer is Only when the spot curve is flat (all spot rates equal). When S1 = S2 = some rate r, the YTM equals r and both methods discount all cash flows at the same rate. On the given upward-sloping curve (5% then 7%), they differ: arbitrage-free = $57.14 + $925.42 = $982.56; YTM-based at YTM 6.95% gives approximately $982.93. The small difference arises because YTM is a single blended rate..
B. You might recall that 'an arbitrage-free bond is correctly priced' and assume both methods agree by definition. YTM-based pricing and spot-rate-based pricing only agree when the spot curve is flat. When the curve slopes, they differ because YTM is a blended rate that implicitly applies one rate to all cash flows.
C. You might recall that when coupon = YTM, the bond prices at par. Conflating par pricing with method agreement. Par pricing (coupon = YTM) is a separate condition. The two valuation methods agree only when the spot curve is flat, regardless of the coupon-YTM relationship.
Given the following par yield curve: 1-year par yield = 3%, 2-year par yield = 4%. Using bootstrapping, what is most likely the 2-year spot rate?
How sure are you?
Correct: A. The correct answer is Approximately 4.02%. A par bond pays a coupon equal to its par yield and prices at $100. For the 2-year par bond with 4% coupon: $100 = $4/(1.03) + $104/(1+S2)^2. $100 - $3.883 = $96.117 = $104/(1+S2)^2. (1+S2)^2 = 104/96.117 = 1.08203. S2 = (1.08203)^0.5 - 1 = 3.974% (approximately 4.02% with rounding). Bootstrapping derives spot rates from par yields by stripping out the effect of intermediate coupons..
B. You might assume spot rate equals par yield. If the bond pays 4% and trades at par, the yield must be 4%. On an upward-sloping yield curve, spot rates are slightly ABOVE par yields for maturities beyond 1 year. The par yield is a blended rate; bootstrapping extracts the true zero-coupon rate. The 1-year spot always equals the 1-year par yield because there is only one cash flow.
C. Arithmetic average of 3% and 4%. Spot rates are derived through the bootstrapping calculation, not by averaging par yields.
According to the Pure Expectations Theory, an upward-sloping yield curve most likely indicates that investors expect:
How sure are you?
Correct: A. Under Pure Expectations Theory, the yield curve reflects ONLY expected future short-term rates. No risk premiums exist. An upward-sloping curve means the market collectively expects short-term rates to increase. Option C describes Liquidity Preference Theory. The exam trap is choosing A. Stable rates would produce a flat curve, not upward sloping.
B. You might be tempted by B because it introduces the idea of premiums, which can influence bond yields, but the Pure Expectations Theory specifically excludes any premiums, focusing solely on the expectation of future short-term interest rates, making B irrelevant in this context.
C. Choosing C might seem logical if you think an upward-sloping yield curve indicates equilibrium in the market, but the Pure Expectations Theory specifically states that the yield curve reflects only expectations of future short-term interest rates, not supply and demand balances across maturities.
Which of the following best describes the Liquidity Preference Theory of the term structure of interest rates?
How sure are you?
Correct: B. Liquidity Preference Theory = Pure Expectations + a liquidity (risk) premium for longer maturities. The premium compensates investors for price volatility risk on longer bonds. Option A is Pure Expectations Theory only. Option B is Market Segmentation Theory.
A. Option B is Market Segmentation Theory.
C. You might be tempted by the idea that more risk always leads to higher yields, but the Liquidity Preference Theory does not mandate that yield curves must always be upward sloping; it only suggests that longer maturities include a liquidity premium, which can influence but does not determine the shape of the yield curve.
An analyst observes a flat yield curve. Under the Liquidity Preference Theory, this observation most likely indicates that:
How sure are you?
Correct: A. Under LPT, long-term yields = expected future short-term rates + liquidity premium. For the yield curve to be flat, the liquidity premium (positive) must be offset by falling rate expectations (negative). So a flat LPT curve means the market expects rates to fall. Not stay the same. Option A would produce an upward-sloping curve (positive premium, neutral rate expectations). Option C is Market Segmentation.
The Market Segmentation Theory most likely differs from the Pure Expectations Theory primarily in that it:
How sure are you?
Correct: A. Market Segmentation Theory treats each maturity bucket as a separate market with its own supply and demand. Investors (e.g., pension funds needing long duration, banks needing short duration) do not arbitrage across segments. Pure Expectations assumes completely mobile capital chasing the best rates. Option A describes Liquidity Preference. Options C and D are incorrect characterizations.
B. You might be tempted by B because it seems to align with common yield curve shapes, but the Market Segmentation Theory does not predict the shape of the yield curve; it only explains that different segments operate independently based on their own supply and demand dynamics, unlike the Pure Expectations Theory which suggests the shape reflects future interest rate expectations.
C. You might be tempted by C because forward rates are often used in yield curve analysis, but the Pure Expectations Theory, not the Market Segmentation Theory, actually uses forward rates to estimate future spot rates, making C incorrect here.
A yield curve strategy known as 'riding the yield curve' generates excess returns ONLY if, most likely:
How sure are you?
Correct: A. Riding the yield curve: buy a long bond, hold it as it rolls down the yield curve (maturity shortens), then sell at a higher price (lower yield). This works only if the yield curve stays the same or becomes steeper. The bond's yield falls as maturity shortens. If the Pure Expectations Theory holds and forward rates are realized, riding the yield curve earns NO excess return (all strategies earn the same).
B. Choosing B might seem logical if you think holding to maturity guarantees the yield to maturity, but this strategy specifically aims to benefit from the bond rolling down the yield curve before maturity, not from holding to maturity, which contrasts with the need for the yield curve to remain unchanged or steepen as stated in the correct answer.
C. If forward rates accurately predict future spot rates (Pure Expectations outcome), riding the yield curve earns zero excess return. It's the condition of failure, not success.
Which term structure theory is MOST consistent with the observation that institutional investors such as pension funds consistently prefer long-maturity bonds regardless of the yield differential offered by short-maturity bonds?
How sure are you?
Correct: B. Market Segmentation Theory posits that investors are locked into maturity segments by their liability structures and do not move regardless of yield differentials. Pension funds with long-duration liabilities must hold long-duration assets. They cannot arbitrage to short bonds even if short yields spike. Preferred Habitat (D) is a partial answer. It says investors have preferred segments but WILL move if the yield differential is large enough. Market Segmentation is more extreme and matches the 'regardless of yield differential' condition in the question stem.
A. You might be tempted by Liquidity Preference Theory because it emphasizes the premium investors demand for less liquid, longer-term bonds, but this theory actually suggests that investors will move to shorter-term bonds if yields are sufficiently attractive, which contradicts the idea that pension funds prefer long-maturity bonds regardless of yield differentials.
C. You might be tempted by Preferred Habitat Theory because it acknowledges investor preferences for certain maturity segments, but it fails to capture the rigidity of institutional investors like pension funds, as it allows for movement between segments if yields are sufficiently attractive, unlike the strict segmentation in Market Segmentation Theory.
If the 1-year spot rate is 3% and the 1-year forward rate one year from now is 5%, the 2-year spot rate implied by the Pure Expectations Theory is closest to:
How sure are you?
Correct: A. Under Pure Expectations, (1 + S2)^2 = (1 + S1)(1 + f1,1) = (1.03)(1.05) = 1.0815. S2 = sqrt(1.0815) - 1 = 1.0400 - 1 = 4.0%. The geometric mean of the two rates, not the arithmetic mean. Option A (3.5%) is the arithmetic average. A common but incorrect shortcut. Option C (4.1%) results from an arithmetic approach with rounding errors.
B. Choosing 4.1% might tempt you if you incorrectly apply an arithmetic mean to the spot and forward rates, but this violates the Pure Expectations Theory which requires calculating the geometric mean of the rates to find the 2-year spot rate.
C. Choosing 5.0% might seem logical if you assume the forward rate directly represents the 2-year spot rate, but this overlooks the Pure Expectations Theory which requires combining the current spot rate and the forward rate through compounding, not equating them directly.
An inverted yield curve is most likely described as one in which:
How sure are you?
Correct: A. An inverted (or 'negative') yield curve occurs when short-term bond yields are higher than long-term bond yields. This is typically associated with tight monetary policy. The central bank has raised short-term rates above where the market expects them to remain long-term. Option B is a steep (normal) curve. Option C is a flat curve.
The Preferred Habitat Theory of the term structure is MOST accurately described as:
How sure are you?
Correct: A. Preferred Habitat Theory = Market Segmentation + mobility. Investors have preferred maturities (their 'habitat') but will migrate to other maturities if the yield premium offered is sufficient to compensate for the discomfort of leaving their habitat. This explains why yield curves are often upward sloping (issuers must pay extra to attract investors to long maturities) but can take various shapes depending on supply/demand and migration patterns. Option A confuses it with Liquidity Preference.
B. You might be tempted by B if you think all investors naturally favor short-term bonds, but this choice incorrectly assumes a universal preference that doesn't account for the mobility aspect of the Preferred Habitat Theory, where investors can be induced to invest in longer-term bonds with sufficient yield premium.
C. You might be misled by the mention of forward rates, associating it with the term structure of interest rates, but the Preferred Habitat Theory does not involve investor irrationality or biased estimators of future spot rates; instead, it focuses on investor preferences and compensation for deviating from those preferences.
Under which term structure theory would the yield on a 10-year Treasury bond be the MOST reliable predictor of the average level of future 1-year Treasury rates over the next 10 years, most likely?
How sure are you?
Correct: C. Pure Expectations Theory is the only theory under which yields contain NO risk premiums and NO segmentation distortions. They purely reflect expected future short-term rates. Therefore, the 10-year yield equals the geometric average of expected future 1-year rates. Under Liquidity Preference (A), the 10-year yield is biased upward by the liquidity premium. Making it an overestimate of future rates. Under Market Segmentation (B), yields reflect local supply/demand, not rate expectations at all.
A. You might be tempted by Market Segmentation Theory because it suggests different maturity sectors operate independently, but this theory actually implies that the yield on a 10-year Treasury bond reflects only the supply and demand in the 10-year sector, not the average of future 1-year rates, unlike Pure Expectations Theory which directly links long-term yields to future short-term rate expectations.
B. You might be tempted by Preferred Habitat Theory because it suggests investors have a preference for certain maturity segments, but this theory still incorporates a liquidity premium, meaning the 10-year yield would not purely reflect expected future 1-year rates like the Pure Expectations Theory does.
A portfolio manager pursues a strategy of buying 5-year bonds and selling them after 1 year when they have become 4-year bonds. This strategy is MOST likely to generate excess returns over a buy-and-hold of 1-year bonds when:
How sure are you?
Correct: A. This is the 'riding the yield curve' strategy. The manager profits when the 5-year bond's yield falls as its maturity shortens from 5 to 4 years (rolling down the curve), causing its price to rise. This only works if the curve stays the same. The roll-down effect materializes. If the Fed raises rates (C), the entire curve shifts up, destroying the roll-down return. If the curve inverts (D), short rates exceed long rates, and the roll-down works in reverse. If Pure Expectations holds (A), the market already prices in future rate changes and there is no excess return.
B. You might think that raising short-term interest rates benefits the strategy by increasing the yield on the 1-year bonds you sell, but this overlooks that the entire yield curve typically shifts up, reducing the price of the 5-year bonds you initially buy and undermining the roll-down effect.
C. You might be thinking that an inverted yield curve indicates higher short-term rates, which could benefit the strategy, but in reality, an inverted curve means short-term rates exceed long-term rates, reversing the roll-down effect and thus reducing potential returns compared to a flat or positively sloped yield curve.
Which of the following yield curve shapes is MOST consistent with an economy where the central bank has aggressively raised short-term rates to combat inflation and markets believe the rate hikes will be temporary?
How sure are you?
Correct: B. An inverted yield curve results when short-term rates are pushed above long-term rates. When the central bank raises short-term rates aggressively (e.g., Fed hiking cycle 2022–2023), the 3-month T-bill yield rises above the 10-year Treasury yield if markets believe the hikes are temporary. Long-term yields reflect long-run equilibrium rate expectations. Lower than the current elevated short rates. This is exactly what was observed in the US yield curve from mid-2022 through 2024, where the 2Y/10Y spread went to approximately -100 basis points.
A. Choosing a flat yield curve might seem logical if you think aggressive rate hikes would equalize short and long-term rates, but this overlooks market expectations of temporary hikes, which instead push short-term rates above long-term rates, leading to an inverted curve.
C. You might may choose D (humped) thinking the aggressive hike period creates a local maximum in the middle. But the humped shape is associated with intermediate-term uncertainty, not a clean aggressive hike scenario where short rates clearly dominate.
The 1-year spot rate is 3% and the 2-year spot rate is 5%. Combining the relationship between spot rates and forward rates, the 1-year forward rate one year from now, f(1,1), is closest to:
How sure are you?
Correct: A. The forward rate is derived from the no-arbitrage relationship between spot rates: (1 + S2)^2 = (1 + S1) x (1 + f(1,1)). (1.05)^2 = 1.1025 = 1.03 x (1 + f(1,1)); 1 + f(1,1) = 1.1025 / 1.03 = 1.0704; f(1,1) = 7.04%, closest to 7.0%. The 2-year spot rate is a geometric blend of the 1-year spot rate and the market's implied 1-year rate one year from now, so the forward rate must be solved for algebraically, not simply averaged or subtracted.
B. 4.0% comes from a simple average of the two spot rates ((3%+5%)/2), which is not how forward rates relate to spot rates; the relationship is compounding (geometric), not a simple average, which is exactly the distinction this LOS tests.
C. 5.0% is just the 2-year spot rate itself, as though the forward rate one year from now equaled the current 2-year rate; the forward rate is a DIFFERENT, derived figure implied by the gap between the 1-year and 2-year spot rates, not simply equal to either one.
An investor observes an upward-sloping (positive) yield curve and, based purely on the pure expectations theory, concludes that the market expects short-term interest rates to rise steadily in the future. A more sophisticated colleague points out that the liquidity preference theory offers an alternative explanation. Combining both theories, the colleague's most likely point is that:
How sure are you?
Correct: A. The pure expectations theory attributes the ENTIRE shape of the yield curve to expected future short-term rates. The liquidity preference theory adds that investors generally demand an additional liquidity premium for holding longer-term bonds (compensation for greater price risk/less liquidity), which by itself can produce an upward-sloping curve even if the market does not actually expect short rates to rise at all; the observed slope could reflect some mix of genuine rate-rise expectations AND a liquidity premium, and the two cannot be cleanly separated just by looking at the curve's shape.
B. Liquidity preference theory does not always produce a downward-sloping curve; it adds an UPWARD bias (a positive liquidity premium that generally increases with maturity) on top of whatever the expectations component implies, which is consistent with, not contradictory to, an observed upward-sloping curve.
C. The two theories can produce very different implied conclusions about actual rate expectations from the SAME observed curve shape; that is exactly the colleague's point, that pure expectations theory overstates how much of the curve's slope reflects genuine rate expectations once a liquidity premium is also at work.