Topic 2
Bond Prices, Yields,
Interest Rates and
Term Structure
Bond Pricing
Bond Yields
Coupon Stripping and Arbitrage
Duration
Term Structure of Interest Rates: Forward
Rate
1
Bond Pricing
Bond value = Present value of coupons +
Present value of par value
P
n
t 1
C
1 r
t
F
1 r n
(1)
C: Coupon payment
F: Par Value
r: interest rate used to discount the future
cash flow
T-period annuity factor: ( 1r (1 1 1r ). n
PV factor: 1
1 r n
Bond Price
2
= Coupon * 1
r
(1
1
1 r n
)+ Par Value * 1
1 r n
= Coupon * Annuity factor (r, T) +Par
Value * PV factor(r, T)
3
Bond Yields
Yield to maturity: the interest rate that
makes the present value of the cash flow
(coupon payments) equal to the market
value (price) of the instrument (bond).
P
n
t 1
C
1 y
t
F
1 y n
(2)
Given the coupons, face value, and bond
price, the yield is just the IRR (internal
rate of return) that makes the above hold
as an equality.
4
Bonds with semiannual coupon payments,
i.e., every six months:
C
n
P
t 1
2
F
y y
, t
(3) n
1 2 1 2
n: is now the number of semiannual periods.
Bonds with default risk:
pC 1 P C pF (1 p) F
P
1 y t
(1 y ) n
,
p: probability of default
: fraction of the promised amount
recovered.
5
Coupon Stripping
zero-coupon bond: makes only a single
payment at maturity. Also called “pure
discount” bond.
Pricing zero-coupon bonds: setting C=0
Price of zero-coupon bond = the present
value of F.
Treasury “Strip”— a Treasury bond that
has been unbundled and sold in pieces with
each piece being a coupon payment or the
principal payment.
6
Unbundling coupon bonds are called
“coupon-stripping.”
Any coupon-bearing bond: as a portfolio of
zero-coupon bonds, with each separate
payment on the whole bond representing a
zero-coupon bond.
For any bond with price P, maturity n, and
coupon rate C/F, the price can be
represented by:
C
n
P z t Fz n
t 1 2
,
zt : current price of a zero-coupon security
with a $1 face value and a maturity of t
periods from now.
7
Arbitrage
Treasury bonds can be reconstituted
from Treasury strips.
Treasury bond and the replicating
portfolio of strips should sell at the same
price. Otherwise there is an arbitrage
opportunity.
8
Example Of Arbitrage
A Treasury note with an 8 1/8% coupon
that matures in February 1998.
Coupons are payable on February 15 and
August 15 of each year.
On June 9, 1994 the asking price of the
bond was 105:29, i.e., 105 29/32 or
105.90625 per $100 face value.
9
Example Of Arbitrage
U.S. Treasury bonds are quoted on a
“skip-day settlement” basis.
This means that a trade occurring on June
9 would be settled on the second business
day after the trade, or on June 13.
June 13 is 118 days from the last coupon
date, while the entire coupon period,
from February 15 to August 15, is 181
days.
Thus, in addition to the quoted price, the
bond buyer must pay the seller 118/181
of a coupon payment.
10
Example Of Arbitrage
Invoice Price = Quoted Price + Coupon
Payment
Therefore, the invoice price of the bond
is
$105.90625+(118/181)(4.0625)=108.554
73.
Arbitrage Opportunity: use Treasury
strip prices from June 9 to see if we could
sell the pieces for more than the whole
bond.
11
Example Of Arbitrage
Date of Payment Amount Bid Price Per $1 of Proceeds
Payment
Aug. 1994 4.0625 0.993125 4.03457
Feb. 1995 4.0625 0.9690625 3.93682
Aug. 1995 4.0625 0.9421875 3.82764
Feb. 1996 4.0625 0.910625 3.69941
Aug. 1996 4.0625 0.8815625 3.58135
Feb. 1997 4.0625 0.8515625 3.45947
Aug. 1997 4.0625 0.821875 3.33887
Feb. 1998 4.0625 0.79375(coupon) 3.22461
Feb. 1998 100.00 0.7940625(Principal 79.40625
)
Total 108.50898
Whole Bond Ask 108.55473
Price
Arbitrage Profit -0.04575
Calculation shows: the prices of the
notes and the strips are very close. A
trader who bought the notes and stripped
them would actually lose about a nickel
per $100 face value.
12
Duration
Interest rate risk to a bondholder: interest
rate changes.
A change in interest rates has two effects.
Price Risk: the value of the bond moves
inversely with interest rates.
Reinvestment Risk: the value of
reinvesting the coupons moves positively
with interest rates.
One definition of duration is that it is
the horizon over which these two risks
would exactly cancel.
13
Duration is a measure of the sensitivity of
a bond’s price to a change in yield.
This gives MacCaulay Duration, DMAC,
as:
n
tC nF
dP
1 y 1 y
t n
P D
d (1 y ) MAC
t 1
P
(4)
(1 y )
MacCaulay duration is the percentage
change in the price of the bond with
respect to the percentage change in 1 plus
the yield.
Duration is a weighted average of all the
time to payment, with weights equal to
the present value of the payment at that
date divided by the bond price.
14
An alternative form of MacCaulay
Duration is:
C
1 y n y
1 y F
DMAC (5)
y C
F
1 y n 1 y
Modified Duration, DMOD :
dP
DMOD
dy
D
P MAC
1 y
(6)
15
Natural interpretation: the percentage
change in a bond’s price when its yield
changes by dy is given by:
dP
P
D dy
MOD (7)
Sometimes this is expressed as
dP/dy= - PDMOD.
Percentage price change =
- Modified duration * Yield change
16
Example of Duration
Consider a bond with 8% annual coupon rate
and yield to maturity of 10% (5% each half
year). Its maturity is two years. We calculate
duration of this bond.
(1) (2) (3) (4) (5)
Time Until Payment Payment PV of (3) (1) times (4)
Payment (yrs) Discounted at divided by
5% bond price
semiannually
0.5 $ 40 $38.095 0.0395 0.0198
1.0 $ 40 36.281 0.0376 0.0376
1.5 $ 40 34.553 0.0358 0.0537
2.0 $ 1040 855.611 0.8871 1.7742
SUM $964.540 1.00 1.8853
The bond has a maturity of two years but a
duration of 1.8853 years.
17
Some Properties of
Duration
1) The duration of a zero-coupon bond
equals its maturity.
2) Holding maturity constant, a bond’s
duration is higher when the coupon rate
is lower.
3) Holding the coupon constant, a
bond’s duration generally increases with
its time to maturity. Duration always
increases with maturity for bonds selling
at par or at a premium to par.
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4) Holding other factors constant, the
duration of a coupon bond is higher when
the bond’s yield is lower.
19
Forward Rates
The future interest rate is typically
uncertain. But we may think that market
participants can form a market’s
consensus for future interest rate.
Consider the following two investment
alternatives for an investor who has a
one-year investment horizon:
Alternative 1: investor buys a one-year
instrument
Alternative 2: investor buys a six-month
instrument and when it matures in six
months buys another six-month instrument.
20
Suppose the six-month spot rate is y1
and the one-year spot rate is y2. We
define f as the six-month interest rate
that prevails six months from now.
Alternative 1: Proceeds are
$100*(1+y2) 2
Alternative 2: Proceeds at the end of six
months are
$100*(1+y1)
If this amount is reinvested at the six-
month rate six months from now at a rate f,
then the total dollars at the end of one year
would be:
$100(1+y1)(1+f)
21
The investor will be indifferent between
the two alternatives if the total dollars are
the same, that is,
$100(1+ y2) 2 = $100(1+y1)(1+f)
Solving it, we get
(1 y ) 2
f = 1 2
(1 y )
2
1
More generally, the forward rate for a
one-period bond to be bought n periods
from now is defined by:
(1 y n ) n
fn = (1 y n 1 ) n 1
1
22
Yield Curve
Yield Curve or Term Structure of Interest
Rate: Relations between yields on
instruments of different maturities
To construct the term structure,
(1) Construct zero-coupon yields for
various maturities
(2) Calculate the yields, y1, y2, …, yn
for zero-coupon bonds with
maturities of 1, 2, …, n periods.
These are called “spot rate”.
Spot rate curve, yield curve or spot yield
curve: time to maturity vs. the yields for
zero-coupon bonds of different maturities
23
Points to Remember
Bond price and yield
Check for arbitrage between strips and
whole bonds
Calculate zero-coupon bond prices and
spot yield curve
Calculate Duration
Understand properties of Duration
Forward rates
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