1. Suppose that the yield curve is flat at 5% per annum with continuous compound
ID: 2701968 • Letter: 1
Question
1. Suppose that the yield curve is flat at 5% per annum with continuous compounding. A swap with a notional principal of $100 million in which 6% is received and six-month LIBOR is paid will last another 15 months. Payments are exchanged every six months. The six-month LIBOR rate at the last reset date (three months ago) was 7%. Answer in millions of dollars to two decimal places.
(i) What is the value of the fixed-rate bond underlying the swap? _ _ _ _ _ _
(ii) What is the value of the floating-rate bond underlying the swap? _ _ _ _ _ _
(iii) What is the value of the payment that will be exchanged in 3 months? _ _ _ _ _ _
(iv) What is the value of the payment that will be exchanged in 9 months? _ _ _ _ _ _
(v) What is the value of the payment that will be exchanged in 15 months? _ _ _ _ _ _
(vi) What is the value of the swap? _ _ _ _ _ _
Explanation / Answer
A.) What is the value of the fixed rate bond underlying the swap?
$102.61 = 3 e^-.05x.25 + 3 e^-.05x.75 + 103 e^-.05x1.25
B.) What is the value of the floating rate bond underlying the swap?
$102.21 = (3.5 + 100) e^-.05x.25
3.5 equals .5 x 7% x 100, where 7% is 6-month LIBOR observed 3 months ago; 3.5 is the
next interest rate payment that will be paid 3 months from now. The floating rate bond
will be worth its par value of 100 immediately after the next interest payment of 3.5.
Since the firm in question receives fixed and pays floating, the value of the swap =
$102.61 %u2013 $102.21 = $0.4
C.) What is the value of the payment that will be exchanged in 3 months?
-0.49 = (3-3.5) e^-.05x.25
Note that, with regard to part C, there is no uncertainty regarding the cash flows that will
be exchanged 3 months from now. All uncertainty was resolved when 6-month LIBOR
was observed 3 months ago at a value of 7%.
D.) What is the value of the payment that will be exchanged in 9 months?
.45 = (3-2.5315) e^-.05x.75. The 5% forward rate continuously compounded is first
restated as an interest rate with semiannual compounding, i.e., 5.6302%. Thus, 2.5315 =
5.6302% x 100 x .5.
The swap cash flows 9 months from now are viewed as a 9-month FRA.
E.) What is the value of the payment that will be exchanged in 15 months? Derivatives Test Bank Dr. J. A. Schnabel Page 15 of 36
.44 = (3-2.5315) e^-.05x1.25. The 5% forward rate continuously compounded is first
restated as an interest rate with semiannual compounding, i.e., 5.6302%. Thus, 2.5315 =
5.6302% x 100 x .5.
Viewing the interest rate swap as portfolio of FRAs with staggered maturities, the value
of the swap to the company that receives fixed and pays floating equals 0.4 = -.49 + .45
+ .44
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