80
McDonald •
Fundamentals of Derivatives Markets
The one year implied forward rate from Year 3 to Year 4 is determined by Equation (7.3):
0
(0, 3)
0.8763
(3, 4)
1
1
0.06515
(0, 4)
0.8227
P
r
P
−=
The par coupon rate for a four year maturity uses Equation (7.6) with
0,
t
=
4,
T
=
and four yearly coupons:
1(
0
,
4
)
(0,1)
(0, 2)
(0, 3)
(0, 4)
1 0.8227
0.96154
0.91573 0.8763 0.8227
0.04958
P
c
PP
−
=
+++
−
=
++
+
=
The four year continuous yield is:
4
ln(0.8227)
0.04879
4
r
−
==
±
Question 7.2
The coupon bond pays a coupon of $60 each year plus the principal of $1,000 after five years. We have
cash flows of 60, 60, 60, 60,1060. To obtain the price of the coupon bond, we multiply each cash flow by
the zero-coupon bond price of that year. Specifically,
Bond Price
60 ( (0,1)
(0, 2)
(0, 3)
(0, 4)
(0, 5)) 1000
(0, 5)
60 (0.96154
0.91573
0.87630
0.8227
0.77611) 1000 0.77611
60 4.3524
776.11
261.14
776.11 1037.25
P
P
=×
+
+
+
+
+
×
+
+
+
+
+
×
+
=
+
=
This yields a bond price of $1,037.25.
±
Question 7.3
This is a straightforward application of Equations (7.1), (7.3), and (7.6). We also need the continuous rate
calculation (derived in Question 7.1):
ln(1/ (0, ))
ln( (0, ))
(0, )
cc
P
TP
T
rT
TT
−
Maturity
Zero-Coupon
Bond Yield
Zero Coupon
Bond Price
One-Year Implied
Forward Rate
Par
Coupon
Cont. Comp.
Zero Yield
1
0.03000
0.97087
0.03000
0.03000
0.02956
2
0.03500
0.93351
0.04002
0.03491
0.03440
3
0.04000
0.88900
0.05007
0.03974
0.03922
4
0.04500
0.83856
0.06014
0.04445
0.04402
5
0.05000
0.78353
0.07024
0.04903
0.04879