FIN 683
Professor Robert Hauswald
Financial-Institutions Management
Kogod School of Business, AU
Solutions 4
Chapter 14: Foreign Exchange Risk
8.
The following are the foreign currency positions of an FI, expressed in the foreign currency.
Currency
Assets
Liabilities
FX Bought
FX Sold
SF134,394
SF53,758
SF10,752
SF16,127
British pound (£)
£30,488
£13,415
£9,146
£12,195
Japanese yen (¥)
¥7,075,472
¥2,830,189
¥1,132,075
¥8,301,887
Swiss franc (SF)
The exchange rate of dollars per SFs is .9301, of dollars per British pounds is 1.6400, and of dollars
per yen is .010600.
The following are the foreign currency positions converted to dollars.
Currency
Assets
Liabilities
FX Bought
FX Sold
Swiss franc (SF)
$125,000
$50,000
$10,000
$15,000
British pound (£)
$50,000
$22,001
$14,999
$20,000
Japanese yen (¥)
$75,000
$30,000
$12,000
$88,000
a. What is the FI’s net exposure in Swiss francs stated in SFs and in dollars?
Net exposure in stated in SFs = SF134,394 - SF53,758 + SF10,752 - SF16,127 = SF75,261
Net exposure in stated in $s = $125,000 - $50,000 + $10,000 - $15,000 = $70,000
1
FIN 683
Professor Robert Hauswald
Financial-Institutions Management
Kogod School of Business, AU
b. What is the FI’s net exposure in British pounds stated in pounds and in dollars?
Net exposure in British pounds
= £30,488 - £13,415 + £9,146 - £12,195= £14,024
Net exposure in $s = $50,000 - $22,001 + $15,000 - $20,000 = $22,999
c. What is the FI’s net exposure in Japanese yen stated in yen and in dollars?
Net exposure in Japanese yen = ¥7,075,472 - ¥2,830,189 + ¥1,132,075 - ¥8,301,887= - ¥2,924,529
Net exposure in $s = $75,000 - $30,000 + $12,000 - $88,000 = -$31,000
d. What is the expected loss or gain if the SF exchange rate appreciates by 1 percent? State you
answer in Swiss francs and dollars.
If assets are greater than liabilities, then an appreciation of the foreign exchange rates will
generate a gain = SF75,261 x 0.01 = SF752.61 or $70,000 x 0.01 = $700.00.
e. What is the expected loss or gain if the £ exchange rate appreciates by 1 percent? State you
answer in pounds and dollars.
Gain = £14,024 x 0.01 = $140 or $22,999 x 0.01 = $230
f. What is the expected loss or gain if the ¥ exchange rate appreciates by 2 percent? State you
answer in yen and dollars.
Loss = - ¥2,924,529 x 0.02 = -$58,491 or -$31,000 x 0.02 = -$620
2
FIN 683
Professor Robert Hauswald
10.
Financial-Institutions Management
Kogod School of Business, AU
City Bank issued $200 million of one-year CDs in the United States at a rate of 6.50 percent. It
invested part of this money, $100 million, in the purchase of a one-year bond issued by a U.S. firm
at an annual rate of 7 percent. The remaining $100 million was invested in a one-year Brazilian
government bond paying an annual interest rate of 8 percent. The exchange rate at the time of
the transaction was Brazilian real 1/$.
a. What will be the net return on this $200 million investment in bonds if the exchange
rate between the Brazilian real and the U.S. dollar remains the same?
Cost of funds
=
0.065 x $200 million
= $13 million
Return on U.S. loan
=
0.07 x $100 million
= $ 7,000,000
Return on Brazilian bond =
(.08 x Real 100 m)/1.00 = $ 8,000,000
Total interest earned
= $15,000,000
Net return on investment = $15 million - $13 million/$200 million = 1.00 percent.
b. What will be the net return on this $200 million investment if the exchange rate
changes to real 1.20/$?
Cost of funds
=
0.065 x $200 million
= $13,000,000
Return on U.S. loan
=
0.07 x $100 million
= $ 7,000,000
Return on Brazilian bond = (0.08 x Real 100m)/1.20
= $ 6,666,667
Total interest earned
= $13,666,667
Net return on investment = $13,666,667 - $13,000,000/$200,000,000 = 0.33 percent.
Consideration should be given to the fact that the Brazilian bond was for Real100 million. Thus, at
maturity the bond will be paid back for Real100 million/1.20 = $83,333,333.33. Therefore, the
strengthening dollar will have caused a loss in capital ($16,666,666.67) that far exceeds the
interest earned on the Brazilian bond.
3
FIN 683
Professor Robert Hauswald
Financial-Institutions Management
Kogod School of Business, AU
c. What will be the net return on this $200 million investment if the exchange rate
changes to real 0.80/$?
Cost of funds
=
0.065 x $200 million
= $13,000,000
Return on U.S. loan
=
0.07 x $100 million
= $ 7,000,000
Return on Brazilian bond = (.08 x Real 100m)/0.80
= $10,000,000
Total interest earned
= $17,000,000
Net return on investment = $17,000,000 - $13,000,000/$200,000,000 = 2.00 percent.
Consideration should be given to the fact that the Brazilian bond was for Real100 million. Thus, at
maturity the bond will be paid back for Real100 million/0.80 = $125,000,000. Therefore, the
strengthening Real will have caused a gain in capital of $25,000,000 in addition to the interest
earned on the Brazilian bond.
11.
Sun Bank USA purchased a 16 million one-year euro loan that pays 12 percent interest annually.
The spot rate of U.S. dollars per euro is 1.40. Sun Bank has funded this loan by accepting a British
pound-denominated deposit for the equivalent amount and maturity at an annual rate of 10
percent. The current spot rate of U.S. dollars per British pound is 1.60.
a. What is the net interest income earned in dollars on this one-year transaction if the spot
rates of U.S. dollars per euro and U.S. dollars per British pound at the end of the year
are 1.50 and 1.85?
.
Loan amount = €16 million x 1.40 = $22.4 million
Deposit amount = $22.4m/1.60 = £14,000,000
Interest income at the end of the year = €16m x 0.12 = €1.92m x 1.50 = $2,880,000
Interest expense at the end of the year = £14,000,000 x 0.10 = £1,400,000 x 1.85 = $2,590,000
Net interest income = $2,880,000 - $2,590,000 = $290,000
4
FIN 683
Professor Robert Hauswald
Financial-Institutions Management
Kogod School of Business, AU
b. What should be the spot rate of U.S. dollars per British pound at the end of the year in order
for the bank to earn a net interest margin of 4 percent?
A net interest margin of 4 percent would imply $22,400,000 x 0.04 = $896,000.
The net cost of deposits should be $2,880,000 - 896,000 = $1,984,000.
Pound rate = $1,984,000/£1,400,000 = $1.4171/£.
Thus, the pound should be selling at $1.4171/£ in order for the bank to earn 4 percent.
c.
Does your answer to part (b) imply that the dollar should appreciate or depreciate against the
pound?
The dollar should depreciate against the pound. Each pound gives fewer dollars.
d. What is the total effect on net interest income and principal of this transaction given the
end-of-year spot rates in part (a)?
Interest income and loan principal at year-end = (€16m x 1.12) x 1.50 = $26,880,000
Interest expense and deposit principal at year-end = (£14m x 1.10) x 1.85 = $28,490,000
Total income = $26,880,000 - $28,490,000 = -$1,610,000
12.
Bank USA just made a one-year $10 million loan that pays 10 percent interest annually. The loan
was funded with a Swiss franc-denominated one-year deposit at an annual rate of 8 percent. The
current spot rate is SF1.60/$1.
a.
What will be the net interest income in dollars on the one-year loan if the spot rate at
the end of the year is SF1.58/$1?
Interest income at year-end = $10m x 0.10 = $1,000,000.
5
FIN 683
Professor Robert Hauswald
Financial-Institutions Management
Kogod School of Business, AU
Interest expense at year-end = (SF16,000,000 x 0.08)/1.58
= SF1,280,000/1.58 = $810,126.58.
Net interest income = $1,000,000 - $810,810.58 = $189,873.42.
b.
What will be the net interest return on assets?
Net interest return on assets = $189,873.42/$10,000,000 = 0.0190 or 1.90 percent.
c. How far can the SF/$ appreciate before the transaction will result in a loss for Bank USA?
Exchange rate = SF1,280,000/$1,000,000 = SF1.28/$, appreciation of 18.99 percent.
d. What is the total effect on net interest income and principal of this transaction given the endof-year spot rates in part (a)?
Interest income and loan principal at year-end = $10m x 1.10 = $11,000,000.
Interest expense and deposit principal at year-end = (SF16,000,000 x 1.08)/1.58
= SF17,280,000/1.58 = $10,936,708.86
Total income = $11,000,000 - $10,936,708.86 = $63,291.14
14.
What are the two primary methods of hedging FX risk for an FI? What two conditions are
necessary to achieve a perfect hedge through on-balance-sheet hedging? What are the
advantages and disadvantages of off-balance-sheet hedging in comparison to on-balance-sheet
hedging?
The manager of an FI can hedge using on-balance-sheet techniques or off-balance-sheet techniques. Onbalance-sheet hedging requires matching currency positions and durations of assets and liabilities. If the
duration of foreign-currency-denominated fixed rate assets is greater than similar currency
6
FIN 683
Professor Robert Hauswald
Financial-Institutions Management
Kogod School of Business, AU
denominated fixed rate liabilities, the market value of the assets could decline more than the liabilities
when market rates rise and therefore the hedge will not be perfect. Thus, in matching foreign currency
assets and liabilities, not only do they have to be of the same currency, but also of the same duration in
order to have a perfect hedge.
Advantages of off-balance-sheet FX hedging:
The use of off-balance-sheet hedging devices, such as forward contracts, enables an FI to reduce or
eliminate its FX risk exposure without forfeiting potentially lucrative transactions. On-balance-sheet
transactions result in immediate cash flows, whereas off-balance-sheet transactions result in contingent
future cash flows. Therefore, the up-front cost of hedging using off-balance-sheet instruments is lower
than the cost of on-balance-sheet transactions. Moreover, since on-balance-sheet transactions are fully
reflected in financial statements, there may be additional disclosure costs to hedging on the balance
sheet.
Off-balance-sheet hedging instruments have been developed for many types of risk exposures. For
currency risk, forward contracts are available for the majority of currencies at a variety of delivery dates.
Moreover, since the forward contract is negotiated over the counter, the counterparties have maximum
flexibility to set terms and conditions.
Disadvantages of off-balance-sheet FX hedging:
There is some credit risk associated with off-balance-sheet hedging instruments since there is some
possibility that the counterparty will default on its obligations. This credit risk exposure is exacerbated in
negotiated markets such as the forward market, but mitigated for exchange-traded hedging instruments
such as futures contracts.
7
FIN 683
Professor Robert Hauswald
15.
Financial-Institutions Management
Kogod School of Business, AU
North Bank has been borrowing in the U.S. markets and lending abroad, thus incurring foreign
exchange risk. In a recent transaction, it issued a one-year, $2 million CD at 6 percent and funded
a loan in euros at 8 percent. The spot rate for the euro was €1.45/$1 at the time of the
transaction.
a. Information received immediately after the transaction closing indicated that the euro
will change to €1.47/$1 by year-end. If the information is correct, what will be the
realized spread on the loan inclusive of principal? What should have been the bank
interest rate on the loan to maintain the 2 percent spread?
Amount of loan in € = $2 million x 1.45 = €2.9 million.
Interest and principal at year-end = €2.9m x 1.08 = €3.132m/1.47 = $2,130,612.24
Interest and principal of CDs = $2m x 1.06 = $2,120,000
Net interest income = $2,130,612.24 – $2,120,000 = $10,612.24
Net interest margin = $10,612.24/2,000,000 = 0.0053 or 0.53 percent.
In order to maintain a 2 percent spread, the interest and principal earned at €1.47/$ should be:
€2.9m.(1 + x)/1.47 = $2.16m. (Because ($2.16m. - $2.12m.)/$2.00m. = 0.02, or 2%).
Therefore, (1 + x) = ($2.16m. x 1.47)/ €2.9m. = 1.0949, and x = 0.0949 or 9.49 percent,
or the bank should have charged a rate of 9.49 percent on the loan.
b. The bank had an opportunity to sell one-year forward euros at €1.46/$. What would
have been the spread on the loan if the bank had hedged forward its foreign exchange
exposure?
Net interest income if hedged
= €2.9m. x 1.08 = €3.132m./1.46 = $2.1452m. - $2.12m.
= $0.0252 million, or $25,205.48
Net interest margin = $0.0252m./$2m. = 0.0126, or 1.26 percent
c. What would have been an appropriate change in loan rates to maintain the 2 percent
spread if the bank intended to hedge its exposure using the forward contracts?
To maintain a 2 percent spread: €2.9m.(1 + x)/1.46 = $2.16m. => x = 8.74 percent
8
FIN 683
Professor Robert Hauswald
Financial-Institutions Management
Kogod School of Business, AU
The bank should increase the loan rate to 8.74 percent and hedge with the sale of forward €s to
maintain a 2 percent spread.
9
FIN 683
Professor Robert Hauswald
Financial-Institutions Management
Kogod School of Business, AU
Chapter 14: Purchasing Power and Interest Rate Parity
19.
Suppose that the current spot exchange rate of U.S. dollars for Australian dollars, SUS$/A$,
is .7590 (i.e., 0.759 dollars, or 75.9 cents, can be received for 1 Australian dollar). The
price of Australian-produced goods increases by 5 percent (i.e., inflation in Australia, IPA, is 5
percent), and the U.S. price index increases by 3 percent (i.e., inflation in the United States, IPUS,
is 3 percent). Calculate the new spot exchange rate of U.S. dollars for Australian dollars that
should result from the differences in inflation rates.
According to PPP, the 5 percent rise in the price of Australian goods relative to the 3 percent rise in the
price of U.S. goods results in a depreciation of the Australian dollar (by 2 percent). Specifically, the
exchange rate of Australian dollars to U.S. dollars should fall, so that:
iUS - iA = ΔSUS$/A$/SUS$/A$
Plugging in the inflation and exchange rates, we get:
.03 - .05 = ΔSUS$/A$/SUS$/A$ = ΔSUS$/A$/ .759
or:
-.02 = ΔSUS$/A$/.759
and:
ΔSUS$/A$ = -(.02) × .759 = -.01518
Thus, it costs 1.518 cents less to receive an Australian dollar (or it costs 15.98 cents (75.9 cents - 1.518
cents), or .74382 of $1, can be received for 1 Australian dollar). The Australian dollar depreciates in
value by 2 percent against the U.S. dollar as a result of its higher inflation rate.
21.
Assume that annual interest rates are 8 percent in the United States and 4 percent in Japan. An FI
can borrow (by issuing CDs) or lend (by purchasing CDs) at these rates. The spot rate is $0.60/¥.
10
FIN 683
Professor Robert Hauswald
Financial-Institutions Management
Kogod School of Business, AU
a. If the forward rate is $0.64/¥, how could the FI arbitrage using a sum of $1million?
What is the expected spread?
Borrow $1,000,000 in U.S. by issuing CDs
⇒ Interest and principal at year-end = $1,000,000 x 1.08 = $1,080,000
Make a loan in Japan
⇒ Interest and principal = $1,000,000/0.60 = ¥1,666.667 x 1.04 = ¥1,733,333
Purchase U.S. dollars at the forward rate of $0.64 x 1,733,333 = $1,109,333.33
Spread = $1,109,333.33 - $1,080,000 = $29,333.33/1,000,000 = 2.93%
b. What forward rate will prevent an arbitrage opportunity?
The forward rate that will prevent any arbitrage is given by solving the following equation:
Ft =
(1 + r Dust )
* St
(1 + r Ljpt )
Ft = [(1 + 0.08) * 0.60]/(1.04) = $0.6231/¥
11
FIN 683
Professor Robert Hauswald
Financial-Institutions Management
Kogod School of Business, AU
Chapter 14: FX VAR
24.
An FI has $100,000 of net positions outstanding in British pounds (£) and -$30,000 in Swiss francs
(SF). The standard deviation of the net positions as a result of exchange rate changes is 1 percent
for the SF and 1.3 percent for the £. The correlation coefficient between the changes in exchange
rates of the £ and the SF is 0.80.
a. What is the risk exposure to the FI of fluctuations in the £/$ rate?
Since the FI has a positive £ position, an appreciation of the £ will increase the value of its £denominated assets more than its liabilities, providing a net gain. The opposite will occur if the £
depreciates.
b. What is the risk exposure to the FI of fluctuations in the SF/$ rate?
Since the FI has a negative net position in SFs, the value of its Swiss-denominated assets will
increase in value, but not as much as the value of its liabilities. Hence, an appreciation of the SF
will lead to a net loss. The opposite will occur if the currency depreciates.
c. What is the risk exposure if both the £ and the SF positions are aggregated?
Use the DEAR formula for a portfolio:
2
2
2
2
DEAR p = ( 100 ) ( 0.013 ) + (-30 ) ( 0.01 ) + 2( 0.01 )( 0.013 )( 100 )(-30 )( 0.8 ) = $1,075.20
The FI’s net position is actually only $1,075.20. Without including correlation, the exposure would
have been estimated at $100,000 - $30,000 = $70,000.
12
FIN 683
Professor Robert Hauswald
Financial-Institutions Management
Kogod School of Business, AU
Chapter 22: Micro-Hedge
8.
In each of the following cases, indicate whether it would be appropriate for an FI to buy or sell a
forward contract to hedge the appropriate risk.
a. A commercial bank plans to issue CDs in three months.
The bank should sell a forward contract to protect against an increase in interest rates.
b. An insurance company plans to buy bonds in two months.
The insurance company should buy a forward contract to protect against a decrease in interest
rates.
c. A savings bank is going to sell Treasury securities it holds in its investment portfolio next
month.
The savings bank should sell a forward contract to protect against an increase in interest rates.
d. A U.S. bank lends to a French company; the loan is payable in euros.
The bank should sell francs forward to protect against a decrease in the value of the euro, or an
increase in the value of the dollar.
e. A finance company has assets with a duration of six years and liabilities with a duration of 13
years.
The finance company should buy a forward contract to protect against decreasing interest
rates that would cause the value of liabilities to increase more than the value of assets, thus
causing a decrease in equity value.
13
FIN 683
Professor Robert Hauswald
9.
Financial-Institutions Management
Kogod School of Business, AU
The duration of a 20-year, 8 percent coupon Treasury bond selling at par is 10.292 years. The
bond’s interest is paid semiannually, and the bond qualifies for delivery against the Treasury bond
futures contract.
a. What is the modified duration of this bond?
The modified duration is 10.292/1.04 = 9.896 years.
b. What is the impact on the Treasury bond price if market interest rates increase 50 basis
points?
∆P = -MD(∆R)$100,000 = -9.896 x 0.005 x $100,000 = -$4,948.08.
c. If you sold a Treasury bond futures contract at 95 and interest rates rose 50 basis points, what
would be the change in the value of your futures position?
∆P = - MD(∆R ) P = - 9.896(0.005)$95,000 = - $4,700.67
d. If you purchased the bond at par and sold the futures contract, what would be the net value
of your hedge after the increase in interest rates?
Decrease in market value of the bond purchase
-$4,948.08
Gain in value from the sale of futures contract
$4,700.67
Net gain or loss from hedge
10.
-$247.41
What are the differences between a microhedge and a macrohedge for a FI? Why is it
generally more efficient for FIs to employ a macrohedge than a series of microhedges?
14
FIN 683
Professor Robert Hauswald
Financial-Institutions Management
Kogod School of Business, AU
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, because the use of a series of microhedges ignores the FI’s
internal hedges that are already on the balance sheet. That is, if a long-term fixed-rate asset position is
exposed to interest rate increases, there may be a matching long-term fixed-rate liability position that
also is exposed to interest rate decreases. Putting on two microhedges to reduce the risk exposures of
each of these positions fails to recognize that the FI has already hedged much of its risk by taking
matched balance sheet positions. The efficiency of the macrohedge is that it focuses only on those
mismatched positions that are candidates for off-balance-sheet hedging activities.
15
FIN 683
Professor Robert Hauswald
Financial-Institutions Management
Kogod School of Business, AU
Chapter 22: Macro-Hedge
12.
Hedge Row Bank has the following balance sheet (in millions):
Assets
$150
Liabilities
$135
Equity
Total
$150
15
Total
$150
The duration of the assets is six years and the duration of the liabilities is four years. The
bank is expecting interest rates to fall from 10 percent to 9 percent over the next year.
a. What is the duration gap for Hedge Row Bank?
DGAP = DA – k DL = 6 – (0.9)(4) = 6 – 3.6 = 2.4 years
b. What is the expected change in net worth for Hedge Row Bank if the forecast is accurate?
Expected ∆E = -DGAP[∆R/(1 + R)]A = -2.4(-0.01/1.10)$150m = $3.272 million
c. What will be the effect on net worth if interest rates increase 100 basis points?
Expected ∆E = -DGAP[∆R/(1 + R)]A = -2.4(0.01/1.10)$150 = -$3.272.
d. If the existing interest rate on the liabilities is 6 percent, what will be the effect on net worth
of a 1 percent increase in interest rates?
16
FIN 683
Professor Robert Hauswald
Financial-Institutions Management
Kogod School of Business, AU
Solving for the impact on the change in equity under this assumption involves finding the
impact of the change in interest rates on each side of the balance sheet, and then
determining the difference in these values. The analysis is based on the equation:
Expected ∆E = ∆A - ∆L
∆A = -DA[∆RA/(1 + RA)]A = -6[0.01/1.10]$150m = -$8.1818 million
and ∆L = -DL[∆RL/(1 + RL)]L = -4[0.01/1.06]$135m = -$5.0943 million
Therefore, ∆E = ∆A - ∆L = -$8.1818m – (-$5.0943m) = - $3.0875 million
16.
Tree Row Bank has assets of $150 million, liabilities of $135 million, and equity of $15
million. The asset duration is six years and the duration of the liabilities is four years.
Market interest rates are 10 percent. Tree Row Bank wishes to hedge the balance sheet with
Treasury bond futures contracts, which currently have a price quote of $95 per $100 face
value for the benchmark 20-year, 8 percent coupon bond underlying the contract.
Calculation of Duration for Problem 16
$1,000 bond, 8% coupon, R = 8.5295% and R = 8.2052%, n = 20 years
Cash
Price = $95
Time
Flow
PV of CF
PV of CF x t
1
80
73.71268
73.71268
2
80
67.91950
135.83900
3
80
62.58161
187.74482
4
80
57.66323
230.65291
5
80
53.13139
265.65695
6
80
48.95572
293.73430
7
80
45.10822
315.75751
8
80
41.56301
332.50477
9
80
38.29659
344.66933
10
80
35.28681
352.86807
17
FIN 683
Professor Robert Hauswald
Financial-Institutions Management
Kogod School of Business, AU
11
80
32.51357
357.64923
12
80
29.95828
359.49933
13
80
27.60381
358.84957
14
80
25.43439
356.08145
15
80
23.43546
351.53196
16
80
21.59364
345.49819
17
80
19.89656
338.24155
18
80
18.33286
329.99152
19
80
16.89206
320.94906
20
1080
210.12054
4202.41084
Total950.00000
Duration =
9853.84304
10.3725
a. Should the bank go short or long on the futures contracts to establish the correct
macrohedge?
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. How many contracts are necessary to fully hedge the bank?
If the market value of the underlying 20-year, 8 percent benchmark bond is $95 per $100, the
market rate is 8.5295 percent (using a calculator) and the duration is 10.3725 as shown on the last
page of this chapter solutions. The number of contracts to hedge the bank is:
18
FIN 683
Professor Robert Hauswald
NF =
Financial-Institutions Management
Kogod School of Business, AU
− ( D A − kDL ) A (6 − (0.9)4)$150m
=
= − 365 contracts
DF x PF
10.3725 x $95,000
c. Verify that the change in the futures position will offset the change in the cash balance sheet
position for a change in market interest rates of plus 100 basis points and minus 50 basis
points.
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 = -10.37255(-0.005/1.085295)$95,000 = $4,539.71 and the
change in 365 contracts is $4,539.71 x -365 = -$1,656,993.13. Since the futures contracts
were sold, they could be repurchased for a loss of $1,656,993.13. The sum of the two
values is a loss of $20,629.49.
d. If the bank had hedged with Treasury bill futures contracts that had a market value of $98 per
$100 of face value, how many futures contracts would have been necessary to hedge fully the
balance sheet?
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:
NF =
− ( D A − kDL ) A − (6 − (0.9)4)$150m − $360,000,000
=
=
= − 1,469 contracts
DF x PF
0.25 x $980,000
$245,000
e. What additional issues should be considered by the bank in choosing between T-bond or T-bill
futures contracts?
19
FIN 683
Professor Robert Hauswald
Financial-Institutions Management
Kogod School of Business, AU
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.
17.
Reconsider Tree Row Bank in problem 16 but assume that the cost rate on the liabilities is
6 percent.
a. How many contracts are necessary to fully hedge the bank?
In this case, the bank faces different average interest rates on both sides of the balance
sheet. Further, the yield on the bonds underlying the futures contracts is a third interest rate.
Thus, the hedge also has the effects of basis risk. Determining the number of futures
contracts necessary to hedge this balance sheet must consider separately the effect of a
change in rates on each side of the balance sheet, and then consider the combined effect on
equity. Estimating the number of contracts can be determined with the modified general
equation shown on the next page.
b. Verify that the change in the futures position will offset the change in the cash balance sheet
position for a change in market interest rates of plus 100 basis points and minus 50 basis
points.
For an increase in rates of 100 basis points, ∆E = 0.01[(4/1.06)$135 m – (6/1.10)$150 m] = $3,087,478.56. The change in the bond value is –10.3725(.01/1.085295)$95,000 = -$9,079.41, and the
change for -340 contracts = $3,087,000.89. The sum of the two values is a net gain of $477.67.
For a decrease in rates of 50 basis points, ∆E = -0.005[(4/1.06)$135 m – (6/1.10)$150 m] =
$1,543,739.28. The change in the bond value is –10.3725(-.005/1.085295)$95,000 =
$4,539.71, and the change for -340 contracts = -$1,543,500.45. The sum of the two values
is a net loss of $238.83.
Modified Equation Model for part (b):
20
FIN 683
Professor Robert Hauswald
Financial-Institutions Management
Kogod School of Business, AU
∆F + ∆E = 0
∆F + ∆A − ∆L = 0
{
− DF ( N F * PF ) *
∆R
(1+ RF )
}= {
− DA *


∆R
∆R
* A − − DL *
* L
(1+ R A )
(1+ RL ) 

}=

∆R
} + { + DL * ∆R * L − D A * ∆R * A } =
− DF ( N F * PF ) *
(1 + RF )
(1+ RL )
(1+ R A )

{ − DF ( N F * PF ) }+ { (1+ RF ) * ( MDL * L − MD A * A) } =
{ N F * PF }+ {
NF =
=
−
1
* ( MDL * L − MD A * A) } = 0
MDF
− { − MDL * L + MD A * A
PF * MDF
− { MD A * A − MDL * L
PF * MDF
}
}
6
4
*150,000,000 −
*135,000,000
1.06
= −( 1.10
10.3725
95,000 *
1.085295
− ( $818,181,818.18 − $509,433,962.26 )
=
$907,944.38
− $308,747,855.92
=
$907,944.38
= − 340 contracts
)
c. If the bank had hedged with Treasury bill futures contracts that had a market value of $98 per
$100 of face value (implying a discount rate of 8 percent), how many futures contracts would
have been necessary to fully hedge the balance sheet?
A market value of $98 per $100 of face value implies a discount rate of 8 percent on the
underlying T-bills. Therefore, the equation developed above in part (a) to determine the number
of contracts necessary to hedge the bank can be adjusted as follows for the use of T-bill contracts:
21
FIN 683
Professor Robert Hauswald
NF =
Financial-Institutions Management
Kogod School of Business, AU
− ( MD A * A − MDL * L
PF * MDF
)
6
4
*150,000,000 −
*135,000,000
1
.
10
1
.
06
=− (
0.25
980,000 *
1.08
− ( $818,181,818.18 − $509,433,962.26 )
=
$226,851.85
− $308,747,855.92
=
$226,851.85
= − 1,361contracts
19.
)
How would your answer for part (b) in problem 16 change if the relationship of the price
sensitivity of futures contracts to the price sensitivity of underlying bonds were br = 0.92?
The number of contracts necessary to hedge the bank would increase to 397 contracts. This can be
found by dividing $360,000,000 by (10.3725 x $95,000 x 0.92).
21.
Consider the following balance sheet (in millions) for an FI:
Assets
Liabilities
Duration = 10 years
$950
Duration = 2 years
$860
Equity
$90
a. What is the FI's duration gap?
The duration gap is 10 - (860/950)(2) = 8.19 years.
22
FIN 683
Professor Robert Hauswald
Financial-Institutions Management
Kogod School of Business, AU
b. What is the FI's interest rate risk exposure?
The FI is exposed to interest rate increases. The market value of equity will decrease if
interest rates increase.
c. How can the FI use futures and forward contracts to put on a macrohedge?
The FI can hedge its interest rate risk by selling future or forward contracts.
d. What is the impact on the FI's equity value if the relative change in interest rates is an
increase of 1 percent? That is, ∆R/(1+R) = 0.01.
∆E = - 8.19(950,000)(.01) = -$77,800
e. Suppose that the FI in part (c) macrohedges using Treasury bond futures that are currently
priced at 96. What is the impact on the FI's futures position if the relative change in all interest
rates is an increase of 1 percent? That is, ∆R/(1+R) = 0.01. Assume that the deliverable
Treasury bond has a duration of nine years.
∆E = - 9(96,000)(.01) = -$8,640 per futures contract. Since the macrohedge is a short hedge, this
will be a profit of $8,640 per contract.
f.
If the FI wants to macrohedge, how many Treasury bond futures contracts does it need?
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 equity value. This will necessitate the sale of $77,800/8,640
= 9.005 contracts. Rounding down, to construct a macrohedge requires the FI to sell 9 Treasury
bond futures contracts.
22.
Refer again to problem 21. How does consideration of basis risk change your answers to problem
21?
23
FIN 683
Professor Robert Hauswald
Financial-Institutions Management
Kogod School of Business, AU
In problem 21, 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 risk, then (∆R/(1+R)) is not necessarily equal to (∆Rf/(1+Rf)). If the FI wants to
fully hedge its interest rate risk exposure in an environment with basis risk, the required number of
futures contracts must reflect the disparity in volatilities between the futures and cash markets.
a. Compute the number of futures contracts required to construct a macrohedge if
[∆Rf/(1+Rf) / ∆R/(1+R)] = br = 0.90
If br = 0.9, then: N f =
− ( D A - k D L ) A − 8.19(950,000)
=
= − 10 contracts
(9)(96,000)(.90)
D F P F br
b. Explain what is meant by br = 0.90.
br = 0.90 means that the implied rate on the deliverable bond in the futures market moves by 0.9
percent for every 1 percent change in discounted spot rates (∆R/(1+R)).
c. If br = 0.90, what information does this provide on the number of futures contracts needed to
construct a macrohedge?
If br = 0.9 then the percentage change in cash market rates exceeds the percentage change in
futures market rates. Since futures prices are less sensitive to interest rate shocks than cash
prices, the FI must use more futures contracts to generate sufficient cash flows to offset the cash
flows on its balance sheet position.
24.
Village Bank has $240 million worth of assets with a duration of 14 years and liabilities
worth $210 million with a duration of 4 years. In the interest of hedging interest rate risk,
Village Bank is contemplating a macrohedge with interest rate T-bond futures contracts
now selling for 102-21 (32nds). The T-bond underlying the futures contract has a duration
of nine years. If the spot and futures interest rates move together, how many futures
contracts must Village Bank sell to fully hedge the balance sheet?
24
FIN 683
Professor Robert Hauswald
NF =
25.
Financial-Institutions Management
Kogod School of Business, AU
− ( D A − kDL ) A − (14 − (0.875)4)$240m
=
= − 2728 contracts
DF x PF
9 x $102,656
Assume an FI has assets of $250 million and liabilities of $200 million. The duration of the
assets is six years and the duration of the liabilities is three years. The price of the futures
contract is $115,000 and its duration is 5.5 years.
a. What number of futures contracts is needed to construct a perfect hedge if br = 1.10?
Nf=
− ( D A - k D L )A − [6 − (0.8 x3)]$250,000,000 − $900,000,000
=
=
= − 1,293.57 contracts
( D f x P f xbr)
5.5 x$115,000 x1.10
$695,750
b. If ∆Rf/(1+Rf) = 0.0990, what is the expected ∆R/(1+R)?
∆R/(1 + R) = (∆Rf/(1+Rf))/br = 0.0990/1.10 = 0.09
25
FIN 683
Professor Robert Hauswald
Financial-Institutions Management
Kogod School of Business, AU
Chapter 24: Swap Mechanics
4.
An insurance company owns $50 million of floating-rate bonds yielding LIBOR plus 1 percent.
These loans are financed with $50 million of fixed-rate guaranteed investment contracts (GICs)
costing 10 percent. A finance company has $50 million of auto loans with a fixed rate of 14
percent. The loans are financed with $50 million of CDs at a variable rate of LIBOR plus 4 percent.
a. What is the risk exposure of the insurance company?
The insurance company (IC) is exposed to falling interest rates on the asset side of the balance
sheet.
b. What is the risk exposure of the finance company?
The finance company (FC) is exposed to rising interest rates on the liability side of the balance
sheet.
c. What would be the cash flow goals of each company if they were to enter into a swap
arrangement?
The IC wishes to convert the fixed-rate liabilities into variable-rate liabilities by swapping the fixedrate payments for variable-rate payments. The FC wishes to convert variable-rate liabilities into
fixed-rate liabilities by swapping the variable-rate payments for fixed-rate payments.
d. Which FI would be the buyer and which FI would be the seller in the swap?
The FC will make fixed-rate payments and therefore is the buyer in the swap. The IC will make
variable-rate payments and therefore is the seller in the swap.
e. Diagram the direction of the relevant cash flows for the swap arrangement.
26
FIN 683
Professor Robert Hauswald
Financial-Institutions Management
Kogod School of Business, AU
Please see the diagram at the top of the next page. Note that the fixed-rate swap payments from
the finance company to the insurance company will offset the payments on the fixed-rate
liabilities that the insurance company has incurred. The reverse situation occurs regarding the
variable-rate swap payments from the insurance company to the finance company. Depending on
the rates negotiated and the maturities of the assets and liabilities, both FIs now have durations
much closer to zero on this portion of their respective balance sheets.
Finance Company
Fixed-rate
assets
Insurance Company
Variable-rate assets
Swap Cash Flows
Fixed-rate swap payments
Variable-rate swap payments
Variable-rate
liabilities @ LLIBOR+4%
+ 4%
f.
Cash
Financing
Markets
Fixed-rate
liabilities @ 10%
What are reasonable cash flow amounts, or relative interest rates, for each of the payment
streams?
Determining a set of reasonable interest rates involves an analysis of the benefits to each FI. That
is, does each FI pay lower interest rates with the swap than contractually obligated without the
swap? Clearly, the direction of the cash flows will help reduce interest rate risk.
One feasible swap is for the IC to pay the FC LIBOR + 2.5 percent, and for the FC to pay the IC 12
percent. The net financing cost for each firm is given below.
Cash market liability rate
Minus swap rate
Plus swap rate
Finance
Insurance
Company
Company
LIBOR + 4%
10.0%
-(LIBOR + 2.5%)
-12.0%
+ 12%
+(LIBOR + 2.5%)
27
FIN 683
Professor Robert Hauswald
Net financing cost (rate)
Financial-Institutions Management
Kogod School of Business, AU
13.5%
LIBOR + 0.5%
Whether the two firms would negotiate these rates depends on the relative negotiating power of
each firm, and the alternative rates for each firm in the alternate markets. That is, the fixed-rate
liability market for the finance company and the variable-rate liability market for the insurance
company.
6.
A commercial bank has $200 million of floating-rate loans yielding the T-bill rate plus 2 percent.
These loans are financed with $200 million of fixed-rate deposits costing 9 percent. A savings bank
has $200 million of mortgages with a fixed rate of 13 percent. They are financed with $200 million
of CDs with a variable rate of the T-bill rate plus 3 percent.
a. Discuss the type of interest rate risk each FI faces.
The commercial bank is exposed to a decrease in rates that would lower interest income, while
the savings bank is exposed to an increase in rates that would increase interest expense. In either
case, profit performance would suffer.
b. Propose a swap that would result in each FI having the same type of asset and liability cash
flows.
One feasible swap would be for the commercial bank to send variable-rate payments of the T-bill
rate + 1 percent (T-bill + 1%) to the savings bank and to receive fixed-rate payments of 9 percent
from the savings bank.
c. Show that this swap would be acceptable to both parties.
Cash market liability rate
Minus swap rate
Savings Bank
Commercial Bank
T-bill + 3%
9%
-(T-bill + 1%)
-9%
28
FIN 683
Professor Robert Hauswald
Financial-Institutions Management
Kogod School of Business, AU
Plus swap rate
Net financing cost rate
+ 9%
+(T-bill + 1%)
11%
T-bill + 1%
The net interest yield on is 2 percent (13% - 11%) for the savings bank and 1 percent [(T-bill + 2%)
– (T-bill + 1%)] for the commercial bank. An adjustment to make the net interest yield equal at 1.5
percent would be to have the savings bank pay a fixed rate of 9.5 percent or receive a variable
rate of T-bill + 0.5 percent. Obviously, many rate combinations could be negotiated to achieve
acceptable rate spreads and to achieve the desired interest rate risk management goals.
Savings Bank
Fixed-rate
assets
13%
Swap Cash Flows
Fixed-rate swap payments
9.0%
Commercial Bank
Variable-rate assets
T-bill + 2%
T-bill + 1%
Variable-rate swap payments
Variable-rate
liabilities @ TT-bill
+ 3%+ 3%
Cash
Financing
Markets
Fixed-rate
liabilities @ 9%
d. What are some of the practical difficulties in arranging this swap?
The floating rate assets may not be tied to the same rate as the floating rate liabilities. This
would result in basis risk. Also, if the mortgages are amortizing, the interest payments would
not match those on the notional amount of the swap.
7.
Bank 1 can issue five-year CDs at an annual rate of 11 percent fixed or at a variable rate of LIBOR
plus 2 percent. Bank 2 can issue five-year CDs at an annual rate of 13 percent fixed or at a variable
rate of LIBOR plus 3 percent.
a. Is a mutually beneficial swap possible between the two banks?
A mutually beneficial swap exists because comparative advantage exists.
29
FIN 683
Professor Robert Hauswald
Financial-Institutions Management
Kogod School of Business, AU
b. Where is the comparative advantage of the two banks?
Bank 1 has a comparative advantage in the fixed-rate market because the difference in fixed rates
is 2 percent in favor of Bank 1. Bank 2 has the comparative advantage in the variable-rate market
because the difference in variable rates is only –1 percent against Bank 1. One way to compare the
rate alternatives is to utilize the following matrix.
Fixed
Variable
Rate
Rate
Bank 1
11%
LIBOR + 2%
Bank 2
13%
LIBOR + 3%
-2%
-1%
Difference
c. What is the quality spread in the fixed versus variable interest rates for the two FIs?
The quality spread is the difference between the fixed-rate versus variable-rate differential. Thus,
the net quality spread = -2% - (-1%) = -1 percent. This amount represents the net amount of gains
(interest savings) to be allocated between the firms.
d. What is an example of a feasible swap?
Many rate combinations are possible to achieve the quality spread or reduced interest charge. The
following is a framework to achieve the outside boundaries of acceptable interest rates using the
matrix of possible rates shown in part (b).
Using the rates shown for Bank 1 as the negotiated swap rates will give the entire quality spread
to Bank 2. The diagram and payoff matrix below verifies this case.
30
FIN 683
Professor Robert Hauswald
Financial-Institutions Management
Kogod School of Business, AU
Swap Cash Flows
Fixed-rate swap payments
Bank 2
Fixed-rate
assets
Bank 1
Variable-rate assets
11.0%
LIBOR+2%
Variable-rate swap payments
Cash
Financing
Markets
Variable-rate
liabilities @ L+3%
LIBOR+3%
Fixed-rate
liabilities @ 11%
The relative payoffs are given below:
Cash market liability rate
Minus swap rate
Plus swap rate
Net financing cost (rate)
Bank 2
Bank 1
LIBOR+3%
11%
-(LIBOR+2%)
-11%
+ 11%
+(LIBOR+2%)
12.0%
LIBOR+2%
Bank 1 is paying the rate it could achieve in the variable rate market, thus Bank 1 receives no
benefit to these swap rates. Now consider the rates shown for Bank 2 in the matrix of rates in part
(b).
Bank 2
Fixed-rate
assets
Variable-rate
liabilities @ L+3%
LIBOR+3%
Swap Cash Flows
Fixed-rate swap payments
Bank 1
Variable-rate assets
13.0%
LIBOR+3%
Variable-rate swap payments
Cash
Financing
Markets
Fixed-rate
liabilities @ 11%
In this case, Bank 2 is receiving the exact rate it owes on the liabilities and it is paying the rate
necessary if it was in the fixed-rate market. Bank 1 receives the entire 1 percent benefit as it is
paying net 1 percent less than it would need to pay in the variable-rate market.
31
FIN 683
Professor Robert Hauswald
Financial-Institutions Management
Kogod School of Business, AU
The relative payoffs are given below:
Cash market liability rate
Minus swap rate
Plus swap rate
Net financing cost rate
Bank 2
Bank 1
LIBOR+3%
11%
-(LIBOR+2%)
-13%
+ 11%
+(LIBOR+3%)
12%
LIBOR+1%
Any swap rate combination between these two boundaries that yields a total saving in combined
interest cost becomes a feasible set of negotiated swap rates. The exact set of rates will depend
on negotiating position of each bank and the expected interest rates over the life of the swap. As
an example, consider the average of the two fixed-rate payments and the average of the two
variable-rate payments. The relative payoffs are given below:
Cash market liability rate
Minus swap rate
Plus swap rate
Net financing cost rate
Bank 2
Bank 1
LIBOR+3.0%
11.0%
-(LIBOR+2.5%)
-12.0%
+ 12.0%
+(LIBOR+2.5%)
12.5%
LIBOR+1.5%
In each case, the banks are paying 0.5 percent less than they would in the relative desired cash
markets.
8.
First Bank can issue one-year, floating-rate CDs at prime plus 1 percent or fixed-rate CDs at 12.5
percent. Second Bank can issue one-year, floating-rate CDs at prime plus 0.5 percent or fixed-rate
at 11.0 percent.
a. What is a feasible swap with all of the benefits going to First Bank?
The possible interest rate alternatives faced by each firm are given below:
32
FIN 683
Professor Robert Hauswald
Financial-Institutions Management
Kogod School of Business, AU
Fixed
Variable
Rate
Rate
First Bank
12.5%
Prime+1.0%
Second Bank
11.0%
Prime+0.5%
1.5%
0.5%
Difference
The quality spread is 1.5 – 0.5 = 1.0 percent. Second Bank has the comparative advantage in the
fixed-rate market and First Bank has the comparative advantage in the variable-rate market. A set
of swap rates within the feasible boundaries that will give all the benefits to First Bank is 11
percent fixed rate and Prime + 0.5 percent variable rate.
b. What is a feasible swap with all of the benefits going to Second Bank?
A set of rates within the feasible boundaries that will give all the benefits to Second Bank is 12.5
percent fixed rate and Prime + 1.0 percent variable rate.
c. Diagram each situation.
Diagram of all the benefits going to First Bank.
First Bank
Fixed-rate
assets
Variable-rate
liabilities @ P+1%
Prime+1%
Swap Cash Flows
Fixed-rate swap payments
Second Bank
Variable-rate assets
11.0%
Prime+0.5%
Variable-rate swap payments
Cash
Financing
Markets
Fixed-rate
liabilities @ 11%
The payoff matrix that demonstrates that all of the benefits go to First Bank follows.
First Bank
33
Second Bank
FIN 683
Professor Robert Hauswald
Cash market liability rate
Minus swap rate
Plus swap rate
Net financing cost rate
Financial-Institutions Management
Kogod School of Business, AU
Prime+1%
11.0%
-(Prime+0.5%)
-11.0%
+ 11%
+(Prime+0.5%)
11.5%
Prime+0.5%
The net cost for First Bank is 11.5 percent, or 1 percent less than it would pay in the fixed-rate
cash market. The net cost for Second Bank is exactly the same as it would pay in the variable-rate
cash market.
Diagram of all the benefits going to Second Bank.
First Bank
Fixed-rate
assets
Variable-rate
liabilities @ P+1%
Prime+1%
Swap Cash Flows
Fixed-rate swap payments
Second Bank
Variable-rate assets
12.5%
Prime+1.0%
Variable-rate swap payments
Cash
Financing
Markets
Fixed-rate
liabilities @ 11%
The net cost for First Bank is 12.5 percent, which is exactly what it would pay in the fixed-rate cash
market. The net cost for Second Bank is Prime - 0.5 percent, or 1 percent less than it would pay in
the variable-rate cash market. The payoff matrix that illustrates that all of the benefits go to
Second Bank follows.
Cash market liability rate
Minus swap rate
Plus swap rate
Net financing cost rate
First Bank
Second Bank
Prime+1%
11.0%
-(Prime+1%)
-12.5%
+ 12.5%
+(Prime+1%)
12.5%
Prime-0.5%
34
FIN 683
Professor Robert Hauswald
Financial-Institutions Management
Kogod School of Business, AU
d. What factors will determine the final swap arrangement?
The primary factor that will determine the final distribution of the swap rates is the present value
of the cash flows for the two parties. The most important no-arbitrage condition is that the
present value of the expected cash flows made by the buyer should equal the present value of the
expected cash flows made by the seller. Secondary factors include the negotiating strengths of
either party to the transaction.
9.
Two multinational FIs enter their respective debt markets to issue $100 million of two-year notes.
FI A can borrow at a fixed annual rate of 11 percent or a floating rate of LIBOR plus 50 basis points,
repriced at the end of the year. FI B can borrow at a fixed annual rate of 10 percent or a floating
rate of LIBOR, repriced at the end of the year.
a. If FI A is a positive duration gap insurance company and FI B is a money market mutual fund,
in what market(s) should each firm borrow so as to reduce its interest rate risk exposure?
FI A will prefer to borrow in the fixed-rate debt market in order to generate positive cash flows
when interest rates increase. This will offset the impact of an increase in interest rates, which
would cause the market value of the insurance company's equity to decline. FI B will prefer to
borrow in the floating rate debt market so as to better match the duration of its short-term assets.
b. In which debt market does FI A have a comparative advantage over FI B?
The matrix of possible interest rates is given below.
Fixed
Variable
rate
rate
FI A 11.0%
LIBOR+0.5%
FI B 10.0%
LIBOR %
Difference
1.0%
0.5%
35
FIN 683
Professor Robert Hauswald
Financial-Institutions Management
Kogod School of Business, AU
FI A has a comparative advantage in the floating-rate market and FI B has a comparative
advantage in the fixed-rate market. This is because the default risk premium of FI A over FI B is 50
basis points in the floating-rate market and 100 basis points in the fixed-rate market.
c. Although FI A is riskier than FI B and therefore must pay a higher rate in both the fixed-rate
and floating-rate markets, there are possible gains to trade. Set up a swap to exploit FI A's
comparative advantage over FI B. What are the total gains from the swap? Assume a swap
intermediary fee of 10 basis points.
The total gains to the swap are 50 basis points (the price differential on FI A's default risk premium
over FI B) less 10 basis points (the swap intermediary fee). Both FI A and B can exploit this price
differential by issuing debt in the debt market in which they have comparative advantage and
then swapping the interest payments. The 40 basis points can be allocated to either FI A and/or FI
B according to the terms of the swap.
A possible set of feasible swap rates that give all of the gains to FI A (see part (d) below) is
illustrated here.
FI A
Fixed-rate
assets
Variable-rate
liabilities @ L+0.5%
LIBOR+0.5%
Swap Cash Flows
Fixed-rate swap payments
FI B
Variable-rate assets
10.0%
LIBOR %
Variable-rate swap payments
Cash
Financing
Markets
Fixed-rate
liabilities @ 10%
Evidence that FI A receives all of the benefits is given in the payoff matrix below.
Cash market liability rate
Minus swap rate
Plus swap rate
Net financing cost rate
FI A
FI B
LIBOR+0.5%
10.0%
-(LIBOR %)
-10.0%
+ 10.0%
+(LIBOR %)
10.5%
LIBOR %
36
FIN 683
Professor Robert Hauswald
Financial-Institutions Management
Kogod School of Business, AU
Less intermediary fee
0.1%
Financing cost rate net of fee
10.6%
FI A is paying the intermediary fee, since FI B is receiving no benefits from this swap transaction.
The 40 basis point net differential could be shared in a number of other combinations where FI A
received most (exploited) of the benefit.
d. The gains from the swap can be apportioned between FI A and FI B through negotiation. What
terms of swap would give all the gains to FI A? What terms of swap would give all the gains to
FI B?
All the gains go to FI A if FI B pays LIBOR for FI A's floating rate debt. Then FI A must pay 10
percent for FI B's fixed-rate debt plus 50 basis points on FI A's floating rate debt plus 10 basis
points for the swap intermediary's fee. The total fixed annual interest cost to FI A is 10.6 percent,
a savings of 40 basis points over the cash-market fixed rate of 11 percent. This swap rate
apportionment is illustrated in part (c) above.
All the gains go to FI B if FI A pays 11 percent for FI B’s fixed-rate, 10 percent debt. Then FI B pays
LIBOR plus 50 basis points on FI A's floating rate debt for a net savings of 50 basis points. The
savings occurs because FI B receives an excess 1 percent from FI A, but must pay 50 basis points
more to FI A than it would pay in the cash floating-rate market. FI A must pay 11 percent against FI
B's fixed-rate debt, but receives its exact liability payment from FI B. A diagram of this allocation is
given below.
FI A
Fixed-rate
assets
Variable-rate
liabilities @ LIBOR+0.5%
Swap Cash Flows
Fixed-rate swap payments
FI B
Variable-rate assets
11.0%
LIBOR+0.5%
Variable-rate swap payments
Cash
Financing
Markets
37
Fixed-rate
liabilities @ 10%
FIN 683
Professor Robert Hauswald
Financial-Institutions Management
Kogod School of Business, AU
In this example, FI B would pay the swap intermediary fee of 10 basis points, and thus would
realize a net, after-fee savings of 40 basis points. The payoff matrix is given below.
Cash market liability rate
Minus swap rate
FI A
FI B
LIBOR+0.5%
10.0%
-(LIBOR+0.5 %)
-11.0%
+ 11.0%
+(LIBOR+0.5 %)
11.0%
LIBOR-0.5%
Plus swap rate
Net financing cost rate
Less intermediary fee
0.1%
Financing cost rate net of fee
LIBOR-0.4%
e. Assume swap pricing that allocates all gains from the swap to FI A. If FI A buys the swap from
FI B and pays the swap intermediary's fee, what are the realized net cash flows if LIBOR is 8.25
percent?
FI A
(in millions of dollars)
Pays out fixed rate
Receives LIBOR from B
FI B
($10.00)
Pays out LIBOR
$8.25
Receives fixed rate from A
Pays floating-rate
Pays fixed-rate to creditors
($8.25)
$10.00
($10.00)
to creditors (LIBOR+0.5%) ($8.75)
Pays intermediary fee
Net cash inflow
($0.10)
($10.60)
Net cash inflow
($8.25)
This solution is an extension of the diagram in part (c) and the explanation at the beginning of part
(d) above where LIBOR is 8.25 percent. The summary shows the effective cost rate converted to
dollars for the total cash flows of each FI. However, the cash flows in a swap arrangement include
only the differential cash flows between the two parties. Thus, at the end of the year, FI A would
pay $1.75m ($10.00m - $8.25m) to FI B and $0.10m to the intermediary for a total cash flow on
the swap arrangement of $1.85m. FI B receives $1.75m from FI A.
38
FIN 683
Professor Robert Hauswald
f.
Financial-Institutions Management
Kogod School of Business, AU
If FI A buys the swap in part (e) from FI B and pays the swap intermediary's fee, what are the
realized net cash flows if LIBOR is 11 percent? Be sure to net swap payments against cash
market payments for both FIs.
FI A
(in millions of dollars)
Pays out fixed rate
Receives LIBOR from B
FI B
($10.00)
Pays out LIBOR
$11.00
Receives fixed rate from A
Pays floating-rate
($11.00)
$10.00
Pays fixed-rate to creditors
($10.00)
Net cash inflow
($11.00)
to creditors (LIBOR+0.5%)($11.50)
Pays intermediary fee
Net cash inflow
($0.10)
($10.60)
Even though LIBOR has increased to 11 percent, FI A’s total effective cost rate has not changed.
The rate remains at 10.60 percent, or a total of $10.60 million. However, the cost rate for FI B has
increased because LIBOR has increased. Thus, the actual cash flows in the swap transaction now
become that FI B pays $1.00m ($11m - $10m) to FI A, and that FI A receives $1.00m and pays out
$0.10m to the intermediary. Each FI, of course, must pay the cash market liability rates.
g. If all barriers to entry and pricing inefficiencies between FI A's debt markets and FI B's debt
markets were eliminated, how would that affect the swap transaction?
If relative prices are the same in the markets of both FI A and FI B, then there are no potential
gains to trade and therefore no swap transactions can take place. Each FI will issue debt in their
respective debt markets.
39
FIN 683
Professor Robert Hauswald
Financial-Institutions Management
Kogod School of Business, AU
Chapter 24: Macro-Hedge
12.
An FI has $500 million of assets with a duration of nine years and $450 million of liabilities with a
duration of three years. The FI wants to hedge its duration gap with a swap that has fixed-rate
payments with a duration of six years and floating rate-rate payments with a duration of two
years. What is the optimal amount of the swap to effectively macrohedge against the adverse
effect of a change in interest rates on the value of the FI’s equity?
Using the formula, NS = [(DA - kDL)A]/(DFixed – DFloating) = [(9 – 0.9x3)$500 million]/(6 – 2) = $787.5 million.
15.
Bank A has the following balance sheet (in millions):
Assets
$50 Rates-sensitive liabilities $75
Fixed-rate liabilities
100
Net worth
25
Total liabilities and equity $200
Liabilities and Equity
Fixed-rate assets
Total assets
Rate-sensitive assets
150
$200
Rate-sensitive assets are repriced quarterly at the 91-day Treasury bill rate plus 150 basis points.
Fixed-rate assets have five years until maturity and are paying 9 percent annually. Rate-sensitive
liabilities are repriced quarterly at the 91-day Treasury bill rate plus 100 basis points. Fixed-rate
liabilities have two years until maturity and are paying 7 percent annually. Currently, the 91-day
Treasury bill rate is 6.25 percent.
a. What is the bank's current net interest income? If Treasury bill rates increase 150 basis points,
what will be the change in the bank's net interest income?
Interest income = 50(.0625 + .015) + 150(.09) = $17.375 million, and interest expense = 75(.0625 +
.01) + 100(.07) = $12.4375 million. Thus, net interest income = $4.9375 million.
After the interest rate increase, interest income = 50(.0775 + .015) + 150(.09) = $18.125 million,
interest expense = 75(.0775 + .01) + 100(.07) = $13.5625 million, and net interest income =
$4.5625 million for a decline of $375,000.
40
FIN 683
Professor Robert Hauswald
Financial-Institutions Management
Kogod School of Business, AU
b. What is the bank's repricing or funding gap? Use the repricing model to calculate the change
in the bank's net interest income if interest rates increase 150 basis points.
Funding gap = Rate sensitive assets - Rate sensitive liabilities = 50 - 75 = - $25 million. The repricing
model states that ∆NII = GAP(∆ R) = -25(.015) = - $0.375 million. The bank is exposed to interest
rate increases since interest expenses increase more than interest income.
c. How can swaps be used as an interest rate hedge in this example?
A short hedge can be used to hedge the bank's interest rate risk exposure. The short hedge can be
implemented by selling futures or forward contracts, buying put options, or buying a swap of
liabilities. Swapping liabilities allows the institution to make fixed-rate liability payments in
exchange for a counter-party making the floating rate payments. Similarly, the FI could also swap
assets. The short hedge can be accomplished by swapping out fixed-rate asset payments in
exchange for floating-rate asset payments.
16.
Use the following information to construct a swap of asset cash flows for the bank in problem 14.
The bank is a price taker in both the fixed-rate market at 9 percent and the rate-sensitive market
at the T-bill rate plus 1.5 percent. A securities dealer has a large portfolio of rate sensitive assets
funded with fixed-rate liabilities. The dealer is a price taker in a fixed-rate asset market paying 8.5
percent and a floating-rate asset market paying the 91-day T-bill rate plus 1.25 percent. All
interest is paid annually.
a. What is the interest rate risk exposure to the securities dealer?
The securities dealer is exposed to interest rate declines.
b. How can the bank and the securities dealer use a swap to hedge their respective interest rate
risk exposures?
41
FIN 683
Professor Robert Hauswald
Financial-Institutions Management
Kogod School of Business, AU
The two counterparties can use a swap of asset cash flows to hedge their respective interest rate
risk exposures. The bank would swap out fixed-rate asset payments in exchange for floating-rate
asset payments to yield positive cash flows when interest rates increase. The securities dealer
would swap out floating-rate asset payments in exchange for fixed-rate asset payments to yield
positive cash flows when interest rates decrease.
c. What are the total potential gains to the swap?
The total gains to the swap trade are 25 basis points. This is because the bank earns a 25 basis
point premium in the floating-rate market and a 50 basis point premium in the fixed-rate market.
d. Consider the following two-year swap of asset cash flows: An annual fixed-rate asset cash flow
of 8.6 percent in exchange for a floating-rate asset cash flow of T-bill plus 125 basis points.
The swap intermediary fee is 5 basis points. How are the swap gains apportioned between the
bank and the securities dealer if they each hedge their interest rate risk exposures using this
swap?
Bank
Securities Dealer
Swap Cash Flows
Variable-rate Fixed-rate swap payments
Fixed-rate
liabilities
8.6%
liabilities
T-bill+1.25%
Variable-rate swap payments
Fixed-rate
assets @ 9.0%
Cash market asset rate
Minus swap rate
Plus swap rate
Net financing cost rate
Minus swap intermediary fee
Cash
Financing
Markets
Variable-rate
assets @ T-bill+1.25%
Bank
Securities Dealer
9.00%
T-bill+1.25%
-8.60%
(T-bill+1.25%)
T-bill+1.25%
8.60%
T-bill+1.65%
8.60%
-0.05%
42
FIN 683
Professor Robert Hauswald
Financial-Institutions Management
Kogod School of Business, AU
T-bill+1.60%
Minus cash market rate
T-bill+1.50%
8.50%
Net gain from swap
0.10%
0.10%
The securities dealer gains 10 basis points because it obtains fixed-rate cash inflows at 8.6 percent
instead of the 8.5 percent available in its cash market. The bank gains 10 basis points because it
obtains floating rate cash inflows at T-bill + 1.60 percent instead of the T-bill + 1.50 percent
available in its cash market. The remaining 5 basis points goes to the swap intermediary.
e. What are the realized cash flows if T-bill rates at the end of the first year are 7.75 percent and
at the end of the second year are 5.5 percent? Assume that the notional value is $107.14
million.
At the end of the first year (in millions of dollars):
Bank Cash Flows
Swap cash inflows
107.14(0.0775 + 0.016) =
Securities Dealer Cash Flows
$10.0176
107.14(0.086) =
$9.214
Cash market cash flows
107.14(0.09) =
$9.6426
Net swap gain (loss)
$0.375
107.14(.0775 + 0.0125) =
$9.6426
($0.4286)
The dealer pays the bank $375,000 to offset the decline in net interest income when interest rates
increase [see part a] and pays the swap intermediary $53,600 (5 basis points), for a total cost of
$428,600 when interest rates increase 150 basis points.
At the end of the second year, interest rates decline to 5.5%.
Bank Cash Flows
Swap cash inflows
107.14(0.0550 + 0.016) =
Dealer Cash Flows
$7.607
107.14(0.086) =
$9.214
107.14(.055 + 0.0125) =
$7.232
Cash market cash flows
107.14(0.09) =
$9.6426
43
FIN 683
Professor Robert Hauswald
Net swap gain (loss)
Financial-Institutions Management
Kogod School of Business, AU
($2.036)
$1.982
The bank pays the dealer $1.982 million and pays the swap intermediary $53,600 (5 basis points),
for a total cost of $2.036 million when interest rates decrease 75 basis points.
f.
What are the sources of the swap gains to trade?
The gains to the swap trade emanate from the pricing discrepancy in the two cash markets. That
is, the bank earns a 50 basis point premium in the fixed-rate asset market, while only a 25 basis
point premium in the floating rate asset market. The swap allows both the bank and the securities
dealer to exploit their own comparative advantage in their respective cash market.
g. What are the implications for the efficiency of cash markets?
There must be some barrier that prevents the two firms from directly transacting in the other's
cash market (or equivalently raises the costs of these cross-market transactions). This barrier may
consist of regulatory restrictions or tax considerations. If, however, the barrier results from
information asymmetries, these potential gains to trade can be expected to disappear as the swap
market develops.
17.
Consider the following currency swap of coupon interest on the following assets:
5 percent (annual coupon) fixed-rate U.S. $1 million bond
5 percent (annual coupon) fixed-rate bond denominated in Swiss francs (SF)
Spot exchange rate: SF1.5/$
a. What is the face value of the SF bond if the investments are equivalent at spot rates?
U.S. $1 million is equivalent to SF1.5 million face value.
44
FIN 683
Professor Robert Hauswald
Financial-Institutions Management
Kogod School of Business, AU
b. What are the realized cash flows, assuming no change in spot exchange rates? What are the
net cash flows on the swap?
Interest payments on the U.S. bond are .05(U.S.$1 million) = $50,000. In Swiss francs, interest
payments are .05(SF1.5 million) = SF75,000. At spot exchange rates, these two cash flows are
identical. There are no net swap cash flows.
c. What are the cash flows if the spot exchange rate falls to SF0.50/$? What are the net cash
flows on the swap?
Coupon payments on the U.S. bond are $50,000, which is equivalent to SF25,000. Coupon
payments on the Swiss franc bond are SF75,000, which is equivalent to $150,000. The net cash
flows on the swap are $100,000, or SF50,000. The counterparty that swaps in Swiss franc bond
payments receives the cash flows. The counterparty that swaps in the U.S. dollar payments makes
the payments.
d. What are the cash flows if the spot exchange rate rises to SF2.25/$? What are the net cash
flows on the swap?
Coupon payments on the U.S. bond are $50,000, which is equivalent to SF112,500. Coupon
payments on the Swiss franc bond are SF75,000, which is equivalent to $33,333. The net cash
flows on the swap are $16,667, or SF37,500. The counterparty that swaps in U.S. dollar bond
payments receives the cash flows. The counterparty that swaps in the Swiss franc payments
makes the payments.
e. Describe the underlying cash position that would prompt the FI to hedge by swapping dollars
in exchange for Swiss francs.
The FI is swapping dollar cash flows in exchange for Swiss francs so as to balance a U.S. dollardenominated liability.
18.
Consider the following fixed-floating-rate currency swap of assets: 5 percent (annual coupon)
fixed-rate U.S. $1 million bond and floating-rate SF1.5 million bond set at LIBOR annually.
45
FIN 683
Professor Robert Hauswald
Financial-Institutions Management
Kogod School of Business, AU
Currently LIBOR is 4 percent. The face value of the swap is SF1.5 million. The spot exchange rate is
SF1.5/$.
a. What are the realized cash flows assuming no change in the spot exchange rate? What are the
realized cash flows on the swap at the spot exchange rate?
Coupon payments on the U.S. bond are $50,000, which is equivalent to SF75,000. Coupon
payments on the Swiss franc bond are SF60,000 at the spot rate of LIBOR of 4%, which is
equivalent to $40,000. The net cash flows on the swap are $10,000, or SF15,000. The counterparty
that swaps in U.S. dollar bond payments receives the cash flows. The counterparty that swaps in
the Swiss franc payments makes the payments.
b. If the 1-year forward rate is SF1.538 per US$, what are the realized net cash flows on the
swap? Assume LIBOR is unchanged.
Coupon payments on the U.S. bond are $50,000, which at forward rates, is equivalent to SF76,900.
Coupon payments on the Swiss franc bond are SF60,000, which is equivalent to $39,012. The net
cash flows on the swap are $10,988, or Sf16,900. The counterparty that swaps in U.S. dollar bond
payments receives the cash flows. The counterparty that swaps in the Swiss franc payments
makes the payments.
c. If LIBOR increases to 6 percent, what are the realized cash flows on the swap? Evaluate at the
forward rate.
Coupon payments on the U.S. bond are $50,000, which at forward rates, is equivalent to SF76,900.
Coupon payments on the Swiss franc bond are SF90,000 at the spot rate of LIBOR of 6 percent,
which is equivalent to $58,518. The net cash flows on the swap are U.S. $8,518, or SF13,100. The
counterparty that swaps in Swiss franc bond payments receives the cash flows. The counterparty
that swaps in the U.S. dollar payments makes the payments.
46