Fixed Income Basics - part 1
Spot Interest rates
The zero-coupon yield curve
Bond yield-to-maturity
Default-free bond pricing
Finance 70520, Spring 2002
The Neeley School of Business at TCU
©Steven C. Mann, 2002
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
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
Coupon Bonds
T
Price =
S Ct B(0,t)
+ (Face) B(0,T)
t=1
where B(0,t) is price of 1 dollar to be received at time t
or
T
Price =
S
t=1
1
1
Ct (1+r )t + (Face) (1+r )T
t
t
where rt is discretely compounded rate associated with
a default-free cash flow (zero-coupon bond) at time t.
Define par bond as bond where Price=Face Value
= (par value)
Yield to Maturity
Define yield-to-maturity, y, as:
T
Price 
S
t=1
1
1
t
Ct(1+y) + (Face) (1+y)T
Solution by trial and error [calculator/computer algorithm]
Example: 2-year 7% annual coupon bond, price =104.52 per 100.
by definition, yield-to-maturity y is solution to:
104.52 = 7/(1+y) + 7/(1+y)2 + 100/(1+y)2
initial guess :
second guess:
y = 0.05
y = 0.045
eventually: when y = 0.04584
price = 103.72
price = 104.68
price = 104.52
(guess too high)
(guess too low)
y = 4.584%
If annual yield = annual coupon, then price=face (par bond)
Coupon bond yield is “average” of zero-coupon yields
T
T
1
1
Bond Value   B(0, t )C t  B(0, T )Face  
Ct 
Face
t
T
(1 0 yT )
t 1
t 1 (1 0 yt )
Coupon bond yield-to maturity, y, is solution to:
T
1
1
Bond Value  
Ct 
Face 
t
T
(1  y )
t 1 (1  y )
T
1
1
Ct 
Face

t
T
(1 0 yT )
t 1 (1 0 yt )
bond: $100 par, 3-year, annual coupon =
10%
T
B(0,T)
B(0,t)Ct
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
total:
25.26
75.83
101.09
9.56%
Bond yield =
Bonds with same maturity but different coupons will have different yields.
bond: $100 par, 3-year, annual coupon =
15%
T
B(0,T)
B(0,t)Ct
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
total:
37.89
75.83
113.72
9.52%
Bond yield =
bond: $100 par, 3-year, annual coupon =
5%
T
B(0,T)
B(0,t)Ct
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
total:
12.63
75.83
88.46
9.61%
Bond yield =
Semi-annual Yield-to-Maturity
Define semi-annual yield-to-maturity, ys, as:
Price 
T
S
t=1
1
1
Ct(1+y /2)+t (Face) (1+y /2)T
s
s
Note effective annual yield-to-maturity is yA  (1+ys/2)2 - 1
Example: 2-year 7% semi-annual coupon bond, price =103.79 per 100.
by definition, semi-annual yield-to-maturity ys is solution to:
103.79 =
S 3.50/(1+ys/2)t
+ 100/(1+ys/2)4
eventually: when ys/2 = 0.0249 = 2.49%
effective annual yield-to-maturity is yA = (1 + 0.0249)2 - 1 = 5.04%
If semi-annual yield = semi-annual coupon, then price=face (par bond)
Reinvestment assumptions and yield-to-maturity
Yield-to-maturity (ytm) is holding period rate of return only if
coupons can be reinvested at the same rate as yield-to-maturity
Example: 6% semi-annual coupon Par bond (price=100.00)
yield-to-maturity, ys, is
defined as:
100 
So that ys = 0.06
3
3
100


(1  y s / 2)
(1  y s / 2) 2
(1  y s / 2) 2
6-month coupon re-invested at ytm becomes 3(1+ys/2) = 3(1.03) in one year.
End-of-year value:
103 + 3(1.03) = 106.09.
Holding period return:
(106.09-100)/100 = 6.09%
Effective annual yield: 6% semi-annual yield = (1+0.06/2)2-1 = 6.09%
When re-investment is compounded semi-annually:
re-investment
holding-period
rate
proceeds at one year
return
5.0%
103 + 3.075 = 106.075
6.075%
7.0%
103 + 3.105 = 106.105
6.105%
Treasury bond quotes and prices
Accrued interest = Coupon x [(days since last coupon)/(days in coupon period)]
coupon
coupon
Coupon period
Coupon
11.625%
Bid
129.875
Maturity
11/15/04
Ask
130.000
Par value (Face)
$100,000
Settlement date
1/22/98
days in coupon period
181
days since last coupon
68
accrued interest
$2,183.70
Total purchase price if bought at bid
$132,058.70
Total purchase price if bought at ask
$132,183.70
Quotes are “clean prices” (no accrued interest)
Actual price is “dirty price”
Floating rate notes
Debt contract:
face value, maturity, coupon payment dates
Interest payments (coupons) reset at each coupon date.
Example:
one-year floater, semi-annual payments, Face=$100.00
payment based on six-month simple rate at beginning of coupon period
date zero (today)
six months later
spot six-month rate
5.25%
5.60%
coupon paid: end of period
c = 5.25/2 = 2.625
c = 5.60/2 = 2.80
Six months from now, value of note is:
102.80/[1+ 0.056 x (1/2)] = 102.80/1.028 = $100
In six months bond will be valued at par.
So value of note at time zero is:
(100 + 2.625)/[1 + 0.0525 x (1/2)] = 102.625/1.02625 = $100
Note value is at par each reset date.