Estimated study time: 60 minutes
Content:
The term structure of interest rates describes the relationship between bond yields and time to maturity for otherwise identical bonds. At CFA Level 2, candidates must master the spot rate curve (also called the zero curve), the forward rate curve, the par curve, and the swap rate curve, and understand the theories that explain why the yield curve takes its observed shape. The spot rate for maturity T (denoted z_T) is the yield on a zero-coupon bond maturing at time T. The par rate for maturity T is the coupon rate that makes a T-period bond priced at par. Forward rates are the rates implied by the spot curve for loans beginning in the future.
The relationship between spot rates and forward rates is governed by the no-arbitrage condition. A forward rate f(j,k) represents the annualized rate for a loan beginning in j years and maturing in k years. The key formula: (1 + z_k)^k = (1 + z_j)^j * (1 + f(j,k))^(k-j), where z_j and z_k are the j-year and k-year spot rates. Solving for the forward rate: (1 + f(j,k)) = [(1 + z_k)^k / (1 + z_j)^j]^(1/(k-j)). Bootstrapping is the process of deriving spot rates from par bond prices: starting with the 1-year spot rate from the 1-year par bond, then extracting the 2-year spot rate from the 2-year par bond price given the known 1-year spot rate, and so on. Spot rates and forward rates together allow pricing of any fixed cash flow stream.
Three main theories explain the shape of the yield curve. The Pure Expectations Theory (PET) holds that forward rates are unbiased predictors of expected future spot rates — the yield curve shape is determined entirely by expectations of future short-term rates. An upward-sloping yield curve implies the market expects rates to rise. PET predicts that returns on bonds of all maturities should be equal over any holding period. The Liquidity Preference Theory (LPT) adds a liquidity premium to the pure expectations component — investors demand a higher return for longer-maturity bonds due to greater price sensitivity to interest rate changes. This means the forward curve overstates expected future rates by the liquidity premium. The Market Segmentation Theory holds that different segments of the yield curve are dominated by different institutional investors (short: banks; long: pension funds, insurers) whose supply and demand is independent, creating yield curve shapes driven by institutional preferences.
The preferred habitat theory (a compromise between PET and market segmentation) suggests that institutions have preferred maturity habitats but will venture outside if the yield premium is sufficient. The yield curve slope is determined by the interaction of rate expectations, liquidity premiums, and supply/demand from institutional preferences. At Level 2, candidates must be able to decompose yield changes into rate expectations and term premium components, understand how the Federal Reserve's policies affect the yield curve shape (QE compresses long rates, flattening the curve; rate hikes steepen the curve initially), and apply yield curve analysis to fixed income portfolio positioning.
Swap rate curves (also called LIBOR curves or, post-LIBOR transition, SOFR curves) provide an alternative benchmark to government bond curves. The swap rate is the fixed rate a corporation pays to receive floating (SOFR) on a standard interest rate swap. Swap rates embed credit risk (because swap counterparties may default) and liquidity characteristics different from government bonds. The swap spread = swap rate - Treasury yield of the same maturity; it reflects credit risk in the banking system and is a widely used indicator of financial stress. The CFA curriculum uses the swap rate curve in bond pricing because it better reflects the cost of financing for corporate issuers. Arbitrage-free term structure models (Ho-Lee, Hull-White) fit the current yield curve and model its evolution to price interest rate derivatives without arbitrage.
Key Terms:
Quiz Questions:
Q1. The 1-year spot rate is 4.0% and the 2-year spot rate is 5.0%. What is the implied 1-year forward rate one year from now, f(1,2)?
A) (1.05)^2 / (1.04)^1 - 1 = 1.1025 / 1.04 - 1 = 1.0601 - 1 = 6.01%. B) 5.0% - 4.0% = 1.0%. C) (5.0% + 4.0%) / 2 = 4.5%. D) (1.04)^2 / (1.05)^1 - 1 = 1.0816 / 1.05 - 1 = 3.0%.
Answer: A — The no-arbitrage condition: (1 + z_2)^2 = (1 + z_1)^1 * (1 + f(1,2))^1. Solving for f(1,2): f(1,2) = (1 + z_2)^2 / (1 + z_1) - 1 = (1.05)^2 / (1.04) - 1 = 1.1025 / 1.04 - 1 = 1.06010 - 1 = 6.01%. The forward rate of 6.01% is higher than the 2-year spot (5.0%) because the yield curve is upward sloping — longer rates are higher, implying the market expects rates to rise (or demanding a liquidity premium for longer maturities).
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Q2. The 2-year par rate is 4.5% and the 1-year spot rate is 4.0%. Using bootstrapping, what is the 2-year spot rate?
A) 4.5% (par rate equals spot rate for all maturities). B) The 2-year spot rate z_2 is found by: $4.5/(1.04) + $104.5/(1+z_2)^2 = $100. Solving: $104.5/(1+z_2)^2 = $100 - $4.327 = $95.673; (1+z_2)^2 = 104.5/95.673 = 1.09225; z_2 = sqrt(1.09225) - 1 = 4.513%. C) The 2-year spot rate equals the average of the 1-year spot and the 1-year forward. D) z_2 = 4.0% + 0.5% = 4.5% by simple addition.
Answer: B — Bootstrapping: price a 2-year bond with 4.5% coupon at par = $100. Cash flows: $4.5 at Year 1 (discounted at z_1 = 4.0%) and $104.5 at Year 2 (discounted at z_2). Setting up: $4.5/1.04 + $104.5/(1+z_2)^2 = $100. $4.327 + $104.5/(1+z_2)^2 = $100. $104.5/(1+z_2)^2 = $95.673. (1+z_2)^2 = 1.09225. z_2 = 4.513%. The 2-year spot rate (4.513%) slightly exceeds the 2-year par rate (4.5%) because spot rates are derived from par bonds and the relationship is not linear.
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Q3. According to the liquidity preference theory, forward rates are upward biased predictors of future spot rates because they embed a liquidity premium. If the 1-year forward rate one year from now is 6.0% and the liquidity premium for the 2-year point is estimated at 0.5%, what is the market's expected 1-year spot rate one year from now?
A) 6.0% + 0.5% = 6.5%. B) 6.0% - 0.5% = 5.5%. C) 6.0% * (1 - 0.5%) = 5.97%. D) The liquidity premium equals the forward rate; expected spot = 0%.
Answer: B — Under liquidity preference theory: Forward Rate = Expected Future Spot Rate + Liquidity Premium. Solving: Expected future spot = Forward Rate - Liquidity Premium = 6.0% - 0.5% = 5.5%. The forward rate overstates the expected rate because investors require a premium for holding longer maturities. The market's true rate expectation is lower than the observed forward rate.
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Q4. A 5-year government bond has a yield of 4.5% and a 5-year fixed-for-floating interest rate swap has a fixed rate (swap rate) of 4.9%. What is the 5-year swap spread, and what does it indicate?
A) 5-year swap spread = 4.9% - 4.5% = 0.40% (40 basis points); this reflects the credit risk of swap counterparties (typically AA-rated banks) above the risk-free government rate and is a measure of financial system credit risk. B) 5-year swap spread = 4.5% - 4.9% = -0.40%; negative spread indicates government bonds are riskier. C) The swap spread cannot be calculated without knowing the floating leg rate. D) The swap spread = 4.9% (the entire swap rate is the spread over zero).
Answer: A — Swap spread = Swap rate - Government bond yield = 4.9% - 4.5% = 0.40% or 40 basis points. The swap spread reflects the credit risk of the bank counterparties in the swap (historically AA-rated large banks) relative to the risk-free government. Wider swap spreads indicate higher perceived credit risk in the banking system and are associated with periods of financial stress (e.g., swap spreads widened dramatically in 2008). Swap spreads can also be negative in some markets under QE conditions.
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Q5. An analyst observes an inverted yield curve (short-term rates higher than long-term rates). Using pure expectations theory, what is the most likely implication for future interest rates?
A) The market expects rates to remain unchanged; the inverted curve reflects only liquidity premiums. B) The market expects short-term interest rates to decline in the future; an inverted yield curve implies that expected future short-term rates are lower than current short-term rates. C) The market expects inflation to fall, leading to permanently lower nominal rates. D) An inverted yield curve has no forecasting power for future rates under PET.
Answer: B — Under the Pure Expectations Theory, the shape of the yield curve is determined entirely by expectations of future short-term rates. An inverted curve (short rates > long rates) means the market expects short-term rates to decline. Long-term rates are a geometric average of expected future short-term rates; if long-term rates are below current short-term rates, expected future short-term rates must be pulling the average down. Historically, inverted yield curves have been strong predictors of economic slowdowns and subsequent rate cuts by central banks.
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