1 million for nine months. Given the following rates, andassuming that he can always place cash at LIBID, what is thebest way of doing so?3 months (91 days)6 months (181 days)9 months (273 days)3v6 FRA6v9 FRADecember futuresMarch futures
5.23/5.33%5.25/5.35%5.25/5.35%5.28/5.33%5.24/5.29%94.68/94.6994.77/94.78
Forward interest rates, FRAs and introduction to futures
If the 3v6 period is covered separately, it is better to buyDecember futures at 94.69 (implied interest rate 5.31%) thanto sell an FRA at 5.28%.If the 6v9 period is covered separately, it is better to sell an FRAat 5.24% than to buy March futures at 94.78 (implied interestrate 5.22%).It is important to remember that, even for an investor, an FRAor a futures contract is still settled against LIBOR, not LIBID.If the investor covers the 3v6 period with futures at 5.31%, forexample, his return for that period will not be 5.31% but rather5.31% less the LIBID/LIBOR spread. If this is 10 basis points,his return will be only 5.21% for that period. Similarly, if hecovers the 6v9 period with an FRA, his return would be say5.14%, not 5.24%The choices are therefore reduced to:• Deposit 9-month cash at 5.25%.• Deposit 6-month cash at 5.25%, sell a 6v9 FRA at 5.24%and roll over the deposit at LIBID after 6 months, to give acompound return of 5.259%: 92181× 1 + 0.0514 ×−11 + 0.0525 ×360360×
360= 0.05259 = 5.259%273
• Deposit 3-month cash at 5.23%, buy December futures at5.31%, sell a 6v9 FRA at 5.24% and roll over the depositat LIBID after 6 months and again after 9 months, to give acompound return of 5.262%:  ⎞⎛9091× 1 + 0.0521 ×⎟⎜ 1 + 0.0523 × 360360⎟⎜⎟⎜⎠⎝92−1× 1 + 0.0514 ×360×
360= 0.05262 = 5.262%273
Of these, the last is the best, although the difference is so smallthat the administrative burden, and uncertainty of cashﬂowtiming, arising from the futures trade might well make thestrategy not worth while. On the other hand, there could bean advantage from the reduced credit risk and capital adequacyrequirements compared with a straightforward 9-month deposit.
79
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Foreign exchange and money markets
3v6 FRA 5.28%
6v9 FRA 5.24%
Dec Futures 5.31%
Mar Futures 5.22%
Cash 5.23%Cash 5.25%Cash 5.25%
Hedging an FRA position with futuresWe have already mentioned that a trader is likely to generate an FRAprice from futures prices rather than from cash forward-forwards.Similarly, he is likely to hedge the resulting position by buying orselling futures.If the FRA period coincides exactly with one futures period, thenthe dealer can hedge the FRA with that futures contract – if he is longof FRAs, then he buys futures to hedge; if he is short of FRAs thenhe sells futures to hedge. He can price the FRA as the interest rateimplied by the futures price, together with an appropriate bid/offerspread.Example 10A dealer sells an FRA for GBP 12 million for a 3-month periodfrom mid-March to mid-June.The number of futures contracts used for the hedge should besuch that a 1 basis point change has equal but opposite effectson the FRA position and the futures hedge. Ignoring exact daycounts and the present value component of the FRA settlementformula, the proﬁt or loss on the FRA arising from a 1 basispoint change in rates would be:GBP 12 million × 0.01% ×
3= GBP 30012
The value of 1 basis point on a sterling futures contract isGBP 12.50. Therefore the dealer sells 24 March futures contractsas a hedge:GBP 300= 24GBP 12.50
Forward interest rates, FRAs and introduction to futures
81
If the FRA period coincides with a strip of futures, then he hedgeswith a strip of futures.
Example 11A dealer buys from a customer a GBP 12 million FRA for the 9month period from mid-September to mid-June. Futures pricesat the time are:September:December:March
95.36 (implied rate 4.64%)95.43 (implied rate 4.57%)95.65 (implied rate 4.35%)
The interest rate implied by the strip of futures is 4.57%:  ⎞⎛9091× 1 + 0.0457 ×⎟ 365⎜ 1 + 0.0464 × 365365⎟⎜⎟×⎜⎠ 273⎝92× 1 + 0.0435 ×−1365= 0.0459 = 4.59%The dealer will therefore quote a rate for the FRA of 4.59%less a spread. He hedges the position by buying 24 Septembercontracts, 24 December contracts and 24 March contracts).When the FRA is settled in mid-September, the Septemberfutures contract expires at the same time and he closes outthe December and March contracts.
If the FRA period does not coincide exactly with futures periods,however, the dealer must use overlapping futures contracts. Suppose,for example, that the FRA is for GBP 12 million for a 6-month periodfrom mid-April to mid-October. This could be hedged by a combination of some March futures, some June futures and some Septemberfutures (see Figure 6.1).
FRA (mid-April to mid-October)March FuturesMarch
April
May
June FuturesJune
July
September Futures
August September October
Figure 6.1
November December
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Foreign exchange and money markets
This period overlaps 2 months out of 3 of the March futurescontract (i.e. mid-April to mid-June, 3 months out of 3 of the Junecontract (mid-June to mid-September) and 1 month out of 3 ofthe September contract (mid-September to mid-October). For eachfutures contract where the overlap is complete, the hedge shouldbe complete – i.e. 24 contracts. For each end where the overlap isnot complete, the hedge should be pro-rata. Therefore the hedgeshould be 16 March contracts, 24 June contracts and 8 Septembercontracts.The total number of futures contracts used for the hedge in thiscase should still be such that a 1 basis point change has equalbut opposite effects on the FRA position and the futures hedge. Theapproximate proﬁt or loss on the FRA arising from 1 basis pointchange in rates in this case would be:GBP 12 million × 0.01% ×
6= GBP 60012
The number of futures contracts required for the hedge is therefore 48:GBP 600= 48GBP 12.50This agrees with the hedge of 16 March contracts, 24 June contractsand 8 September contracts.
Example 12In January, a dealer buys a 4v9 FRA from a customer for the5-month period from 12 May to 12 October for EUR 10 millionand needs to hedge this with futures.If the FRA were for a 3-month period, the approximate valueof a 1 basis point change on the FRA would be EUR 250:EUR 10,000,000 × 0.01% ×
3= EUR 25012
The value of a 1 basis point change on a single EUR futurescontract is EUR 25. Therefore 10 contracts would be required.However, the FRA period is in fact 5 months and overlapsapproximately 5 weeks out of 12 of the March futures period, allof the June futures period and approximately 3 weeks out of 12of the September futures period (see Figure 6.2).
Forward interest rates, FRAs and introduction to futures
83
FRA (12 May to 12 October)March FuturesMarch
April
May
June FuturesJune
July
September Futures
August September October
November December
Figure 6.2
5The hedge required is therefore 4.2 = approximately×1012March contracts, 10 June contracts and 2.5 = approximately3× 10 September contracts.12The approximate value of a 1 basis point change on the 5month FRA would be EUR 417:EUR 10,000,000 × 0.01% ×
5= EUR 41712
The total number of contracts required for the hedge is therefore16.7:EUR 417= 16.7EUR 25As futures must be traded in round amounts, the dealer therefore buys 4 March contracts, 10 June contracts and 3 Septembercontracts.In mid-March, the hedge must be adjusted because the Marchfutures contracts expire. They are therefore replaced by 4 moreJune contracts. This creates a bias, however. The FRA is for aperiod starting in mid-May but the futures strip is for a periodstarting in mid-June. The hedge is satisfactory if the forwardyield curve moves up or down in parallel. However, if it alsotwists anti-clockwise, the loss on the FRA position could begreater than the proﬁt on the futures hedge (or the proﬁt on theFRA less than the loss on the futures). In order to reduce thisrisk, the dealer could superimpose a further hedge, by buyinga further 4 June contracts and selling 4 September contracts.This is essentially the same strategy as a spread – see below. Hewould then be left with a hedge of long 18 June contracts andshort 1 September contract.On 10 May, when LIBOR is ﬁxed for the FRA settlement, allthe futures contracts are closed.
The hedge in the previous example is not perfect, for variousreasons:
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Foreign exchange and money markets
• Futures contracts are for standardized notional amounts only.3year, rather than 90, 91• Futures proﬁt/loss is based on exactly 12or 92 calendar days etc.• FRA settlements are discounted but futures settlements are notand also the timing of the futures variation margin cashﬂows isunknown.• There is basis risk because the futures dates do not coincide withthe FRA date.
Quoting for ﬁxed deposits with delayed interestsettlementA ﬁxed deposit for longer than 1 year conventionally pays interest atthe end of each year and then at maturity. Sometimes a customerasks for all the interest to be paid at maturity instead. Effectively inthis case, the bank is keeping the interim interest payments, reinvesting them until maturity, and then giving them to the customer.In quoting to the customer in such a case, account must thereforebe taken of the rate at which it is possible to reinvest the interimcashﬂows. Since this reinvestment rate can be ﬁxed in advance byusing FRAs or futures, the FRA or futures rates can be used for thecalculation.Example 13You are a dealer. You quote a customer 5.13% for a depositof GBP 1 million for 18 months, with interest paid at the endof 1 year and again at maturity. The customer asks to have allthe interest paid at the end and asks for a quote on that basisinstead. What rate do you quote? 12v18 FRAs are currentlyquoted at 4.55/4.65%. The 1-year period has 365 days and the18-month period has 548 days.Effectively, you will need to take the interest amount at theend of 1 year and invest it for 6 months – which you can protectat the FRA bid rate of 4.55% – and then add it to the remaininginterest due at the end of 18 months.The interest amount normally due at the end of 365 days is:GBP 1,000,000 × 0.0513 ×
365= GBP 51,300.00365
Reinvest this amount of GBP 51,300 for the remaining 183 daysat 4.55%, to give a total of:183GBP 51,300 1 + 0.0455 ×= GBP 52, 470.27365
Forward interest rates, FRAs and introduction to futures
85
The interest amount normally due at the end of 548 days is:GBP 1,000,000 × 0.0513 ×
183= GBP 25,720.27365
The total interest therefore payable at the maturity is:GBP 52,470.27 + GBP 25,720.27 = GBP 78,190.54This represents an interest rate of:78,190.54 365×= 0.05208 = 5.208%1,000,000 548
SpeculationThe most basic trading strategy is to speculate on whether the cashinterest rate when the forward period begins is higher or lower thanthe FRA rate or implied futures rate. If the trader expects interestrates to rise, he buys an FRA or sells a futures contract; if he expectsrates to fall, he sells an FRA or buys a futures contract. In example 6,we considered speculation using futures contracts. Here is the samestrategy using FRAs:Example 14On 15 May, a dealer expects interest rates to fall and takes aspeculative position. He therefore sells a 2v3 FRA (from 17 Julyto 17 August) for EUR 60 million at 4.73%. On 15 July, 1-monthEURIBOR is ﬁxed at 4.80%. What is his proﬁt or loss?The interest rate has risen so he makes a loss of EUR 3,601.78:31(0.0473 − 0.048) ×360 = EUR 3,601.78EUR 60,000,000 × 311 + 0.048 ×360
The result is not exactly the same as in example 6 using futures, for2 reasons:(i) The FRA settlement is based on a period of 31 days and a year of360 days, whereas a 1-month futures contract is always based1on exactly 12year.
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Foreign exchange and money markets
(ii) The FRA settlement amount is discounted to a present value, butis received or paid correspondingly earlier
Strip tradingIn the same way, if a trader is using futures and wishes to speculateon a longer-term rate than the 3-month rate, he would sell or buy astrip of futures.
Example 15It is now February. A dealer with no position expects 12-monthinterest rates to rise. He therefore sells a strip of March, June,September and December futures. If he is correct and the 12month rate rises above the 12-month interest rate implied bystripping these four prices together, he will be able to reversethe four contracts at an overall proﬁt.
Calendar spreadA spread is a strategy whereby the trader buys a shorter-dated FRAor futures contract and sells a longer-dated one, because he expectsthe difference between them to change – a twist in the shape of theyield curve – but does not necessarily have any expectation about thewhole yield curve moving up or down.
Example 16Interest rates for euros are now as follows (for simplicity, wehave ignored bid/offer spreads in this example):2 months (61 days)5 months (153 days)8 months (245 days)11 months (337 days)2v5 FRA8v11 FRA
5.00%5.00%5.00%5.00%4.96%4.84%
A trader expects the yield curve to become positive. He thereforesells a 2v5 FRA and buys a 8v11 FRA.
Forward interest rates, FRAs and introduction to futures
87
Later that day, rates are as follows:2 months (61 days)5 months (153 days)8 months (245 days)11 months (337 days)2v5 FRA8v11 FRA
5.01%5.04%5.07%5.10%5.02%5.01%
The dealer has lost 6 basis points on the 2v5 FRA but gained17 basis points on the 8v11 FRA – a net gain of around 11 basispoints because the yield curve has twisted as he expected.
Calendar spreadA dealer expecting the yield curve to twist anti-clockwise sellsa shorter-date FRA and buys a longer-dated FRA (or buys ashorter-dated futures and sells a longer-dated futures).A dealer expecting the yield curve to twist clockwise buysa shorter-date FRA and sells a longer-dated FRA (or sells ashorter-dated futures and buys a longer-dated futures).
A longer-term spread can be taken if a trader has a view onshort-term yields compared with bond yields – for example, a spreadbetween a 3-month Eurodollar futures and long-term USD bondfutures. In this case an adjustment has to be made for the differencein maturity of the underlying instrument. Settlement on the shortterm futures relates to a 90-day instrument, while settlement on thebond futures relates to a notional 15-year bond. For a given changein yield therefore, there will be a far greater proﬁt or loss on thebond futures than on the short-term futures. To balance this, thetrader would buy or sell a much smaller notional amount of the bondfutures than of the short-term futures.
Cross-market spreadA spread can similarly be taken on the difference between twomarkets. For example, if sterling interest rates are above dollar ratesand a trader believes that the spread between them will narrow, hecould sell sterling FRAs and buy dollar FRAs, or buy sterling futuresand sell dollar futures.
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Foreign exchange and money markets
ArbitrageBecause FRAs and futures are essentially the same instrument tradedin different ways, there is a straightforward arbitrage between themif prices move out of line – either buy a FRA and buy futures, or sella FRA and sell futures.Because of the close link between the two instruments, and the factthat dealers will hedge positions in one against the other, FRAs areoften dealt to coincide with futures dates – that is, with effective dateson the third Wednesday of March, June, September or December anda forward period of 3 months. These futures dates are known as IMMdates, after the IMM, the oldest ﬁnancial futures exchange.
Example 17It is mid-March and rates for US dollars are as follows:3v6 FRA:June futures
4.75/4.80%95.17/95.18
A dealer buys a USD 1 million 3v6 FRA at 4.80% and at the sametime he buys 1 futures contract at 95.18. The implied interestrate of the futures contract is 4.82%. The dealer has thereforelocked in a proﬁt of 2 basis points.There are at least two possible problems for the dealer, evenif the effective date of the FRA exactly matches the futures date.Firstly, if interest rates rise at all signiﬁcantly, he will make aproﬁt on the FRA and a loss on the futures contract. The proﬁtwill be discounted to a present value but the loss will not be. Theloss will also arise earlier (because of the mechanics of variationmargin payments), which effectively increases the value of theloss even further.Secondly, if interest rates fall, he will make a loss on the FRAand a proﬁt on the futures. If the FRA is for a 3-month periodof, say, 92 days, this would magnify the loss compared to thefutures proﬁt which is based effectively on 90 days.Either of these effects could offset his apparent proﬁt of 2 basispoints.
Arbitraging and creating FRAsIt is possible to make use of FRAs and futures in covering forward FXpositions, and also to construct synthetically an FRA in one currencyfrom an FRA in another. These topics are covered in the FX section,in Chapter 9.
Forward interest rates, FRAs and introduction to futures
89
Exercises1 Based on the following cash market rates for Swedish kronor,what is the theoretical rate at which a dealer would take a 6v9forward-forward deposit from a customer?3 months (91 days)6 months (182 days)9 months (273 days)
4.87/5.00%5.12/5.25%5.00/5.12%
2
On Friday, you lend EUR 10 million for 1 week at 4.3% andborrow the same amount overnight at 4.25%. What is the breakeven rate for the remaining period?3 You sold a 2v5 GBP 2,000,000 FRA when rates were quotedto you as 7.30/7.40%. Settlement is now due and 3-month(91 days) GBP LIBOR is 7.15%. What amount do you pay orreceive?4 You are a dealer. You are given a deposit of EUR 7 million at4.9% for 9 months (273 days). You also lend EUR 7 million at4.9% for 3 months (92 days). You cover the mismatch with a 3v9FRA, quoted to, you as 4.80/4.85%.a. Do you buy or sell the FRA?b. At what price do you deal?c. What notional principal amount do you deal for the FRA,assuming that you wish to match your exposure as closelyas possible and that you can deal exactly the right amount?d. When the time comes to settle the FRA, the 6-month interestrate is 4.95%/5.05%. What is the settlement amount on theFRA, and who pays whom?e. You always borrow at LIBOR and lend at LIBID. What is youroverall cash proﬁt or loss in euros at the end of nine months?5
6
You sell 10 December euro futures contracts at 95.32. In whichdirection do you expect euro interest rates to move? You buythem back at 95.43. What is your proﬁt or loss?It is now the middle of March. You fund yourself at LIBOR ﬂatand need to borrow euros for 9 months. What is the cheapestway of achieving this, given the following rates?3-month cash:6-month cash9-month cash:3v6 FRA:3v9 FRA:6v9 FRA:
5.10/5.20%5.25/5.35%5.35/5.45%5.34/5.39%5.42/5.47%5.40/5.45%
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Foreign exchange and money markets
June futures:September futures:
94.60/94.6194.57/94.58
7 It is now the middle of October. You are a dealer and sell a 7v12FRA to a customer. Which futures contracts might you use tohedge your position?8 You have taken a deposit from one customer for 6 months andlent to another for 3 months. Assuming all the dates match,what is the best hedge for the mismatched position, given thefollowing prices available to you?3v6 FRA:futures9
6.51/6.56%93.50/93.51
It is now the middle of April. What is the theoretical 2v8 FRA ratein US dollars consistent with the following futures prices?MarchJuneSeptemberDecember
95.2395.6195.8495.93
10 It is now March and the yield curve is negative. You expect it tobecome less negative over the next few months but have no viewon whether interest rates will rise or fall overall. What might bean appropriate trading strategy?
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Part 3Foreign Exchange
7
Spot foreignexchangeIntroductionThe uses of spot foreign exchangeA spot transaction is a straightforward purchase or sale of onecurrency against another. For an end-user, such a transaction isgenerally used simply for converting one currency to another. Forexample, if an industrial company in the UK has received US dollarsfor its exports to the USA, it probably needs to convert them tosterling, which it will do using a spot transaction.Banks and other ﬁnancial institutions, however, often use thespot market for speculation – that is, deliberately taking a risk inthe hope of making a proﬁt. For example, a German bank (with theeuro as its domestic currency) might believe that the Japanese yenwill strengthen during the day. It will therefore undertake a spottransaction to buy yen against euros. As it does not actually need theyen, it will sell them back again into euros later the same day, so thatthere is no net inﬂow or outﬂow of yen, other than possibly a proﬁtor loss between the two transactions.Some banks present themselves to the market as market makers.This means that they are prepared to quote a spot price for buying andselling, to anyone who asks (as long as they are creditworthy) – bothcustomers and other ﬁnancial institutions.In practice, the market is of course less segmented than this. Individuals and corporate users also take speculative positions. Conversely,banks also use spot transactions to convert cash into another currencyother than for speculation. Also, a bank which deals with a customerat the customer’s request – for example, selling Australian dollars tothe customer – has created a risk for itself. If it wishes to remove thisrisk, it will close the position by buying the same amount of Australiandollars from another bank, in another transaction.
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Foreign exchange and money markets
The value dateA spot transaction is for delivery two working days after the dealingdate (the date on which the contract is made). This allows timefor the necessary paperwork and cash transfers to be arranged.Normally therefore, if a spot deal is contracted on Monday, Tuesday orWednesday, delivery will be two days after (i.e. Wednesday, Thursdayor Friday respectively). If a spot deal is contracted on a Thursday orFriday, the delivery date is generally on Monday or Tuesday respectively, as neither Saturday nor Sunday are working days in the majormarkets.There are, however, some exceptions. For example, a price for theUS dollar against the Canadian dollar in the professional interbankmarket generally implies delivery on the next working day after thedealing day. This is referred to as funds rather than spot. A spot price(value two working days after the dealing day, as usual) can generallybe requested as an alternative. The rate for the US dollar against theHong Kong dollar is also often traded for value the next working day.The US dollar/ Turkish lira rate is generally traded for value on eitherthe same day as the deal is done, or the next day. A problem arises intrading Middle East currencies where the relevant markets are closedon Friday but open on Saturday. A spot deal on Wednesday betweenthe US dollar and the Saudi riyal, for example, would need to have asplit settlement date: the dollars would be settled on Friday, but theriyals on Saturday.If the spot date falls on a public holiday in one or both of the centersof the two currencies involved, the next working day is taken as thevalue date. For example, if a spot deal between sterling and the USdollar is transacted on Thursday 31 August, it would normally befor value Monday 4 September. If this date is a holiday in the UK orthe USA, however, all spot transactions on Thursday 31 August arefor value Tuesday 5 September. If the intervening day (between todayand spot) is a holiday in one of the two centers, the spot value date isagain usually delayed by one day.The recognized opening and closing times for the foreign exchangemarket each week are 5:00 a.m. Monday, Sydney time and 5:00 p.m.Friday, New York time.
How spot rates are quotedBase currency and variable currencyThroughout this book, we have generally used ISO codes (also usedby the SWIFT system) to abbreviate currency names. You can ﬁnd alist of the codes in Appendix C.
Spot foreign exchange
95
In the spot market, a variable number of units of one currency isquoted per one unit of another currency. In this book, we have usedthe usual convention that, for example, the US dollar/Japanese yenexchange rate is written as ‘USD/JPY’ if it refers to the number of yenequal to 1 US dollar but as ‘JPY/USD’ if it refers to the number of USdollars equal to 1 yen. The currency code written on the left is thebase currency; there is always 1 of the base unit. The currency codewritten on the right is the variable currency (or counter currency orquoted currency); the number of units of this currency equal to 1 ofthe base currency varies according to the exchange rate.When quoting against the euro, it is the practice in the interbankmarket to quote all currencies in terms of a varying number ofunits of that currency per 1 euro. In other words, the euro is, byconvention, always the base currency if it is one of the two currenciesinvolved. Similarly, apart from the euro, it is the interbank conventionto quote all currencies against sterling using sterling as the basecurrency. Again, apart from the euro, sterling, Australian dollar andNew Zealand dollar, all rates against the US dollar are always quotedinterbank with the US dollar as the base currency.The Canadian dollar is generally quoted as the variable currencyagainst the US dollar, although the other way round is possible.In the currency futures markets, as opposed to the interbankmarket, quotations against the USD usually have USD as the variablecurrency.In other cases, there is not a universal convention for which wayround to quote a cross-rate – that is, which is the base currency andwhich the variable currency.
Cross-ratesAlthough dealing is possible between any two convertible currencies – for example, New Zealand dollars against euros or Swiss francsagainst Japanese yen – the interbank