0.041 − 0.039) × 360 = 1,4211831 + 0.039 ×360At the same time as the FRA is settled, the dealer reversesthe original swap. The dollar amount of this reverse swapcan be calculated to include also the dollar FRA settlementamount. Assuming that the current swap rate is exactly inline with the current sterling and dollar LIBORs of 6.5% and3.9%, based on a spot of 1.4560, the swap rate would be−180 points.The resulting cashﬂows are as follows:7 months
1 month− USD 1,425,000+ GBP 1,000,000FRA settlement: − USD1,421
+ USD 1,454,699− GBP 1,030,459
reverse swap:
+ USD 1,426,421− GBP979,685
− USD 1,454,699+ GBP 1,011,613
net cashﬂows:
+ GBP
− GBP
swap:
The amount of− GBP 18,251:
20,315
− GBP 18,846
has
a
18,846
present
value
of
−18,846 = −18,2511831 + 0.065 ×365The two net sterling cashﬂows therefore have a net present valueof + GBP 2,064 (= +GBP 20, 315 − GBP 18, 251).This net sterling settlement amount implies a sterling FRArate of 6.075%. This can be seen because a sterling FRA at
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Foreign exchange and money markets
6.075% on GBP 1,000,000 would have resulted in exactly thesame sterling settlement amount (allowing for rounding differences):1831,000,000 × (0.06075 − 0.065) ×365 = 2,0641831 + 0.065 ×365The effective sterling FRA rate achieved could have been calculated more simply by using exactly the same covered interestarbitrage formula as before:Sterling interest rate created synthetically⎛⎞ GBP/USD ratedaysUSD×× for near date − 1⎠=⎝ 1+GBP/USD raterateUSD yearfor far dateGBP yeardays3651831.4250−1 ×=1 + 0.041 ××3601.4117183×
= 0.060752 = 6.075%
A long (or short) position in an FRA in one currency can becreated from a long (or short) position in an FRA in a secondcurrency by buying and selling (or selling and buying) the ﬁrstcurrency against the second currency forward-forward.
Creating the variable currency FRA rate synthetically by usingan FRA in the base currencyVariable currency FRA rate created syntheticallybase currencydays=1+×base yearFRA ratevariable yearexchange rate for far date−1 ××exchange rate for near datedaysor
Hedging swaps
171
Creating the base currency FRA rate synthetically by usingan FRA in the variable currencyBase currency FRA rate created syntheticallydays=1 + variable currency FRA rate ×variable yearbase yearexchange rate for near date−1 ××exchange rate for far datedays
Exercises1 You are a dealer and a customer has just sold you GBP 5 millionagainst EUR for value 3 months forward outright. Which of thefollowing are possible ways of hedging your risk, either mostly orcompletely?(a)
Buy EUR against GBP for value 3 months forwardoutright.(b) Buy EUR spot, and also buy and sell EUR against GBP in a3-month swap.(c) Buy EUR spot, and also borrow GBP for 3 months, and alsolend EUR for 3 months.2 You have just bought and sold EUR 10 million against USD, spotagainst 3 months (91 days) at 0.9120 and 0.9163. You hedge theposition approximately by lending EUR 10 million at 3.0% andborrowing USD 9.12 million at 5%. If you are based in EUR, whatadditional small transaction could you undertake to completelyhedge any remaining risk?3
It is now the middle of April. You have bought and soldEUR 10 million against Swiss francs, 2 months against 5 months(a 91-day period), with settlement rates of 1.6000 and 1.5910.You wish to hedge the position. Given the following rates, whatis the cheapest way of hedging, and approximately what proﬁt orloss do you expect on doing so?EUR/CHF
spot:2 months:5 months:
1.6038/4869/65168/162
EURCHF
2v5 FRA:2v5 FRA:
4.0/4.1%1.5/1.6%
EURCHF
June futures:June futures:
95.95/95.9698.39/98.40
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Foreign exchange and money markets
4
An investor has EUR 3 million to invest for one week from spot.He can either invest directly in euros at 4.9%, or he can invest indollars at 4.5%. The following exchange rates are quoted to him:EUR/USD1 week:
spot:
0.9125/291.2/1.1
What yield can he achieve if he invests via dollars?5 A dealer needs to produce overnight sterling on Friday (i.e. for a3-day run from Friday to Monday). He decides that he can do itmost cheaply via euros. If the following rates are quoted to him,what is his effective borrowing cost in sterling?
6
EUR/GBP
spot:O/N:T/N:
0.6250/551.1/1.21.5/1.6
EURGBP
O/N:O/N:
3.2/3.3%6.3/6.4%
You wish to buy a Swiss franc 2v5 FRA (a 92-day period). Giventhe following market prices, at what rate can you create onesynthetically via sterling?GBP/CHF
spot:2 months:5 months:
2.4038/48124/120298/292
GBP
2v5 FRA:
6.0/6.1%
10
Managing thespot risk on aforward FXpositionIt may be helpful to explain why we have included this chapter here,rather than including it within the ‘Bank risk management’ chapterlater in the book. Although it is important for dealers to understandthe wide range of risks faced by a bank, most of these risks fall to becontrolled largely by someone other than the dealer. The particularrisk discussed in this chapter, however, requires continual positiveaction by the dealer which forms an essential part of his day-to-daytrading activity.When a forward dealer takes a position, he is taking a risk deliberately because he is hoping to make a proﬁt out of changes in theforward swap price, which depends essentially on interest rates asdiscussed in an earlier chapter. However, also as mentioned earlier,for each forward foreign exchange position, there is another risk,which arises out of potential spot exchange rate movements. Thisrisk is known as the tail of the forward position.
Hedging a forward swappositionThe valuations, and calculations of proﬁt and loss, in this chapter, aswell as the calculation of the hedge required, depend on the conceptof present value. This is covered in Chapter 2.
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Foreign exchange and money markets
Example 1A forward dealer in a euro-based bank buys and sells USD10 million against Norwegian kroner, spot against 6 monthswhen rates are as follows:USD/NOK
spot:6-month swap:EUR/USDspot:EUR/NOKspot:USD 6-month interest rate:NOK 6-month interest rate:
10.9530+2171 points0.90009.85774.0%8.0%
The settlement rates on the deal are set as 10.9530 and 11.1701.The 6-month period is 182 days.Later the same day, the dollar interest rate rises to 4.15%, NOKinterest rates rise to 8.07%, and the forward swap points movein line to +2126. The spot rate does not change. The dealer nowcloses out his position, in order to take his proﬁt. He thereforesells and buys USD 10 million against kroner at 10.9530 and11.1656.The cashﬂows arising from the two deals are as follows:Spot
6 months
Original deal: + USD 10,000,000− NOK 109,530,000
− USD 10,000,000+ NOK 111,701,000
− USD 10,000,000+ NOK 109,530,000
+ USD 10,000,000− NOK 111,656,000
Close deal:Net result:
+ NOK
45,000
The proﬁt of NOK 45,000 arising after 6 months has a presentvalue of NOK 43,236:NOK 45,000 = NOK 43,2361821 + 0.0807 ×360This proﬁt can be valued in euros by converting at the spot rateof 9.8577, to give a proﬁt of EUR 4,386.
Example 2Now suppose that, having made the original transaction exactlyas in Example 1, the USD/NOK spot rate also changes, at the
Managing the spot risk on a forward FX position
175
same time as the dollar interest rate, to 10.6800. As a result,the forward swap points become +2073 rather than +2126 andthe settlement rates on the closing deal are set as 10.6800 and10.8873.The cashﬂows from the two deals are now as follows:Spot
6 months
Original deal: + USD 10,000,000− NOK 109,530,000Close deal:− USD 10,000,000+ NOK 106,800,000Net result:
− NOK
2,730,000
− USD 10,000,000+ NOK 111,701,000+ USD 10,000,000− NOK 108,873,000+ NOK
2,828,000
Suppose that the euro exchange rates also change as follows:EUR/USDEUR/NOK
spot:spot:
0.91009.7188
Overall, the dealer still appears to show a proﬁt, which is nowNOK 98,000 (=−NOK 2,730,000 + NOK 2,828,000). However,there is now a cashﬂow problem. The cash outﬂow of NOK2,730,000 arises 6 months earlier than the cash inﬂow of NOK2,828,000 and must therefore be funded for this period. Thebank therefore has a funding cost of:NOK 2,828,000 × 0.0807 ×
182= NOK 115,378360
This funding cost is greater than the apparent proﬁt of NOK98,000. As a result, the bank actually makes a real loss.As before, the net result can be calculated as a present value ineuros by ﬁrst calculating the present value of the NOK 2,828,00future cashﬂow, combining this with the NOK 2,730,000 spotcashﬂow, and ﬁnally converting to euros at the spot rate of9.7188, to give a net loss of EUR 1,323:+NOK 2,828,000 = +NOK 2,717,1451821 + 0.0807 ×360− NOK 2,730,000 + NOK 2,717,145 = −NOK 12,855−12,855= −1,3239.7188
176
Foreign exchange and money markets
The spot rate movement used in this example might be ratherlarge for a movement during 1 day, although certainly possible. Overseveral days, it might not be unusual. The important point is thata forward dealer might anticipate correctly the relative change ininterest rates and hence the direction of change in the forward swappoints, but have his proﬁt signiﬁcantly reduced, or even completelyoffset, by a change in the spot rate.Clearly, the forward dealer would like to be able to hedge againstthis effect. To a large extent, this is possible to do, by hedging the netpresent value of the cashﬂows:Example 3With all the same rates as in the previous example, the cashﬂowsarising from the original deal are as follows:SpotOriginal deal: + USD 10,000,000− NOK 109,530,000
6 months− USD 10,000,000+ NOK 111,701,000
The present value of the 6-month krone cashﬂow is as follows:+NOK 111,701,000 = +NOK 107,358,9281821 + 0.08 ×360Therefore the net present value of the spot and forward kronecashﬂows together is:−NOK 109,530,000 + NOK 107,358,928 = −NOK 2,171,072Similarly, the net present value of the spot and forward dollarcashﬂows is:USD 10,000,000 = +USD 198,214+USD 10,000,000 − 1821 + 0.04 ×360Suppose that the dealer hedges both these present value positions (net short NOK 2,171,072 and net long USD 198,214)against euros at the spot rates of 9.8577 and 0.9000. When hecloses his original position, he also closes out the hedges at thenew spot rates of 9.7188 and 0.9100. The cashﬂows are then asfollows:
Managing the spot risk on a forward FX position
177
SpotOriginal deal: + USD 10,000,000− NOK 109,530,000Hedges:+ NOK2,171,072− EUR220,241− USD198,214+ EUR220,238Close deal:− USD 10,000,000+ NOK 106,800,000Close hedges: − NOK2,171,072+ EUR223, 389+ USD198, 214− EUR217,818Net result:
− NOK+ EUR
2,730,0005, 568
6 months− USD 10,000,000+ NOK 111,701,000
+ USD 10,000,000− NOK 108,873,000
+ NOK
2,828,000
In addition to the actual overall net present value loss ofEUR 1,323 as calculated in Example 2. there is now also aproﬁt on the hedges of EUR 5,568. The net result is therefore:−EUR 1,323 + EUR 5,568 = +EUR 4,245Because of the hedges, this is close to the result in Example 1,where there was no change in the spot rate.
The result of Example 3 (where changes in the spot rates were offsetby the result of the spot hedge) is not exactly the same as the resultof Example 1 (where there were no changes in the spot rates).The main reason for this difference is that the hedge protectsagainst the effect of spot rate changes on the net present value of theoriginal deal, but cannot hedge against the effect of spot rate changeson the size of the proﬁt itself. If there had been no proﬁt – becausethere were no interest rate changes – then the hedge would have beencomplete. This can be seen by analyzing the values of the cashﬂowsas follows:Originally:• Spot cashﬂow of +USD 10,000,000 is worth(at 0.9000):
+EUR 11,111,111
• 6-month cashﬂow of −USD 10,000,000 has apresent value at 4.0% of −USD 9,801,786This is worth (at 0.9000): −EUR 10,890,873
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Foreign exchange and money markets
• Spot cashﬂow of −NOK 109,530,000 is worth(at 9.8577):−EUR 11,111,111• 6-month cashﬂow of +NOK 111,701,000 hasa present value at 8.0% of +NOK 107,358,928This is worth (at 9.8577): +EUR 10,890,870After the rate changes:• Spot cashﬂow of +USD 10,000,000 is worth(at 0.9100):+EUR 10,989,011• 6-month cashﬂow of −USD 10,000,000 has apresent value at 4.15% of −USD 9,794,506This is worth (at 0.9100): −EUR 10,763,193• Spot cashﬂow of −NOK 109,530,000 is worth(at 9.7188):−EUR 11,269,910• 6-month cashﬂow of +NOK 111,701,000 has apresent value at 8.07% of +NOK 107,322,424This is worth (at 9.7188): +EUR 11,042,765The spot hedges protect all the original dollar and krone presentvalues (+USD 10,000,000, −USD 9,801,786, −NOK 109,530,000 and+NOK 107,358,928) against changes in the euro spot exchange rates.After the interest rates change, however, these present values themselves change slightly: −USD 9,801,786 becomes −USD 9,794,506and +NOK 107,358,928 becomes +NOK 107,322,424. These differences represent the expected proﬁt or loss arising from having takenthe position in the ﬁrst place. The spot hedges cannot protect theeuro value of these differences, because it is impossible to know whatthe size of any such proﬁts or losses will be.After the interest rates change, therefore, the present value ofthe dollar cashﬂows has increased by USD 7,280 (the differencebetween −USD 9,801,786 and −USD 9,794,506). The euro value ofthis USD 7,280 proﬁt is therefore not protected. As the EUR/USDrate has moved from 0.9000 to 0.9100, the euro value of this proﬁt isdiminished in by EUR 89.Similarly, the present value of the krone cashﬂows has fallen byNOK 36,504 (the difference between +NOK 107,358,928 and +NOK107,322,424). Again, the euro value of this NOK 36,504 loss isnot protected and the exchange rate change from 9.8577 to 9.7188increases this loss by EUR 53.The dealer’s proﬁt is therefore diminished by EUR 142 (= EUR 89 +EUR 53). In normal circumstances, this consideration is a relativelyminor one.
Managing the spot risk on a forward FX position
179
A hedged position will therefore not remain completely hedged,so that the required amounts of the hedges change as interestrates change. In the example, once the interest rates have changed,the amount of the dollar hedge should be increased by USD 7,280and the amount of the krone hedge should be increased by NOK36,504.This approach applies not just to a single position, but to a portfolio of positions. In order to hedge his book, a forward dealerneeds to calculate the net present value of all spot and futurecashﬂows of all his positions in a particular currency – includingany spot hedge he already has in place. This net position canthen be hedged spot against his own reporting currency. Afterrehedging, the net present value of all his cashﬂows in each foreigncurrency, including his spot hedges, should have been brought backto zero.
After hedging, the net present value of all cashﬂows in eachforeign currency, including the spot hedges themselves, shouldbe zero.
Two further points are worth mentioning. Firstly, the hedge will alsoonly be exact to the extent that the interest rates used to discount thecashﬂows reﬂect the interest rates implicit in the forward exchangerates.Secondly, there is a further, less important, reason for the discrepancy between the results in Examples 1 and 3 above. The forwardpoints used in the examples are consistent with the theoretical ratesderived from the interest rates used. However, because the forwardsare only quoted to four decimal places, there is a certain amount ofrounding involved which makes a small difference.
Measuring the spot risk on aforward outright positionSo far, we have considered a forward swap position. The same considerations apply to a forward outright position. If a bank wants tomeasure the risk, valued at the spot date, of a particular openforward outright position, this risk can, as above, be measured as thepresent value of the outright position.
180
Foreign exchange and money markets
Example 4A euro-based bank buys USD 10 million 6 months forwardoutright (182 days) at 11.1701 against NOK 111,701,000 whenrates are as follows:USD/NOK
spot:6-month swap:EUR/USDspot:EUR/NOKspot:USD 6-month interest rate:NOK 6-month interest rate:
10.9530+2171 points0.90009.85774.0%8.0%
Later the same day, the spot rates change as follows:USD/NOKEUR/USDEUR/NOK
spot:spot:spot:
10.68000.91009.7188
and the forward swap points move in line to +2117. The dealernow closes out his position by selling USD 10 million 6 monthsforward at 10.8917 against NOK 108,917,000.The dealer has made a loss of NOK 2,784,000, value 6 monthsforward. The present value of this loss is EUR 275,320:−NOK 2,784,000 = −NOK 2,675,7801821 + 0.08 ×360− 2, 675, 780 ÷ 9.7188 = −275, 320The risk of the position could have been measured originally asthe present value of the forward outright position, which is longUSD 9,801,786 and short NOK 107,358,928:USD 10,000,000 = USD 9, 801, 7861821 + 0.04 ×360NOK 111, 701, 000 = NOK 107, 358, 9281821 + 0.08 ×360
Managing the spot risk on a forward FX position
181
The effect on this position, of the changes in exchange rateswould be EUR 275,331:9, 801, 786 9, 801, 786−= −119, 6800.91000.9000−107, 358, 928 107, 358, 928+= −155, 6519.71889.8577− 119, 680 − 155, 651 = −275, 331Apart from rounding, this is the same as above, illustrating thatthe forward outright position of USD 10 million dollars againstNOK can be seen as having the same spot risk as a spot positionof the present values of the outright position.
Exercise1
You are a forward dealer in a sterling-based bank. You have takena position by selling and buying EUR 5 million against Swissfrancs, 2 months against 6 months, at a swap rate of −100 points.The rates are set at 1.5950 and 1.5850. What would be an appropriate hedge to put in place against the spot risk? Current ratesare as follows:EUR/GBP spot:GBP/CHF spot:
0.62502.5600
EUR
2 months (61 days):6 months (183 days):
4.1%4.3%
CHF
2 months (61 days):6 months (183 days):
2.2%2.4%
11
CurrencyrelationshipsOverviewIn considering exchange rate movements, both historical and prospective, it is useful to see different currencies within some structure. Thisis not an exact science, but broadly speaking there are at least twomajor currency blocs – the US dollar bloc and the euro bloc – andseveral major currency pairs which are traded in great volume andaround which other relationships revolve – USD/EUR, USD/JPY andEUR/JPY. For many exchange rates other than these, it is oftenhelpful to consider these few rates as central and other rates assatellites around them.Consider, for example, the exchange rate between the Canadiandollar and the Danish krone. In practice, the market tends to think interms of the Canadian dollar relative to the US dollar. This does notimply that the USD/CAD rate is, or is expected to be, stable but ratherjust that dealers tend to focus on changes in the USD/CAD rate whenthey think about the Canadian dollar. This is not surprising, giventhe close historical and economic relationship between the USA andCanada. Similarly, when considering the Danish krone, the markettends to focus on its rate against the euro, rather than against thedollar. Therefore in analysing the CAD/DKK rate, one would tendto analyse separately 3 rates – USD/CAD, EUR/DKK and EUR/USD.The movement in the CAD/DKK rate is in practice a combination ofthese three.In the same way, the dollar generally serves as a convenient benchmark for considering currencies in Latin America, the Middle East,Africa and the Far East, such as the Brazilian real, the SaudiArabian riyal, the Somalian shilling and the Singapore dollar. Aswith the Canadian dollar, this does not imply stability against the
Currency relationships
183
dollar – often quite the contrary – but merely gives a starting point.The euro bloc is composed not only of the other currencies in the EU(Danish krone, Swedish krona and to some extent sterling) but alsoof such other European currencies as the Swiss franc and Norwegiankrone.The yen, the most important currency outside these 2 groupings,is likely to become increasingly the centre of a third major bloc(Figure 11.1).
Latin American currencies
Various MiddleEastern currenciesHong Kong dollarSingapore dollarVarious other Asian currencies
Danish krone
Sterling
Canadian dollar
Swedish kronaUS dollar
Swiss franc
Euro
Norwegian kroneYen
Various EasternEuropean currencies
Australian dollar
Figure 11.1 The three major currency blocs.
Exchange rate regimesGovernment policy with regard to how it chooses to manage or notmanage the exchange rate of its currency, is clearly important toexchange rate movements. Again, distinctions between the followingcategories are in practice generally not clear-cut, but they nevertheless offer a helpful framework:• Fixed-rate regime Under a ﬁxed-rate system, the exchange rateof a currency is ﬁxed by the government for the foreseeable futurewith respect to some other currency.The Hong Kong dollar, for example, is ﬁxed against the US dollarat 7.80, although in practice it generally trades slightly strongerthan this. The monetary authorities try to correct it if it movessigniﬁcantly away from this level. The Saudi Arabian riyal is ﬁxedvery tightly to the dollar at 3.75. The CFA franc is ﬁxed againstthe euro, following its former link to the French franc.• Floating-rate regime Under a freely ﬂoating system, the valueof a currency is determined by the market, with no governmentintervention. Exchange rates are determined entirely by forces of
184
Foreign exchange and money markets
supply and demand. The three major currencies, the US dollar,euro and yen, for example, essentially ﬂoat freely against oneanother. Although there is occasionally intervention by centralbanks to put a limit to, or even reverse, very large changes inthese rates, this is comparatively rare.A problem with such a regime is that the free-market exchangerate at any moment might not be consistent with longer-term equilibrium. In practice therefore, governments sometimes interveneto smooth movements in the foreign exchange market. This is adirty ﬂoating or managed ﬂoating system. Although there is noofﬁcial rate between