is 365. One year gives an exact fractionof 1.. Actual/actual: This is an extension of the previous basis. Thedenominator is 366 when the period bridges over February 29 ina leap year; it is 365 otherwise.. 30/360: The number of days is computed assuming that all monthshave exactly 30 days. For example, in that format the number of daysbetween January 18, 1999, and February 18, 1999, is 30, whereas inreality it is 31. In practice, this system is almost equivalent to theactual/365.These different ``accounting'' systems have historically been used indifferent countries, and each of them still remains prevalent in certainparts of the world. In addition, different maturities or instruments do
170
THEORY and TOOLS
not use the same basis. For example, short-term regular deposits inJapan20 and France21 use actual/365, whereas longer-termÐthat is,above one yearÐuse 30/360. In the United States all deposits carryan actual/360 basis, whereas most bonds use a 30/360 basis. Certaininterest rate swaps in France use an actual/actual basis, while swaps inthe United States use 30/360. The enumeration is convincing: Tradersneed to make sure that they are familiar with the conventions beforetrading any money market products.22Conversion from one basis to another is relatively straightforward. If t represents the time fraction on a given number of days,then:t30=360  tact=365
tact=360  tact=365 
365360
tact=act  tact=365
Accrual Convention
The accrual convention refers to the mathematical formulation of interestaccumulation over time. The ®nancial literature in general uses anexponential accrual: $1,000 invested at 5 percent over a year yields$1,000  (1 + 5 percent)  $1,050. If the same amount is investedover six months and renewed at the same rate, the ®nal ¯ow is$1,000  (1 + 5 percent/2)2  $1,050.63. In general, for an investmentover d days, compounded every n days at an annual rate r, the accrualfactor is23:n  rd=nf  1360If we consider a yearly compounding over two yearsÐn  360,d  720Ðf becomes: f  1  r2 . Conversely, for a semiannual compounding over a year, n  180, d  360, and f  1  r=22 . At the limitwhere n is one single day, that is, for an overnight compounding,f  1  r=360d .A second convention, continuous compounding, is used in continuous ®nance theory, and it is an extension of the above formalism. It20
Interestingly in Japan short-term deposits are actual/365 when quoted domestically,actual/360 when quoted internationally.21France, pre-euro introduction. After January 1, 1999, all euro market participantsadopted similar conventions.22Act/360 and act/365 are the two systems used most widely. Act/act was used inFrance exclusively and will probably disappear with the introduction of the euro.23Note that we implicitly use an act/360 basis.
CHAPTER 4 MONEY MARKETS
171
rests on the assumption that compounding is effected more and morefrequently over shorter and shorter periods of timeÐthat is, n getssmaller and smaller. Taken to the limit, interest compounds continuously, and the algebraic expression of f becomes24:f  edr=360Under this convention, a one-year investment of $1,000 at 5 percentyields $1,000  e5%  $1; 051:27, more than an annual and semiannualcompounding.25There is, however, a much more practical accrual method, commonly referred to as zero coupon.26 The idea is to substitute a linearrelationship in place of an exponential expression and eliminateinterim compounding. Thus, with the same notations, we want:d  r0f  1360By construction, r0 is different from r, and we have to establish therelationships between the two, which we will do shortly. It is important, however, to note that none of these notations is right or wrong.Different needs require different tools, and many traders ®nd it easierto manipulate zero coupon rates, whereas academics usually rely oncontinuous compounding because it is convenient from an algebraicand theoretical point of view.27Out of the practicalities we derive from zero coupons, four areparticularly interesting: homogeneity within the term structure, inclu24
This accrual factor is based on the following result:1  n  h1=n ! ehn!0
which applied to:

f  1gives:

n  rd=n360
n  rd=n! edr=360n!036025This is a general result: Everything else being equal, more frequent compoundingyields a higher payoff at maturity. Therefore, continuous compounding yields thehighest possible payoff at maturity.26The term zero coupon was originally used to characterize bonds that do not pay anyinterim interest, but rather a lump sum upon redemption. By extension, it nowdescribes all money market transactions in which no interim ¯ows occur, regardlessof the nature of the transaction. There are, for example, zero-coupon swaps.27Note that the time basis is entirely independent from the linear or exponential natureof the accrual formulation.1
172
THEORY and TOOLS
sion of the reinvestment risk, simpli®ed treatment of nonstandardmaturities, and homogeneity across currencies.28Transactions maturing before one year usually do not have anyinterim reinvestment, in which case n  d  maturity of the transactionand the accrual factor is simply: f  1  d  r=360. In these situations,zero coupon rates are exactly equal to ``standard'' ratesÐr0  rÐandtherefore all rates below one year can be considered zero coupon andin practice are quoted as such. In particular, the LIBORs are quoted aszero coupons.If we take a maturity beyond one year, say, two years, an annualcompounding has a known exponential form f  1  r2 because inmost cases reinvestment occurs yearly. A zero coupon, on the otherhand, has the form: f  1  2  r0  by de®nition. Interestingly, this second expression is homogeneous to the shorter-term linear notation.Therefore, if we can convert long-term rates to zero-coupon equivalents, the resulting term structure is homogeneous below and aboveone year and easier to manipulate.In addition, because there is no physical ¯ow before maturity, thezero-coupon rate, if calculated properly, accounts for the reinvestmentrate as opposed to the implicit assumption in the expression (1  r2that we can reinvest for the second period at the same rate we did forthe ®rst.As far as nonstandard maturities are concerned, note that theLIBOR term structure shows 13 discrete maturities from 1 week to 12months. However, if now is exactly three months before a futuresexpiry, two weeks later we will ®nd ourselves 21/2 months before,and we still need to compute an interest rate accurately. We wouldtherefore appreciate a continuous-term structure, which would provideaccess to rates not-included in the standard set of LIBORs.The natural inclination is to state that the 21/2-month rate is somewhere between the 2- and 3-month rates. Market practice has institutionalized this reasoning, and traders customarily compute linearestimations for broken maturities. The problem gets more complex,however, if we consider a 21/2-year rate for which we have to accountfor annual reinvestments when rates are derived from an exponentialaccrual method. Therefore, this averaging is truly simple only whenapplied to zero-coupon rates.Zero coupons are also helpful in comparing across markets becausewithout consistent conventions it is not possible to effectively assess28
Practically speaking, the absence of interim payments before maturity also translatesinto a much simpler treasury management.
CHAPTER 4 MONEY MARKETS
173
relative funding costs. In particular, when pricing multicurrency products such as foreign exchange swaps, the adoption of zero-couponrates on each side signi®cantly simpli®es the calculation.Unfortunately, however, zero coupons are rarely quoted directly inthe marketplace, and they must be deducted from other rates. Thereare typically two main extraction processes: from swap rates or fromshort-term interest rate futures contracts.29
Holidays
Finally, it is necessary to say a word about holidays. All countries havetraditional customs associated with national holidays. It is not alwayseasy to know in advance all the days that markets will be closed in theyears ahead, and the further we look, the more dif®cult it gets. For thatreason, money market traders have adopted a general convention thatsimpli®es date calculations related to unexpected holidays. This rule isknown as the next business-day convention: If a transaction generates a¯ow on a nonbusiness day, this ¯ow is automatically delayed until thenext business day. A variant includes the possibility of bridging the endof a month when postponing the ¯ow. For ®scal, legal, or accountingreasons, this may not be bene®cial to one of the parties. Thus themodi®ed next-business-day rule states the ¯ow has to be postponed if itfalls on a national holiday unless it crosses the end of a month, inwhich case it has to be settled on the ®rst business day prior to itsoriginal date.
Conventions: Why Bother?
This whole discussion about conventions may seem somehow unnecessary, but it is needed because money market transactions presuppose a deep knowledge of the marketplace. Who, as a junior trader, hasnot confused actual/360 with 30/360 or actual/365 and taken a humiliating loss as a result? The intrinsic value of these conventions willbecome even more apparent in the following example: Suppose that asa treasurer, you need to ®nance $1,000,000 for three years. If this deal istaking place in the United States, chances are that the basis is by default29
In the United States many government bonds can be ``stripped'' from their couponpayments, which in practice means that the notional payment on expiry is tradeddistinctly from the stream of coupon payments. The zero coupon is then implicit inthe stripped structure although it is quoted as a discount, not as an outright rate. Weconcentrate on swaps and short-term interest rate futures because they have widermarket usageÐthat is, higher liquidity.
174
THEORY and TOOLS
30/360. You do not have to worry about any holiday within the nextthree years because you and your counterpart will default to the samecalculations. Finally, if you want to avoid a reinvestment risk, you maywant to trade a zero coupon, and the trader in front of you knowsexactly what you mean and will quote one at your request. In theprocess, both parties have saved considerable time. There is notmuch theory behind these conventions, only a practical standardization.
NOTATIONSParadoxically, despite the seemingly complex nature of the above conventions, we will generally not indicate which basis or accrual methodis used in a particular instance from now on. Most often we will usezero coupons, but there may be situations in which exponential compounding is warranted. The context dictates which method is moreappropriate, and the notations re¯ect the speci®cs of that method.With respect to the calculation basis, it mostly depends on the countrywe operate in and possibly on the structure of the transaction weexamine. Therefore, to remain relatively unspecialized, we have toleave this particularity aside, except in numerical applications inwhich we will take the U.S. conventions.Furthermore, for the sake of simpli®cation, we adopt the following shortcut: r alternatively describes the actual interest rate or thesame rate already adjusted by the necessary time fraction. For example, assume we consider a three-month arbitrage opportunity andwant to compute the associated ®nancing cost. If the three-monthcurrent rate is 5 percent, and the number of days to maturity is 90(actual/360), the interest calculation incorporates a time fraction of90/360. In our convention, the symbol r indifferently refers to 5 percent or 90/360  5 percent depending on the context. For example, ifwe note (1  r) the accrual factor, r clearly has to be interpreted as90/360  5 percent.Once we know how to obtain zero-coupon rates, this shortcutmeans that our accrual factor will almost always be (1  r). If we consider a two-year 5 percent zero-coupon rate, this notation would translate into (1  2  5 percent)  1:10. If we considered a one-week depositat 4 percent, it would be (1  7  4 percent/360)  1:00077. All thesemanipulations may appear painful and useless, but they will be quiteuseful in keeping the notations simple when we get to more complexformulas.
CHAPTER 4 MONEY MARKETS
175
SETTLEMENTIt should not be surprising to discover that different currencies tend tohave different settlement conventions. The most common is T  2: Thecash ¯ow resulting from a transaction is exchanged two days after thetransaction has been agreed upon. This effective date marks the beginning of interest accrual calculation. The British pound is the only majorexception with transactions settled at T  1:30Obviously, overnight transactions do not follow these rules and aresettled on the exact day they are traded. To allow the necessary wiretransfers to take place properly, most players try to complete them asearly as possible, typically before noon.It is important to notice here that a rate quoted today does notaccrue from today. If we take the general case of a T  2 settlement, aone-week 5 percent rate indicates the cost of money for a period of oneweek from today's settlement date. For example, if today isWednesday, this 5 percent indicates an operation from Friday toFriday next week. If we were to trade on a Thursday, the operationwould start next Monday to the following Monday. Note also that anovernight contracted on a Friday is effective on three days, until thenext Monday. All this goes without saying for those familiar with theconventions and calculations. For everybody else, it is important torealize the amount of details necessary to manage such transactions.Despite the high level of standardization, it is extremely easy to misprice or mismanage one.
SPREAD AND MIDMARKETThe LIBOR is by de®nition an offered rate, a rate at which bankers offerliquidity to the marketÐthat is, lend. The price at which they are willing to borrow is sometimes referred to a ``LIBID,'' which is not anacronym but a convenient shortcut.As a general rule, the same way stocks trade with a bid-offerspread, interest rates exhibit spreads between the prices at which atrader is ready to borrow or lend. The typical spread on the NYSE,30
All this is very ¯exible, however. It is usually possible in practice to negotiate nonstandard terms with a counterpart, for example, a U.S. dollar±denominated operationlasting 21/ 2 months and effective three days from now. In that case the rate quoted isnot a standard rate, and it may be different from a LIBOR, for example.
176
THEORY and TOOLS
for example is 1/8, or $0.125.31 The typical LIBOR spread is also 1/8Ðthat is, 0.125 percent. This is recognized as ``market practice'': There isno obligation for a treasury manager to quote a 1/8 spread, and he orshe may not do so if the market conditions do not allow him or her todo so. On the other hand, most participants are used to that ®gure. Thespread obviously depends in a large part on the liquidity in the underlying currency, and the difference between the domestic and the international markets plays a role as well.32For all practical purposes, the spread is a transaction cost, and thereare basically two ways to handle it. Consider an index arbitrage situation: As we will see, such positions are usually long and have to bere®nanced regularly, typically every three months at the expiration ofthe futures contract. Because of this structural characteristic, an indexarbitrageur would tend to use an offered rate. Exceptionally, if he orshe is in a position to lend, he or she has to be aware that the rate he orshe uses most often is not appropriate and should be corrected by 0.125percent. This is a conservative approach, in which the activity's treasury management is based on the side of the spread that is most frequently used. Any transaction on the opposite side creates a loss equalto the spread.The second way to consider the situation is to base all interest ratecalculations on the midmarket rateÐthat is, the average between thebid and offer. In that case, every treasury operation incurs a transactioncost equal to 0.0625 percent because the rate at which it is effecteddiffers from the rate used for evaluation by this amount. Naturallythe choice of one convention or the other is based on a relative appreciation of the activity and of the magnitude of the risk associated withinterest rates in general.3331
What we call ``typical spread'' is different from tick value: The tick value on the NYSEis 1/16 ($0.0625), the minimum difference in price accepted by the system. Large capstrade with a 1/16 spread, but the vast majority in general still has a spread of 1/8.32To put 0.125 percent in perspective, consider what happens with a bank account.Most banks in the United States pay currently between 2 and 4 percent interest onsavings, while general-purpose loans are priced somewhere between 8 and 10 percent. The resulting spread is somewhere between 4 and 6 percent.33A natural question here is why didn't we mention the spread on stock prices as beinga transaction cost. In reality it is, but the situation is fundamentally different from thatof interest rates. Because equity markets close every day with a unique referencepriceÐtypically the price of the last transactionÐthere is no spread attached to this®nal price. In contrast, money market operations always involve a spread because themarket virtually never closes on a unique rate. If we ever get to a point where stockstrade 24 hours a day with no interruption and no discontinuity in liquidity, theinclusion of market spread in formal valuations will probably be advisable. A second
CHAPTER 4 MONEY MARKETS
177
NONSTANDARD MATURITYWe already mentioned that a simple linear estimation was a convenientway to compute a nonstandard rate, assuming we are dealing withzero-coupon rates.34 Speci®cally, consider that we are looking for therate r of a transaction on n days. We know the rates r1 and r2 for thetwo closest standard maturities n1 and n2 , respectively, withn1 < n < n2 . A linear interpolation gives:r
n2
n  r1  nn2 n1
n1   r2
This result happens to be independent from the time basis used as longas it is homogeneous among all components and can be used betweenany two rates in the term structure.35Note, however, that this calculation is by de®nition an approximation: If r1 and r2 are quoted directly in the marketplace, nothing indicates that anybody will be willing to trade at the rate r derived from theinterpolation.36 Therefore, this method should be applied to obtain a``best'' estimate of a missing rate, not to recon®rm an existing quotedmaturity.
INTEREST RATE SWAPSDe®nitions and Markets
A swap is a ®nancial instrument whereby two parties enter an agreement to exchange a certain number of ¯ows at speci®ed dates in the
(Footnote continued)reason deals with the magnitude of the cost. A 12.5 cents spread on a $20 stock is 0.63percent, whereas 0.125 percent is 2.5 percent of a 5 percent rate. If transaction costs onstocks were to reach the 2.5 percent level, it would be absolutely necessary to includethem in any formal valuation.34It is a common estimation but not necessarily the best one. In practice, cubic splinesare also frequently used because they are more accurate. The general problem ofinterpolating rates is not clearcut as we have intentionally presented it. There areseveral other interpolation methods possible, polynomial, for example. A weightedaverage has the important advantage of being simple.35For example, if r1 is quoted as actual/365, and r2 as actual/360, one of them has to betransformed for the formula to be coherent.36If this equation held consistently, the two-year rate would always be the arithmeticaverage of one- and three-year rates. It is not always the case in practice, and thedifference indicates the extent to which this interpolation is reliable.
178
THEORY and TOOLS
future. The amounts of these ¯ows are not necessarily known at thetime of the transaction, but both parties have agreed on their calculations and on the relevant parameters entering into these calculations.Swaps are derivative products, similar to futures in that there is noexchange of money at initiation, and transactions are governed byover-the-counter contracts between the parties.37As an illustration of the widespread uses of swaps, Table 4-3gives the details of OTC outstanding positions and gross marketvalues for 1998 and 1999.38 Swaps are clearly traded on a diverseset of underlying securities, from foreign exchange to gold. In eachcategory they represent a sizable fraction of the outstanding notional,for example, 80 and 70 percent, respectively, in foreign exchange andinterest rate activity.Interest rate swaps fall under the generic de®nition: Each partypays the other a leg of the swap computed from a notional amountand a reference interest rate. For example, one side might pay annual¯ows while the other will pay monthly payments. The rates used maybe ®xed at the time of the transaction or reset regularly. If reset duringthe life of the swap, they match the frequency associated with theschedule: In the above example, we would use a variable one-monthrate to compute the monthly ¯ows. The notional amount is traditionally ®xed throughout the life of the transaction, although ``amortizing''swapsÐin which the notional is amortized progressivelyÐare alsotraded.As was clearly visible in Table 4-3, interest rate derivatives areby far the most liquid instruments in the OTC market. Tables 4-4and 4-5 give more details about the volume and characteristics ofthe market.All together, the four most traded currencies account for 89 percentof the total outstanding notional in June 1999, and their aggregatetransactions represent a little less than $54 billion a day in 1997.Interestingly, swaps are still mostly used by ®nancial institutions (8937
However, this is probably less true nowadays. For example, realizing the fact thatmany swaps transactions could be standardized, the London International FinancialFuture Exchange listed a futures contract on swap in 1998Ðsee www.liffe.com fordetails. In general, the swap market is certainly mature enough to undergo a wave ofstandardization, which would possibly transfer some of the traditional OTC productsto exchange ¯oors.38The gross market value is the pro®t or loss generated by a variation in the marketparameters. Swaps, like futures, are zero-sum instruments; therefore the gross marketvalues indicate the amount of pro®t transferred from one of the parties to the otherfrom the transaction's inception date.
CHAPTER 4 MONEY MARKETS
179
TABLE 4-3 The Global Over-the-Counter (OTC) Derivatives Marketsa(Amounts outstanding in billions of U.S. dollars)Notational Amounts
Gross Market Values
EndJune1998
EndDec.1998
EndJune1999
EndJune1998
EndDec.1998
EndJune1999
A. Foreign exchange contractsOutright forwardsand forex swapsCurrency swapsOptions
18,719
18,011
14,899
799
786
582
12,1491,9474,623
12,0632,2533,695
9,5412,3503,009
476208115
49120096
32919261
B. Interest rate contractsbFRAsSwapsOptions
42,3685,14729,3637,858
50,0155,75636,2627,997
54,0727,13738,3728,562
1,160331,018108
1,675151,509152
1,357121,222123
C. Equity-linked contractsForwards and swapsOptions
1,2741541,120
1,4881461,342
1,5111981,313
19020170
23644192
24452193
D. Commodity contractscGoldOtherForwards and swapsOptions
451193258153106
41518223313797
441189252127125
381028ÐÐ
431330ÐÐ
442322ÐÐ
9,331
10,388
10,536
393
492
400
72,143
80,317
81,458
2,580
3,231
2,628
1,203
1,329
1,119
Ð
Ð
Ð
E. OtherdGrand totalGross credit exposureeMemorandum Item:Exchange-traded contractsf
14,256
13,549
17,262
a
All ®gures are adjusted for double-counting. Notational amounts outstanding have been adjusted by halvingpositions vis-aÁ-vis other reporting dealers. Gross market values have been calculated as the sum of the totalgross positive market value of contracts and the absolute value of the gross negative market value of contractswith non-reporting counterparties.bSingle-currency contracts only.cAdjustments for double-counting estimated.dFor end-June 1998: positions reported by non-regular reporting institutions in the context of the triennial CentralBank Survey of Foreign Exchange and Derivatives Market Activity at end-June 1998; for subsequent periods;estimated positions of non-regular reporting institutions.eGross market values after taking into account legally enforceable bilateral netting agreements.fSources: FOW TRADEdata; Futures Industry Association; various futures and options exchanges.(Source: Bank of International Settlements, ``The Global OTC Derivatives Market at end of June 1999,''www.bis.org.)Note: For an exhaustive introduction to OTC derivatives, see, for example, Robert McLaughlin, Over-theCounter Derivative Products: A Guide to Business and Legal Risk Management and Documentation (New York:McGraw-Hill, 1998).
180
THEORY and TOOLS
TABLE 4-4
Currency
NotionalVolume, 1997,$US billions
USDJPYDEMFRFGBPOtherTotal
$4,386$2,899$2,593$2,518$1,034$3,637$17,067
(Source:
Share of Total
OutstandingNotional,Year-End 1997,$US billions
Share of Total
25.7%17.0%15.1%14.8%6.1%21.3%100%
$6,078$4,313$3,278$2,377$1,456$4,789$22,291
27.3%19.3%14.7%10.7%6.5%21.5%100%
International Swaps and Derivatives Association, www.isda.org.)
TABLE 4-5 The Global OTC Interest Rate Derivatives Marketsa(Amounts outstanding in billions of U.S. dollars)Notational Amounts
Gross Market Values
EndJune1998
EndDec.1998
EndJune1999
EndJune1998
EndDec.1998
EndJune1999
Total contractsWith other reporting dealersWith other ®nancial institutionsWith non®nancial customersUp to one yearbBetween one and ®ve yearsbOver ®ve yearsbU.S. dollarEurocJapanese yenPound sterlingSwiss francCanadian dollarSwedish kronaOther
42,36818,24418,6945,43017,42216,8058,14113,21413,5767,1643,2881,0557708262,475
50,01524,44219,7905,78318,18521,40510,42013,76316,4619,7633,9111,3207479393,113
54,07227,05921,1495,86320,28721,98511,80016,07317,48310,2074,3981,4048141,1222,570
1,160463515182ÐÐÐ3114761945919141968
1,675748683244ÐÐÐ370786212130311825102
1,357634559164ÐÐÐ33758419210327162177
Memorandum item:Exchange-traded contractsd
13,107
12,305
13,810
Ð
Ð
Ð
a
See footnote a to Table 4-3.Residual maturity.cBefore end-June 1999; legacy currencies of the euro.dSee footnote f to Table 4-3.b
(Source:1999.'')
Bank of International Settlements, ``The Global OTC Derivatives Market at End of June
CHAPTER 4 MONEY MARKETS
181
percent), for maturities over one year (62 percent). Note as well thetotal outstanding notional for listed products, one quarter the size ofthe OTC market.
Structures
Interest rate swaps can be seen as two money market operations backto back: Party A lends the notional amount