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##### The Term Structure of Interest Rates

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**Overview of Term Structure of Interest Rates**• The relationship between yield to maturity and maturity. • Information on expected future short term rates (short rate) might be implied from the yield curve. (short rate vs. spot (average) rate) • The yield curve is a graph that displays the relationship between yield and maturity. • Three major theories are proposed to explain the observed yield curve.**Spot Rates on Yield Curves**Yields Upward Sloping Flat Downward Sloping Maturity**Expected Interest Rates in Coming Years**Expected One-Year Short Rates in Coming Years YearInterest Rate 0 (today) 8% 1 10% 2 11% 3 11%**Pricing of Zero Coupon Bonds using Expected Rate**PVn = Present Value of $1 in n periods r1 = One-year short rate for period 1 r2 = One-year short rate for period 2 rn = One-year short rate for period n yn= yield for a zero coupon security with a maturity of n**Thus**Short rates Spot rates • But what about the other way around?**Forward Rates from Observed Long-Term Rates**fn = one-year forward rate for period n yn = yield for a security with a maturity of n**Example of Forward Rates**4 yr = 9.993% 3yr = 9.660% fn = ? (1.0993)4 = (1.0966)3 (1+fn) (1.46373) / (1.31870) = (1+fn) fn = .10998 or 11% Note: this is expected rate that was used in the prior example.**Downward Sloping Spot Yield Curve**Zero-Coupon RatesBond Maturity 12% 1 11.75% 2 11.25% 3 10.00% 4 9.25% 5**Forward Rates for Downward Sloping Yield Curve**1yr Forward Rates 1yr [(1.1175)2 / 1.12] - 1 = 0.115006 2yrs [(1.1125)3 / (1.1175)2] - 1 = 0.102567 3yrs [(1.1)4 / (1.1125)3] - 1 = 0.063336 4yrs [(1.0925)5 / (1.1)4] - 1 = 0.063008**Theories of Term Structure**• Expectations • Liquidity Preference • Upward bias over expectations • Market Segmentation • Preferred Habitat**Expectations Theory**• Observed long-term rate is a function of today’s short-term rate and expected future short-term rates. • Long-term and short-term securities are perfect substitutes. • Forward rates that are calculated from the yield on long-term securities are market consensus expected future short-term rates.**Liquidity Premium Theory**• Long-term bonds are more risky. • Investors will demand a premium for the risk associated with long-term bonds. • The yield curve has an upward bias built into the long-term rates because of the risk premium. • Forward rates contain a liquidity premium and are not equal to expected future short-term rates.**Liquidity Premiums and Yield Curves**Yields Observed Yield Curve Short Rates Liquidity Premium Maturity**Liquidity Premiums and Yield Curves**Yields Observed Yield Curve Short Rates Liquidity Premium Maturity**Market Segmentation and Preferred Habitat**• Short- and long-term bonds are traded in distinct markets. • Trading in the distinct segments determines the various rates. • Observed rates are not directly influenced by expectations. • Preferred Habitat: • Modification of market segmentation • Investors will switch out of preferred maturity segments if premiums are adequate.