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Introduction to the Term Structure
The term structure of interest rates describes the relationship between yield and maturity for bonds that are similar in all respects except maturity. This relationship is typically illustrated through a yield curve, which is fundamental to understanding bond pricing and risk management.
Types of Yield Curves
| Yield Curve Shape | Description | Economic Interpretation |
|---|---|---|
| Normal/Upward Sloping | Long-term yields > Short-term yields | Expected economic growth and inflation |
| Inverted/Downward Sloping | Short-term yields > Long-term yields | Expected economic slowdown or recession |
| Flat | Similar yields across maturities | Economic transition period |
| Humped | Medium-term yields highest | Uncertainty about future economic conditions |
Key Insight
The yield curve is constructed using government bonds (risk-free securities) to isolate the effect of maturity on yield, removing credit risk considerations.
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Spot Rates and Forward Rates
Spot Rates (Zero Rates)
A spot rate is the yield on a zero-coupon bond (pure discount bond) for a given maturity. Spot rates represent the annualized return for investing in a security for a specific period, with no intermediate cash flows.
Zero-Coupon Bond Pricing Formula
Bond Price = Face Value / (1 + Spot Rate)^Time to Maturity
Where:
Face Value = Par value at maturity
Spot Rate = Zero-coupon yield for the maturity
Time to Maturity = Years until bond matures
Bond Price = Face Value / (1 + Spot Rate)^Time to Maturity
Where:
Face Value = Par value at maturity
Spot Rate = Zero-coupon yield for the maturity
Time to Maturity = Years until bond matures
Forward Rates
A forward rate is the interest rate that links two spot rates over two consecutive periods. It represents the rate that will make an investor indifferent between investing for the longer period versus investing for the shorter period and then reinvesting.
Forward Rate Calculation Formula
(1 + s₂)² = (1 + s₁) × (1 + f₁,₁)
Where:
s₂ = 2-year spot rate
s₁ = 1-year spot rate
f₁,₁ = 1-year forward rate starting in 1 year
(1 + s₂)² = (1 + s₁) × (1 + f₁,₁)
Where:
s₂ = 2-year spot rate
s₁ = 1-year spot rate
f₁,₁ = 1-year forward rate starting in 1 year
Example: Calculating Forward Rates
Given:
(1.04)² = (1.03) × (1 + f₁,₁)
1.0816 = 1.03 × (1 + f₁,₁)
f₁,₁ = (1.0816/1.03) - 1 = 5.01%
- 1-year spot rate (s₁) = 3%
- 2-year spot rate (s₂) = 4%
(1.04)² = (1.03) × (1 + f₁,₁)
1.0816 = 1.03 × (1 + f₁,₁)
f₁,₁ = (1.0816/1.03) - 1 = 5.01%
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Traditional Term Structure Theories
Pure Expectations Theory
The pure expectations theory states that forward rates are unbiased predictors of future spot rates. Long-term interest rates are geometric averages of expected future short-term rates.
Key Assumptions
- Investors are risk-neutral regarding maturity
- Bonds of different maturities are perfect substitutes
- Forward rates equal expected future spot rates
Liquidity Preference Theory
This theory suggests that investors demand a premium (liquidity premium) for holding longer-term bonds due to increased interest rate risk. Forward rates exceed expected future spot rates by this premium.
Liquidity Preference Theory Formula
Forward Rate = Expected Future Spot Rate + Liquidity Premium
Where:
Forward Rate = Rate implied by term structure
Expected Future Spot Rate = Market expectation
Liquidity Premium = Compensation for maturity risk
Forward Rate = Expected Future Spot Rate + Liquidity Premium
Where:
Forward Rate = Rate implied by term structure
Expected Future Spot Rate = Market expectation
Liquidity Premium = Compensation for maturity risk
Preferred Habitat Theory
Investors and borrowers have preferred maturity ranges (habitats) based on their needs and risk tolerance. Interest rates reflect supply and demand imbalances in different maturity segments, and investors require premiums to move outside their preferred habitats.
Market Segmentation Theory
An extreme version of preferred habitat theory suggesting that different maturity segments are completely separate markets with no substitution between them. Yields are determined solely by supply and demand within each segment.
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Modern Term Structure Models
Equilibrium vs. Arbitrage-Free Models
| Model Type | Approach | Examples | Key Feature |
|---|---|---|---|
| Equilibrium Models | Start with economic fundamentals | Vasicek, Cox-Ingersoll-Ross (CIR) | Derive theoretical yield curve |
| Arbitrage-Free Models | Start with observed yield curve | Ho-Lee, Hull-White | Fit to market prices exactly |
Interest Rate Volatility Models
Modern models incorporate interest rate volatility to better price bonds and derivatives:
- One-Factor Models: Interest rates driven by a single risk factor
- Multi-Factor Models: Multiple sources of uncertainty affect the term structure
- Mean Reversion: Interest rates tend to revert to a long-term average level
Vasicek Model
dr = a(b - r)dt + σdW
where: a = speed of mean reversion, b = long-term mean, σ = volatility, dW = random shock
where: a = speed of mean reversion, b = long-term mean, σ = volatility, dW = random shock
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Yield Curve Strategies
Active Yield Curve Strategies
Portfolio managers can implement various strategies based on expected changes in the yield curve:
| Strategy | Expectation | Implementation | Risk |
|---|---|---|---|
| Duration Management | Parallel yield curve shifts | Adjust portfolio duration | Non-parallel shifts |
| Yield Curve Positioning | Non-parallel shifts | Concentrate in specific maturities | Unexpected curve movements |
| Carry Trade | Stable yield curve | Borrow short, lend long | Curve flattening or inversion |
Yield Curve Trades
Example: Bullet vs. Barbell Strategy
Bullet Strategy: Concentrate holdings in intermediate maturities
Barbell Strategy: Hold short and long-term bonds, avoid intermediate maturities
When to use Bullet: Expect yield curve to steepen
When to use Barbell: Expect yield curve to flatten or become more volatile
Barbell Strategy: Hold short and long-term bonds, avoid intermediate maturities
When to use Bullet: Expect yield curve to steepen
When to use Barbell: Expect yield curve to flatten or become more volatile
Riding the Yield Curve
When the yield curve is upward sloping and stable, buying bonds with maturities longer than the investment horizon can generate excess returns as the bonds "roll down" the curve and their yields decline.
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Factors Affecting the Term Structure
Macroeconomic Factors
- Monetary Policy: Central bank actions affect short-term rates directly
- Inflation Expectations: Higher expected inflation steepens the yield curve
- Economic Growth: Strong growth expectations increase long-term yields
- Fiscal Policy: Government borrowing affects the supply of bonds
Market Factors
- Supply and Demand: Bond issuance and investor demand in different maturity segments
- Flight to Quality: Economic uncertainty increases demand for government bonds
- International Flows: Foreign investment affects domestic yield curves
- Market Volatility: Higher volatility increases demand for liquidity premiums
Yield Curve Inversions
Inverted yield curves have historically been reliable predictors of economic recessions. However, the relationship is not perfect, and other factors such as central bank policies and global conditions must be considered.