1. Introduction to Fixed Income – part 2
Forward Interest rates
yield curves
spot
par
forward
Introduction to Term Structure
Finance Fall 2003
Advanced Investments
Associate Professor Steven C. Mann
The Neeley School of Business at TCU
S. Mann, 2003.

2. Term structure “Term structure” may refer to various yields:
7.0
6.5
6.0
5.5
5.0
Term structure
Typical interest rate
term structure
Maturity (years)
“Term structure” may refer to various yields:
“spot zero curve”: yield-to-maturity for zero-coupon bonds
source: current market bond prices (spot prices)
“forward curve”: forward short-term interest rates: “short rates”
source: zero curve, current market forward rates
“par bond curve”: yield to maturity for bonds selling at par
source: current market bond prices
S. Mann, 2003.

3. one-year zero yield : 0y1 =5.85% ; B(0,1) = 1/(1.0585) = 0.944733
Forward rates
Introductory example (annual compounding) :
one-year zero yield : 0y1 =5.85% ; B(0,1) = 1/(1.0585) =
two-year zero yield: 0y2 =6.03% ; B(0,2) = 1/(1.0603)2 =
$1 investment in two-year bond produces $1( )2 = $ at year 2.
$1 invested in one-year zero produces $1( ) = $ at year 1.
What “breakeven” rate at year 1 equates two outcomes?
( )2 = ( ) [ 1 + f (1,2) ]
breakeven rate = forward interest rate from year 1 to year 2 = f (1,2)
(one year forward, one-year rate)
1 + f (1,2) = (1.0603)2/(1.0585) =
f (1,2) = = 6.21%
and $ (1.0621) = $

4. Forward and spot rate relationships : annualized rates

5. Example: Using forward rates to find spot rates
Given forward rates, find zero-coupon bond prices, and zero curve
Bond paying $1,000:
maturity Price yield-to-maturity
year 1 $1,000/(1.08) = $ y1=[1.08] (1/1) -1 =8%
year 2 $1,000/[(1.08)(1.10)] = $ y2 = [(1.08)(1.10)](1/2)- 1 =8.995%
year 3 $1,000/[(1.08)(1.10)(1.11)] = $ y3 =[(1.08)(1.10)(1.11)] (1/3) = 9.660%
year 4 $1,000/[(1.08)(1.10)(1.11)(1.11)] = $ y4 =[(1.08)(1.10)(1.11)(1.11)] (1/4) = 9.993%

6. Yield curves rate Forward rate zero-coupon yield coupon bond yield
Typical
upward sloping
yield curve
maturity
rate
Typical
downward sloping
yield curve
Coupon bond yield
zero-coupon yield
forward rate
maturity
S. Mann, 2003.

7. Coupon bond yield is “average” of zero-coupon yields
Coupon bond yield-to maturity, y, is solution to:

8. Bonds with same maturity but different coupons will have different yields.

9. Determination of the zero curve
B(0,t) is discount factor: price of $1 received at t; B(0,t) = (1+ 0yt)-t .
Example:
find 2-year zero yield
use 1-year zero-coupon bond price
and 2-year coupon bond price:
bond price per $100: yield
1-year zero-coupon bond %
2-year 6% annual coupon bond %
B(0,1) = Solve for B(0,2):
6% coupon bond value = B(0,1)($6) + B(0,2)($106)
$100 = ($6) + B(0,2)($106)
100 = B(0,2)($106)
= B(0,2)(106)
B(0,2) = /106 =
so that 0y2 = (1/B(0,2))(1/2) -1 = (1/0.8897)(1/2) -1
= %

10. “Bootstrapping” the zero curve from Treasury prices
Example:
six-month T-bill price B(0,6) =
12-month T-bill price B(0,12) =
18-month T-note with 8% coupon paid semi-annually price =
find “implied” B(0,18):
= 4 B(0,6) + 4 B(0,12) + (104)B(0,18)
= 4 ( ) B(0,18)
= B(0,18)
= 104 B(0,18)
B(0,18) = /104 =
24-month T-note with 7% semi-annual coupon: Price =
= 3.5B(0,6) + 3.5B(0,12) + 3.5B(0,18) B(0,24)
= 3.5( ) B(0,24)
B(0,24) = ( )/103.5 =

11. One-year holding period returns of zero-coupons: invest $100:
Holding period returns under certainty (forward rates are future short rates)
One year later:
f (0,1) = 0y1 = 10%
f (1,2) = %
f (2,3) = %
One-year holding period returns of zero-coupons:
invest $100:
one-year zero: $100 investment buys $100/ = $ Face value.
At end of 1 year, value = $ ; return = (108/100)-1 = 8.0%
two-year zero: $100 investment buys $100/ = $ Face value.
at end of 1 year, Value = $118.80/1.10 = $ ;
return = (108/100) -1 = 8.0%
three-year zero: $100 investment buys $100/ = $ face value
at end of 1 year, value = $131.87/[(1.10)(1.11)] = $ ;
If future short rates are certain, all bonds have same holding period return

12. Holding period returns when future short rates are uncertain
One year holding period returns of $100 investment in zero-coupons:
one-year zero: $100 investment buys $100/ = $ Face value.
1 year later, value = $ ; return = (108/100)-1 = 8.0% (no risk)
two-year zero: $100 investment buys $ face value.
1 year later: short rate = 11%, value = /1.11 = % return
short rate = 9%, value = /1.09 = % return
Risk-averse investor with one-year horizon holds two-year zero
only if expected holding period return is greater than 8%:
only if forward rate is higher than expected future short rate.
Liquidity preference: investor demands risk premium for longer maturity

13. Term Structure Theories
1) Expectations: forward rates = expected future short rates
2) Market segmentation: supply and demand at different maturities
3) Liquidity preference: short-term investors demand risk premium
rate
Forward rate = expected short rate + constant
Par Bond yield curve is upward sloping
Expected short rate is constant
maturity
Yield Curve: constant expected short rates
constant risk premium
S. Mann, 2003.

14. Yield curves with liquidity preference
rate
Forward rate
Liquidity premium
increasing with maturity
Par bond yield curve
Expected short rate is declining
maturity
rate
Forward rate
Humped par bond yield curve
Constant Liquidity premium
Expected short rate is declining
maturity
S. Mann, 2003.