Chapter 7 Swaps
Interest Rate Discussion for Swap Valuation problems. The rule is:
For discounting use CCAR
For calculating CF's use APR (S-A for interest rate swaps and annual for currency swaps.)
All rates (LIBOR, the treasury rate, loan rates…) can be stated in CCAR or APR (with any compounding
Required conversions depend how the rates are given in the problem.
In the example from the lecture (also Example 7.3 p. 162 in the text), some of the rates are APR S-A
and some are CCARs:
We are given 6 month LIBOR from 3 months ago as 10.2% APR S-A.
We are given 3, 9 and 15 month CCARs (10%, 10.5% and 11%).
So the yield curve is not flat (all the discount rates are not the same).
We are also told the fixed payments are 8% APR S-A.
For the bond method, we use the 8% APR S-A to calculate the fixed payments (k).
We use the 10.2% APR S-A to calculate the next floating payment (k*).
We use the CCARs to discount the payments.
For the forward method for the problem in class:
We need to figure out what the 2nd and 3rd floating payments will be (we already know the 1st is
10.2% APR S-A).
First we need to first calculate the 6 month fwd rates in 3 months and 9 months (10.75% and 11.75%)
Then we need to convert them to APR S-A (11.04% and 12.10%) to calculate the expected payments.
Then we discount the cash flow differences by the CCARs (10%, 10.5% and 11%)
Note: None of the questions in this HW assignment are directly from the end-of-chapter
questions so the question numbers do not correspond to the text question numbers.
1. (This is a modification of Question 7.9. In the text these are lending rates, not
borrowing rates. )
Companies X and Y have been offered the following borrowing rates per annum on a
$5 million 10-year investment:
Company Fixed Rate Floating Rate
X 8.0% LIBOR + 100
Y 8.8% LIBOR + 100
Company X requires a floating rate loan; company Y requires a fixed rate loan. Design
a swap that will net a bank, acting as intermediary, 0.2% per annum and that will
appear equally attractive to X and Y.
First determine if there is a gain from a swap and the total amount of the potential gain:
X borrows Fixed 8.0% X borrows Float LIBOR + 100
Y borrows Float LIBOR + 100 Y borrows Fixed 8.8%
Total Cost LIBOR + 900 Total Cost LIBOR + 980
Since their desired borrowing structures produce the higher total cost, there is a gain from a swap
and the gain is 80bps. Alternatively, X’s comparative advantage is in Fixed since it can pay an
80bps less than Y and Y’s comparative advantage is in Float since it can borrow at the same rate
The bank receives 20 bps so X and Y gain (80 – 20)/2 = 30 bps each.
Company X will borrow at the fixed rate and swap to pay a floating rate.
Without the swap, X would have paid LIBOR + 100.
With the swap X will pay LIBOR + 100 – 30 = LIBOR + 70
Company Y will borrow at the floating rate and swap to pay a fixed rate.
Without the swap, Y would have paid 8.8%.
With the swap Y will pay 8.8% – 0.3% = 8.5%
X: Y: FI:
Loan: Pay 8.0% Loan: Pay LIBOR + 100 X Swap: Get LIBOR
Swap: Pay LIBOR Swap: Get LIBOR Pay 7.3%
Get 7.3% Pay 7.5% LIBOR Y Swap: Get 7.5%
Net: Pay LIBOR + 70 Net Pay 8.5% Pay LIBOR
Net Get 0.2%
LIBOR LIBOR LIBOR+100
X FI Y
8.0% 7.3% 7.5%
Company X pays LIBOR + 70 instead of LIBOR + 100
Company Y pays 8.5% instead of 8.8%
2. This is the currency swap example from the lecture with dollar values.
Borrowing rates: USD AUD
GE 5% 12.6%
Qantas 7% 13%
GE wants to borrow AUD and Q ants to borrow USD. Assume the current exchange rate is
0.60 USD/AUD. The Loan amounts (and therefore the swap principal amounts) are 60
USD and 100 AUD. Note that the loan amounts are equal at the current exchange rate.
Construct a swap with an FI incurring all the exchange rate risk and receiving 20bps of the
gain, with the remainder of the gain split between Q and GE.
Look at total borrowing costs:
GE borrows USD 5.0% GE borrows AUD 12.6%
Q borrows AUD 13.0% Q borrows UUD 7.0%
Therefore there is a gain of 160bps if each company borrows their home currency and swaps.
Loan Pays 5% on 60 USD = –3 USD
Swap Gets 5% on 60 USD = +3 USD
Pays 11.9% on 100 AUD = –11.9 AUD
Net = –11.9 AUD/100AUD = 11.9%
Loan Pays 13% on 100 AUD = –13 AUD
Swap Gets 13% on 100 AUD = +13 AUD
Pays 6.3% on 60 USD = –3.78 USD
Net = –3.78 USD/60 USD = 6.3%
GE Swap Pays 5% on 60 USD = –3 USD
Gets 11.9% on 100 AUD = +11.9 AUD
Q Swap Pays 13% on 100 AUD = –13 AUD
Gets 6.3% on 60 USD = +3.78 USD
USD Net = 3.78 – 3 = 0.78 USD
AUD Net = 11.9 – 13 = –1.1 AUD
Each period, the FI needs to “buy” 1.1 AUD using the 0.78 USD
The current cost is 1.1 AUD x (.60 USD/AUD) = 0.66 USD
Profit = 0.78 – 0.66 = 0.12 USD
0.12 USD /60 USD = 0.002 = 20 bps
3. Company A would like to Borrow Euros, while Company B would like to Borrow US
dollars. The quoted annualized borrowing rates are:
Company Dollars Euros
A 8% 8%
B 6% 5%
a. What are the potential gains from the swap? Which company has the comparative
advantage when borrowing in Euros?
$: 8% - 6% = 2% €: 8% - 5% = 3% 3% - 2% = 1%
Company B has the comparative advantage borrowing Euros. It has 3% spread under Company A in
Euros and only a 2% spread under Company A in Dollars.
b. Design a swap with a financial institution in which only the financial institution bears
foreign exchange risk, the gains from the swap are split equally although the financial
institution is compensated by taking a cut of 0.1% of company B’s split.
A: Loan: Pays 8% $ B: Loan: Pays 5% € FI: A Swap: Pays 8% $
Swap: Gets 8% $ Swap: Gets 5% € Gets 7.5% €
Pays 7.5% € Pays 5.6% $ B Swap: Pays 5% €
Net = -7.5% € Net = -5.6% $ Gets 5.6% $
Net = -8% + 7.5% - 5% + 5.6% = 0.1%
7.5% € 5.0% € 5.0% €
A FI B
8.0% $ 8.0% $ 5.6% $
4. Moments ago, an FI initiated the two swaps shown in the diagram below. In the swap
with Company A, the FI receives 6.8% on US dollars and pays 6.5% on euros. In the
swap with Company B, it receives 4.8% on euros and pays 5.0% on dollars. The
notional amounts are $250 million and €200 million. Assume these two swaps are
considered separately from the rest of the FI’s obligations.
6.8% $ 5.0% $
A FI B
6.5% € 4.8% €
a) Calculate the annual NET number of dollars paid or received by the FI each period
Dollar rates: 6.8% - 5.0% = 1.8%
Rates applied to $250m nominal amount:
Dollars Received = 1.8% x $250m = $4.5m
b) Calculate the annual NET number of euros paid or received by the FI each period.
Euro rates: 4.8% - 6.5% = -1.7%
Rates applied to €200m nominal amount:
Euros paid = -1.7% x €200m = -€3.4m
c) Calculate FI’s annual profit in dollars at the current exchange rate. Also calculate the profit in
dollars as a percent of the notional dollar amount of the dollar amount of the swap.
At each payment time, the FI gets $4.5m and owes €3.4m
The cost in dollar of €3.4m is -€3.4m x 1.25 $/€ = -$4.25m
Profit in dollars: $4.50m - $4.25m = $0.25m
Profit as a percent of the nominal swap value = $0.25/$250m = 0.0010 = 0.10%
Note that 1.8% - 1.7% = 0.10%
d) At what exchange rate is the FI’s profit equal to zero? Does this happen when the dollar
strengthens or weakens?
Calculate the exchange that equates €3.4m equal $4.5m:
$4.5m/€3.4m = 1.3235 $/€
So at 1.3235 $/€ it takes all of the $4.5m to buy the €3.4m
The dollar weakens
5. A company can borrow at 6% fixed rate or LIBOR + 100 floating rate.
The LIBOR swap rates are 4.40% at 4.50%.
a) Assume the firm wants to borrow at a fixed rate. Calculate the net borrowing rate if it borrows
at floating rate from the market and then swaps for a fixed rate payment. Should it borrow
and swap or just borrow? Why?
Borrows at a fixed rate from the market: Cost = -6.00%
Borrows at a float rate from the market and “buy LIBOR” from the FI:
Cost = - (LIBOR + 100) – 4.50% + LIBOR = -5.50%
5.50% < 6.00% so borrow float and swap for fixed
b) Now assume the firm wants to borrow at a floating rate. Calculate the net borrowing rate if it
borrows fixed from the market and then swaps for a floating rate. Should it borrow and swap
or just borrow? Why?
Borrow at a floating rate from the market: Cost = -(LIBOR + 100)
Borrow at a fixed rate from the market and “sell LIBOR” to the FI:
Cost = -6.00% + - LIBOR + 4.40% = -(LIBOR + 160)
LIBOR + 100 < LIBOR + 160 so just borrow at a floating rate
6. Under the terms of an interest-rate swap, a financial institution has agreed to pay 6%
APR Semi-Annual and receive LIBOR APR Semi-Annual on a notional principal of
$100 million with payments exchanged every six months. The swap has remaining life
of 16 months. Two months ago, six-month LIBOR was 5.5% APR. The current term
structure of interest rates is flat so LIBOR for all maturities is 5.21% CCAR.
(a) Value the swap (using the Forward Rate Method).
Calculate timing of payments:
14 months remaining so final payment is at time 16.
6 months earlier is time 10.
6 months earlier is time 4.
Fixed Payments are $100(0.06/2) = $3
The 1st floating payment (at 4 months) = $100(0.055/2) = $2.75
The 2nd and 3rd payments are calculated using the implied forward rates.
Since the yield curve is flat, all forward rates are equal to the current rate of 5.21% CCAR.
But CFs are calculated using APR S-A, so convert 5.21% CCAR to APR S-A:
rm = m(erccar/m – 1) = 2(e0.0521/2 – 1) = 5.28% APR S-A
So use 5.28% APR to calculate the 2nd and 3rd floating payments. (Note that this is not a prediction of what the rates
- and therefore payments - will be. It is the rates that can be locked-in today using forward rate agreements.)
2nd pmt = $100 (0.0528/2) = $2.64
3nd pmt = $100(0.0528/2) = $2.64
The PV of each payment = f = (F0 - K)e-r(T)
For each payment
K = the fixed payment = $3
r = 5.21% CCAR
T = 4 and then 10 and then 16
F0 = $2.75 and then $2.64 and then $2.64
f4 = (2.75 – 3)e-.0521(4/12) = -$0.246
f10= (2.64 – 3)e-.0521(10/12) = -$0.345
f16 = (2.64 – 3)e-.0521(16/12) = -$0.336
VSwap = - $0.246 - $345 - $0.336 = -$0.927
(b) What has happened to the relevant interest rates since the swap was initiated? Is the FI
paying or receiving the floating rate? Has the value of the swap increased or
When the swap was initiated, LIBOR was (approximately) 6%. It has since decreased. The FI is receiving the now
lower floating rate. The value of the swap has decreased.
(c) What payment must he FI make to the counterparty in order to exit the swap? Or
what payment must he FI make today (along with offsetting forward rate agreements)
to offset the swap?
Under the terms of the swap, the FI must pay:
2.75 – 3.00 = 0.25 in 4 months
2.64 – 3.00 = 0.36 in 10 months
2.64 – 3.00 = 0.36 in 16 months
The PV of these payments is $0.927. Therefore the FI may be able to pay $0.927 to the counterparty to exit the
swap. Or the FI can use forward-rate-agreements today to lock in these payments pay $0.927 to offset the swap
7. Suppose the term structure of interest rates is flat in both the US and Sweden and the
current risk free is for US dollars (USD) is 2.00% CCAR for all maturities and 5.00%
CCAR for Swedish Krona (SEK) for all maturities.
At some point in the past, an FI entered into a swap in which it receives annual
payments of 6% on 20 million USD and makes annual payments of 8% on 100 million
SEK. There are exactly 30 months remaining in the swap. The current exchange rate
is 0.15 USD/SEK. Use the steps below to value the swap from the FI’s perspective.
(a) Calculate the timing and amounts of the remaining payments.
The final payment is in 30 months with payments at 18 months and 6 months.
In 6 months the FI receives 6% on 20.00 USD (1.20 USD) and pays 8.00% on 100 SEK (8.00 SEK)
In 18 months the FI receives 6% on 20.00 USD (1.20 USD) and pays 8.00% on 100 SEK (8.00 SEK)
In 30 months the FI receives 6% on 20.00 USD and the notional amount (21.20 USD) and pays 8.00% on 100 SEK
and the notional amount (108.00 SEK)
(b) Calculate the forward price of 1 SEK in 6, 18 and 30 months
The spot value price is 0.15 USD/SEK
Value of 1 SEK in 6 months: F0,6 = S0e(r-rf)T = 0.15e(0.02-0.05)(6/12) = 0.1478 USD/SEK
Value of 1 SEK in 18 months: F0,18 = S0e(r-rf)T = 0.15e(0.02-0.05)(18/12) = 0.1434 USD/SEK
Value of 1 SEK in 30 months: F0,30 = S0e(r-rf)T = 0.15e(0.02-0.05)(30/12) = 0.1392 USD/SEK
(c) Calculate the dollar value of the SEK to be exchanged in 6, 18 and 30 months
Dollar Value of 8.00 SEK in 6 months: 8.00 SEK x 0.1478 USD/SEK = 1.1821 USD
Dollar Value of 8.00 SEK in 18 months: 8.00 SEK x 0.1434 USD/SEK = 1.1472 USD
Dollar Value of 108.00 SEK in 30 months: 108.00 SEK x 0.1392 USD/SEK = 15.0294 USD
(d) Calculate the PV of the exchanges (in USD terms) to be made in 6, 18 and 30 months.
Calculate the value of the swap.
f = [1.20 – 1.1821]e-0.02(6/12) = [0.0179]e-0.02(6/12) = 0.0177 USD
f = [1.20 – 1.1472]e-0.02(18/12) = [0.0528]e-0.02(18/12) = 0.0512 USD
f = [21.20 – 15.0294]e-0.02(30/12) = [6.1706]e-0.02(30/12) = 5.8696 USD
Vswap = 0.0177 + 0.0512 + 5.8696 = 5.9385 USD
(e) Did the USD strengthen or weaken relative to the SEK since the swap was initiated? Is
the FI paying or receiving USD? Did the value of the swap increase or decrease since
The notional values must have been equal at the inception of the swap:
Exchange rate at the inception of the swap = 20 USD/ 100 SEK = 0.20 USD/SEK
Current Exchange rate = 0.15 USD/SEK
The dollar strengthened (fewer dollars needed to buy one krona). The FI is receiving USD. The
value of the swap increased from a value of zero when it was created to 5.9385 million USD.
8. An FI pays 8% on $10m (pmt = $0.8m) and receives 5% on ¥1,200m (pmt = ¥60m)
annually for the next 3 years. The current exchange rate is 110 ¥/$. (S = 1/110 =
CCAR on $ for all maturities is 9%. (r12 = r24 = r36 = 0.09)
CCAR on ¥ for all maturities is 4% (r12 = r24 = r36 = 0.04)
(a) Value the swap:
Value the swap as the sum of the values of four forwards:
Three for ¥60m at 12, 24 and 36 months and one for ¥1,200m at 36 months.
In general, the value of a forward at time 1 is f 1 = [F – K]e–r1T1
Where: K = the delivery price (not to be confused with k, the coupon value above) = 0.08
F = the currency forward price if set today = Se(r-rf)T
Note that F and K (and therefore f) are in terms of 1 yen, so they must be multiplied to account
for the size of each theoretical forward contract.
f12 = [F12 (¥60m) – K($10m)]e–.09(12/12)
f24 = [F24 (¥60m) – K($10m)]e–.09(24/12)
f36 = [F36 (¥60m) – K($10m)]e–.09(36/12)
f36 = [F36 (¥1,200m) – K($10m)]e–.09(36/12)
Note: F is $/¥ and is multiplied by ¥ to yield $. K is $ and has been set at 8%
So what is the value of 1 yen in 12 months (F12)?
F12 = Se(r-rf)T = 0.009091e(0.09 – 0.04)(12/12) = 0.009557 $/¥
F24 = Se(r-rf)T = 0.009091e(0.09 – 0.04)(24/12) = 0.010047 $/¥
F36 = Se(r-rf)T = 0.009091e(0.09 – 0.04)(36/12) = 0.010562 $/¥
f12 = [0.009557 $/¥ (¥60m) – $0.8m]e–.09(12/12) = -$0.2071
f24 = [0.010047 $/¥ (¥60m) – $0.8m]e–.09(24/12) = -$0.1647
f24 = [0.010562 $/¥ (¥60m) – $0.8m]e = -$0.1669
f36 = [0.010562 $/¥ (¥1,200m) – $10m]e–.09(36/12) = $2.0418
VSwap = $1.5431
(b) Has the dollar strengthened or weakened since the swap was initiated? Is the FI paying
or receiving dollars? Has the swap’s value from the perspective of the FI increased or
Exchange rate when swap was initiated: $10/¥1200 = 0.008333 $/¥
Current exchange rate = 0.009091 $/¥
Since it takes more of a dollar to buy one yen, the dollar has weakened.
The FI is paying (less valuable) dollars and receiving (more valuable) yen.
The swap has increased in value.
9. An FI pays 6 month LIBOR and receives 8% APR S-A on $100m
There is 15 months remaining on the swap. The CCAR spot rates for 3, 9 and 15
months are r3 = 10%, r9 = 10.5% and r15 = 11%.
Three months ago 6 month LIBOR was 10.2% APR S-A.
(Recall that all cash flows are determined from APRs and all discount rates are CCARs.)
First determine the remaining cash flows (CFs). If the swap has 15 month remaining, it must have a CF
at that time. Since it is a domestic interest rate swap, the CF must occur every 6 months. Therefore there
must be a CF 6 months prior at 9 months and 6 months prior to that at 3 months. Note that in this
example, you are given the CCAR discount rates for these times.
The amount of the floating CF in three months was determined 3 months ago and is calculated using
10.2% APR S-A.
0 3 9 15
-10.20% -LIBOR -LIBOR
+8% +8% +8%
Value the swap as the sum of the values of three forwards at 3, 9 and 15 months.
In general, the value of a forward at time 1 is f 1 = [K – F]e–r1T1
K = the delivery price = (0.08/2)$100
F = the forward price if set today = (Expected LIBOR/2)($100)
So the current value of the three FRAs (or forwards) are:
f3 = [(0.08 – LIBOR)/2($100)]e–.010(3/12)
f9 = [(0.08 – LIBOR)/2($100)]e–.0105(9/12)
f15 = [(0.08 – LIBOR)/2($100)]e–.011(15/12)
We know the proper value of LIBOR in 3 months. This was set 3 months ago at 10.2% APR.
We still need the value of LIBOR in 9 and 15 months.
Since we know the CCAR spot rates, we can calculate the CCAR forward rates and then convert those to
APR forward rates since CFs are calculated using APRs.
The formula for implied forward rates is:
rF = [r2t2 – r1t1]/[t2 – t1]
The forward rates from 3 months to 9 months and from 9 to 15 months are:
r3,9 = [0.105(9/12) – 0.10(3/12)]/ [(9/12) – (3/12)] = 0.1075 CCAR
r9,15 = [0.11(15/12) – 0.105(9/12)]/ [(15/12) – (9/12)] = 0.1175 CCAR
Convert these to APRs to calculate the CFs:
rm = m[erc/m – 1]
rm = APR with m compounding periods
rc = CCAR
r3,9 = 2[e0.1075/2 – 1] = 0.1104 APR
r9,15 = 2[e0.1175/2 – 1] = 0.1210 APR
So now we can value the three FRAs:
f3 = [(0.08 – 0.102)/2($100)]e–.10(3/12) = -$1.07
f9 = [(0.08 – 0.1104)/2($100)]e–.105(9/12) = -$1.41
f15 = [(0.08 – 0.1210)/2($100)]e–.11(15/12) = -$1.79
VSwap = -$4.27
10. The spot price of a euro is 1.25 USD per EUR. An American-based investment manager
is considering buying a five-year Triple-A rated US Dollar (USD) denominated bond
that pays a 4.00% coupon. Also available is a five-year Triple-A rated Euro (EUR)
denominated bond that pays a 3% coupon. A Triple-A rated FI is offering a five-year
USD-EUR swap. The swap terms are that the investment manager will pay 3.00% on
80 million EUR and receive 4.10% on 100 million USD.
(a) Describe the investment manager’s cash flows at times 0 through time 5 if it swaps with
the FI and buys 80 million EUR worth of the 3.00% bond.
Time € Bond CF € Swap CF $ Swap CFs
0 -€ 80.00 € 80.00 -$100.00
1 € 2.40 -€ 2.40 $4.10
2 € 2.40 -€ 2.40 $4.10
3 € 2.40 -€ 2.40 $4.10
4 € 2.40 -€ 2.40 $4.10
5 € 82.40 -€ 82.40 $104.10
The investment manager’s net CFs are the same as if it had bought a $100 bond with a 4.10%
coupon – a coupon greater than the 4.00% available on the USD bond.
(b) Why would the investment manager buy the EUR bond and swap as opposed to buying
the USD bond?
The combination of the EUR bond and EUR for USD Swap allows the investment manager to earn 4.10%
on a $100 investment. The USD bond pays only 4.00%.
Note: In theory this opportunity would not persist for very long since market participants would purchase
the EUR bond driving up its price and therefore driving down its yield and then sell EUR to the FI for
USD causing the FI to pay fewer USD per EUR.
(c) Describe an additional risk incurred by the investment manager associated with this
trade (relative to just buying the 4.00% USD bond).
The FI, to mitigate its credit risk with the investment manager, would require the investment manager to
post collateral (cash) if the swap value decreased from the perspective of the investment manager.
Therefore the investment manager has risk associated with having to commit capital to post collateral if
the USD weakens relative to the EUR – a risk it does not incur if it just buys the USD bond.
(d) Assume the investment manager buys the EUR bond and swaps with the FI. Exactly
three years later – when there are two payments remaining – the spot exchange rate
moves to 1.30 USD per EUR. In addition, the risk free rate on USD is 4.00% CCAR for
all maturities and the risk free rate on EUR is 3.00% CCAR for all maturities.
Calculate the value of the swap from the perspective of the investment manager. (This
equals the cash the investment manager must post.)
The rule for valuing a swap is to calculate the amount you have to pay (or get) today to lock-in
the required swap payments. The investment manager is committed to paying 2.40 EUR in one
year and 82.40 EUR in two years. The manager will also receive 4.10 USD in one year and
104.10 USD in two years.
The question is how much in USD would it cost for the manager to lock in the purchase of 2.40
EUR in one year and 82.40 EUR in two years? In other words, what is the one-year forward
price of 2.40 EUR and what is the two-year forward price of 82.40 EUR given the current
exchange rate and risk-free rates?
F0,1 year = S0e(r-rf)T = 1.30e(0.04 – 0.03)(1) = 1.3131 USD/EUR
F0,2 years = S0e(r-rf)T = 1.30e(0.04 – 0.03)(2) = 1.3263 USD/EUR
The Investment Manager has agreed to
Pay €2.40 and get $4.10 in one year
Pay €82.40 and get $104.10 in two years
Today’s cost of those EUR (which the investment manager does not really have to buy since they will be
paid by the bond it owns) would be:
2.40 EUR x 1.3131 USD/EUR = 3.1514 USD
82.40 EUR x 1.3263 USD/EUR = 109.2840 USD
f1 year = [$4.10 – $3.1514]e–.04(1) = [0.9486]e–.04(1) = $0.9114
f2 year = [$104.10 – $109.2840]e–.04(2) = [-$5.1840]e–.04(2) = -$4.7854
VSwap = $0.9114 – $4.7854 = -$3.8740
The investment manager must post as collateral $3,874,000 in cash. If the investment manager does not
have (or cannot raise) the cash, it must sell the bond (or another asset) in-order to close the swap.
Note: In the absence of the requirement by the FI to mitigate its counter party risk relative to the
investment manager, the cash flows of the bond and swap payments will continue to off-set. See part (a).
The trouble is the FI is not entitled to the CFs from that bond and therefore will require the investment
manager to post collateral of $3,874,000.