In-Class Problems
Problem 1 [Forward Premium/Discount]:
Suppose,
• S0RUB/USD = RUB 5/USD → S0USD/RUB = USD 0.2000/RUB
• F1RUB/USD = RUB 6/USD → F1USD/RUB = USD 0.1667/RUB
Is the USD/ RUB selling at a forward premium or discount?
Solution:
To answer this correctly, follow Rule #2: “Always think of the currency in the denominator of a foreign
exchange quote.”
The dollar (in the denominator) is selling at a forward premium.
Dollar’s forward premium = (RUB 6/USD – RUB 5/USD) / (RUB 5/USD) = +20%.
The ruble (in the numerator) must be selling at a forward discount.
Ruble’s forward premium = (USD 0.1667/RUB – USD 0.2/RUB) / (USD 0.2/RUB) = -0.1667,
or -16.67 percent.
Problem 2 [Law of One Price with Transaction Costs]
This problem is associated to the slides 6-8 in the handout titled International Parity Conditions.
It shows an example with transactions costs in both gold and currency.
A London dealer quotes “GBP 930.00/oz bid and GBP 940/oz offer.” The spot exchange rate between
the GBP and the USD is USD 1.600/GBP. A dealer in New York wants to set comparable euro bid and
offer prices. What are the bid and offer prices in New York?
Solution:
Bid PUSD = PGBP SUSD/GBP = (GBP 930.00/oz) (USD 1.60/GBP) = USD 1,488.00/oz
Offer PUSD = PGBP SUSD/GBP = (GBP 940.00/oz) (USD 1.60/GBP) = USD 1,504.00/oz
The dollar quotes should be “USD 1,488/oz bid and USD 1,504/oz ask”. If the New York dealer’s bid
quote is higher than USD 1,504/oz, then there may be an arbitrage opportunity to buy gold in London
and sell it in New York.
Problem 3 [Cross Exchange Rate Equilibrium]
Calculate the following cross exchange rates:
(a) If exchange rates are 200 yen per dollar and 50 U.S. cents per Swiss franc, what is the exchange
rate of yen per franc?
(b) The dollar is trading at ¥100/$ and at Sfr1.60/$. What is the yen per franc rate?
Solution:
Parts (a) and (b) are both an application of the Cross Exchange Rate Equilibrium formula. For both
problems, the cross product of the three currencies should be one,
S JPY / USD×S USD /CHF ×S CHF / JPY =1
(a)
In the problem, we have S JPY / USD=JPY 200/USD and S USD /CHF =USD 0.5/ CHF . Note that from
the formula above,
S JPY / USD×S USD /CHF =
1
S
CHF / JPY
=S JPY / CHF =200
JPY
USD
JPY
×0.5
=100
USD
CHF
CHF
or JPY 100 per CHF.
(b)
Similarly, we have S JPY / USD=JPY 100 /USD and S CHF /USD=CHF 1.6/USD .
1
1
USD
S USD /CHF = CHF / USD =
=0.625
CHF
CHF . Using this latter value in the original
Note that since
S
1.6
USD
formula,
S JPY / USD×S USD /CHF ×S CHF / JPY =1 →100
JPY
USD
1
JPY
×0.625
= CHF / JPY =S JPY /CHF =62.5
.
USD
CHF S
CHF
Problem 4 [Covered Interest Arbitrage]:
Currency exchange rates and Eurocurrency interest rates are as follows:
•
•
•
•
Current Singapore dollar (SGD) spot rate:
1-year Singapore dollar (SGD) forward rate:
1-year Singapore dollar (SGD) interest rate:
1-year USD interest rate:
USD 0.50 / SGD
USD 0.51 / SGD
4.0 percent
6.0 percent
Is covered interest arbitrage worthwhile? If so, explain the steps and compute the profit based on an
initial (time = 0) transaction of USD 1 million. Calculate your profit in USD in one period.
Solution:
First, we need to check if the Interest Rate Parity formula is an identity. As it can be seen below, it is
not an identity. Therefore there is opportunity for a covered interest arbitrage.
USD
USD / SGD
USD
SGD F 1
(1+i ) (1+0.06)
1.02=
= USD /SGD ≠
=
=1.0192307692
SGD
USD S 0
(1+0.04)
(1+i
)
0.50
SGD
USD /SGD
/ SGD
USD
F1
↓ F USD
(1+i )
(1+i USD )↑
1
>
>
, to get back to equilibrium
. Therefore,
SGD
/SGD
S USD/
(1+i SGD )
↑ S USD
(1+i SGD)↓
0
0
0.51
Given that
the strategy will be,
1. Borrow USD 1,000,000 at i USD . You will be paying the bank this amount plus interest in
USD in one year.
/SGD
2. Buy SGD at S USD
. To buy SGD, you are selling the USD you borrowed from the bank.
0
SGD
3. Invest SGD at i
. The money you are lending will be yours in one year plus the interest
you are earning.
/ SGD
4. Sell SGD 1-year forward at F USD
. Today (at time 0), you agree with the bank to sell the
1
specific amount of SGD you are getting from your investment to the bank. The bank is
committed to buying your SGD and selling the equivalent USD to you.
The next table offers the details of the transactions and the arbitrage profit in one year,
Steps
1. Borrow USD 1 million
2. Sell USD and buy SGD at the spot
3. Invest SGD
4. Sell SGD one-year forward
Today
In one year
1,000,000.00 USD 1,060,000.00 USD
-1,000,000.00 USD
2,000,000.00 SGD
-2,000,000.00 SGD 2,080,000.00 SGD
-2,080,000.00 SGD
1,060,800.00 USD
Arbitrage Profit
800.00 USD
Problem 5 [Rationale for Hedging Currency Risk]:
Suppose corporate income up to $250,000 is taxed at a rate of 25 percent. Income over $250,000 is
taxed at 40 percent. The taxable income of Quack Poultry will be either $200,000 or $300,000 with
equal probability. Quack’s income variability arises entirely from an exposure to currency risk.
(a) What is Quack’s expected tax liability if it does not hedge its currency risk?
(b) What is Quack’s expected tax liability if it is able to completely hedge its currency risk
exposure and lock in taxable income of $250,000 with certainty?
(c) In what way does hedging have value for Quack Poultry?
Solution:
(a)
Not hedging implies volatility in the expected tax liability so we need to calculate the expected value of
the tax liability,
Tax Income=$ 200,000 =$ 200,000×0.25=$ 50,000
Tax Income=$ 300,0000=$ 250,000×0.25+$ 50,000×0.4=$ 62,500+$ 20,000=$ 82,500
E [Tax]=0.5×$ 50,000+0.5×$ 82,500=$ 66,250
(b)
If the company hedges its currency risk exposure and the taxable income is $250,000 with certainty,
Tax=$ 62,500
(c)
There is a benefit of $(66,250 – 62,500) or $3,750 if a hedging policy is implemented to eliminate
currency risk. If the cost associated to the hedging policy is less than $3,750, the managers should
decide to adopt a hedging policy.
Problem 6 [Multinational Netting]
H.K.
Japan
H.K. Dollars
(HK$)
U.K.
10.000
Japanese Yen (¥)
0.100
U.K. Pound (£)
15.000
150.000
U.S. Dollar ($)
10.000
100.000
U.S.
0.067
0.100
0.007
0.010
1.500
0.667
$20
U.S
parent
H.K Affiliate
$40
HK$100
HK$600
$80
£40
HK$100
£20
¥4,000
¥4,000
U.K.
Affiliate
£20
Japanese
Affiliate
¥2,000
Solution:
H.K.
Receiving
Affiliate
H.K.
Japan
U.K.
U.S.
Total Payments
Japan
0
20
30
20
70
Paying Affiliate
U.K.
U.S.
10
60
0
40
30
0
40
80
80
180
10
40
60
0
110
Total Receipts Total Payments Net Receipts Net Payments
80
70
10
0
100
80
20
0
120
180
0
60
140
110
30
0
440
440
The result implies,
• U.K. affiliate paying $10 to the H.K. affiliate
• U.K. affiliate paying $30 to the U.S. parent
• U.K. affiliate paying $20 to the J.P. Affiliate
Problem 7 [Transaction Exposure to Currency Risk]:
Suppose S0$/£ = $1.25/£, F1$/£ = $1.2/£, i£ = 11.56%, and i$ = 9.82%. You are to receive £100,000 on a
shipment of Madonna albums in one year. You want to fix the amount you must pay in dollars to avoid
foreign exchange risk.
Solution:
(a) Form a forward market hedge. Identify which currency you are buying and which currency you
are selling forward. When will currency actually change hands – today or in one year?
This is the case of a multinational firm importing albums from U.K. In its balance sheet, you
have a liability (Account Payable) in the foreign currency (£100,000). Since you need to pay
£100,000 in 1 year, you need to “buy pounds forward”. This a long position in the derivatives
market (forward contract) given that you are short in your underlying position (balance sheet).
Buying sterling pounds forward is equal to selling USD forward. The cash is exchanged in 1
year.
(b) Form a money market hedge that replicates the payoff on the forward contract by using the spot
currency and Eurocurrency markets. Identify each contract in the hedge. Does this hedge
eliminate your exposure to foreign exchange risk?
In this case, a money market hedge implies making sure using the money markets that you have
enough sterling pounds in 1 year to pay your liability. The strategy is the following:
1. Borrow USD 112,047.33. You have the commitment to pay USD 123,050.38 in 1 year.
2. Exchange USD 112,047.33 for GBP at S0$/£ . You get GBP 89,637.86
3. Invest GBP 89,637.86 at i£. You have in 1 year GBP100,000 which is enough to cover your
liability.
The implicit exchange rate in
(=USD123,050.38/GBP100,000).
this
money
market
hedge
is
USD1.2305/GBP
The following table shows you the cash flows at time zero and in one year. Note that the
exposure to the GBP is zero today and in one year.
Steps
0. Exposure
3. Invest GBP at iGBP
2. Sell USD at the spot exchange
1. Borrow USD at iUSD
Today
In one year
-100,000.00 GBP
-89,637.86 GBP 100,000.00 GBP
89,637.86 GBP
-112,047.33 USD
112,047.33 USD -123,050.38 USD
(c) Are these currency and Eurocurrency markets in equilibrium? In case they are not in
equilibrium, how would you arbitrage the difference from the parity condition?
Currency and money markets are not in equilibrium since the exchange rate in the forward
contract is different from the exchange rate in the money market hedge. We use the interest rate
parity condition to know the strategy to arbitrage away the disequilibrium.
USD /GBP
F1
(1.0982) (1+i USD )
1.20
=
=0.96<0.9844=
=
USD /GBP
1.25
(1.1156) (1+i GBP )
S1
USD /GBP
↑ F1
↓ (1+i USD)
>
USD /GBP
GBP
↓ S1
↑ (1+i )
(d) Borrow GBP at iGBP
(e) Sell GBP spot at S0USD/GBP. (You are buying USD spot)
(f) Invest USD at iUSD
(g) Sell USD 1-year forward at F1USD/GBP. (You are buying GBP 1-year forward).
Problem 8 [Managing Transaction Exposure to Currency Risk]
Suppose that Boeing Corporation exported a 747 to British Airways and billed £10 million payable in
one year. The money market interest rates and foreign exchange rates are given as follows:
• The U.S. interest rate:
6.10% per year
• The U.K. interest rate:
9.00% per year
• The spot exchange rate:
$1.50/£
• The forward exchange rate: $1.46/£ (1-year maturity)
Solution:
(a) Show how to use a forward contract to totally hedge against currency risk.
i. What is the strategy?
Sell GBP 10 million one-year forward
ii. Graph the value of the proceeds to be received in one year against the future value of the
exchange rate ($/£) at that moment, for the hedged and unhedged position.
The value of the proceeds in USD in one year will follow the following formula:
Proceeds
USD
USD /GBP
=Hedged Amount in GBP×S 1
USD /GBP
=GBP 10,000,000×S 1
Value of the Proceeds
USD 25,000,000
USD 20,000,000
USD
USD 15,000,000
Unhedged Position (Real
Assets)
Hedged Position (Real +
Derivatives)
USD 10,000,000
USD 5,000,000
USD 0
0.1 0.3 0.5 0.7 0.9 1.1 1.3 1.5 1.7 1.9
0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
Future Spot Exchange rate (USD/GBP)
iii. What is the ex-post profit function associated to the forward hedging strategy?
USD / GBP
Π=(F 1
USD /GBP
−S 1
)×Hedged Amount in GBP
/GBP
Π=(USD1.46 /GBP−S USD
)×GBP 10,000,000
1
iv. Suppose you are Boeing's CFO (You wish!). You have some expectations about what is
going to be the spot exchange rate in one year ( E ( S 1 ) ). Will Boeing hedge if,

E ( S 1 )≈F 1 → If the forward rate is approximately equal to the expectations for the
future spot exchange rate, the CFO will be willing to hedge given that the expected
profit/loss from the hedging strategy will be zero but there is a reduction in the riskiness
of the cash flows.

E ( S 1 )<F 1 → If the forward rate is greater than the CFO's expectations for the future
spot exchange rate, the CFO will hedge for sure given that the expected profit from the
hedging strategy is positive and additionally there is a reduction in the riskiness of the
cash flows.

E ( S 1 )>F 1 → If the forward rate is less than the CFO's expectations for the future
spot exchange rate, the expected profit from the hedging strategy is negative (bad for the
company) but there is a reduction in the riskiness of the cash flows (good for the
company). The CFO may still hedge if he/she thinks that the reduction in risk will more
than compensate the reduction in value coming from the negative profit from the
forward position.
(b) Show how to use a money markets to totally hedge against currency risk.
i. What is the strategy?
 Borrow today the present value of GBP 10 million
 Exchange this amount in GBP into USD at the spot
 Deposit the USD amount today in an account earning the U.S. interest rate
1. In one year you have the equivalent USD amount to GBP 10 million
ii. Show the cash flows associated to the strategy
Transaction
1. Borrow pounds
2. Buy dollars spot with pounds
3. Invest in United States
Collect Pounds receivables
Net cash flow
Cash Flow today Cash Flow at Maturity
9,174,312 GBP
-10,000,000 GBP
13,761,468 USD
-9,174,312 GBP
-13,761,468 USD
14,600,917 USD
10,000,000 GBP
0 USD
14,600,917 USD
(c) Show how to use options to hedge against currency risk.
i. What type of option you need to use to cover the downside currency risk Boeing is facing?
Given that you have down-side risk associated to the depreciation of the GBP, you need to
buy a put option on the GBP.
ii. Suppose you have the option to identify in (I) with an exercise price of $1.46/£ and a 1-year
expiration. Further, assume that the option premium (price) is $0.02/£. What is the total cost
today associated to hedging using options?
The total cost associated to hedging with the put option, assuming full hedging is,
USD 0.02
×GBP 10,000,000=USD 200,000 .
GBP
iii. What if the ex-post profit function associated to the option hedging strategy?
In one year, the profit will be given by the following formula,
{Max [ K
USD/ GBP
USD/ GBP
−S 1
]−Premium
USD /GBP
×(1+i
USD
)}× Hedged Amount in GBP
/GBP
{Max [USD 1.46 /GBP −S USD
]−USD 0.02 /GBP ×(1+0.061)}×GBP10 ,000,000
1