Answers to Chapter 23
Questions:
1. The major differences between futures and forward contracts are:
i. Futures contracts are traded in open exchanges in standardized units, with fixed maturities. Forward contracts are
bilateral agreements between two counter parties. Hence, they can be tailor-made to the buyer’s satisfaction.
2. a. Sell forward
b. Buy forward
3. A hedge involves protecting the price of or return on an asset from adverse changes in price or return in the
market. A naive hedge usually involves the use of a derivative instrument that has the same underlying asset as the
4. a. You are obligated to take delivery of a $100,000 face value 20-year Treasury bond at a price of $95,000 at
some predetermined later date.
5. A microhedge uses a derivative contract such as a forward or futures contract to hedge the risk exposure of a
specific transaction, while a macrohedge is a hedge of the duration gap of the entire balance sheet. FIs that attempt
to manage their risk exposure by hedging each balance sheet position will find that hedging is excessively costly,
6. Basis risk is the lack of perfect correlation between changes in the yields of the on-balance-sheet assets or
7. a. The FI can either (i) buy a call option, or (ii) sell a put option on interest rate instruments, such as T-bonds, to
generate positive cash flows in the event that interest rates decline.
c. An FI is better off purchasing calls as opposed to writing puts. This is for two reasons. First, regulatory
8. a. March US Treasury Bond calls at 13400: 3 24/64 or $3,375.00 per $100,000 contract
9. a. 1. The value of the call decreases, 2. the value of the call increases.
10. A hedge with futures contracts produces symmetric gains and losses with interest rate increases and decreases.
That is, if the FI loses value on the bond resulting from an interest rate increase, it enjoys a gain on the futures
By comparison, a hedge with an option contract completely offsets losses but only partly offsets gains. That is,
gains and losses from hedging with options are no longer symmetric for interest rate increases and decreases. For
example, if the FI loses value on the bond due to an interest rate increase, a gain on the options contract offsets the
11. a. The bank faces the risk that interest rates will increase. The FI should buy a put option. If rates rise, the CDs
can be issued only at a lower price. But, the increase in interest rates also lowers the price of the security underlying
b. The insurance company (IC) is concerned that interest rates will fall, and thus the price of the bonds will rise. The
IC should buy call options on bonds. As rates fall, the underlying bond prices increase, but can be bought for less
c. The thrift will incur a loss on the sale if rates rise and the value of the bonds falls. The thrift should buy a put
d. The U.S. bank will incur a loss on the loan if the dollar appreciates (euros depreciate). Thus, the bank should buy
f. The FI is concerned that interest rates will fall, causing the value of the liabilities to rise more than the value of
12. In the case of writing a call option, the manager is obligated to sell the interest rate futures contract to the call
option buyer at the price of $114,000 per $100,000 contract. If the call option buyer chooses to exercise the option
13. a. The pension fund manager is exposed to interest rate declines (price increases).
futures.
14. Buying a cap means buying a call option or a succession of call options on interest rates. Specifically, if interest
rates rise above the cap rate, the seller of the cap—usually a bank—compensates the buyer—for example, another FI
—in return for an up-front premium. As a result, buying an interest rate cap is like buying insurance against an
15. A forward contract requires delivery or taking delivery of some commodity or financial security at a specified
17. First, FIs remain more likely to fail because of credit risk than either interest rate risk or FX risk. Second, credit
18. A total return swap involves swapping an obligation to pay interest at a specified fixed or floating rate for
payments representing the total return on a loan or a bond of a specific amount. The swap can be designed to cover
19. The total return swap includes an element of interest rate risk, while the pure credit swap has stripped this risk
from the contract. In a pure credit swap, the lender makes a fixed fee or payment premium to the counterparty in
20. The credit risk on a swap is lower than that of a loan for the following reasons:
b) In most cases, payments are made through netting by novation, which nets all payments with one
counterparty, further reducing the possibility of default.
Problems:
1. a. Interest plus principal expense on six-month CD = $1m x (1 + 0.065/2) = $1,032,500
Interest and principal earned on Swedish bond = $1,000,000/0.18 = SKr5,555,555.56 x (1 + 0.075/2) = SKr5,763,889
b. Net interest income should be = 0.005 x $1,000,000 = $5,000
2. The finance company will pay a fixed rate, while the insurance company pays a LIBOR based rate. One such
feasible swap would be for the insurance company to pay the finance company LIBOR + 2.5% and the finance
3. a. The commercial bank is at risk for a drop in rates that would lower interest income, while the savings
association is at risk of an interest rate increase, thus raising the cost of funds.
c. With this swap, the bank receives (T-bill + 0.02) – 0.09 + 0.09 – (T-bill + 0.01) = 0.01, the savings association
d. It is possible that the floating rate asset might not be tied to the same rate as the floating rate liability. This would
4. a. Ex ante, this is a profitable transaction since the spread is 2%. The 2% spread on $100 million (£150 million)
Eurodollar CD British Loan
tCash Outflow (U.S.$) (£) Cash Inflow (£) Spread (£)
1 7m 10.5m 13.5m 3m
However, this spread will be reduced or eliminated if the pound depreciates relative to the U.S. dollar. That is, if it
takes more pounds to purchase U.S. dollars, it will be more costly for the bank to repay the Eurodollar CD using
b. Expected future exchange rates are:
End of year 1: £1.65/U.S.$
Eurodollar CD British Loan
t Cash Outflow(U.S.$) (£) Cash Inflow (£) Spread (£)
c. t Cash Flow Swap Payments Net Swap Cash Flow Total Cash Flow
(£) (£) (£)
In the last column, the cash flows of the underlying cash position (in part b) are added to the cash flows from the
swap hedge. That is, at the end of the first year, the spread on the loan versus the CD is £1.95m. The swap generates
5. a. There is a feasible swap because of the comparative advantage inherent in the two sets of borrowing rates.
c. There is more than one feasible swap, but one such swap would be the following:
Bank 2 can issue floating rate CDs at (LIBOR + 3%), pay 13%, and receive (LIBOR + 3.5%). Its net cost is 12.5%
which is less than its current fixed payment.
The following problems are related to Appendix 23A, 23B, and 23C material.
6. a. The 20-year 8% coupon $100,000 Treasury bond has a duration of 10.292 years.
Discount
Time Exponent CF Factor PVCF PVCF x t
0.5 1 4,000 0.9615 3,846.2 1,923.1
1.0 2 4,000 0.9246 3,698.2 3,698.2
1.5 3 4,000 0.8890 3,556.0 5,334.0
2.0 4 4,000 0.8548 3,419.2 6,838.4
2.5 5 4,000 0.8219 3,287.7 8,219.3
11.5 23 4,000 0.4075 1,622.9 18,663.0
12.0 24 4,000 0.3901 1,560.5 18,726.0
12.5 25 4,000 0.3751 1,500.5 18,756.0
Duration = 1,029,200/100,000 = 10.292
c. A bid-ask quote of 101 – 130 = $101 13/32 per $100 face value. Since the Treasury bond futures contracts are for
7. The expected change in the spot position = -9.4 x (0.01/1.07) x 10,400,000 = -$913,645. This would mean a price
8. a. DGAP = DA – k DL = 6 – (0.9)(4) = 6 – 3.6 = 2.4 years
d. Solving for the impact on the change in equity under this assumption involves finding the impact of the change in
Expected E = A – L
9. a. The bank should sell futures contracts since an increase in interest rates would cause the value of the equity
and the futures contracts to decrease. But the bank could buy back the futures contracts to realize a gain to offset the
decreased value of the equity.
b. The number of contracts to hedge the bank is:
contracts365
$95,000x10.3725
0m(0.9)4)$15(6
F
Px
F
D
)A
L
kD
A
(D
F
N
c. For an increase in rates of 100 basis points, the change in the cash balance sheet position is:
Expected E = -DGAP[R/(1 + R)]A = -2.4(0.01/1.10)$150m = -$3,272,727.27. The change in bond value =
-10.3725(0.01/1.085295)$95,000 = -$9,079.41, and the change in 365 contracts is -$9,079.41 x -365 =
$3,313,986.25. Since the futures contracts were sold, they could be repurchased for a gain of $3,313,986.25. The
sum of the two values is a net gain of $41,258.98.
For a decrease in rates of 50 basis points, the change in the cash balance sheet position is:
Expected E = -DGAP[R/(1 + R)]A = -2.4(-0.005/1.10)$150m = $1,636,363.64. The change in each bond value =
d. If Treasury bill futures contracts are used, the duration of the underlying asset is 0.25 years, the face value of the
contract is $1,000,000, and the number of contracts necessary to hedge the bank is:
contracts1,469
$245,000
00$360,000,0
$980,000x0.25
0m(0.9)4)$15(6
F
Px
F
D
)A
L
kD
A
(D
F
N
e. In cases where a large number of Treasury bonds are necessary to hedge the balance sheet with a macrohedge, the
FI may need to consider whether a sufficient number of deliverable Treasury bonds are available. The number of
Treasury bill contracts necessary to hedge the balance sheet is greater than the number of Treasury bonds, the bill
market is much deeper and the availability of sufficient deliverable securities should be less of a problem.
10. The number of contracts necessary to hedge the bank would increase to 397 contracts. This can be found by
11. a. The mutual fund needs to enter into a contract to buy Treasury bonds at 98-24 in four months. The fund
b. The number of contracts can be determined by using the following equation:
contracts6.88
$98,750x 8.5
$481,250x 12
F
x P
F
D
x P D
F
N
Rounding this up to the nearest whole number is 7.0 contracts.
c. In this case the value of br = 1.12, and the number of contracts is 6.88/1.12 = 6.14 contracts. This may be
adjusted downward to 6 contracts.
12. a. The duration gap is 10 – (860/950)(2) = 8.19 years.
d. E = – 8.19(950,000)(0.01) = -$77,800
f. To macrohedge, the Treasury bond futures position should yield a profit equal to the loss in equity value (for any
given increase in interest rates). Thus, the number of futures contracts must be sufficient to offset the $77,800 loss in
13. In problem 12, we assumed that basis risk did not exist. That allowed us to assert that the percentage change in
interest rates (R/(1+R)) would be the same for both the futures and the underlying cash positions. If there is basis
a. If br = 0.9, then:
contracts 10,005 =
)(0.90)(9)(96,000
00,000)8.19(950,0
=
brx
F
x P
F
D
A )
L
kD –
A
(D
=
NF
c. If br = 0.9 then the percentage change in cash market rates exceeds the percentage change in futures market rates.
14.
contracts2728
$102,656x9
240m(0.875)4)$(14
F
Px
F
D
)A
L
kD
A
(D
F
N
15. a.
b. A $100,000 20-year, eight percent bond selling at $96,157 implies a yield of 8.4 percent.
c. B = -5 x $100,000,000 x 0.01/1.08 = -$4,629,630
d P = 824 x 3,250 = 824 x (-0.625) x (-10.1) x $96,157 x R /1.084
Solving for the change in interest rates gives
e. B = 3,250 x 824 = -5 x $100,000,000 x R /1.08
16. a. The duration gap for the bank is [12 – (720/840)7] = 6. Therefore, the bank is concerned that interest rates
b. The bonds underlying the put options have a market value of $104,531.25. Thus,
s
contractor14,75414,753.75
5$104,531.2x 8.17x 0.4
00$840,000,0x 6
BxDxδ
DGAPxA
p
N
c. The change in equity value is
d. P = Np( || x D x B x R/(1+R)) = 14,754 x 0.4 x 8.17 x $104,531.25 x 0.005/1.0756 = $23,429,185 gain
f. The cost of the put options = $12,909,750
g. Use the equation in part (d) above and solve for R. Then,
17. a. MD = D/(1 + 0.10) = 7/1.10 = 6.3636 years
BDx x δ
A]x
DL
k –
DA
[
=
Np
c. The change in equity value is
d. P = Np(|| x D x B x R/(1=R)) = 2,877 x 0.3 x 7 x $96,000 x 0.005/(1.10) = $2,636,378 gain
f. Use the equation in part (d) above and solve for R. Then
18. a. The mutual fund is concerned about interest rates falling which would imply that bond prices would increase.
Therefore, the FI should buy call options to guarantee a certain purchase price.
b.
s
option call 237or 236.75
$103,250x 9x 0.5
0$10,000,00x 11
x B x D δ
Ax
D
=
NC
c. The quote for T-bond options is 1-25, or 1 25/64 =1.390625 per $100 face value. This converts to $1,390.625 per
$100,000 option contract. The total cost of the hedge is 237 x $1,390.625 = $329,578.125.
d. For a rate increase, the B = -11 x $10,000,000 x (0.005)/1.0768 = -$510,773. If rates decrease, the value of the
e. If interest rates decrease, the value of the underlying bonds, and thus the option value, increases.
If rates rise, the option values would theoretically fall by $511,312. However, the option holder (the FI) is not
19. a. Premium for purchasing the cap = 0.0065 x $200 million = $1,300,000. If interest rates rise to 10 percent,
cap purchasers receive $200 million x 0.01 = $2,000,000. The net savings is $700,000.
If interest rates rise to 11 percent, the cap purchaser receives 0.02 x $200m = $4,000,000, and the net savings =
$4,000,000 – $2,680,000 = $1,320,000.
c. If the FI sells the floor, it receives net $1,380,000 minus the cost of the cap of $1,300,000 = +$80,000.
If interest rates fall to 3 percent, floor purchasers receive 0.01 x $200 million = $2,000,000. The net savings to the FI
d. The FI needs to sell: NVf x 0.0069 = $1,300,000, or NVf = $188,405,795 worth of 4 percent floors.