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Unformatted text preview: CHAPTER 7
Swaps Problem 7.1.
Companies A and B have been oﬂ'ered the following rates per annum on a $20 million ﬁve—year loan: ________________________.___——#————— Fibced Rate Floating Rate
____________,_________—————————
Company A 5.0% LIBOR + 0.1%
Company B 6.4% LIBOR + 0.6% M CompanyA requires a ﬂoatingrate loan; company B requires a ﬁxedrate loan. Design
a swap that will net a bank, acting as intermediary, 0.1% per annum and that will appear
equally attractive to both companies. A has an apparent comparative advantage in ﬁxedrate markets but wants to borrow
ﬂoating. B has an apparent comparative advantage in ﬂoatingrate markets but wants to
borrow ﬁxed. This provides the basis for the swap. There is a 1.4% per annum differential
between the ﬁxed rates offered to the two companies and a 0.5% per annum diiferential
between the ﬂoating rates offered to the two companies. The total gain to all parties horn
the swap is therefore 1.4 —— 0.5 = 0.9% per annum. Because the bank gets 0.1% per annum
of this gain, the swap should make each of A and B 0.4% per arinum better oﬁ. This means
that it should lead to A borrowing at LIBOR — 0.3% and to B borrowing at 6.0%. The
appropriate arrangement is therefore as shown in Figure 87.1. 5% Company '
' . LIBOR+0.6% Figure S7 .1 Swap for Problem 7.1 Problem 7.2. Company X wishes to borrow U.S. dollars at a ﬁxed rate of interest. Company Y
wishes to borrow Japanese yen at a ﬁxed rate of interest. The amounts required by the
two companies are roughly the same at the current exchange rate. The companies have
been quoted the following interest rates, which have been adjusted for the impact of taxes: 50 ____________________,__,_____._———————— Yen Dollars
__________________.___..._#__————————
Company X 5.0% 9.6%
Company Y 6.5% 10.0% __________________n____...__————————— Design a swap that will not a bank, acting as intermediary, 50 basis points per annum.
Make the swap equally attractive to the two companies and ensure that all foreign exchange risk is assumed by the bank. X has a comparative advantage in yen markets but wants to borrow dollars. Y has
a comparative advantage in dollar markets but wants to borrow yen. This provides the
basis for the swap. There is a 1.5% per annum differential between the yen rates and a
0.4% per annum differential between the dollar rates. The total gain to all parties from
the swap is therefore 1.5 — 0.4 = 1.1% per annum. The bank requires 0.5% per annum,
leaving 0.3% per annum for each of X and Y. The swap should lead to X borrowing dollars
at 9.6 — 0.3 = 9.3% per annum and to Y borrowing yen at 6.5  0.3 = 6.2% per annum.
The appropriate arrangement is therefore as shown in Figure 87.2. All foreign exchange risk is borne by the bank. Yen 6.2% Financial ; Institution Dollars n1"/ Dollars 9.3% Dollars 10% Figure 87.2 Swap for Problem 7.2 Problem 7.3.
A $100 million interest rate swap has a remaining life of 10 months. Under the terms of the swap, six—month LIBOR is exchanged for 7% per annum (compounded semiannu—
ally). The average of the bid—oﬁ'er rate being exchanged for sixmonth LIBOR in swaps of
all maturities is currently 5% per annum with continuous compounding. The six—month
LIBOR rate was 4.6% per annum two months ago. What is the current value of the swap
to the party paying ﬂoating? What is its value to the party paying ﬁxed? In four months $3.5 million (= 0.5 x 0.07 x $100 million) will be received and $2.3
million (= 0.5 x 0.046 X $100 million) will be paid. (We ignore day count issues.) In 10
months $3.5 million will be received, and the LIBOR rate prevailing in four months’ time
will be paid. The value of the ﬁxedrate bond underlying the swap is 3.5e005x4/ ‘2 + 103.5eOO5xm/12 = $102.718 million 51 The value of the ﬂoatingrate bond underlying the swap is (100 + 2.3)6‘0‘05X4/ 12 = $100609 million The value of the swap to the party paying ﬂoating is $102.718 —— $100609 = $2.109 million.
The value of the swap to the party paying ﬁxed is —$2.109 million. These results can also be derived by decomposing the swap into forward contracts.
Consider the party paying ﬂoating. The ﬁrst forward contract involves paying $2.3 million
and receiving $3.5 million in four months. It has a value of 1.25035“! 12 = $1.180
million. To value the second forward contract, we note that the forward interest rate
is 5% per annum with continuous compounding, or 5.063% per annum with semiannual
compounding. The value of the forward contract is 100 x (0.07 x 0.5 — 0.05063 x 0.5)e"°'05"1°/ 12 = $0.929 million
The total value of the forward contracts is therefore $1.180 + $0.929 = $2.109 million. Problem 7.4.
Explain what a swap rate is. What is the relationship between swap rates and par
yields? A swap rate for a particular maturity is the average of the bid and offer ﬁxed rates
that a market maker is prepared to exchange for LIBOR in a standard plain vanilla swap
with that maturity. The swap rate for a particular maturity is the LIBOR/ swap par yield
for that maturity. Problem 7.5. _ A currency swap has a remaining life of 15 months. it involves exchanging interest
at 10% on £20 million for interest at 6% on $30 milliOn once a year. The term structure
of interest rates in both the United Kingdom and the United States is currently ﬂat, and
if the swap were negotiated today the interest rates exchanged would be 4% in dollars
and 7% in sterling. All interest rates are quoted with annual compounding. The current
exchange rate (dollars per pound sterling) is 1.8500. What is the value of the swap to the
party paying sterling? What is the value of the swap to the party paying dollars? The swap involves exchanging the sterling interest of 20 x 0.10 = 2.0 million for the
dollar interest of 30 x 0.06 = $1.8 million. The principal amounts are also exchanged at
the end of the life of the swap. The value of the sterling bond underlying the swap is 2 22 (1001/4 + (1.07)5/4 = ”'18? million pounds The value of the dollar bond underlying the swap is 1.8 31.8 (1_04)1/4 + W = $32.06}. million 52 lion. rots.
[lion
. 180
rate
uual par ates
wap
'ield .Irest
ture
and
liars
rent the the
:1 at erling is therefore The value of the swap to the party paying st 32.061  (22.182 x 1.85) = w$8.976 million
wap to the party paying dollars is +$8.976 million. The results can also
be obtained by viewing the swap as a portfolio of forward contracts. The continuously corn— poundedi terling and dollars are 6.766% per annum and 3.922% per annum.
GS are 1.856(033922036766)x025 = 18369 nterest rates in s
The 3month and 1 
and 1.856003922436766)“25 = 1.7854. The values of the two forward contracts corre
change of interest for the party paying sterling are therefore (1.8  2 x 1.8369)e"°'°3922x°'25 = —$1.855 million no.03922x125 = —$1.686 million (1.8 — 2 x 1.7854)c
the exchange of principals is ward contract corresponding to The value of the for
)c3’("°35‘:’22"1'25 = ——$5.435 million (30 — 20 x 1.7854 The total value of the swap is —$1.855  $1.686 — $5.435 = —$8.976 million. Problem 7.6.
the market risk in a ﬁnancial con Explain the diﬁer tract. Credit risk arises from the possibility of a default by the
arises from movements in market variables
complication is that the credit risk in a swap is contingent on the
A company’s position in a swap has credit risk only when the ence between the credit risk and Problem 7.7.
r tells you that he has just negotiated
that he achieved the 5.2% A corporate treasure
of 5.2%. The treasurer explains
d swapping LIBOR for petitive ﬁxed rate of interest
rate by borrowing at sixmonth LIBOR plus 150 basis points an
omparative 3.7%. He goes on to say t sibIe because his company has a c advantage in the ﬂoating— ket. What has the treasurer overlooked?
truly ﬁxed bec ating declines, it will The rate is not ause, if the company’s credit I not be able to roll OVer its ﬂoating rate borrowings at LIBOR plus 150 basis points. The
effective ﬁxed borrowing rate then increases. Suppose for example that the treasurer’s
Spread over LIBOB. increases from ' ‘ ts. The borrowing 150 basis points to 20
rate increases from 5.2% to 5.7%. t to credit risk when it enters into two offsetting swap 53 At the start of the swap, both contracts have a value of approximately zero. As time
passes, it is likely that the swap values will change, so that one swap has a positive value
to the bank and the other has a negative value to the bank. If the counterparty 0n the
Other side of the positivevalue swap defaults, the bank still has to honor its contract with the other counterparty. It is liable to lose an amount equal to the positive value of the
swap. Problem 7.9. Companies X and Y have been offered the following rates per annum on a $5 million
10year investment: l 1 Fixed Rate Floating Rate a;  m l‘ “ Company X 8.0% LIBOR
Company Y 8.8% LIBOR Company X requires a ﬁxedrate investment; company Y requires a ﬂoatingrate in
vestment. Design a swap that will net at bank, acting as intermediary, 0.2% per annum
and will appear equally attractive to X and Y. The spread between the interest rates offered to X and Y is 0.8% per annum on ﬁxed
rate investments and 0.0% per annum on ﬂoating rate investments. This means that the
total apparent beneﬁt to all parties from the swap is 0.8% per annum. Of this 0.2% per
i, annum will go to the bank. This leaves 0.3% per annum for each of X and Y. In other
iii, words, company X should be able to get a ﬁxedrate return of 8.3% per annum while
‘ company Y should be able to get a ﬂoatingrate return LIBOR + 0.3% per annum. The
in: required swap is shown in Figure 87.3. The bank earns 0.2%, company X earns 8.3%, and
company Y earns LEBOR + 0.3%. Company
X l Figure 87.3 Swap fer Problem 7.9 Problem 7. 10. A ﬁnancial institution has entered into an interest rate swap with company X. Un der the terms of the swap, it receives 10% per annum and pays sixmonth LIBOR on a . principal of $10 million for ﬁve years. Payments are made every six months. Suppose
‘ that company X defaults on the sixth payment date (end of year 3) when the interest
1‘ rate (with semiannual compounding) is 8% per annum for all maturities. What is the loss to the ﬁnancial institution? Assume that sixmonth LIBOR was 9% per annum halfway
a}, through year 3. 54 At the end of year 3 the ﬁnancial institution was due to receive $500,000 (= 0.5 x 10%
of $10 million) and pay $450,000 (= 0.5 x 9% of $10 million). The immediate loss is
therefore $50,000. To value the remaining swap we assume than forward rates are realized.
All forward rates are 8% per annum. The remaining cash ﬂows are therefore valued on
the assumption that the ﬂoating payment is 0.5 x 0.08 x 10, 000,000 = $400, 000 and the
net payment that would be received is 500, 000 — 400,000 = $100, 000. The total cost of
default is therefore the cost of foregoing the following cash ﬂows: year 3: $50,000
year 3%: $100,000
year 4: $100,000
year 4%: $100,000
year 5: $100,000 default as $413,000. Problem 7.11. impact of taxes): A B
ff,”—
U.S. dollars (ﬂoating rate) LIBOR + 0.5% LIBOR + 1.0%
Canadian dollars (ﬁxed rate) 5.0% .. 6.5% /——————— Assume that A wants to borrow US. dollars at a ﬂoating rate of interest and B wants
to borrow Canadian dollars at a ﬁxed rate of interest. A ﬁnancial institution is planning
to arrange a swap and requires a 50basispoint spread. If the swap is equally attractive
to A and B, what rates of interest will A and B end up paying? Company A has a comparative advantage in the Canadian dollar fixedrate market.
Company B has a comparative advantage in the US. dollar ﬂoatingrate market. (This
may be because of their tax positions.) However, company A wants to borrow in the
US. dollar floating—rate market and company B wants to borrow in the Canadian dollar
fixedrate market. This gives rise to the swap Opportunity. The differential between the US. dollar ﬂoating rates is 0.5% per annum, and the
diﬁerential between the Canadian dollar ﬁxed rates is 1.5% per annum. The difference
between the differentials is 1% per annum. The total potential gain to all parties from the
swap is therefore 1% per annum, or 100 basis points. If the ﬁnancial intermediary requires
50 basis points, each of A and B can be made 25 basis points better off. Thus a swap can
be designed so that it provides A with US. dollars at LIBOR + 0.25% per annum, and B
with Canadian dollars at 6.25% per annum. The swap is shown in Figure 37.4. 55 M USS: LIBOR+1% C$: 5% C$z 6.25%
Company ' Financial  Company
A , Institution B US$. LIBOR+O.25% USS: LIBOR+1% Figure 87.4 Swap for Problem 7.11 Principal payments ﬂow in the opposite direction to the arrows at the start of the life
of the swap and in the same direction as the arrows at the end of the life of the swap.
The ﬁnancial institution would be exposed to some foreign exchange risk which could be
hedged using forward contracts. Problem 7.12. jj‘ I A ﬁnancial institution has entered into a 10year currency swap with company Y.
1': . Under the terms of the swap, the ﬁnancial institution receives interest at 3% per annum
j; in Swiss francs and pays interest at 8% per annum in US. dollars. Interest payments are
H . exchanged once a year. The principal amounts are 7 million dollars and 10 million francs.
 . Suppose that company Y declares bankruptcy at the end of year 6, when the exchange
ill . rate is $0.80 per franc. What is the cost to the ﬁnancial institution? Assume that, at the
end of year 6, the interest rate is 3% per annum in Swiss francs and 8% per annum in US.
.i dollars for all maturities. All interest rates are quoted with annual compounding. l When interest rates are compounded annually 1+ T
FO=SO(1+TTI) where F0 is the T—year forward rate, .90 is the spot rate, 1' is the domestic riskfree rate,
1 ' and T'f is the foreign riskfree rate. As 1" = 0.08 and 'T'f = 0.03, the spot and forward
exchange rates at the end of year 6 are spot: 0.8000
1 year forward: 0.8388
2 year forward: 0.8796
'3‘ 3 year forward: 0.9223
,1: 4 year forward: 0.9670 The value of the swap at the time of the default can be calculated on the assumption
i. that forward rates are realized. The cash ﬂows lost as a result of the default are therefore
i as follows: 56 //’———————— Dollar Swiss Franc Forward Dollar Equivalent Cash Flow Year Paid Received Rate of Swiss Franc Received Lost
6 560,000 300,000 0.8000 ' 240,000 320,000
7 560,000 300,000 0.8388 251,600 308,400
8 560,000 300,000 0.8796 263,900 296,100
9 560,000 300,000 0.9223 276,700 283,300 10 7,560,000 10,300,000 0.9670 9,960,100 2,400,100 / Discounting the numbers in the ﬁnal column to the end of year 6 at 8% per annum, the cost of the default is $679,800.
Note that, if this were the only contract entered into by company Y, it would make no sense for the company to default at the end of year six as the exchange of payments
at that time has a positive value to company Y. In practice company Y is likely to be
defaulting and declaring bankruptcy for reasons unrelated to this particular contract and
payments on the contract are likely to stop when bankruptcy is declared. Problem 7.13.
After it hedges its foreign exchange risk using forward contracts, is the ﬁnancial in
stitution’s average spread in Figure 7.10 likely to be greater than or less than 20 basis points? Explain your answer. The ﬁnancial institution will have to buy 1.1% of the AUD principal in the forward
market for each year of the life of the swap. Since AUD interest rates are higher than
dollar interest rates, AUD is at a discount in forward markets. This means that the AUD
purchased for year 2 is less expensive than that purchased for year 1; the AUD purchased
for year 3 is less expensive than that purchased for year 2; and so on. This works in favor
of the ﬁnancial institution and means that its spread increases with time. The spread is
always above 20 basis points. , Problem 7 .14.
“Companies with high credit risks are the ones that cannot access ﬁxedrate markets directly. They are the companies that are most likely to be paying ﬁxed and receiving
ﬂoating in an interest rate swap.” Assume that this statement is true. Do you think it
increases or decreases the risk of a ﬁnancial institution’s swap portfolio? Assume that
companies are most likely to default when interest rates are high. Consider a plainvanilla interest rate swap involving two companies X and Y. We
suppose that X is paying ﬁxed and receiving ﬂoating while Y is paying ﬂoating and receiving
ﬁxed. The quote suggests that company X will usually be less creditworthy than company
Y. (Company X. might be a BBBrated company that has difﬁculty in accessing ﬁxed
rate markets directly; company Y might be a AAArated company that has no difﬁculty
accessing ﬁxed or ﬂoating rate markets.) Presumably company X wants ﬁxedrate funds
and company Y wants ﬂoatingrate funds. The ﬁnancial institution will realize a loss if company Y defaults when rates are high
or if company X defaults when rates are low. These events are relatively unlikely since (a) 57 Y is unlikely to default in any circumstances and (b) defaults are less likely to happen When
rates are low. For the purposes of illustration, suppose that the probabilities of various
events are as follows: Default by Y: 0.001
Default by X: 0.010
Rates high when default occurs: 0.7
Rates low when default occurs: 0.3 The probability of a loss is
0.001 x 0.7 + 0.010 x 0.3 = 0.0037 if the roles of X and Y in the swap had been reversed the probability of a loss would
be
0.001 x 0.3 + 0.010 x 0.7 = 0.0073 Assuming companies are more likely to default when interest rates are high, the above
argument shows that the observation in quotes has the effect of decreasing the risk of a
ﬁnancial institution’s swap portfolio. It is worth noting that the assumption that de
faults are more likely when interest rates are high is open to question. The assumption
is motivated by the thought that high interest rates often lead to ﬁnancial diﬂiculties for
corporations. However, there is often a time lag between interest rates being high and the
resultant default. When the default actually happens interest rates may be relatively low. Problem 7.15.
Why is the expected loss from a default on a swap less than the expected loss from
the default on a loan with the same principal? In an interestrate swap a ﬁnancial institution’s exposure depends on the difference
between a ﬁxedrate of interest and a ﬂoatingrate of interest. It has no exposure to the
notional principal. In a loan the Whole principal can be lost. Problem 7.16.
A bank ﬁnds that its assets are not matched with its liabilities. It is taking ﬂoating
rate deposits and making ﬁxedrate loans. How can swaps be used to offset the risk? The bank is paying a. ﬂoatingrate on the deposits and receiving a ﬁxedrate on the
loans. It can offset its risk by entering into interest rate swaps (with other ﬁnancial
institutions or corporations) in which it contracts to pay ﬁxed and receive ﬂoating. Problem '7 .17.
Explain how you would value a swap that is the exchange of a ﬂoating rate in one
currency for a ﬁxed rate in another currency. The ﬂoating payments can be valued in currency A by (i) assuming that the forward
rates are realized, and (ii) discounting the resulting cash ﬂows at appropriate currency
A discount rates. Suppose that the value is VA. The ﬁxed payments can be valued in 58 the appropriate currency B discount rates. Suppose
(number of units of currency A per
QVB. Alternatively, it is currency B by discounting them at
that the value is VB. If Q is the current exchange rate unit of currency B), the value of the swap in currency A is VA 
VA/Q ~ V3 in currency B. Problem 7.18.
The LIBOR zero curve is ﬂat at 5% (continuously co rates for 2 and 3year semiannual pay swaps are 5.4%
the LIBOR zero rates for maturities of 2.0, 2.5, and 3.0 years.
swap rate is the average of the 2 and 3year swap rates.) This means that a t...
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 Spring '08
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