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Bond Price Implied Forward
0.05164 Question 7.8.
We have to take into account the interest we (or our counterparty) can earn on the FRA
settlement if we settle the loan on initiation day, and not on the actual repayment day. Therefore,
we tail the FRA settlement by the prevailing market interest rate of 5%. The dollar settlement is: 49 Part 2 Forwards, Futures, and Swaps (r − rFRA ) annually 1 + rannually × notional principal = (0.05 − 0.06) × $500,000.00 = −$4,761.905
1 + 0.05 b)
If the FRA is settled on the date the loan is repaid (or settled in arrears), the settlement
amount is determined by: (r annually − rFRA )× notional principal = (0.05 − 0.06 ) × $500,000.00 = −$5,000 We have to pay at the settlement, because the interest rate we could borrow at is 5%, but we have
agreed via the FRA to a borrowing rate of 6%. Interest rates moved in an unfavorable direction.
We can find the implied forward rates using the following formula:
[1 + r (t, t + s )] = P(P(,0,+)s )
0 This yields the following rates on the synthetic FRAs:
− 1 = 0.010884
r0 (90, 270 ) =
− 1 = 0.025734
r0 (90, 360 ) =
− 1 = 0.039596
r0 (90,180) = Question 7.12.
We can find the implied forward rate using the following formula:
[1 + r (t, t + s )] = P(P(,0,+)s )
0 With the numbers of the exercise, this yields:
r0 (270, 360 ) = 0.96525
− 1 = 0.0135135
0.95238 The following table follows the textbook in looking at forward agreements from a borrower’s
perspective, i.e. a borrower goes long an FRA to hedge his position, and a lender is thus short the
FRA. Since we are the counterparty for a lender, we are in fact the borrower, and thus long the
forward rate agreement.
50 Chapter 7 Interest Rate Forwards and Futures Transaction today
enter long FRAU t=0 t=270
− 10 M × 1.013514
= −10.13514 M Sell 9.6525M Zero Coupons
maturing at time t=180
Buy (1+0.013514)*10M *0.95238
Zero Coupons maturing at time t=360
TOTAL 9.6525M -10M − 10 M × 1.013514
× 0.95238 = −9.6525M
0 + 10.13514 M 0 0 By entering in the above mentioned positions, we are perfectly hedged against the risk of the
FRA. Please note that we are making use of the fact that interest rates are perfectly predictable.
We would like to guarantee the return of 6.5%. We receive payments 6.95485 after year one and
year two, and a payment of 106.95485 after year three. If interest rates are uncertain, we face an
interest rate risk for the investment of the first coupon from year one to year two and for the
discounting of the final payment from year three to year one.
Suppose we enter into a forward rate agreement to lend 6.95484 from year one to year two at the
current forward rate from year one to year two, and we enter into a forward rate agreement to
borrow 106.95485 tailed by the prevailing forward rate for year two to year three, at the
prevailing forward rate. This leads to the following cash-flow table:
Buy 3-year par bond
Receive first coupon
Enter short FRAU
Receive second coupon
enter long FRA for tailed position t=0
0 Receive final coupon and principal
-6.95485 0 t=2 6.95485 × 1.0700237
=99.025804 113.4225 t=3 -106.95485
0 We see that we can secure the same gross return as in the previous question,
113.4225 ÷ 100 = 1.065 . By entering appropriate FRAs, we secured the desired return of 6.5%.
Please note that we made use of the fact that we knew that we wanted to undo the position at t=2. 51 Part 2 Forwards, Futures, and Swaps Question 7.16.
a) The implied LIBOR of the September Eurodollar futures of 96.4 is: 100 − 96.4
= 0 .9 %
As we want to borrow money, we want to buy protection against high interest rates,
which means low Eurodollar future prices. We will short the Eurodollar contract.
One Eurodollar contract is based on a $1 million 3-month deposit. As we want to hedge a
loan of $50M, we will enter into 50 short contracts.
A true 3-month LIBOR of 1% means an annualized position (annualized by market
conventions) of 1%*4 = 4%. Therefore, our 50 short contracts will pay: [96.4 − (100 − 4) × 100 × $25]× 50 = $50,000
The increase in the interest rate has made our loan more expensive. The futures position that we
entered to hedge the interest rate exposure, compensates for this increase. In particular, we pay
$50,000,000 × 0.01 − payoff futures = $500,000 − $50,000 = $450,000 , which corresponds to the
0.9% we sought to lock in.
Question 7.18. a)
We face the classic problem of asset mismatch. We are interested in locking in an interest
rate for a 150-day investment, 60 days from now. However, while the Eurodollar futures matures
60 days from now, it secures a lending rate for 90 days....
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This document was uploaded on 03/11/2014 for the course FIN 402 at FPT University.
- Spring '14