Adjust the bank discounted rate to make it comparable
n
P
P
r
BEY
365
000
,
10
×
−
=
%
28
.
8
90
365
800
,
9
800
,
9
000
,
10
=
×
−
=
BEY
r
BD
BEY
r
P
r
=
×
×
365
360
000
,
10
•
Example: same as before
•
BDR versus BEY

38
Spot Zero-Coupon (or Discount)
Rate
!
Spot Zero-Coupon (or Discount) Rate is the annualized
rate on a pure discount bond
!
where
B(0,t)
is the market price at date 0 of a bond paying off $1 at date t
!
General pricing formula
(
)
t
B
R
t
t
,
0
)
1
(
1
,
0
=
+
(
)
∑
∑
=
=
=
+
=
T
t
t
T
t
t
t
t
t
B
F
R
F
P
1
1
,
0
0
,
0
)
1
(
Bond Par Yield
!
Recall that a par bond is a bond with a coupon
identical to its yield to maturity
!
The bond's price is therefore equal to its principal
!
Then we define the par yield
c(n)
so that a n-year
maturity fixed bond paying annually a coupon rate of
c(n)
with a $100 face value quotes at par
!
Typically, the par yield curve is used to determine the
coupon level of a bond issued at par
∑
∑
=
=
+
+
−
=
⇒
+
+
+
=
n
i
i
i
n
n
n
n
n
i
i
i
R
R
n
c
R
R
n
c
1
,
0
,
0
,
0
1
,
0
)
1
(
1
)
1
(
1
1
)
(
)
1
(
100
)
1
(
)
(
100
100

39
Zero-Coupon Bonds
!
Zero-Coupon Bond
!
Does not make coupon payments
!
Always sells at a
discount
(a price lower than face value),
so they are also called
pure discount bonds
!
For example,
!
In USA,
Treasury Bills
are U.S. government zero-coupon
bonds with a maturity of up to one year.
!
In France
BTF
:
(bon du Trésor à taux fixe)
are France
government
zero-coupon
bonds
(prepaid
interest)
with
a
maturity of up to one year.
Zero-Coupon Bonds
!
Suppose that a one-year, risk-free, zero-coupon bond
with a $100,000 face value has an initial price of
$96,618.36. The cash flows would be:
!
Although
the
bond
pays
no
“interest,”
your
compensation is the difference between the initial price
and the face value.

40
Zero-Coupon Bonds
!
Yield to Maturity
!
The discount rate that sets the present value of the
promised bond payments equal to the current market
price of the bond.
!
Price of a Zero-Coupon bond
(1
)
=
+
n
n
FV
P
YTM
Zero-Coupon Bonds
!
Yield to Maturity
!
For the one-year zero coupon bond:
!
Thus, the YTM is 3.5%.
1
100,000
96,618.36
(1
)
=
+
YTM
1
100,000
1
1.035
96,618.36
+
=
=
YTM

41
Zero-Coupon Bonds
!
Yield to Maturity
!
Yield to Maturity of an
n
-Year Zero-Coupon Bond
1
1
=
−
n
n
FV
YTM
P
Zero-Coupon Bonds Example
!
Problem
!
Suppose
that
the
following
zero-coupon
bonds
are
selling at the prices shown below per $100 face value.
Determine the corresponding yield to maturity for each
bond.
Maturity
1 year
2 years
3 years
4 years
Price
$98.04
$95.18
$91.51
$87.14

42
Zero-Coupon Bonds Example
!
Solution:
1/2
1/3
1/4
YTM
(100 / 98.04)
1
0.02
2%
YTM
(100 / 95.18)
1
0.025
2.5%
YTM
(100 / 91.51)
1
0.03
3%
YTM
(100 / 87.14)
1
0.035
3.5%
=
−
=
=
−
=
=
=
−
=
=
=
−
=
=
Zero-Coupon Bonds
!
Risk-Free Interest Rates
!
A default-free zero-coupon bond that matures on date
n
provides a risk-free return over the same period.
!
Thus,
the
Law
of
One
Price
guarantees
that
the
risk-free interest rate equals the yield to maturity on
such a bond.
!
Risk-Free Interest Rate with Maturity
n
=
n
n
r
YTM

43
Zero-Coupon Bonds
!

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