Final - The Yield Curve

Kelley School of Business: Intermediate Investments F303

Professor Mathias S. Kruttli Spring 2026

10 - Fixed Income II: The Yield Curve

Agenda

  • Spot rates

  • Forward rates

  • The yield curve (term structure of interest rates)

  • Theories of the term structure of interest rates

The Yield Curve

  • Definition: The yield curve is a graph that shows the yield to maturity (YTM) as a function of maturity. It is also referred to as the term structure of interest rates.

  • Focus: Typically, the yield curve refers to U.S. Treasuries or corporate bonds with the same credit rating.

Shapes of the Yield Curve

  • Inverted Yield Curve: A falling yield curve, where yields decrease as maturity increases.

  • Normal Yield Curve: A rising yield curve, where yields increase with maturity.

  • Current Shape: An analysis of today's yield curve shape can be found on the Treasury Department website.

Spot Rates

  • Definition: The yield to maturity (YTM) on a zero-coupon bond is called the spot rate.

  • Notation: The n-year spot rate is denoted as yny_n.

  • Yield Curve Representation: The yield curve displays various maturities' spot rates, with YTM expressed as an annual rate.

Finding Spot Rates

  • Example Calculation: A five-year zero-coupon Treasury priced at 80% of face value.

  • Objective: Calculate the annualized five-year spot rate.

Finding the Yield Curve

  • Market Prices for Zero-Coupon Bonds:   - 1-year zero: P=925.93P = 925.93   - 2-year zero: P=841.75P = 841.75   - 3-year zero: P=758.33P = 758.33   - 4-year zero: P=683.18P = 683.18

  • Mathematical Formulation:   - For each period, find YTM using:     rac1000(1+yn)n=Prac{1000}{(1+y_n)^n} = P where P is the market price.

Yield to Maturity for Zero-Coupon Bonds

  • Years to Maturity and Corresponding Yields:   - 1 Year: y_1 = 8.000\%   - 2 Years: y_2 = 8.995\%   - 3 Years: y_3 = 9.660\%   - 4 Years: y_4 = 9.993\%

Pricing Bonds with Spot Rates

  • Example Problem: Price of a 2-year 10% coupon bond with a face amount of $1,000, where:   - 1-year spot rate is 7%   - 2-year spot rate is 12%

  • Calculation:   - P=rac100(1.07)+rac1100(1.12)2=970.371P = rac{100}{(1.07)} + rac{1100}{(1.12)^2} = 970.371

  • Yield to Maturity (YTM) on the bond: 11.75%.

Finding the Spot Rates - Case Study 2

  • Investment Strategies:   - Expected Returns: 8% on initial investment (Year 1) and 10% on subsequent investment (Year 2)   - Cumulative Returns Calculation:     1.08imes1.10=1.1881.08 imes 1.10 = 1.188

  • Investment Yield:   Y2=8.995 extor 1.089952=1.188Y_2 = 8.995\, ext{or} \, 1.08995^2 = 1.188

Finding Spot Rates - Case Study 3

  • Information Provided:   - 2-year annual coupon bond with:     - Coupon: 8%     - Maturity Value: $100     - Price: $99.10     - 1-year spot rate: 4%

  • Objective: Find the 2-year spot rate, y2y_2.

  • Equation:   - 99.10=rac8(1.04)+rac108(1+y2)299.10 = rac{8}{(1.04)} + rac{108}{(1 + y_2)^2}

  • Solution:   - y2=ext((108/(99.1−(8/(1.04))))0.5)−1=8.698%y_2 = ext{((108/(99.1 - (8/(1.04))))^0.5) - 1 = 8.698\%}

Forward Rates

  • Definition: The forward rate ( extit{f_n}) is the interest rate you can contract on today to lend in the future.

  • Example:   - extit{f_2} is the forward rate from period t+1 to t+2.

  • Relationship with Spot Rates:   - The forward rates can be computed using the formula:     (1+yn)n=(1+yn−1)n−1(1+fn)(1+y_n)^n = (1+y_{n-1})^{n-1}(1+f_n).

Forward Rates: Examples

  • Example 1: For the 1-year spot rate (y1) at 4% and 2-year spot rate (y2) at 8%, find forward rate in year 2 (f2).

  • Example 2: If the 5-year spot rate (y5) is 8% and the 6-year spot rate (y6) is 10%, find the forward rate in year 6 (f6).

The Relationship Between Forward Rates and Future Interest Rates

  • Concept: Are forward rates indicative of future interest rate expectations?

  • Implication Question: Does a forward rate of 6% (f2) suggest anything about future interest rates a year from now?

Causes of an Upward Sloping Yield Curve

  • Expectations Driving Yield Changes:   1. Investors expect future short-term interest rates to be higher than the current rates.   2. Long-term bonds carry higher risk, thus higher YTM.

Predicting Spot Rates Based on Future Interest Rates

  • Calculation of 2-Year Spot Rate:

  • Given 1-year spot rate of 5% and expected 1-year rate a year from now being 7%:   - (1+y2)2=(1+y1)(1+E(r2))(1+y_2)^2 = (1+y_1)(1+E(r2))   - Result: y2=6%y_2 = 6\%

Analyzing Yield Curve during Interest Rate Expectations

  • Further prediction with expected 1-year rates:   - 1-year spot rate at 5% and anticipated future rates.   - Find 3-year spot rate (y3).   - Result: y3 = 6.49% indicating an upward sloping yield curve.

Investor Sentiment and Economic Growth

  • Reasons for Beliefs:   - Expectations of high economic growth can lead to higher inflation and hence higher future interest rates.   - The Federal Reserve may lower interest rates to stimulate the economy, leading to a flat or inverted yield curve leading to recession risks.

The Expectations Theory of Yield Curve

  • Concept: The expectations theory states that the yield curve's slope is determined solely by expectations regarding future short-term interest rates.

  • Forward Rates Correlation: Forward rates (e.g., f2) reflect expected future short-term rates; thus:     - fn=E(rn)f_n = E(r_n) where f is the forward rate and E(r_n) is the expected rate.

Under the Expectations Theory

  • Understanding Term Structure:   - Rising term structures imply expectations of rising short-term rates.   - Flat term structures represent stagnant future short-term rate expectations.   - Falling term structures indicate expectations of declining short-term rates.

The Liquidity Preference Theory of Yield Curve

  • Definition: Suggests that there is a liquidity premium which increases with the time to maturity due to investor preferences for shorter maturities.

  • Implications: The yield curve may still slope upwards even when future interest rates are stable or declining due to this premium.

  • Pricing Framework: Forward rates can be adjusted to include liquidity premiums.

Liquidity Preference Theory Example Calculation

  • Given specific rates, liquidity premiums:   - Example: If y1=4%y_1 = 4\%, with future expectations and a liquidity premium of 1%, calculate the two-year spot rate y2y_2.

Indications of Term Structures and Liquidity Preference

  • Rising Term Structure: Unclear prediction on future short-term rates without more data.

  • Flat Term Structure: Uncertain predictions regarding future short-term outcomes.

  • Downward Sloping Term Structure: Generally indicates expectations of decreasing short-term rates.

Yield Curve and Risk in Bond Markets

  • Comparative Market Traits:   - Stocks and derivatives trade on exchanges providing visibility across price ranges.   - Bonds trade predominantly on OTC markets, experiencing illiquidity as maturity approaches.

  • Price Spread Rationale: Differences between on-the-run and off-the-run treasury bonds stem from relative liquidity conditions affecting yield calculations.

Practicing Calculations: Example Problems

  • 2-year 6% coupon bond:   - With 1-year and 2-year spot rates, compute price based on market rates.

  • 4-year forward rate assessment for given spot rates.

Upcoming Topics

  • Duration

  • Bond portfolio management.