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List of contents .................................................................................................. III
1. List of exhibits ............................................................................................. III
2. Table index ................................................................................................. III
3. List of abbreviations ................................................................................... III
4. List of symbols ............................................................................................ III
1.
Introduction ................................................................................................. 1
2.
Arguments for and against hedging ............................................................ 2
2.1.
Risk exposures ...................................................................................... 2
2.2.
Currency hedging .................................................................................. 3
2.3.
Cross-currency hedging ......................................................................... 4
2.4.
Hedging currency gaps .......................................................................... 5
3.
The Basic Approach: Hedging the Principle................................................ 7
3.1.
Minimum-Variance hedge ratio .............................................................. 9
3.1.1.
Translation risk ................................................................................. 10
3.1.2.
Economic risk ................................................................................... 10
3.1.3.
Hedging total currency risk ............................................................... 11
3.2.
Optimal hedge ratio ............................................................................. 11
3.2.1.
Basis risk .......................................................................................... 11
3.2.2.
Expected hedged return and the basis ............................................. 13
4.
Hedging example of an import-export firm ................................................ 13
Bibliography ..................................................................................................... 15
II

Exhibit 1.0: Hedging foreign currency receivable
Table 1.0: Importance of interest rate differentials to futures prices
Table 4.0: Current exchange rates on April 11
CBOT The Chicago Board of Trade
CME Chicago Mercantile Exchange
LIFFE London International Financial Futures Exchange
US United States
SF Swiss franc
V t
The value of the portfolio of foreign assets to hedge, measured in foreign currency at time t
V t
* The value of the portfolio of foreign assets, measured in domestic curr-
ency at time t
S t
The spot exchange rate: domestic currency value of one unit of foreign currency quoted at time t
F t
The futures exchange rate: domestic currency value of one unit of foreign currency quoted at time t
R The rate of return of the portfolio measured in foreign currency terms
R * The rate of return of the portfolio measured in domestic currency terms
R
H
Rate of return on the hedged position measured in foreign currency
terms s The percentage movement in the spot exchange rate
III

h Hedge ratio
R
F
Percentage change in the futures price h * Optimal hedge ratio
2
F
Variance of the return on the futures r
$
Interest rate in dollars r £ Interest rate in pounds
IV

A future is a financial derivative. A derivative is a financial instrument whose value depends on the value of more basic underlying variables.
These variables are often assets such as stocks, bonds, or currencies. In the last 20 years derivatives have become increasingly important in the world of finance.
A future contract is an agreement between two parties to buy or sell an asset at a certain time in the future for a certain price. The Chicago Board of Trade (CBOT) and the Chicago Mercantile Exchange (CME) and the
London International Financial Futures Exchange (LIFFE) are the largest exchanges on which futures contracts are traded. The exchange specifies certain standardized features of the contract to make trade possible.
1
To hedge a portfolio, futures or forward currency contracts may be used.
They differ in several ways. The main difference is that futures are exchange-traded contracts and forwards are over-the-counter contracts. Anyways, forward and futures contracts allow a manager to take the same economic position. Throughout this paper the term futures will denote both futures and forward contracts.
2
There are three important groups of participants in futures markets. Hedgers, speculators and arbitrageurs.
Hedgers want to avoid an exposure to adverse movements in the price of an asset. In history, futures markets were originally set up to meet the needs of hedgers. Farmers wanted to lock in an assured price for their goods. Merchants wanted to lock in a price they would pay for the produce of the farmers. Using futures contracts enabled both parties to achieve their objectives.
Speculators wish to take a position in the market. They are either betting that the price of an asset will go up or down.
Arbitrageurs are the third important group of participants in futures markets. They are seeking for a riskless profit by simultaneously entering into transactions in two or more markets.
3
1 Hull, John C. (2006): Options, Futures, and other Derivatives, 6 th Ed. New Jersey, pp. 1-6
2 Solnik, B. / McLeavey, D. (2006): International Investments, 5 th Ed. London, p. 573
3 Hull, John C. (1995): Introduction to futures and options markets, 2 nd Ed. New Jersey, pp. 6-11
1

There are many obvious arguments in favor of hedging. Companies in the business of e.g. manufacturing, retailing, wholesaling or providing services do not have particular skills in predicting variables such as interest rates, commodity prices, or currency movements. It makes sense for them to hedge the risk associated with these variables. That way the companies can focus on their main activities where they do have particular skills and expertise. By hedging, they avoid unpleasant surprises such as a sharp rise in the price of a commodity or an unexpected appreciation or depreciation of the domestic currency compared to a foreign currency. But still, in practice, many risks are left unhedged.
4
Future financial cash flows deriving from financial derivates, have to be measured considering the current interest rates and the future interest rates. Multinational financial management distinguishes between accounting exposures that deal with the translation of foreign currency denominated assets into the financial statements and the future net cash flow effects of financial activities denominated in foreign currencies.
The risks associated with payments relating to financial activities are classified as transaction exposure and operating exposure . Together transaction and operating exposures are referred to as economic exposures .
Transaction exposures derive from existing contractual obligations denominated in foreign currencies. Since they relate to activities that are already contracted, transaction exposures are relatively short-term in nature.
Operating exposures derive from future effects on cash flows. They have not yet been recorded in the books and, therefore, constitute unrealized future projections. Operating exposures may comprise stipulations of the longer-term effects of the institutions` strategic position in a global market context.
Summarizing economic exposures refer to all potential future cash flow implications deriving from changes in the foreign exchange rates.
The recorded transaction exposure and the projected operating exposure cash flow effects from changes in foreign exchange rates deriving from financial activities will be effectuated at different points in time in the future and may include activities in many different currencies. There is a need to develop a reasonable overview of the corporation`s economic exposures for different future time intervals. Besides the overview it may be worthwhile to develop a more detailed analysis for each of the major currency areas in which the corporation is active.
The distinction between transaction and operational exposures is potentially important. There is a tendency to focus more on transaction exposures because they appear more real once they have been recorded in the books. Financial assets are typically included in the transaction exposure calculations whether they constitute short- or long-term instruments
4 Hull, John C. (1995): Introduction to futures and options markets, 2 nd Ed. New Jersey, p. 87
2

because the structure of their future cash flows is well defined. Any changes in the interest rate level will have a direct impact on their valuation, since all asset values are determined as the present value. The present value is found by discounting the future cash flows accruing from the assets. An increase in the interest rate will lead to a reduction of asset values, whereas a drop in the interest rate will increase the assets value.
5
Transactions involving the exchange of currencies may constitute payments for trade transactions involving delivery of goods and services across national borders. The focus in this paper is on capital transactions involving debt service payments on foreign loan commitments or settlements relating to short-term currency positions. There are many examples where different institutions would be interested to lock-in the future foreign exchange rate. However, this paper focuses on a few transactions to show how financial futures can be used to hedge.
Different hedging techniques apply whether the investor is an US resident or non-US resident:
An US resident who receives payments in a foreign currency can hedge by selling futures of that currency against US dollars.
An US resident who makes payments in a foreign currency can hedge by buying futures contracts of that currency against US dollars.
A non-US resident receiving payments in US dollars can hedge by buying futures contracts on the home currency.
A non-US resident making payments in US dollars can hedge by selling futures contracts on the home currency.
Hedging example of an American company:
In April the American company has shipped goods to a company in Switzerland. The price has been settled in Swiss francs (SF). The payment of
10,000,000 Swiss francs is expected in mid-June.
The following exchange rates exist: spot foreign exchange rate → 0,3466 US $ / SF
Swiss franc futures contract → 0,3400 US $ / SF
(delivery in June)
The American exporter is worried that the Swiss franc will weaken over the two-month period. That means that he would receive fewer dollars for the incoming Swiss franc amount at that time. To lock-in the futures foreign exchange rate, he sells 80 Swiss franc futures contracts for delivery in
June (contract size SF 125,000).
In June it turns out that the spot exchange rate has dropped to 0,3350 US
$ / SF.
5 Andersen, T. J. (2006): Global Derivatives. A Strategic Risk Management Perspective, London, pp. 55-58
3

The following scenario appears in June:
Price of Swiss franc futures contract sold
Spot exchange rate in June
Price gain per contract sold
Profit per contract sold
(125,000 SF x 0,0050 US $ / SF)
0,3400 US $ / SF
0,3350 US $ / SF
0,0050 US $ / SF
$ 625,00
Loss on spot sale of Swiss francs
(10,000,000 SF x (-0,0050 US $ / SF))
Profit from sale of 80 futures contracts
(80 x $ 625,00)
- $ 50,000
$ 50,000
As a result we find out that the loss made in the cash exchange market is compensated by the profit gained on the sale of the futures contracts.
6
Cross-currency hedging is a trade in foreign exchange derivatives denominated in one currency to reduce an exposure in another currency.
The receivable or payable is denominated in a non-US currency, and the exporter or importer is located in a currency area outside the United states. In that case, the US dollar exchange rate is not directly involved in the transaction.
Two cross-currency hedging principles based on futures contracts with US dollar-based foreign exchange quotes apply:
A non-US resident receiving payment in a foreign currency can hedge by selling futures contracts on the foreign currency against dollars and buying futures contracts on the domestic currency against dollars.
A non-US resident making payments in a foreign currency can hedge by buying futures contracts on the foreign currency against dollars and selling futures contracts on the domestic currency against dollars.
In this type of cross-currency hedge, the hedge often is close to being complete. Due to the standard size on the futures contracts, a 100 % hedge is rarely possible.
To avoid the problems of imperfect hedging in cross-currency exchanges, today several futures exchanges offer futures contracts directly in the major cross-currency quotes.
7
6 Andersen, T. J. (2006): Global Derivatives. A Strategic Risk Management Perspective, London, pp. 268-271
7 Andersen, T. J. (2006): Global Derivatives. A Strategic Risk Management Perspective, London, pp. 271-272
4

Consider a € 10,000,000 receivable due in June. The future value of this receivable in US dollar terms can be locked-in by selling 80 contracts with a € 125,000 denomination. This way, the full amount of the receivable is covered (80 x €125,000 = € 10,000,000).
The following exchange rates exist: spot foreign exchange rate → 0,8771 $ / €
June euro futures contract → 0,8721 $ / €
In June 2 possible scenarios may take place:
Scenario 1: The euro foreign exchange rate increases to 0,8800 $ / €
Sold futures
Bought futures
80 x € 125,000 x 0,8721 $ / €
80 x € 125,000 x 0,8800 $ / €
= $ 8,721,000
= $ -8,800,000
Loss $ -79,000
Spot market conversion € 10,000,000 x 0,8800 $ / € = $ 8,800,000
Net $ 8,721,000
Explanation: In the case that the euro foreign exchange rate increases to
0,8800 $ / € we receive a sales amount with the sale of 80 euro June futures contracts of US $ 8,721,000 from selling the futures contracts. The futures position can be closed out by buying the futures contracts back for
US $ 8,800,000. We suffer a loss on the futures position of $ 79,000. Converting the euro receivables at the spot foreign exchange rate for $
8,800,000. Subtracting the loss on the futures position leads to the net of $
8,721,000.
Scenario 2: The euro foreign exchange rate decreases to 0,8700 $ / €
Sold futures
Bought futures
80 x € 125,000 x 0,8721 $ / €
80 x € 125,000 x 0,8700 $ / €
= $ 8,721,000
= $ -8,700,000
Profit $ 21,000
Spot market conversion € 10,000,000 x 0,8700 $ / € = $ 8,700,000
Net $ 8,721,000
Explanation: In the case that the euro foreign exchange rate decreases to
0,8700 $ / € we receive the same sales amount with the sale of 80 euro
June futures contracts of US $ 8,721,000. Now the futures position can be closed out by buying the futures contracts back for US $ 8,700,000. We gain a profit on the futures position of $ 21,000. Converting the euro receivables at the decreased spot foreign exchange rate for $ 8,700,000.
Adding the profit on the futures position leads to the net of $ 8,721,000.
This example shows that the futures euro foreign exchange rate has been effectively locked in. It has been locked in by engaging into a futures position with an inverse payoff profile of the underlying long euro position of the June receivables.
5

Exhibit 1.0 shows the hedging position with the foreign currency receivable.
8
Source: Andersen, T. J. (2006): Global Derivatives. A Strategic Risk Management Perspective, London, p. 278
Exhibit 1.0: Hedging foreign currency receivable.
On the horizontal axis we find the $ /
€ exchange rate. The vertical axis expresses the value in US dollars. The curve of the currency receivables
(dashed line) has a positive slope. The reason is that an increase of the $ /
€ exchange rate leads to a higher $ value. The curve of the short futures
(solid line) has a negative slope. The short futures position is a futures sale of the underlying asset (here the expected € 10,000,000 in June). If the current foreign exchange rate (a) is lower than the futures exchange rate (c) we gain a profit on the futures position (in the example profit of $
21,000). On the other hand if the current foreign exchange rate increases
(b) we suffer a loss on the futures position (in the example loss of $
79,000).
Hence, by selling the futures position we locked-in the future euro foreign exchange rate (c).
8 Andersen, T. J. (2006): Global Derivatives. A Strategic Risk Management Perspective, London, pp. 277-278
6

To explain the approach of hedging the principle (“the market value”) we have a look at an example where a citizen of country A wants to hedge a portfolio assets denominated in currency B. The citizen of country A would sell a futures contract to exchange currency B for currency A. The size of the contract should equal the principal of the assets to be hedged.
We consider an US investor with £ 1 million invested in British bonds. To hedge the position the investor would sell £ 1 million worth of dollars.
On the Chicago Mercantile Exchange (CME), investors are able to buy and sell futures contracts of £ 62,500 wherein the price is expressed in dollars per pound (so c alled “indirect quotation”). On the London International Financial Futures Exchange (LIFFE) the same size of contracts is also found.
Assuming that on September 12, an US investor can buy or sell futures with delivery in December for a futures exchange rate of 1.95 dollars per pound. The spot exchange rate is 2 dollars per pound. The investor must sell a total of 16 contracts (
£ 1,000,000 : £ 62,500 = 16), in order to hedge the £ 1 million principle. A few weeks later the futures exchange rate drops to 1.85 dollars per pound and the spot exchange rate drops to 1.90 dollars per pound. The value of the British assets rises to £ 1,010,000. The hedge is undertaken at time 0, and we are going to calculate the rate of return on the portfolio from time 0 to a future time t.
First we want to calculate the impact of the dollar value (domestic currency) on the portfolio caused by the appreciation by 1 percent of the pound value of the British assets and the drop by 5 percent of the pound exchange rate:
Formula 2.01 shows that the variation in dollar value is calculated by subtracting the value of the portfolio in domestic currency at time 0 (V
0
* ) from the value of the portfolio in domestic currency at time t (Vt * ).
V t
S t
– V
0
S
0
= Vt * - V
0
* (formula 3.01)
In absolute dollar terms:
(£ 1,010,000 x $ 1.90 / £) – (£ 1,000,000 x $ 2 / £)
= $ 1,919,000 – $ 2,000,000 = $ -81,000
The loss in portfolio value is $ 81,000.
On the other hand, the realized gain on the futures contract sale is calculated as followed:
Using formula 2.02 we calculate the realized gain by subtracting the futures exchange rate at time t (F t
) from the futures exchange rate at time 0
(F
0
). Multiplying that with the value of the portfolio in foreign currency at time 0 (V
0
).
Realized gain = V
0
(F
0
– F t
) (formula 3.02)
In absolute dollar terms:
£ 1,000,000 ( $ 1.95 / £ - $ 1.85 / £ ) = $ 100,000
The gain on the futures contract sale is $ 100,000.
7

Adding up the loss in portfolio value and the realized gain on the futures contract gives us the net profit on the hedged position:
Net profit = (V t
S t
– V
0
S
0
) + V
0
(F
0
– F t
) (formula 3.03)
= - $ 81,000 + $ 100,000 = $ 19,000.
The net profit on the hedged position is $ 19,000.
The rate of return in dollars on the hedged position (R
H
) can easily be calculated by dividing the profit calculate in equation 2.03 by the original portfolio value (V
0
S
0
).
R
H
= (V t
S t
– V
0
S
0
) + V
0
(F
0
– F t
) / V
0
S
0
(formula 3.04)
= $ 19,000 / $ 2,000,000 = 0.0095 = 0.95 %
As a result we can say that this position is almost perfectly hedged. The 1
% return on the British asset is transformed into a 0.95 % return in US dollars, despite the drop in value of the British pound. The slight difference, between 1 % and 0.95 %, is caused by the fact that the investor hedged only the principle of £ 1 million, and not the unexpected return on the British investment which equals 1% in the example. The 5 % drop in sterling value applied to this 1 % return exactly equals the 0.05 % difference ( 1 % x 5 % = 0.0005 = 0.05 % ).
To find out the exact relationship between dollar and pound returns on the foreign portfolio we can calculate the rate of return of the portfolio measured in domestic currency terms (R * ):
R * = (Vt * - V
0
* ) / V
0
* (formula 3.05)
= ($ 1,919,000
– $ 2,000,000) / $ 2,000,000 = - 0,0405 = - 4,05 %
Comparing this with the rate of return of the portfolio measured in foreign currency terms (R):
R = (Vt - V
0
) / V
0
(formula 3.06)
= (£ 1,010,000 - £ 1,000,000) / £ 1,000,000 = 0,01 = 1 %.
In this case the currency contribution, R * - R, is equal to the exchange rate variation plus the cross-product sR.
To show that I am going to calculate the percentage movement in exchange rate (s) first: s = (St - S
0
) / S
0
= (
$ 1,90 / £ - $ 2 / £) / ($ 2 / £) = - 0,05 = - 5 %
Now we can see that the currency contribution
R * - R = - 4,05 % - 1 % = - 5,05 % is equal to the exchange rate variation + cross product sR = - 5 % + (- 0,05 %) = - 5,05
%.
Over long periods the difference can become significant, when the value of the portfolio in foreign currency fluctuates widely. To reflect movements in the value of the position to be hedged, the amount of currency hedging
8

should be adjusted periodically. One possibility is that a portfolio manager could decide to hedge the expected future value of the portfolio rather than its current principal value. This method is risky if expectations do not materialize. Therefore this approach is only applied for fixed-income securities to hedge both the principal value and the yield to be accrued. But still, periodic adjustment of the currency hedge would be necessary to cover unexpected capital gains or losses on the price of the fixed-income securities.
9
The ratio of the size of the position taken in futures contracts to the size of the exposure is called the hedge ratio (h).
10 It can easily be calculated using the following formula: h = number of contracts x size x spot price
market value of asset position 11
In the case that the asset underlying the futures contract is the same as the asset being hedged, it is natural to use a hedge ratio of 1.0 to hedge the entire position.
When cross-currency hedging is used, as explained in chapter 2.3., setting the hedge ratio equal to 1.0 is not always optimal. In that case the hedger should choose a value for the hedge ratio that minimizes the variance of the value of the hedged position.
12
A problem appears when the foreign currency value of a foreign investment reacts systematically to an exchange rate movement. For example, a drop in the value of the British pound could lead to an increase in the value of a British company, regarding the company is measured in British pounds. Once, the pound depreciation will increase the pound value of cash flows received from abroad, and it also makes the company`s products more attractive abroad. In that case there is a covariance between the asset return measured in local currency and the exchange rate movements. The covariance between these two variables is equal to the correlation times the product of the standard deviations of the two variables.
Investors usually care about the rate of return on their investment and the variance thereof. One goal is to search for minimum variability in the value of the hedged portfolio. If an investor decides to hedge, he would like to set the hedge ratio that way, that he minimizes the variance of the return on the hedged portfolio. Modifying formula 3.04 using the hedge ratio leads to the following formula for the rate of return on the hedged portfolio:
R
H
= R * - h x R
F
(formula 3.07)
9 Solnik, B. / McLeavey, D. (2006): International Investments, 5 th Ed. London, pp. 573-575
10 Hull, John C. (2006): Options, Futures, and other Derivatives, 6 th Ed. New Jersey, p. 56
11 Solnik, B. / McLeavey, D. (2006): International Investments, 5 th Ed. London, p. 524
12 Hull, John C. (2006): Options, Futures, and other Derivatives, 6 th Ed. New Jersey, pp. 56-57
9

The rate of return on a hedged portfolio is equal to the rate of return on the original unhedged portfolio (R * ), minus h times the percentage change on the futures price (R
F
).
To calculate the optimal hedge ratio (h * ), which minimizes the variance of
R
H
, we use the following equation: h * = cov (R * ,R
F
)
2
F
( formula 3.08)
The optimal hedge ratio is equal to the covariance of the rate of return on the original unhedged portfolio with the percentage change on the futures price, divided by the variance of the return on the futures ( 2
F
).
The minimum variance hedge ratio can be expressed as the sum of two components, linked to different aspects of currency risk: Translation risk and economic risk.
Translation risk occurs by translating the value of the asset from the foreign currency to the domestic currency. Even if the foreign currency value of the asset was constant, for example a deposit in foreign currency, translation risk would still be present. The hedge ratio of translation risk is 1.
Usually this is taken to mean that a currency hedge should achieve on a foreign asset the same rate of return in domestic currency as can be achieved on the foreign market in foreign currency terms. Consider an US investor would try to achieve a dollar rate of return on a British bonds portfolio equal to what he could have achieved in terms of pounds. That way creating a perfect currency hedge is equivalent to nullifying a currency movement and translating a foreign rate of return directly into a similar domestic rate of return.
When the foreign currency value of a foreign investment reacts systematically to an exchange rate movement economic risk exists. Economic risk is in addition to translation risk. Consider a country that has an exchange rate target. If the local currency depreciates that will lead the country to raise its interest rates to push down local bond prices. That means there is a positive covariance between bond returns, measured in local currency, and the exchange rate movement. At the end an investor from abroad will lose twice from the foreign currency depreciation. First, the percentage translation loss equals the percentage depreciation of the foreign currency. Second, the value of the investment measured in foreign currency itself will drop.
10

The investor should hedge both, translation and economic currency risk if he worries about the total influence of a foreign exchange rate depreciation on the portfolio value, measured in domestic currency.
In practice, most portfolio managers only care about translation risk. They adopt a unitary hedge ratio if they try to minimize the impact of currency risk. Minimizing accounting currency losses is an objective choice. Looking at the portfolio accounting standpoint, currency loss is simply stated as the difference in return when measured in domestic and foreign currencies.
Also, the sensitivity of an asset value to an exchange rate movement has to be estimated either by using economic models or from historical data.
Even though estimates might be imprecise. Hedging only translation risk might not be optimal from an economic point of view.
The minimum-variance hedge ratio, as described in chapter 3.1., is not necessarily optimal in a risk-return framework. So far we have not considered transaction costs. Transaction costs can also affect hedging strategies.
Futures and spot exchange rates differ by a basis. Changes in the basis can affect hedging strategies creating basis risk. Basis risk will be explained in chapter 3.2.1..
The percentage movement in the futures and in the spot exchange rate will differ by the basis over time. That means that in the long run, the return on the hedged portfolio will differ from the portfolio return achieved in foreign currency by the basis, even with a hedge ratio of 1. This aspect will be discussed in chapter 3.2.2.
Futures exchange rates are directly determined by two factors. The spot exchange rate and the interest rate differential between two currencies.
The premium, which is the percentage difference between the futures and the spot exchange rates, equals the same maturity as the forward contract.
The interest parity theory says that the spot rate, forward rate and interest rates are linked through arbitrage. This leads to the following relations:
F = (1 + r
$
) ( formula 3.09)
S (1 + r £ )
F – S = (1 + r
$
) – (1 + r
£ ) = r
$
- r £ (formula 3.10)
S (1 + r £ ) (1 + r £ )
Explanation: In both formulas the spot exchange rate (S) and the futures exchange rate (F) are expressed as dollar value of one pound. The interest rates are expressed in dollars (r
$
) and pounds (r £ ), with the same ma-
11

turity as the futures contract. Both interest rates are period rates, not annualized. To get the annualized rates they simply have to be multiplied by the number of days until maturity and divided by 360 days.
Changes in the basis have an impact on the quality of the currency hedge.
Although currency risk is removed by hedging, some additional risk is taken in form of basis risk. The basis risk is quite small. Futures prices for contracts near maturity closely follow spot exchange rates because at that point the interest rate differential is a small component of the futures price.
To show that we use Table 1.0, which shows the importance of interest rate differentials to futures prices.
Maturity
Pound interest rate (%)
One
Month
14,0
Dollar interest rate (%)
Futures price (dollars per pound)
10,0
1,993
Interest rate component ($) -0,007
Three
Months
13,5
10,0
Twelve
Months
13,0
10,0
1,983
-0,017
1,947
-0,053
Source: Solnik, B. / McLeavey, D. (2006): International Investments, 5 th Ed.
London, p. 578
Table 1.0: Importance of interest rate differentials to futures prices
In the table we find the futures price of British pound contracts with 1, 3, and 12 months left until maturity. The spot exchange rate is currently $
2,00 per pound. The pound and dollar interest rates (annualized) are given for the different months left until maturity.
As an example we will calculate the one month futures price in dollars per pound:
F = S + S r
$
- r £ = 2,00 + 2,00 - 4 / 12 % = 1,993
(1 + r £ ) 1 + 14 / 12 %
The formula for calculating the futures price simply results from rearranging formula 3.10. Notice, that the interest differential for one month
(r
$
- r £ = 10 % - 14 % = -4 %) equals the annualized rate differential of -4 percent divided by 12. The interest rate component expressed in dollars is calculated by subtracting the spot exchange rate from the futures price.
Even though the interest rate differential is very large, the effect on the one-month futures price is minimal. That is, because the spot exchange rate is the driving force behind short-term forward exchange rate movements. We can draw the conclusion that basis risk is very small compared to the currency risk that is being hedged.
12

So far the focus was on minimizing the risk. We found out that a hedge ratio of 1 will minimize transaction risk. But considering the long run, even with a hedge ratio of 1, the return on the hedged portfolio will differ from the portfolio return achieved in foreign currency by the interest rate differential. The goal was to eliminate the variance of the difference of the return on the hedged position and the return on the unhedged position (R
H
-
R) in foreign currency terms. However, it is impossible to set them equal.
The reason is we can only hedge with futures contracts with a price F different from S. Over time, the percentage change in the futures price (R
F
) will differ from the percentage movement in the spot exchange rate (s), which affects the portfolio by the interest rate differential.
13
Consider a small US import-export firm that is negotiating a large purchase of Japanese watches from a firm in Japan. The Japanese firm has demanded that payment will be made in yen upon delivery of the watches. In that case the US firm bears the exchange risk. The price of the watches is agreed today to be yen 2,850 per watch for 15,000 watches. Delivery will take place in seven months. The American firm will have to pay 15,000 x
2,850 yen = yen 42,750,000 in seven months.
Table 4.0 shows the current exchange rates on April 11.
Spot rate
June Futures
Foreign exchange rates, April 11, in $ / yen
0.004173
0.004200
September Futures
December Futures
0.004237
0.004265
Source: Kolb, Robert W. (1996): Financial derivatives, 2 nd Ed. Oxford, p.206
Table 4.0: Current exchange rates on April 11
With the current spot rate the purchase price of the 15,000 watches would be 42,750,000 yen x 0.004173 $ / yen = $ 178,396. If the futures prices on
April 11 are traded as a forecast of future exchange rates, it seems that the dollar is expected to lose ground against the yen. With the December futures trading, the actual dollar cost might be closer to 42,750,000 yen x
0.004265 $ / yen = $ 182,329. If delivery and payment are to occur in December, the importer might reasonably estimate the actual dollar outlay to be about $ 182,000 instead of $ 178,000.
13 Solnik, B. / McLeavey, D. (2006): International Investments, 5 th Ed. London, pp. 576-581
13

To avoid any worsening of the exchange position, the importer decides to hedge the transaction by trading foreign currency exchange futures. Since delivery is expected in November, the importer decides to trade the December futures. By this choice, the hedger avoids having to roll over a nearby contract, thereby reducing transaction costs. Also, the December contract has the advantage of being the first contract with maturity after the hedge horizon, so the December futures exchange rate should be close to the spot exchange rate prevailing in November, when the yen are needed.
The importer`s next difficulty stems from the fact that the futures contract size is yen 12.5 million. If he trades three contracts, the transaction will be worth 3 x 12,500,000 yen = yen 37,500,000. If he trades four contracts, the transaction will be worth 4 x 12,500,000 yen = yen 50,000,000. However, coverage really is needed for the amount of only yen 42,750,000. No matter which way he decides to trade, the American firm will be left with some unhedged exchange risk.
The importer decides to trade three contracts.
1. Cash market position
In April the importer anticipates a need for yen 42,750,000 in November, which have an expected value of $ 182,329 in November.
In November he receives the watches and buys the yen 42,750,000 on the spot market for a total of $ 182,671.
The spot market result is
Anticipated costs $ 182,329
Actual costs $ -182,671
$ - 342
2. Futures market position
In April the importer buys 3 December yen futures contracts for a total commitment of 37,500,000 yen x 0.004265 $ / yen = $ 159,938.
In November he sells the 3 December yen futures contracts at 0.004270 $
/ yen for a total value of 37,500,000 yen x 0.004270 $ / yen = $ 160,125.
The futures market result is a profit of $ 160,125 - $ 159,938 = $ 187.
Because the futures has moved only 0.000005 $ / yen, the futures profit is only $ 187. This gives a total loss of the entire transaction of $ -342 + $
187 = $ -155. Had there been no hedge, the loss would have been the full loss generated by the cash market of $ -342.
Conclusion: This hedge was only partially effective for two reasons. First, the futures price did not move as much as the cash price. The cash price changed by 0.000008 $ / yen, but the futures price changed by only
0.000005 $ / yen. Second, the American importer was not able to fully hedge the position due to the contract size. Having a need for yen
42,750,000 and trading futures for only yen 37,500,000, the importer was left with an unhedged exposure of yen 5,250,000.
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14 Kolb, Robert W. (1996): Financial derivatives, 2 nd Ed. Oxford, pp. 206-209
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Andersen, T. J. (2006): Global Derivatives. A Strategic Risk Management
Perspective, London
Hull, John C. (2006): Options, Futures, and other Derivatives, 6 th Ed. New
Jersey
Hull, John C. (1995): Introduction to futures and options markets, 2 nd Ed.
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Kolb, Robert W. (1996): Financial derivatives, 2 nd Ed. Oxford
Solnik, B. / McLeavey, D. (2006): International Investments, 5 th Ed. London
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