00,000 after 4 months, similarto the ﬂows in Example 2. The dealer therefore also buys JPY13,980,039 (the difference between JPY 1,171,019,961 and JPY1,185,000,000) 1-month forward outright. This is the amountwhich would make the yen cashﬂows net to zero if we were usingcash forward-forward borrowings.Also, as the FRAs are in fact contracts for differences ratherthan cash instruments, the dealer will still need to reverse theswap cashﬂows when the ﬁrst leg of the forward-forward swapsettles after one month. This can be done using either a 3-monthswap again, or a borrowing and a deposit. We will assume herethat the dealer simply uses a swap.After 1 month, US dollar LIBOR is set at 4.95% and yen LIBORis set at 0.6%, and the USD/JPY spot rate is 125.00. Assumingthat the USD/JPY 3-month FX swap at the same time is exactlyin line with these interest rates, the 3-month outright rate is123.6425 (this is unrealistically precise, but helps to clarify theﬁgures in this example).The settlements on the FRAs are a payment of USD 3,700 anda receipt of JPY 295,560, as follows:919,880,121 × (0.048 − 0.0495) ×360= −3,700911 + 0.0495 ×360911,171,019,961 × (0.005 − 0.006) ×360= −295,560911 + 0.006 ×360The cashﬂows from the deal and the cover are therefore asfollows:
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157
1 month+ USD10,000,000− JPY 1,185,000,0001-month hedge:− USD117,975+ JPY13,980,039dollar FRA settlement: − USD3,700yen FRA settlement: + JPY295,560reverse deal:− USD10,000,000+ JPY 1,250,000,000original deal:
net cashﬂows:
− USD+ JPY
4 months− USD10,000,000+ JPY 1,172,500,000
+ USD10,000,000− JPY 1,236,425,000
121,67579,275,599 − JPY
63,925,000
At the 1-month date, the ﬁnal net 4-month cashﬂow of – JPY63,925,000 can be converted to – USD 517,015 (using the 3month forward outright rate then prevailing of 123.6425). Thisdollar cashﬂow has a present value on the 1-month date of−USD 510,626:−517,015 = −510,626911 + 0.0495 ×360Also, the cashﬂow of +JPY 79,275,599 can be converted to +USD634,205 (using the spot rate of 125.00 then prevailing).The 3 resulting cashﬂows of – USD 121,675, +JPY 79,275,599and – JPY 63,925,000 therefore represent a net proﬁt after onemonth of:−USD 121, 675 + USD 634, 205 − USD 510, 626 = +USD 1, 904This net proﬁt of USD 1,904 is, as expected, exactly the same asin Example 2.In the last example, the dealer must in practice also take accountof the fact that the swap ultimately used to reverse the ﬂows maynot be exactly in line with the LIBORs. Also, even if it is, he willpresumably be taking the price from someone else and thereforelosing the bid/offer spread.Apart from these problems, the swap formula used previously canbe applied to calculate the forward-forward swap rate created synthetically from FRAs:
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Foreign exchange and money markets
Forward-forward swap rate created synthetically⎛  ⎞daysvariable⎜ currency FRA × variable year ⎟⎜ ⎟⎟⎜days⎠⎝− base currency FRA ×base year= outright to near date × daysbase currency1+×FRAbase year
As covered brieﬂy in Chapter 6, futures are the exchange-tradedequivalent of FRAs. The last example could therefore have usedfutures as a hedge, instead of FRAs. In major currencies at least,futures have the advantage of being liquid and ﬁnely priced. Thedrawbacks for hedging, however, include the mechanical complexityof margin calls and the inﬂexibility regarding periods and dates. Thefollowing example is parallel to the last one, using futures insteadof FRAs.Example 5Today is 5 March, spot is 7 March, the 1-month date is 7 Apriland the 4-month date is 7 July. A dealer buys and sellsUSD 10 million against yen, 1 month against 4 months (91 days)at 118.50 and 117.25.He hedges the position by buying 10 March dollar futurescontracts at 95.30 (representing USD 10 million at 4.70%) andselling 12 March yen futures contracts at 99.55 (representingJPY 1,200 million at 0.45%).As in Example 4, there is another small FX risk to be hedged.Again, consider what the cashﬂows would be if, instead ofusing yen futures, the dealer used a cash forward-forwardyen borrowing for 90 days at 0.45%. There would be a cashinﬂow of 1,200,000,000 after 1 month and a cash outﬂow of JPY1,201,350,000 after 4 months. The dealer therefore also sellsJPY 15,000,000 (the difference between JPY 1,200,000,000 andJPY 1,185,000,000) 1 month forward outright and also buys JPY28,850,0000 (the difference between JPY 1,201,350,000 and JPY1,172,500,000) 4 months forward outright.On 17 March (the last trading day for March futures),the futures positions are closed out and rolled into Junefutures – i.e. the March futures are reversed and replaced bynew June futures positions. The closing prices for the March
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159
futures are 95.25 for dollars (representing a dollar interest rateof 4.75%) and 99.52 for yen (representing a yen interest rate of0.48%). The June futures prices at the same time are 94.90 fordollars and 99.40 for yen.On 5 April, the dealer closes out the June futures contracts.The futures prices are then 94.80 for dollars and 99.35 for yen.As in Example 4, the USD/JPY spot rate is 125.00 and the3-month outright rate is 123.6425.The proﬁts and losses on the futures contracts are as follows:March contracts:
−95.30 + 95.253= −USD 1,250×100123+99.55 − 99.52×= +JPY 90,000JPY 1,200,000,000 ×10012
USD 10,000,000 ×
June contracts:
3−94.90 + 94.80×= −USD 2,500100123+99.40 − 99.35×= +JPY 150,000JPY 1,200,000,000 ×10012
USD 10,000,000 ×
The cashﬂows from the deal and the cover are therefore asfollows:1 month+ USD10,000,000− JPY 1,185,000,000outright hedges: + USD126,582− JPY15,000,000dollar futures: − USD3,750yen futures:+ JPY240,000reverse deal:− USD10,000,000+ JPY 1,250,000,000original deal:
net cashﬂows:
+ USD+ JPY
4 months− USD10,000,000+ JPY 1,172,500,000− USD246,055+ JPY28,850,000
+ USD10,000,000− JPY 1,236,425,000
122,832 − USD50,240,000 − JPY
246,05535,075,000
At the 1-month date, the ﬁnal 4-month cashﬂow of – JPY35,075,000 can be converted to − USD 283,681 (using the3-month forward outright rate then prevailing of 123.6425).This gives a total dollar cashﬂow at 4 months of − USD 529,736
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(= −USD 246, 055 − USD 283, 681). This dollar cashﬂow has apresent value on the 1-month date of − USD 523,190:−529,736 = −523,190911 + 0.0495 ×360Also, the cashﬂow of +JPY 50,240,000 can be converted to +USD401,920 (using the spot rate of 125.00 then prevailing).The four resulting cashﬂows of +USD 122,832, +JPY50,240,000, −USD 246,055 and −JPY 35,075,000 thereforerepresent a net proﬁt after 1 month of:+USD 122, 832 + USD 401, 920 − USD 523, 190 = +USD 1, 562The net proﬁt of USD 1,562 in the above example is a little differentfrom the USD 1,904 in the previous example, because futures are aless exact hedge. In particular:• The amount of the hedge is restricted to a roundnumber – USD 10 million and JPY 1,200 million in this case.• The futures contract period does not coincide exactly with theswap period. In this case, it was also necessary to roll over thefutures when they expired in March.• The proﬁt and loss on these futures contracts is calculated using a90-day period, whereas there are in fact 91 days in this particular3-month period.• There is a basis risk. That is, the movements in the futures pricesdo not necessarily match exactly the movement in the interestrates underlying the swap price. In this example, there was atotal gain on the yen futures contract of 8 basis points (3 basispoints gained on the March contract from 99.55 to 99.52, plus 5basis points gained on the June contract, from 99.40 to 99.35)However, the movement in the forward interest rate underlyingthe swap was 10 basis points, from 0.5% to 0.6%. It is possibleto improve slightly the hedge against this basis risk, but thedetailed use of futures strategies is really outside the scope ofthis book.• The timing of the cashﬂows on the futures also affects the result,because part of the loss on the dollar futures and the proﬁt on theyen futures arose in March – although we did not take accountof this timing complication above, assuming instead that all theproﬁt and loss on the futures arose in April.
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161
A forward-forward FX position to buy and sell one currencyagainst another can be hedged by selling an FRA (or buyingfutures) in the ﬁrst currency and buying an FRA (or sellingfutures) in the second currency, plus small spot or outrighthedges to bring the notional net foreign currency cashﬂows tozero.
Covered interest arbitrageBorrowing or investing via a different currencySuppose that an organization needs to fund itself in one currencybut can borrow relatively more cheaply in another; it can choosedeliberately to borrow in that second currency and use a forwardswap to convert the borrowing to the ﬁrst currency. This might bedone, for example, if the resulting all-in cost of borrowing were slightlyless than the cost of borrowing the ﬁrst currency directly. It mightalso be done if borrowing in the original currency of choice were forsome reason unavailable.Even if it does not need to borrow, a bank can still borrow in thesecond currency, use a forward swap to convert the borrowing to theﬁrst currency and then make a deposit directly in the ﬁrst currency; areason for doing this would be that a proﬁt can be locked in becausethe swap price is slightly out of line with the interest rates.Taking advantage of such a strategy is known as covered interestarbitrage.
Covered interest arbitrage is synthetically creating a loan (ordeposit) in one currency by combining a loan (or deposit) inanother currency with a forward foreign exchange swap.Example 6A dealer needs to produce (i.e. borrow) GBP 1 million for6 months (183 days) from spot. The following rates are quotedto him:GBP/USD
spot:6 months:
1.4250/56133/131
GBPUSD
6 months:6 months:
6.0/6.1%4.0/4.1%
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Foreign exchange and money markets
The simplest approach would be to borrow sterling directly at6.1%. An alternative is to borrow dollars at 4.1% and swap themto sterling as follows.The dealer borrows enough dollars so that when convertedto sterling for value spot, he receives exactly GBP 1 million.Based on a spot rate of 1.4250 (see below for why we havechosen the left side of the spot rate), the amount to borrow isUSD 1,425,000.He calculates the amount of dollars to be repaid after sixmonths:interest = 1,425,000 × 0.041 ×
183= 29,699.38360
principal + interest = 1,425,000.00 + 29,699.38 = 1,454,699.38He transacts a foreign exchange swap to buy and sell sterlingagainst dollars, spot against 6 months. The dollar amounttransacted for value spot is the amount of dollars borrowed(USD 1,425,000); the dollar amount transacted for value6 months is the amount of dollars repaid (USD 1,454,699.38).The swap deal is transacted on the left side of the price (becausethe market-maker who has quoted the price is selling and buyingthe base currency) at −133 points. Therefore if the settlementrate for spot is set at 1.4250, the settlement rate for 6 monthsmust be set at 1.4117 (= 1.4250 − 0.0133).Applying these settlement rates, the cashﬂows are as follows:6 months
spotdollar borrowing: + USD 1,425,000
− USD 1,454,699.38
swap:
− USD 1,425,000+ GBP 1,000,000
+ USD 1,454,699.38− GBP 1,030,459.29
net cashﬂows:
+ GBP 1,000,000
− GBP 1,030,459.29
The net result is effectively a borrowing of GBP 1,000,000 with atotal repayment after six months of GBP 1,030,459.29 – i.e. aneffective interest cost of GBP 30,459.29.This cash interest cost is for a period of 183 days. It can beexpressed as an interest rate per annum of 6.075%:30,459.29 365×= 0.06075 = 6.075%1,000,000 183
Hedging swaps
163
This rate achieved of 6.075% is slightly cheaper than the 6.1%cost of borrowing sterling directly. It is therefore worthwhile tocreate a synthetic sterling borrowing via dollars in this way.Three points to notice in the above Example are:• The Example assumes that the dealer is taking another bank’srates to borrow dollars at 4.1% and another bank’s rates to swapat −133 points. If he were able to trade on his own prices, theresult would be even better.• When a swap is dealt, the amount of the deal (for example,GBP 1,000,000) is usually the same at both ends of the deal,spot and forward. In the example above, however, the amountsare mismatched, with GBP 1,030,459.29 at the far end in order tomatch the swap cashﬂows exactly with the underlying ﬂows arisingfrom the borrowing. It is generally acceptable in the market to usemismatched amounts in this way as long as the mismatch is notgreat.• When dealing a forward swap rather than a forward outright, it isthe swap price that is dealt rather than the spot price; the spotprice is needed only for settlement. The spot dealer is not involvedand the spot spread is not involved. In general, therefore, the spotand forward settlement prices could be 1.4250 (the bid side of thespot price, as used in the example above) and 1.4117, or 1.4256(the offer side of the spot price) and 1.4123, or something betweensuch as 1.4253 and 1.4120. The important point is that the spotrate must be a current market rate and the difference between thespot and forward settlement prices must be the correct swap priceof −133 points.Conventionally for a swap, an approximate mid-price is takenfor the spot. In an example such as this where the amounts aremismatched, however, it is common to use, for the whole deal,whichever side of the spot price would normally be used for themismatch amount. The mismatch in this example is a forwardpurchase of GBP 30,459.29 (= 1, 030, 459.29 − 1, 000, 000). Foran outright alone, the quoting bank would wish to deal on theleft for this, based on a spot of 1.4250, so that the settlement rates would be 1.4250 and 1.4117; commonly therefore,it would choose to deal on these same rates for the wholemismatched swap.Although this approach is common, it does not necessarilybeneﬁt the quoting bank. In the example above, the quoting bankdoes indeed beneﬁt by using a spot of 1.4250, as we have done. In
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Foreign exchange and money markets
general, however, this depends on whether the forward points arepositive or negative. Speciﬁcally in the case of a mismatched swapfor covered interest arbitrage, it is usually the case that:– if the points are negative, the quoting bank beneﬁts slightly byusing the spot rate on the same side as the swap: if the leftswap price is used (as in the example above), the quoting bankbeneﬁts by using the left side of the spot price; if the right swapprice is used, the quoting bank beneﬁts by using the right sideof the spot price.– if the points are positive, the quoting bank beneﬁts slightly byusing the spot rate on the opposite side from the swap: if theleft swap price is used, the quoting bank beneﬁts by using theright side of the spot price; if the right swap price is used,the quoting bank beneﬁts by using the left side of the spotprice.The effect is in any case generally not great, as can be seen inExample 7 below.The formula we saw in the previous chapter for calculating aforward outright from interest rates was:Forward outrightdays1 + variable currency interest rate ×year= spot × days1 + base currency interest rate ×yearThis formula can be turned round to give the result of the coveredinterest arbitrage from the swap price:Covered interest arbitrageCreating the variable currency interest rate synthetically byusing a borrowing or deposit in the base currency:Variable currency interest rate created synthetically⎛⎞exchange rate⎜⎟for far datedaysbase currency⎜⎟×=⎜ 1+− 1⎟×rate⎝⎠base yearexchange ratefor near date×
variable yeardays
Hedging swaps
165
or:Creating the base currency interest rate synthetically by usinga borrowing or deposit in the variable currency:Base currency interest rate created synthetically⎛
exchangerate fornear date
⎞
⎜⎟⎜⎟⎜⎟daysvariable⎜⎟×=⎜ 1+− 1⎟×⎜⎟currency ratevariable yearexchange⎜⎟⎝⎠rate forfar date×
base yeardays
Note that the formulas are very similar – be careful about which wayround the two exchange rates are!Example 7Using the same information as in the last example, the effectiveinterest cost in sterling can be calculated as follows:Sterling interest rate created synthetically⎛⎞ GBP/USD ratedaysUSD×× for near date − 1⎠=⎝ 1+GBP/USD raterateUSD yearfor far dateGBP year×days1.4250365183×−1 ×=1 + 0.041 ×3601.4117183= 0.060752 = 6.0752%Using slightly different settlement rates, as discussed above,would produce only a slightly different synthetic sterling cost.The result based on a spot of 1.4253 would be 6.0748%:Sterling interest rate created synthetically1.4253365183×−1 ×=1 + 0.041 ×3601.4120183= 0.060748 = 6.0748%
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Foreign exchange and money markets
The result based on a spot of 1.4256 would be 6.0744%:Sterling interest rate created synthetically1831.4256365=1 + 0.041 ××−1 ×3601.4123183= 0.060744 = 6.0744%
The following Example shows how covered interest arbitrage can beused in the same way by an investor, rather than a depositor. TheExample also shows the use of the formula with a tom/next swap. Asmentioned above, when using the formula to calculate the effectiverate achieved, be careful about which is the exchange rate for thenear date and which is the exchange rate for the far date.Example 8An investor has SEK 5 million to invest tom/next. He can eitherinvest directly in kronor at 5.3%, or invest in euros at 4.5% andswap them to kronor. The following exchange rates are quotedto him:EUR/SEK
spot:T/N:
11.2360/703/4
Investing via euros would be as follows.Sell SEK 5 million and buy euros, value tomorrow.Invest the euros tom/next at 4.5%.Sell forward the maturing euro deposit amount back into kronor.The swap involves buying and selling the euros tom/next, at+3 points. If the quoting bank determines 11.2370 as the spotrate, the settlement rates are 11.2367 (tomorrow) and 11.2370(spot).The amount of the euro deposit is:5,000,000= 444,970.5011.2367The maturing amount of the deposit is then:1= 55.62360Principal + interest = 444,970.50 + 55.62 = 445,026.12
Interest = 444,970.50 × 0.045 ×
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167
The cashﬂows are as follows:spoteuro deposit: − EUR
444,970.50
6 months+ EUR
445,026.12
+ EUR444,970.50− SEK 5,000,000.00
− EUR 445,026.12+ SEK 5,000,758.51
net cashﬂows: − SEK 5,000,000.00
+ SEK 5,000,758.51
swap:
The net result is effectively a tom/next deposit of SEK 5,000,000with total proceeds of SEK 5,000,758.51 – i.e. effective interestreceived of SEK 758.51.This interest earned is for a period of 1 day. It can be expressedas an interest rate per annum of 5.46%:758.51360×= 0.0546 = 5.46%5,000,0001This yield of 5.46% is higher than the 5.3% quoted for kronor. Itis therefore worth while to create a synthetic krona deposit viaeuros in this way.This synthetic yield can be calculated using the formula asfollows:Krona interest rate created synthetically⎛⎞ EUR/SEK ratedays= ⎝ 1 + EUR rate ×× for far date − 1⎠EUR/SEK rateEUR yearfor near dateSEK yeardays11.23703601×−1 ×=1 + 0.045 ×36011.23671×
= 0.0546 = 5.46%
A borrowing (or lending) in one currency can be created byborrowing (or lending) a second currency and buying and selling(or selling and buying) the ﬁrst currency against the secondcurrency.
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Foreign exchange and money markets
Creating an FRA in onecurrency from an FRA inanother currencyThe concept of covered interest arbitrage can be used for any period,including one that starts in the future. For example, a borrowing insterling from 1 month forward to 7 months forward can be createdsynthetically from a forward-forward borrowing in dollars combinedwith a forward-forward FX swap. This is exactly the same structureas used in Example 7, with the difference that the start date isin 1 month’s time, rather than spot. Because an FRA is the offbalance sheet equivalent of a cash forward-forward borrowing ordeposit, the same structure can be used to create a synthetic FRAin one currency from an FRA in another currency. The followingexample is parallel to Example 6, using the same rates but based ona forward start date. The result is a synthetic sterling FRA at exactlythe same rate as the synthetic sterling borrowing rate achieved inExample 6.Example 9A dealer wishes to create a synthetic long FRA position inGBP 1 million for 1 month against 7 months (183 days). Thefollowing rates are quoted to him:GBP/USD
1-month outright:1 v 7:
1.4250133/131
GBP FRAUSD FRA
1 v 7:1 v 7:
6.0/6.1%4.0/4.1%
The dealer buys a dollar FRA at 4.1% for the 1-month forwarddollar equivalent of GBP 1 million. Based on 1.4250, this isUSD 1,425,000.He calculates the amount of dollars which would be repaid6 months later, on a borrowing of USD 1,425,000 at 4.1%:183= 29,699360Principal + interest = 1,425,000 + 29,699 = 1,454,699
Interest = 1,425,000 × 0.041 ×
He transacts a forward-forward FX swap to buy and sell sterlingagainst dollars, 1 month against 7 months. The dollar amount
Hedging swaps
169
transacted for value 1 month is the amount of the dollarFRA (USD 1,425,000); the dollar amount transacted for value7 months is the amount of dollars which would have beenrepaid on a cash borrowing (USD 1,454,699). The swap deal istransacted on the left side of the price at −133 points. Thereforeif the settlement rate for 1 month is set at 1.4250, the settlementrate for 6 months must be set at 1.4117 (= 1.4250 − 0.0133).After 1 month, GBP/USD spot is 1.4560, sterling 6-monthLIBOR is 6.5% and dollar 6-month LIBOR is 3.9%. The dollarFRA settlement is a loss of USD 1,421:1831,425,000 × (