The Term Structure of Interest Rates

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.

Yields

Spot Rates on Yield Curves

Upward Sloping Flat Downward Sloping Maturity

Expected Interest Rates in Coming Years

Expected One-Year Short Rates in Coming Years 1 2 3 Year 0 (today) Interest Rate 8% 10% 11% 11%

Pricing of Zero Coupon Bonds using Expected Rate

PV n

 ( 1 

r

1 )  ( 1  1

r

2 )...( 1 

r n

)  ( 1  1

y n

)

n

PV n = Present Value of $1 in n periods r 1 = One-year short rate for period 1 r 2 = One-year short rate for period 2 r n = One-year short rate for period n

y n=

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 ( 1 

f n

)  ( 1 ( 1  

y n

)

n y n

 1 )

n

 1

f n = one-year forward rate for period n y n = yield for a security with a maturity of n

( 1 

y n

)

n

 ( 1 

y n

 1 )

n

 1 ( 1 

f n

)

Example of Forward Rates

4 yr = 9.993% 3yr = 9.660% fn = ?

(1.0993) 4 = (1.0966) 3 (1+f n ) (1.46373) / (1.31870) = (1+f n ) f n = .10998 or 11% Note: this is expected rate that was used in the prior example.

Downward Sloping Spot Yield Curve

Zero-Coupon Rates Bond Maturity

12% 1 11.75% 11.25% 2 3 10.00% 9.25% 4 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 = 3yrs [(1.1) 4 / (1.1125) 3 ] - 1 4yrs [(1.0925) 5 / (1.1) 4 ] - 1 0.102567

= 0.063336

= 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.

f n



E

(

r n

)

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.

Yields

Liquidity Premiums and Yield Curves

Observed Yield Curve Short Rates Liquidity Premium Maturity

Yields

Liquidity Premiums and Yield Curves

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.