Estimated study time: 45 minutes
Content:
Yield measures quantify the return characteristics of fixed income securities and provide the basis for pricing and comparing bonds. The yield to maturity (YTM) is the most commonly used measure — it is the IRR of the bond's cash flows, the single discount rate equating the present value of coupons and par to the current price. The bond equivalent yield (BEY) convention is used for U.S. bonds: for semiannual-pay bonds, the BEY = 2 × semiannual YTM. The effective annual yield (EAY) accounts for compounding: EAY = (1 + periodic rate)^m – 1, where m is the number of compounding periods per year. For semiannual bonds: EAY = (1 + YTM/2)^2 – 1. The money market yield (MMY or CD equivalent yield) and bank discount yield (BDY) are used for short-term instruments; they are not directly comparable to bond yields without conversion.
The spot rate (zero-coupon rate) is the yield for a specific maturity assuming a single cash flow at that maturity. Spot rates are derived from the prices of zero-coupon bonds or stripped Treasury securities. The forward rate is the interest rate agreed today for a borrowing or lending that will begin at a future date. The relationship between spot rates and forward rates: (1 + S2)^2 = (1 + S1) × (1 + 1f1), where S1 and S2 are spot rates and 1f1 is the one-period forward rate one year from now. Forward rates implied by the spot curve reflect market expectations of future short-term rates (under the Pure Expectations Theory) plus potential risk premiums. Bootstrapping is the process of extracting spot rates from the prices of coupon bonds with different maturities.
The yield curve (term structure of interest rates) plots spot rates or YTMs against maturity. The normal (upward-sloping) yield curve reflects higher yields for longer maturities, compensating investors for greater interest rate risk, inflation uncertainty, and liquidity risk. An inverted (downward-sloping) yield curve — where short-term rates exceed long-term rates — historically predicts recessions, as it reflects either current restrictive monetary policy or expectations that rates will fall sharply. A flat yield curve offers similar yields across maturities. The yield curve shifts in parallel (all maturities move together), twist (the slope changes, with short and long rates moving differently), and butterfly (the curvature changes, with intermediate rates moving differently from short and long).
Three traditional theories explain the shape of the yield curve. The Pure Expectations Theory (or Unbiased Expectations Theory) holds that the yield curve solely reflects market expectations of future short-term rates — an upward slope means rates are expected to rise; inverted means rates are expected to fall. No maturity risk premium exists. The Liquidity Preference Theory adds a liquidity premium to the pure expectations view — investors require additional compensation for longer-maturity bonds' greater price volatility, causing the yield curve to be biased upward. The Market Segmentation Theory argues that investors have specific maturity preferences and markets for different maturities are largely separate, with supply and demand within each segment determining rates. The Preferred Habitat Theory is a hybrid — investors have preferred maturities but can be enticed away with sufficient yield premium.
Key Terms:
Quiz Questions:
Q1. A 1-year spot rate is 4% and a 2-year spot rate is 5%. What is the 1-year forward rate, one year from now (the implied forward rate 1f1)?
A) 5% B) 6% C) 4.5% D) 3%
Answer: B — Using the relationship: (1 + S2)^2 = (1 + S1) × (1 + 1f1). (1.05)^2 = (1.04) × (1 + 1f1). 1.1025 = 1.04 × (1 + 1f1). (1 + 1f1) = 1.1025 / 1.04 = 1.0601. 1f1 ≈ 6.01% ≈ 6%. The forward rate is higher than the 2-year spot rate (5%) because the term structure is upward-sloping, implying the market expects rates to be higher one year from now than today's rates.
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Q2. A bond has a semiannual YTM of 3.5% (per period). What is the effective annual yield (EAY)?
A) 7.0% B) 7.12% C) 7.35% D) 6.75%
Answer: B — EAY = (1 + 0.035)^2 – 1 = (1.035)^2 – 1 = 1.071225 – 1 = 7.1225% ≈ 7.12%. The BEY would be 7.0% (= 3.5% × 2). The EAY is slightly higher because it accounts for the compounding effect of reinvesting the first semiannual coupon and earning a return on it for the second half of the year. The difference matters when comparing semiannual bonds to annual-pay bonds or bank deposits.
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Q3. The yield curve is currently inverted (short-term rates exceed long-term rates). According to the Pure Expectations Theory, this MOST likely indicates:
A) Investors require a higher premium for short-term bonds due to their higher liquidity B) The market expects interest rates to decline in the future, so long-term rates are lower than short-term rates C) Demand for long-term bonds is lower than for short-term bonds D) The central bank has stopped setting short-term interest rates
Answer: B — Under the Pure Expectations Theory, the yield curve shape reflects only expected future short-term rates. An inverted curve means the market expects short-term rates to fall in the future — today's restrictive monetary policy (high short-term rates) is expected to ease. Empirically, inverted yield curves are strongly associated with subsequent recessions, making them one of the most reliable leading economic indicators.
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Q4. According to the Liquidity Preference Theory, the yield curve should normally be:
A) Downward-sloping (inverted) to reflect time preference for current consumption B) Upward-sloping, even if no change in future rates is expected, due to the liquidity premium C) Flat if current rates are expected to remain stable D) Downward-sloping if current rates are expected to remain stable
Answer: B — The Liquidity Preference Theory adds a risk (liquidity) premium to the pure expectations component. Even if interest rates are expected to remain flat forever, longer-maturity bonds carry more interest rate risk (price volatility) and deserve compensation. This premium causes the yield curve to slope upward in the absence of rate change expectations. Only when rates are expected to fall sharply enough would the curve invert under this theory.
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Q5. A Treasury strip (zero-coupon bond) with 3 years to maturity is priced at $864.00 per $1,000 face value. What is the 3-year spot rate?
A) 4.5% B) 5.0% C) 4.95% D) 5.1%
Answer: B — $864 = $1,000 / (1 + S3)^3. (1 + S3)^3 = 1,000 / 864 = 1.1574. (1 + S3) = (1.1574)^(1/3) = 1.05003. S3 ≈ 5.0%. Spot rates from Treasury strips are the "pure" time value rates for each maturity — they are unaffected by coupon reinvestment assumptions and form the foundation of the spot curve used to value all other fixed income securities.
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