Example•Consider three bonds with the following characteristics:Determine spot rates for the 6-month (1st period), 1-year (2nd period), and 18-month (3rd period), expressed as semiannual APRs.BondMaturityCoupon Rate (Semiannual) Price16 month ࠵?1=0%$98212 month ࠵?2=0%$95318month ࠵?3=8%$102

Investment Theory, COMM 371, Dr. Farid Novin
Example
Bond
Maturity
Coupon Rate
(Semiannual)
Price
1
6 month
࠵?
1
=0%
$98
2
3
18month
8%
$102
࠵?
1
=
࠵?
(1+
࠵?
1
2
)
1
•
Isolate
࠵?
1
⇒
1+
࠵?
1
2
=
࠵?
࠵?
1
⇒
࠵?
1
=
100
98
− 1
∗ 2 = 4.0 %

Investment Theory, COMM 371, Dr. Farid Novin
Example
࠵?
2
=
࠵?
2
2
(1+
࠵?
2
2
)
1
+
࠵?
2
2
+࠵?
(1+
࠵?
2
2
)
2
⇒
95 =
0
(1+
࠵?
2
2
)
1
+
100
(1+
࠵?
2
2
)
2
⇒
(1 +
࠵?
2
2
)
=
(
100
95
)
1/2
࠵?
2
= 5.20
Bond
Maturity
Coupon Rate
(Semiannual)
Price
1
2
12 month
࠵?
2
=0%
$95
3

Investment Theory, COMM 371, Dr. Farid Novin
Example
Maturity
Coupon Rate
(Semiannual)
Price
1
(6 month)
2 (12 month)
3 (18month)
࠵?
3
=8%
102
࠵?
3
=
࠵?
3
2
(1+
࠵?
3
2
)
1
+
࠵?
3
2
(1+
࠵?
3
2
)
2
+
࠵?
3
2
+࠵?
(1+
࠵?
3
2
)
3
⇒
102 =
4
(1+
࠵?
3
2
)
1
+
4
(1+
࠵?
3
2
)
2
+
104
(1+
࠵?
3
2
)
3
⇒ ࠵?
3
=
6.58

Investment Theory, COMM 371, Dr. Farid Novin
Bootstrapping
•Calculate the spot rates for 3 treasury securities with face values of $100 and the following characteristics
•
࠵?
1
= 3%
•
࠵?
2
=
࠵?
2
(1+
࠵?
1
2
)
1
+
࠵?
2
+࠵?
(1+
࠵?
2
2
)
2
=>
100 =
2
(1+ .015)
+
100+2
(1+ ࠵?
2/2
)
2
=>
࠵?
2
= 4.01%
•
࠵?
3
=
࠵?
3
(1+
࠵?
1
2
)
1
+
࠵?
3
(1+
࠵?
2
2
)
2
+
࠵?
3
+࠵?
(1+
࠵?
3
2
)
3
=>
100 =
2.5
(1+ .015)
+
2.5
(1+ .02)
2
+
102.5
(1+
࠵?
3
2
)
3
Maturity
Market rates
Price
6-months
3%
100
12-months
4%
100
18-monts
5%
100
=>
࠵?
3
= 5.00%

Investment Theory, COMM 371, Dr. Farid Novin
Synthetic Replication of a Zero-Coupon Bond•Two bonds with face values of $100, are available for trading, and have the following characteristics:•How can one create $100, one year from now by trading this portfolio?•Questions:(i) How do we synthetically create the one-year zero by trading in the two-year bonds?(ii) What is the no-arbitrage price or cost of this synthetic one-year zero ?MaturityCoupon Rate(Annual rate)Price2-year4%$96.372-year8%$103.74

Investment Theory, COMM 371, Dr. Farid Novin
Synthetic Replication of a Zero-Coupon Bond
•Suppose that portfolio consists of n units of the first bond and m units of the second.
Year
Portfolio
Zero-Coupon Bond
1
$4 n + $8 m
$100
2
$104 n + $108 m
0
Maturity
Coupon Rate
(Annual rate)
Price
2-year
4%
$96.37
2-year
8%
$103.74

Investment Theory, COMM 371, Dr. Farid Novin