Introduction to Fixed Income – part 2
Forward Interest rates
yield curves
spot
par
forward
Introduction to Term Structure
Finance 30233 - Fall 2003
Advanced Investments
Associate Professor Steven C. Mann
The Neeley School of Business at TCU
Term structure
yield 7.0
6.5
6.0
Typical interest rate
term structure
5.5
5.0
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22
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
Forward rates
Introductory example (annual compounding) :
one-year zero yield : 0y1 =5.85% ; B(0,1) = 1/(1.0585) = 0.944733
two-year zero yield: 0y2 =6.03% ; B(0,2) = 1/(1.0603)2 = 0.889493
$1 investment in two-year bond produces $1(1+0.0603)2 = $1.1242 at year 2.
$1 invested in one-year zero produces
$1(1+0.0585) = $1.0585 at year 1.
What “breakeven” rate at year 1 equates two outcomes?
(1 + 0.0603)2 = (1 + 0.0585) [ 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) = 1.0621 - 1
and $1.0585 (1.0621) = $1.1242.
= 1.062103
= 6.21%
Forward and spot rate relationships : annualized rates
f (1,2) 
(1 0 y2 ) 2
(1 0 y1 )
 1
f ( n, n  1) 
(1 0 yn1 )n1
(1 0 yn )n
 1

 (1 0 yn1 )n1 


1
1



 B(0, n)  1
f ( n, n  1)  

1

n 

1
 B(0, n  1) 

 (1 0 yn ) 
B(0,1)
1
f (1,2) 
 1 ; B(0,1)  B(0,1)
B(0,2)
1  f (1,2)
(1 0 yn 1 ) n1
f ( n, n  1) 
 1
n
(1 0 yn )

B(0, n)
 1 ;
B(0, n  1)
1
B(0, n  1)  B(0, n)
1  f ( n, n  1)
Example: Using forward rates to find spot rates
Given forward rates, find zero-coupon bond prices, and zero curve
n
forward rate
(year)
0
1
2
3
f
f
f
f
f (n,n+1)
(0,1) =
8.0%
(1,2) = 10.0%
(2,3) = 11.0%
(3,4) = 11.0%
bill price
B(0,n+1)
B(0,1) = 0.92593
B(0,2) = 0.84175
B(0,3) = 0.75833
B(0,4) = 0.68318
spot rate
0yn+1
8.000%
8.995%
9.660%
9.993%
Forward rates
Spot rates
0
2
12.0%
11.0%
10.0%
9.0%
8.0%
7.0%
6.0%
1
3
Bond paying $1,000:
maturity Price
yield-to-maturity
(1/1)
year 1
$1,000/(1.08)
= $925.93
0y1=[1.08]
-1
year 2
$1,000/[(1.08)(1.10)]
= $841.75
0y 2 =
year 3
$1,000/[(1.08)(1.10)(1.11)]
= $758.33
0y3 =[(1.08)(1.10)(1.11)]
year 4
$1,000/[(1.08)(1.10)(1.11)(1.11)]
= $683.18
=8%
[(1.08)(1.10)](1/2)- 1
=8.995%
(1/3)
0y4 =[(1.08)(1.10)(1.11)(1.11)]
= 9.660%
(1/4)
= 9.993%
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
Coupon bond yield is “average” of zero-coupon yields
T
T
1
1
C

Face
t
t
T
(1 0 yT )
t 1 (1 0 yt )
Bond Value   B(0, t )C t  B(0, T )Face  
t 1
Coupon bond yield-to maturity, y, is solution to:
T
1
1
Bond Value  
C

Face 
t
t
T
(
1

y
)
(
1

y
)
t 1
T
1
1
C

Face

t
t
T
(
1

y
)
(
1

y
)
t 1
0 t
0 T
bond: 3-year $100 face; annual coupon = 10%
B(0,t)Ct
T
B(0,T)
B(0,3)$100
0y T
1
0.92593 8.00%
9.26
2
0.84175 9.00%
8.42
3
0.75833 9.66%
7.58
75.83 Bond Value
Bond yield =
9.56%
25.26
75.83
101.09
Bonds with same maturity but different coupons will have different yields.
bond: 3-year $100 face; annual coupon = 15%
B(0,t)Ct
T
B(0,T)
B(0,3)$100
0y T
1
0.92593 8.00%
13.89
2
0.84175 9.00%
12.63
3
0.75833 9.66%
11.37
75.83 Bond Value
Bond yield =
9.52%
37.89
75.83
113.72
bond: 3-year $100 face; annual coupon = 5%
B(0,t)Ct
T
B(0,T)
B(0,3)$100
0y T
1
0.92593 8.00%
4.63
2
0.84175 9.00%
4.21
3
0.75833 9.66%
3.79
75.83 Bond Value
Bond yield =
9.61%
12.63
75.83
88.46
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:
1-year zero-coupon bond
94.7867
2-year 6% annual coupon bond
100.0000
yield
5.500%
6.000%
B(0,1) = 0.9479. Solve for B(0,2):
6% coupon bond value = B(0,1)($6) + B(0,2)($106)
$100
= 0.9479($6) + B(0,2)($106)
100
= 5.6872 + B(0,2)($106)
94.3128 = B(0,2)(106)
B(0,2) = 94.3128/106 = 0.8897
so that
0y2
= (1/B(0,2))(1/2) -1 = (1/0.8897)(1/2) -1
= 6.0151%
“Bootstrapping” the zero curve from Treasury prices
Example:
six-month T-bill price
12-month T-bill price
B(0,6) = 0.9748
B(0,12) = 0.9493
18-month T-note with 8% coupon paid semi-annually price = 103.77
find “implied” B(0,18):
103.77 =
=
=
96.0736 =
B(0,18) =
4 B(0,6) + 4 B(0,12) + (104)B(0,18)
4 (0.9748+0.9493) + 104 B(0,18)
7.6964 + 104 B(0,18)
104 B(0,18)
96.0736/104 = 0.9238
24-month T-note with 7% semi-annual coupon: Price = 101.25
101.25 = 3.5B(0,6) + 3.5B(0,12) + 3.5B(0,18) + 103.5B(0,24)
= 3.5(0.9748+0.9493+0.9238) + 103.5B(0,24)
B(0,24) = (101.25 - 9.9677)/103.5 = 0.9016
Holding period returns under certainty (forward rates are future short rates)
n
forward rate
(year)
0
1
2
3
f
f
f
f
f (n,n+1)
(0,1) =
8.0%
(1,2) = 10.0%
(2,3) = 11.0%
(3,4) = 11.0%
bill price
B(0,n+1)
B(0,1) = 0.92593
B(0,2) = 0.84175
B(0,3) = 0.75833
B(0,4) = 0.68318
spot rate
0yn+1
8.000%
8.995%
9.660%
9.993%
One year later:
f (0,1) = 0y1 = 10%
f (1,2) =
11%
f (2,3) =
11%
One-year holding period returns of zero-coupons:
invest $100:
one-year zero: $100 investment buys $100/92.92593 = $108.00 Face value.
At end of 1 year, value = $108.00 ;
return = (108/100)-1 = 8.0%
two-year zero: $100 investment buys $100/84.175 = $118.80 Face value.
at end of 1 year, Value = $118.80/1.10 = $108.00 ;
return = (108/100) -1 = 8.0%
three-year zero: $100 investment buys $100/75.833 = $131.87 face value
at end of 1 year, value = $131.87/[(1.10)(1.11)] = $108.00 ;
return = (108/100) -1 = 8.0%
If future short rates are certain, all bonds have same holding period return
Holding period returns when future short rates are uncertain
n
forward rate
(year)
0
1
2
3
f
f
f
f
f (n,n+1)
(0,1) =
8.0%
(1,2) = 10.0%
(2,3) = 11.0%
(3,4) = 11.0%
bill price
B(0,n+1)
B(0,1) = 0.92593
B(0,2) = 0.84175
B(0,3) = 0.75833
B(0,4) = 0.68318
spot rate
0yn+1
8.000%
8.995%
9.660%
9.993%
possible short rate (0y1) evolution:
now
one year later
11.00%
8.00%
9.00%
One year holding period returns of $100 investment in zero-coupons:
one-year zero: $100 investment buys $100/92.92593 = $108.00 Face value.
1 year later, value = $108.00 ;
return = (108/100)-1 = 8.0% (no risk)
two-year zero: $100 investment buys $118.80 face value.
1 year later: short rate = 11%, value = 118.80/1.11 = 107.03
short rate = 9%, value = 118.80/1.09 = 108.99
7.03% return
8.99% 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
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
Yield Curve:
constant expected short rates
constant risk premium
maturity
Yield curves with liquidity preference
rate
Forward rate
Par bond yield curve
Liquidity premium
increasing with maturity
Expected short rate is declining
maturity
rate
Forward rate
Humped par bond yield curve
Constant Liquidity premium
Expected short rate is declining
maturity