1. The Term Structure of Interest Rates
Chapter 15

2. Overview of Term Structure
The relationship between yield to maturity and maturity.
Information on expected future short term rates can be implied from yield curve.
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.

3. Yield Curves
Yields
Maturity
Upward Sloping
Downward Sloping
Flat

4. Expected Interest Rates (Table 15.1)
Expected One-Year Rates in Coming Years
Year Interest Rate
0 (today) 8%
1 10%
2 11%
3 11%

5. Pricing of Bonds using Expected Rates
PVn = Present Value of $1 in n periods
r1 = One-year rate for period 1
r2 = One-year rate for period 2
rn = One-year rate for period n

6. Bond Prices using Expected Rates
Time to Maturity Price of Zero* Yield to Maturity
1 $ %
* $1,000 Par value zero

7. Forward Rates from Observed Rates
fn = one-year forward rate for period n
yn = yield for a security with a maturity of n

8. Example of Forward Rates using Table 15.2
4 yr = yr = fn = ?
(1.0993)4 = (1.0966)3 (1+fn)
( ) / ( ) = (1+fn)
fn = or 11%
Note: this is expected rate that was used in the prior example.

9. Downward Sloping Spot Yield Curve
Zero-Coupon Rates Bond Maturity
12% 1
11.75% 2
11.25% 3
10.00% 4
9.25% 5

10. Forward Rates Downward Sloping Y C
1yr Forward Rates
1yr [(1.1175)2 / 1.12] =
2yrs [(1.1125)3 / (1.1175)2] - 1 =
3yrs [(1.1)4 / (1.1125)3] =
4yrs [(1.0925)5 / (1.1)4] =

11. Theories of Term Structure
Expectations
Liquidity Preference
Upward bias over expectations
Market Segmentation
Preferred Habitat

12. 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.

13. 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.

14. Liquidity Premiums and Yield Curves
Yields
Observed Yield Curve
Forward Rates
Liquidity Premium
Maturity

15. Liquidity Premiums and Yield Curves
Yields
Maturity
Liquidity Premium
Forward Rates
Observed Yield Curve

16. 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.

17. Using Spot Rates to Price Coupon Bonds
A coupon bond can be viewed as a series of zero coupon bonds.
To find the value each payment is discount at the zero coupon rate.
Once the bond value is found, one can solve for the yield.
It’s the reason that similar maturity and default risk bonds sell at different yields to maturity.

18. Sample Bonds A B Maturity 4 years 4 years Coupon Rate 6% 8%
Par Value 1,000 1,000
Cash Flow in
Cash Flow in 4 1,060 1,080
Assuming Annual compounding

19. Price Using Spot Rates Bond A
Period
Spot Rate
Cash Flow
PV of Flow 1
.05 60
57.14 2
.0575
53.65 3
.063
49.95 4
.067
1,060
817.80
Total
978.54

20. Price Using Spot Rates Bond B
Period
Spot Rate
Cash Flow
PV of Flow 1
.05 80
76.19 2
.0575
71.54 3
.063
66.60 4
.067
1,080
833.23
Total
1,047.56

21. Solving For Yield to Maturity
Bond A
Bond Price
YTM %
Bond B
Price 1,047.56
YTM %