4.913%921 + 0.0491 ×360
Forward rate agreement (FRA)An FRA is an off-balance sheet agreement to make a settlement in thefuture which, when combined with a cash borrowing in the future,can be used to have the same economic effect as a forward-forward.It is an agreement to pay or receive, on an agreed future date, thedifference between a ﬁxed interest rate agreed at the outset andthe interest rate actually prevailing at that future date, calculatedon an agreed notional principal amount. The principal itself is nottransferred in either direction.Example 3I will need to borrow cash for a 3-month period, starting in2 months’ time. I have an arrangement with a bank which willallow me to borrow at LIBOR plus 0.50%.I believe that interest rates will rise over the next 2 months, sothat if I wait, I will have a higher borrowing cost than if I couldborrow now.I therefore arrange a 2v5 FRA, whereby I will pay a ﬁxed rate,which is agreed with the bank at 5.22%, and receive LIBOR forthe 2v5 period.
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The net result will be a cost ﬁxed at 5.72%, as follows:pay to the lending bankreceive in the FRApay in the FRA
LIBOR + 0.50%LIBOR5.22%.
net cost
5.72%
Although I do not know what LIBOR will be when the forward3-month period starts, I know that the LIBOR I pay will be offsetexactly by the LIBOR I receive.Pay LIBOR + 0.50%to cash lender
Receive LIBOR in FRA
Borrower
Pay 5.22% in FRA
PricingIn calculating a theoretical FRA price, we can apply exactly the sameideas as in the forward-forward calculation earlier. As we are not actually borrowing cash with the FRA, however, we could calculate usingmiddle rates for both the 5-month and 2-month periods – ratherthan separate bid and offered rates. This calculation produces atheoretical FRA rate which is comparable to LIMEAN. By convention, however, FRAs are always settled against LIBOR rather thanLIMEAN. We therefore need to add the difference between LIMEANand LIBOR – generally around 0.05% – to this theoretical rate. Analternative approach is to base the theoretical price calculation onLIBOR for both periods, rather than LIMEAN. The result is different,but generally only very slightly.
Theoretical FRA rate⎛ ⎞LIBOR fordays in longer period1+×⎜⎟yearlonger period − 1⎟=⎜⎝⎠LIBOR fordays in shorter period1+×shorter periodyearyear×days difference
Forward interest rates, FRAs and introduction to futures
65
In practice, a dealer generally prices an FRA from the futuresmarket (see below) rather than from cash forward-forwards, becausefutures are very liquid and cash is expensive on the balance sheet.However, futures pricing theory still comes back to forward-forwardpricing and in any case, FRAs in minor currencies where there isno futures market do need to be priced from, and hedged by, cashforward-forwards.Having established a theoretical FRA price, a dealer would thenbuild a spread round this price to give a two-way price with a bid andan offer. Based on this two-way price, a customer or bank wishingto pay the ﬁxed FRA rate (as in Example 3) would pay the offer (thehigher rate). This is known as buying an FRA. A customer wishingto receive the ﬁxed FRA rate would receive the bid (the lower rate).This is known as selling an FRA. The terminology of buying or sellingan FRA has nothing to do with the customer/bank relationship; it ismerely a way of distinguishing who is the ﬁxed rate payer from whois the ﬁxed rate receiver.
The buyer in an FRA is the party paying the FRA rate andreceiving LIBOR.The seller in an FRA is the party receiving the FRA rate andpaying LIBOR.
SettlementThe settlement on an FRA – of the difference between the FRA rateand LIBOR at the beginning of the period – is in fact settled netrather than gross. That is, the difference is paid in one direction orthe other, rather than having two payments, one in each direction. Byconvention, this difference is paid or received at the beginning of theforward period to which it relates, rather than at the end as is normalfor interest payments. It is therefore discounted to a present value(see Chapter 2) in order to calculate the actual settlement amountpaid. Taking this into account, the settlement amount is as follows:
FRA settlement amount = notional principal amountdays in FRA period(FRA rate-LIBOR) ×year× days in FRA period1 + LIBOR ×year
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The direction of payment depends simply on which of the two ﬂowswould be bigger if they were instead paid gross. This means:
The buyer pays the seller if LIBOR is ﬁxed lower than theFRA rate.The seller pays the buyer if LIBOR is ﬁxed higher than the FRArate.
The LIBOR used to calculate the settlement amount depends onthe currency. With a sterling FRA, it is generally the LIBOR for theforward period that prevails on the effective date of the FRA – that is,the ﬁrst day of the forward period. LIBOR is therefore ﬁxed on thesettlement date itself. This is also sometimes the case with Canadiandollars and Hong Kong dollars.With an FRA in other currencies, it is generally the LIBOR thatprevails two business days earlier. This is because the LIBOR for acurrency other than sterling is for value spot. In domestic markets inother currencies, as opposed to the international market, FRAs canbe traded for settlement against a same-day value reference, as withsterling.Suppose that an FRA is traded for a regular period – for example,7 months against 10 months. In the case of foreign currency FRAs,both the effective date (7 months) and the end date (10 months) aregenerally taken from the spot date. Because of non-business days,however, it might be that, when LIBOR is ﬁxed, the period betweenthe two dates turns out not to be a regular period. The followingexample demonstrates this.Example 4It is now Thursday 13 March 2003. I will need to borrowUSD 1 million for a 3-month period, starting in 2 months’ time.I purchase a 2v5 FRA at 5.22% on a notional amount ofUSD 1 million.Spot value date is Monday 17 March. The 2-month value dateis Monday 19 May (because 17 and 18 May are non-businessdays). The 5-month value date is Monday 18 August (because17 August is a non-business day). I have therefore purchasedan FRA from 19 May to 18 August (a 91-day period).
Forward interest rates, FRAs and introduction to futures
67
LIBOR on 15 May for the period from 19 May to 18 Augustis not exactly the 3-month LIBOR ﬁxing on that day, because3-month LIBOR would be from 19 May to 19 August. It is therefore necessary on 15 May to interpolate between the 2-monthLIBOR and 3-month LIBOR ﬁxings, although the differencewould be rather slight. Assume that the LIBOR so calculated forthe period 19 May to 18 August is 5.05%. The FRA settlementamount is then USD 424.31:91(0.0522 − 0.0505) ×360 = USD 424.31USD 1,000,000 × 911 + 0.0505 ×360Because I purchased the FRA and rates went down, I will paythe FRA settlement rather than receive it.Assuming that on 15 May, I arrange a 3-month borrowingof USD 1 million, this would normally be from 19 May to 19August, so that, as well as being based on a slightly differentLIBOR from the FRA settlement, the borrowing would be forone day longer than the FRA. It would have been possibleinstead to have arranged the FRA originally for the exact dates Iwanted.
FRA periods longer than one yearIf the period of the FRA is longer than one year, the correspondingLIBOR rate used for settlement relates to a period where interest isconventionally paid at the end of each year as well as at maturity. A6v24 FRA, for example, covers a period from 6 months to 24 monthsand will be settled against an 18-month LIBOR rate at the beginningof the FRA period.An 18-month deposit would, however, typically pay interest at theend of one year and again after 18 months. As FRA settlements arepaid at the beginning of the period on a discounted basis, each ofthese amounts needs to be discounted separately. Strictly, the netsettlement payment calculated for the end of 18 months should bediscounted at an appropriate compounded 18-month rate and the netsettlement amount calculated for the end of the ﬁrst year should bediscounted at 1-year LIBOR. In practice, the FRA settlement LIBORis generally used for both. The ﬁnal discounted settlement amount istherefore as follows:
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For periods longer than a year but less than 2 yearsFRA settlement amount = notional principal amount⎛⎞days of FRA period in ﬁrst yearrate-LIBOR)×(FRA⎜⎟year⎜⎟⎜⎟days of FRA period in ﬁrst year⎜⎟⎜⎟1 + LIBOR ×⎜⎟year⎜⎟⎜ ⎛⎞⎟⎜⎟⎜⎟⎟⎟⎜ ⎜×⎜ ⎜⎟⎟⎟⎟⎜ ⎜⎜ ⎜ (FRA rate-LIBOR) × days of FRA period in second year ⎟ ⎟⎟⎟⎜ ⎜year⎟⎟⎜ ⎜ ⎟ ⎟⎜+⎜ ⎟⎟⎜ ⎜days of FRA period in ﬁrst year⎟⎟⎜ ⎜1 + LIBOR ×⎟⎟⎜ ⎜year⎜ ⎜  ⎟ ⎟⎠⎠⎝ ⎝days of FRA period in second year× 1 + LIBOR ×year
This same principle can be applied to the settlement of FRAs coveringany period.
Introduction to futuresIn general, a futures contract in any market is a contract in which thecommodity being bought and sold is considered as being delivered(even though this may not physically occur) at some future daterather than immediately – hence the name. The signiﬁcant differencesbetween a futures contract and a forward arise in two ways.Firstly, a futures contract is traded on a particular exchange(although two or more exchanges might trade identically speciﬁedcontracts). A forward, however, which is also a deal for delivery ona future date, is dealt over the counter (OTC) – a deal made betweenany two parties, but not on an exchange.Secondly, futures contracts are generally standardized, whileforwards are not. The speciﬁcations of each futures contract are laiddown precisely by the relevant exchange and vary from instrumentto instrument and from exchange to exchange. Some contracts, forexample, speciﬁcally do not allow for the instrument to be delivered;although their prices are calculated as if future delivery does takeplace, the contracts must be reversed before the notional delivery
Forward interest rates, FRAs and introduction to futures
69
date, thereby capturing a proﬁt or a loss. Short-term interest ratefutures (STIR futures), for example, cannot be delivered.The theory underlying the pricing of a futures contract dependson the underlying instrument on which the contract is based. Fora futures contract based on 3-month interest rates, the pricing istherefore based on the same forward-forward pricing theory explainedearlier.Example 53-month EURIBOR futures contract traded on LIFFE:Exchange: LlFFE (London International Financial Futures andOptions Exchange).Underlying: The basis of the contract is a 3-month deposit ofEUR 1 million based on an ACT/360 year.Delivery: It is not permitted for this contract to be delivered;if a trader buys such a contract, he cannot insist that, on thefuture delivery date, his counterparty makes arrangements forhim to have a deposit for 3 months from then onwards at theinterest rate agreed. Rather, the trader must reverse his futurescontract by delivery, thereby taking a proﬁt or loss.Delivery months: The nearest 3 months following dealing andMarch, June, September and December thereafter.Delivery day: First business day after the Last Trading Day.Last trading day: 10.00 a.m. 2 business days prior to thethird Wednesday of the delivery month.Settlement price: On the last day of trading – usually the thirdMonday of the month – LIFFE declares an ‘exchange deliverysettlement price’ (EDSP) which is the closing price at whichany contracts still outstanding will be automatically reversed.The EDSP is 100 minus the European Bankers Federation/ACI3-month offered rate (EURIBOR) at 11:00 a.m. Brussels time.Price: The price is determined by the free market and is quotedas an index rather than an as interest rate. The index isexpressed as 100 minus the implied interest. Thus a price of94.52 implies an interest rate of 5.48% (100 − 94.52 = 5.48).
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Price movement: Prices are quoted in units of 0.005. Thisminimum movement is called a tick.Proﬁt and loss value: The P&L is deﬁned as being calculatedon exactly 3/12 of a year, regardless of the number of days ina calendar quarter. The proﬁt or loss on a single contract istherefore:Contract amount × price movement ×
312
Therefore the value of a 1 basis point movement is EUR 25.00and the value of a 1 tick movement (the tick value) is EUR 12.50:3= EUR 25.00123= EUR 12.50EUR 1 million × 0.005% ×12EUR 1 million × 0.01% ×
There are relatively minor differences between futures exchanges, andeven between different STIR contracts on the same exchange:• Underlying The typical contract speciﬁcation for short-terminterest rate futures is for a 3-month interest rate, although1-month contracts also exist in some currencies on someexchanges (for example, US dollar and euro futures contracts).The precise speciﬁcation varies from exchange to exchange butis in practice for exactly one quarter of a year (or exactly onetwelfth of a year for 1-month contracts). Thus settlement on thesterling 3-month interest rate futures contract traded on LIFFE,is effectively equivalent to 91 14 days’ settlement because sterling isbased on a year of 365 days:91 143=36512The size of the contract varies. For example, the US dollar andeuro 3-month contracts are USD 1 million and EUR 1 million, theUS dollar and euro 1-month contracts are USD 3 million andEUR 3 million, the sterling 3-month contract is GBP 500,000 andthe yen 3-month contract is JPY 100 million.
Forward interest rates, FRAs and introduction to futures
71
• Delivery date STIR contracts worldwide are generally based onthe delivery month cycle of March, June, September and Decemberas described above. Some contracts include serial months (asin the example above) but others do not. Serial months areadditional delivery months other than these four, added so thatthe three nearest possible months are always available. Forexample, on 5 June, the delivery months available would be June,July, August, September, December, March. . . . On 25 July, thedelivery months available would be August, September, October,December, March. . . .In the case of sterling futures, both the last trading day and theLIBOR ﬁxing used for settlement are the third Wednesday of thedelivery month (or the next following business day), rather thantwo business days earlier.• Trading Trading times vary. Some contracts are tradedby open outcry (see below) – notably on the IMM (theInternational Monetary Market, the ﬁnancial sector of theChicago Mercantile Exchange (CME)) – and some are tradedelectronically.• Price movement Tick sizes, and hence tick values, vary. Forexample, the minimum price movement on sterling is 1 basispoint. The minimum price movement for US dollars variesfrom 14 basis point for the nearest-dated futures contact,through 12 basis point for subsequent ones, to 1 basis point forlater ones.• Settlement price The settlement price varies according to bothcurrency and exchange.
Example 6A dealer expects interest rates to fall (futures to rise) and takes aspeculative position. He therefore buys 20 EUR 1-month futurescontracts at 95.27. He closes them out subsequently at 95.20.What is his proﬁt or loss?The price has fallen, so he makes a loss of EUR 3,500:Number of contracts × contract amount × price movement ×= 20 × EUR 3,000,000 × 0.07% ×
1= EUR 3,50012
112
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Foreign exchange and money markets
Short-term interest rate futuresPrice = 100 − implied forward-forward interest rate × 100Proﬁt on a long position in a 3-month contractsale price − purchase price3×= contract amount ×10012
The mechanics of futuresMarket participantsThe users of an exchange are its members and their customers. Anexchange also has locals, private traders dealing only for their ownaccount. Each member may enter into transactions either on his ownaccount or, if he is a public order member, also on behalf of customers.Certain members may also act as ﬂoor dealers executing transactionson behalf of other members.Open outcry versus screen-tradingThere are two methods of dealing. The ﬁrst, traditional, method isopen outcry, whereby the buyer and seller deal face to face in public inthe exchange’s trading pit. This should ensure that a customer’s orderis always transacted at the best possible rate and ensures completetransparency of all dealing. The second method is electronic screentrading, designed to simulate open outcry but with the advantage oflower costs and wider access. Some exchanges use only screen-basedtrading and there is a general trend away from open outcry becauseof its higher costs.ClearingThe futures exchange is responsible for administering the market, butall transactions are cleared through a clearing house, which may beseparately owned. On LIFFE, for example, this function is performedby the London Clearing House (LCH). Following conﬁrmation of atransaction, the clearing house substitutes itself for each counterparty and becomes the seller to every buyer and the buyer to everyseller.Only clearing members of an exchange are entitled to clear theirtransactions directly with the clearing house. In order to becomea clearing member, certain conditions set by the exchange andthe clearing house have to be satisﬁed, including a minimum net
Forward interest rates, FRAs and introduction to futures
73
worth requirement. Non-clearing members have to clear all theirtransactions through a clearing member.Margin requirementsIn order to protect the clearing house, clearing members are requiredto place collateral with it for each deal transacted. This collateral iscalled initial margin.Members are then debited or credited each day with variationmargin which reﬂects the day’s loss or proﬁt on contracts held.Customers, in turn, are required to pay initial margin and variationmargin to the member through whom they have cleared their deal.The initial margin is intended to protect the clearing house for theshort period until a position can be revalued and variation margincalled for if necessary. As variation margin is paid each day, the initialmargin is relatively small in relation to the potential price movementsover a longer period.The calculation of variation margin is equivalent to marking tomarket – that is, revaluing a futures contract each day at the closingprice. The variation margin required is the tick value multiplied by thenumber of ticks price movement since the close the previous day. Forexample, if the tick value is EUR 12.50 on each contract, and the pricemoves from 94.730 to 94.215 (a fall of 103 ticks), the loss on a longfutures contract is EUR (12.50 × 103) = EUR 1,287.50. Dependingon the size of initial margin already placed with the exchange, andthe exchange’s current rules, a variation margin may not be calledfor below a certain level.Initial margin requirements need to be funded by members for aslong as positions are held open. It is therefore in the interests ofpromoting market liquidity to keep this margin as low as possiblewhile maintaining adequate protection for the clearing house. Forexample, the LCH’s system for calculating margin, known as SPAN(standard portfolio analysis of risk), recognizes that a combination offutures (and options) may have less risk than the sum of the risksattached to each element of the position individually, and thereforerequire less margin.Closing outA futures position can be closed out by means of an exactly offsetting transaction. Contracts which are not settled before maturityare required to be reversed automatically at maturity and the pricedifference paid or received.All contracts are completely fungible; once a purchase of a certainquantity of a particular contract is added to an existing holdingof the same contract, there is no distinction whatever between the
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Foreign exchange and money markets
two purchases; they simply become a single, larger holding of thecontract.Limit up/downSome markets impose limits on trading movements in an attempt toprevent wild price ﬂuctuations and hence limit risk to some extent.BasisIn practice, the actual futures price trading in the market is unlikelyto be exactly the same as the fair theoretical value which can becalculated according to the underlying cash market forward-forward.The difference between the actual price and the theoretical fair priceis termed the value basis.Basis is the difference between what the futures price would be,based on the current cash interest rate, and the actual futures price.This difference should tend towards zero on the last trading day forthe futures contract.Basis risk is the risk arising from the basis on a futures position.Suppose, for example, that on 2 April a futures trader sells a 1v4 FRAto a customer which will settle on 4 May and that he hedges this byselling futures contracts for delivery on 18 June. The trader cannot beperfectly hedged because on 4 May the cash market 3-month LIBORagainst which the FRA will be settled will not necessarily have movedsince 2 April to the same extent as the futures price. He thus has abasis risk.Value basis = theoretical futures price – actual futures priceBasis = implied cash price – actual futures priceBasis risk is the risk arising from the basis
Example 7It is now mid-February and the current 3-month GBP LIBOR is5.32%; the current June futures price is 94.37 and the theoretical June futures price based on the current cash market is94.39.Value basis = 94.39 − 94.37 = 0.02 = 2 basis pointsBasis = (100 − 5.32) − 94.37 = 94.68 − 94.37 = 0.31= 31 basis points
Forward interest rates, FRAs and introduction to futures
75
Futures compared with FRAsIt is worth summarizing the differences between an FRA and a futurescontract:• Standardization versus ﬂexibility For each futures contract,the amount, delivery date and period are all standardized. Withan FRA, they are entirely ﬂexible.• Margin versus credit risk Dealing in futures requires thepayment of collateral (initial margin) as security. In addition,variation margin is paid or received each day to reﬂect the day’sloss or proﬁt on contracts held. The margin system ensures thatthe exchange clearing house is generally fully protected againstthe risk of default. There is therefore usually virtually no creditrisk in dealing futures. When trading OTC, however, professionaltraders usually deal on the basis of credit lines, with no securityrequired.• Settlement An important practical difference between FRAs andfutures is in the settlement mechanics. An FRA settlement ispaid at the beginning of the forward period, and the settlementformula incorporates discounting to a present value. The futuressettlement – the proﬁt or loss on the contract – is also all settled bythe same date, via the variation margin payments made during theperiod from transaction until the futures delivery date. However, inmost futures markets, this settlement is not discounted. A 90-dayFRA is not therefore exactly matched by an offsetting futurescontract even if the amounts and dates are the same.• Liquidity and spread Standardization and transparency generally ensure a liquid market in futures contracts, together withnarrower spreads between bid and offer than in FRAs. For deliverydates far in the future, on the other hand, there may be insufﬁcient liquidity in the futures market, whereas an FRA price mightbe available.• Underlying commodity Futures contracts are not available inall currencies, whereas an FRA might be available, albeit at anunattractive price.• Taking a position FRAs and futures are in opposite directions.A buyer of an FRA will proﬁt if interest rates rise. A buyer of afutures contract will proﬁt if interest rates fall. If a trader sellsan FRA to a counterparty, he might therefore also sell a futurescontract to cover his position.
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Foreign exchange and money markets
The yield curveIn Chapter 3 we discussed the yield curve – how interest rates varywith term to maturity.Rather than this normal yield curve, a more useful curve to lookat might be one that shows what the market expects rates to be atcertain times in the future. For example, what is the 3-month ratenow, what will the 3-month rate be after 1 month, after 2 months,after 3 months etc.? Given enough such rates, it is in fact possible towork backwards to construct the normal yield curve.Suppose, for example, that the 3-month rate now is 5.0% andthe market expects that rates will fall slightly during the next3 months – so that at the end of 3 months, the new 3-month ratewill be 4.94%. Given these data, what should the 6-month rate benow?The answer must be the rate achieved by taking the 3-month ratenow, compounded with the expected 3-month rate in 3 months’ time;otherwise there would be an expected proﬁt in going long for 3 monthsand short for 6 months, or vice versa, and the market would tend tomove the rates. In this way, the 6-month rate now can be calculatedusing a strip (see Chapter 2): 9236091× 1 + 0.0494 ×−1 ×1 + 0.0500 ×360360183= 0.0500 = 5.00%If we now work in the other direction, we would ﬁnd that the 3v6forward-forward rate is 4.94% as expected:⎛⎞1831 + 0.0500 ×⎜⎟ 360360⎜ − 1⎟⎝⎠ × 92 = 0.0494 = 4.94%911 + 0.0500 ×360This shows that a ﬂat short-term yield curve – in our example, the3-month and 6-month rates are the same at 5.00% – does not implythat the market expects interest rates to remain stable. Rather, itexpects them to fall.An important point here is to consider the question of which comesﬁrst. Are forward-forward rates (and hence FRA rates and futuresprices) the mathematical result of the yield curve? Or are the market’sexpectations of future rates (i.e. forward-forwards, FRAs and futures)the starting point, and from these it is possible to create the yieldcurve? The question is a circular one to some extent, but market
Forward interest rates, FRAs and introduction to futures
77
traders increasingly look at constructing a yield curve from expectedfuture rates for various periods and maturities – in other words, thefutures market is the driving force.
Applications of FRAs andfuturesAs with any instrument, FRAs may be used for hedging, speculatingor arbitrage, depending on whether they are taken to offset an existingposition or taken as new positions themselves.
HedgingAs we have discussed above, a future borrower wishing to hedgeagainst the possibility of higher interest rates would buy an FRA orsell futures. A future investor or lender wishing to hedge against thepossibility of lower interest rates would sell an FRA or buy futures.Example 8A dealer has lent USD 3 million for 1 month and borrowedUSD 3 million for 9 months. He is therefore over-borrowed andwishes to hedge this position with an FRA.1v9 FRAs are quoted to him as 4.50/4.60%. He therefore sellsan FRA at 4.50% and, after 1 month rolls over the lending atLIBID. The net result will be as follows:earn on rolloverpay in the FRAreceive in the FRA.net income
LIBIDLIBOR4.50%.4.50% – spread between LIBID and LIBOR
In this case, the result is not known exactly because FRAs arealways priced and settled against LIBOR but the rollover will beat LIBID. The dealer can estimate, however, what the spread islikely to be.In practice, the amount of the FRA dealt will probably beUSD 3 million. If the dealer wishes to hedge the position exactlyhowever, he can calculate the maturing amount of the lendingat the end of 1 month, including principal and interest. This isthe amount which he will have available to relend at that time,and is therefore the amount to be protected by the FRA if hewishes to be precise.
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Foreign exchange and money markets
Receive LIBID on rollover
Pay LIBOR in FRA
Dealer
Receive 4.50% in FRA
In Chapter 2 we used a strip to calculate a borrower’s overall costwhen he rolls over a loan. We have repeated the strip formula below.Exactly the same approach works when using FRAs or futures, as inthe next example.Creating a stripThe simple interest rate for a period up to one year⎞⎛days in ﬁrst period1+ﬁrstinterestrate×⎜⎟year⎟⎜ ⎟⎜ ⎟⎜days in second period⎟⎜= ⎜ × 1 + second interest rate ×⎟year⎟⎜⎜⎟×. . . . . . .⎟⎜⎠⎝days in last period−1× 1 + last interest rate ×yearyear×total days
Example 9It is now mid-September and an investor wishes to depositUSD