Unformatted text preview: We face the problem that the 90-day and
150-day interest rates may not be perfectly correlated. (For example, the term structure could,
over the next 60 days, move from upward sloping to downward sloping).
As we want to lend money, we want to buy protection against low interest rates, which
means high Eurodollar future prices. We will therefore go long the Eurodollar contract.
100 − 94
= 1 .5 % .
Under the assumption that the 3-month LIBOR rate and the 150-day interest rate are based on the
same annualized interest rate of 6%, we are able to lock in an interest rate of:
1 .5 % ×
= 2.5% . Please note that this is a rather strong assumption.
90 c) The implied LIBOR of the September Eurodollar futures of 94 is: d)
One Eurodollar futures contract is based on a $1 million 3-month deposit. As we want to
hedge an investment of $100M, we will enter into 100 long contracts. Again, we are making the
strong assumption that the annualized 3-month LIBOR rate and the annualized 150 day rate are
identical and perfectly correlated. 52 Chapter 7 Interest Rate Forwards and Futures Question 7.20. We will use the Excel functions Duration and Mduration to calculate the required durations:
They are of the form:
MDURATION(Start Date; Terminal Date; Coupon; Yield; frequency)
DURATION(Start Date; Terminal Date; Coupon; Yield; frequency),
where frequency determines the number of coupon payments per year. In order to use the
function, we have to give Excel a start date and terminal date, but we can just pick two dates that
are exactly the requested number of years apart. Plugging in the values of the exercises yields:
a) Macaulay Duration = 4.59324084
Modified Duration = 4.3983078 b) Macaulay Duration = 5.99377496
Modified Duration = 5.73566981 c)
We need to find the yield to maturity of this bond first. We can do so by using the YIELD
function of Excel. Plugging in the relevant values, we get: Yield = 0.07146759.
Now we can again use the Mduration and duration formulas. This yields:
Macaulay Duration = 7.6955970
Modified Duration = 7.1822957
Question 7.22. We will exploit equation (7.13) of the main text to find the optimal hedge ratio:
N =− D1 B1 ( y1 ) / (1 + y1 ) 6.631864 × 106.44 / (1.05004 )
D2 B2 ( y 2 ) / (1 + y 2 ) 7.098302 × 112.29 / (1.05252 )
757.2951883 Therefore, we have to short 0.887707 units of the nine-year bond for every eight-year bond to
obtain a duration-matched portfolio. 53 Chapter 8
We first solve for the present value of the cost per three barrels, based on the forward
(1.07 )3 We then obtain the swap price per barrel by solving: ⇔ b) x
x = 55.341
= 20.9519 We first solve for the present value of the cost per two barrels (year 2 and year 3):
(1.065) (1.07 )3 We then obtain the swap price per barrel by solving: ⇔ x
(1.065) (1.07 )3
= 21.481 Question 8.4.
The fair swap rate was determined to be $20.952. Therefore, compared to the forward curve
price of $20 in one year, we are overpaying $0.952. In year two, this overpayment has increased
to $0.952 × 1.070024 = 1.01866 , where we used the appropriate forward rate to calculate the
interest payment. In year two, we underpay by $0.048, so that our total accumulative
underpayment is $0.97066. In year three, this overpayment has increased again to
$0.97066 × 1.08007 = 1.048 . However, in year three, we receive a fixed payment of 20.952,
which underpays relative to the forward curve price of $22 by $22-$20.952 = 1.048. Therefore,
our cumulative balance is indeed zero, which was to be shown. 54 Chapter 8 Swaps Question 8.6.
In order to answer this question, we use equation (8.13.) of the main text. We assumed that the
interest rates and the corresponding zero-coupon bonds were:
8 interest rate
0.12 zero-coupon price
0.8929 Using formula 8.13., we obtain the following per barrel swap prices:
4-quarter swap price:
8-quarter swap price: $20.8532
$20.4275 The total costs of prepaid 4- and 8-quarter swaps are the present values of the payment
obligations. They are:
4-quarter prepaid swap price:
8-quarter prepaid swap price: $80.4190
$153.2460 Question 8.8.
We use formula (8.4), and replace the forward interest rate with the forward oil prices. In
particular, we calculate: ∑ P (0, t )F
∑ P (0, t )
6 i i =3 0
i =3 0 0 ,ti = $20.3807 i Therefore, the swap price of a 4-quarter oil swap with the first settlement occurring in the third
quarter is $20.3807. 55 Part 2 Forwards, Futures, and Swaps Question 8.10. We use equation (8.6) of the main text to answer this question: ∑ Q P (0, t )F
∑ Q P (0, t )
8 i =1 ti
i =1 i 0 ti 0 0 ,t i , where Q = [1,2,1,2,1,2,1,2] i After plugging in the relevant variables given in the exercise, we obtain a value of $20.4099 for
the swap price.
Question 8.12. With a...
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- Spring '14
- Derivatives, simple strategies