to party B according to a setof negotiated terms, while party B lends the same exact amount back toA with a somewhat different structure. Accordingly, there is no needfor an exchange of notional at any time; only interest payments aresettled regularly.The most common type of swaps involves ®xed versus variablepayments.39 The reasons that two parties may be willing to entersuch an agreement are diverse and depend on the type of institutionsinvolved. A manufacturing corporation and an insurance companycertainly have different needs to ful®ll in their day-to-day business;nevertheless, both may use swaps at some point for a common purpose: to manage an interest rate exposure.Consider, for example, a corporation that has funded its growththrough debt issuance at a time of low interest rates. This corporationmay have been in a situation in which a variable coupon was its bestoption, and thus it carries an exposure to interest rates that mayprove painful: In a context of tightening monetary policy or risinginterest rates, its interest burden will increase signi®cantly. In thatsituation, the company could potentially get rid of this undesiredexposure through an appropriate swap. If it pays ®xed interestsand receives variable payments matching its variable coupon, thevariable risk goes away and the company has secured its ®nancingcost once and for all. Here, the swap provides a protection against arise in interest rates.The opposite situation is also frequent. Consider the case of aninsurance company receiving premiums from its customers on amonthly basis. This money has to be invested at the prevalent interestrate, with the risk that these rates may go down. Lending at a ®xed rateand borrowing the variable side of a swap would offer the moneymanagers a convenient protection, providing a secured rate over thelife of the swap.
39
Other types of swaps: For example, in the United States, LIBOR 1M versus 3M or 3Mversus 6M, LIBOR 6M versus fed funds, prime versus 3M.
182
THEORY and TOOLS
A broad discussion about the intrinsic power of swap transactionsis beyond our scope, but we have to dedicate a signi®cant amount oftime to their pricing methodology. On one side they represent a hugeamount of trading activity in the OTC market, and at the same timethey have direct applications in the world of arbitrage from a riskmanagement perspective.In particular, they have three powerful features: They give easyand cheap access to long-term ®nancing, provide a short path to azero-coupon term structure, and constitute the simplest and cheapestway to manage an exposure to interest rates.
Conventions
The same way established conventions prevail in cash transactions,the swap market is heavily permeated by standards, norms, andconventions. For example, U.S. dollar±denominated swaps areusually semiannual on the ®xed side, indexed on three-monthLIBORs,40 and use the 30/360 basis. In other words, payments onthe ®xed side occur twice a year and four times a year on the variableleg. Following this path, many other markets have adopted similarconventions, as seen in Table 4-6.Settlement is dependent upon the currency, usually T  2; T  1 forthe British pound. It is customary to refer to the ®xed side to design theway the swap has been traded: A ®xed-rate payer has ``bought'' or``borrowed'' the swap. Bid-ask spreads are generally much smaller
TABLE 4-6Country
Short-TermDeposit Basis
Swap Basis
FixedFrequency
VariableReference
U.S.U.K.JapanEuro zone
Act/360Act/365Act/360Act/360
30/360Act/365Act/36530/360
SemiannualSemiannualSemiannualAnnual
3M LIBOR6M LIBOR6M LIBOR6M EURIBOR
(Source:
40
Compiled by author from Bloomberg data.)
Annual and semiannual with respect to the legs of a swap indicate the frequency ofpayments. Rates are still fundamentally annual with respect to their accrual period, aswe de®ned it earlier. For example, if a semiannual payment is due with a rate of 6percent, the payment is 6 percent/2  3 percent of the notional.
CHAPTER 4 MONEY MARKETS
183
than for cash deposits: typically 0.03 to 0.06 percent and we will seespeci®cally why this is so.Regarding the inclusion of holidays in the date calculations, swapsbehave in the exact same way as cash transactions: They follow thenext business day rule or its modi®ed version.41
Long-Term Financing
Consider the following ``standard'' two-year swap: ®xed annual versussix-month variable. We buy this swapÐthat is, borrow the ®xed rate,paying 5 percent every year for two years. Against these ¯ows, wereceive a six-month rate two times a year.In parallel, we borrow an amount equal to the notional of theswap every six months for six months, at a rate as close as possible tothe swap referenceÐthat is, at a six-month LIBOR. The LIBOR beingan offered rate, this operation presents no dif®culty. The resultingstructure is a portfolio in which all variable ¯ows match and disappear: It is a plain two-year borrowing at a rate equal to the ®xed legof the swap.This construction is extremely interesting for two reasons:. To secure a term loan over a period of two years, any serious
lender will request piles of ®nancial information and would pricethe loan according to the perceived credit risk. In addition tobeing expensive, this path may be long and painful becauseevery dollar lent is at risk for a period of two years.Comparatively, consider the swap transaction: The notional isnot at risk, and the credit exposure for both parties is limited tointerest paymentsÐthat is, a fraction of the notional. Credit is stillan issue, but to a much smaller extent. The cash loan still occurs,but every six months for a period of six months: Credit requirements are less stringent, and the borrower will ®nd lenders muchmore easily.42
41
Swaps have been massively standardized over the years by the International Swapsand Derivatives Association (ISDA, www.isda.org). The ISDA has issued genericguidelines on all aspects of these transactions over the years, and most legal con®rmations nowadays reference these guidelines as being the governing defaults.42We see again the distinction between interest rate exposure and cash management.Despite the existence of the swap, a short-term loan is necessary to gain access to cash.
184
THEORY and TOOLS
. The borrower keeps its ®nancial ¯exibility. For example, if for
some reason the loan is not needed anymore, the transactioncan be closed every six months by simply unwinding the swap;and because the swap market is extremely liquid, the total cost toexit is very low. If interest rates have moved from the originaltime of the transaction, exiting may generate a pro®t or a loss, butthis effect would have been identical with a two-year term loan.Therefore, nothing fundamental is changed with a swap structureexcept that the overall funding operation leaves more ¯exibilityand is certainly cheaper.
Naturally, the actual borrowing must occur four times, every sixmonths. This is certainly a constraint, and the inclusion of a swapcomponent is not suited for all ®nancing needs. However, it is a convenient way to achieve equivalent results, provided the overall situation allows it.
Swaps and Zero Coupons
Because swap rates are essentially long-term borrowing rates whenlinked to short-term loans, we will consider them as a direct andmost accurate indication of the prevalent interest rates for money markettransactions of the corresponding maturity. For example, a two-year®xed annual swap price of 5 percent indicates that the cost ofborrowing over two years is 5 percent annually, regardless of thevariable schedule. In fact, because we accept the constraint to adaptto the variable schedule, it totally disappears from our considerations.43Now consider the same swap without the ®xed ¯ow at the end ofthe ®rst year. The rate we would pay to enter this structure is exactlythe two-year zero-coupon rate because we would pay interest onlyonce at maturity. The same reasoning can be applied to longer maturity: Remove the two intermediate payments from a three-year ®xedannual swap and you get a three-year zero coupon. This tweaking is
43
It is very important here again to refer to the context in which we make thisstatement. An arbitrageur has access to an internal swap desk to price and tradethese instruments. In addition, the credit rating of this institution is usually goodenough for him or her to access LIBORs without any credit spread. For all thesereasons entering a two-year swap is exactly equivalent to borrowing on a term oftwo years. For any other participants not in the same environment, the cost ofborrowing for the same term may be signi®cantly higher.
CHAPTER 4 MONEY MARKETS
185
exactly the trick we need to translate a swap term structure into a zerocoupon yield curve. Furthermore, because swaps are so liquid andquoted for a diversity of maturities and currencies, their prices tendto be extremely accurate as to the overall market expectations, makingthem a very reliable source of information.
Swaps and Risk Management
The example of the corporation entering a swap to hedge its interestcharge is a perfect example of the uses of swaps to hedge miscellaneousrisks attached to interest rates. Considering how liquid the swap market has become, and the fact that prices are widely available frominformation providers such as Reuters and Bloomberg, swaps havebecome by far the cheapest way to hedgeÐor createÐa medium- tolong-term exposure. In particular, in the context of index arbitrage, wewill see that they are widely used in conjunction with and complementto short-term futures contracts.
ZERO-COUPON TERM STRUCTURESwap Asymmetries and Bid-Ask Spreads
Swaps are interest rate derivative products, but compared to otherderivatives we have seen, they have an exotic feature: They are notsymmetrical with respect to the price of their underlying. Speci®cally,because swap variable rates are usually indexed against the LIBOR,which is an offered rate, swaps implicitly take the perspective of aborrower of liquidity.In all our discussions so far, we have neglected this point, as wehave systematically described swaps as practical vehicles for borrowers, and only occasionally for lenders. For lenders, there areextra costs associated with trading swapsÐnamely, the bid-askspread on cash deposits. If we want to borrow liquidity for twoyears, for example, we know we can enter a swapÐwhich does notprovide cash in and of itselfÐand renew a short-term borrowingoperation at every reset on the variable leg. The borrowing is alwayspossible, and it is cost free if we can do it at the price of the variablelegÐusually the LIBOR. Now if we were to trade the other way andsell the swap to lend liquidity for two years, we would have to renewa loan regularly at every reset on the variable leg, and this would costus an extra 0.125 percent of the bid-ask spread. This cost of 0.125percent appears in addition to the spread on the swap, which is
186
THEORY and TOOLS
usually relatively low, typically below 1/ 16Ðthat is, 0.0625 percent.44For example, selling a swap with quarterly variable resets and lending cash at the LIBID every three months will generate costs of 0.125percent divided by 4, four times a year.The convention behind this asymmetry is essentially historical.Swaps originally were promoted to non®nancial corporations astools for risk management purposes. Because non®nancial institutionstend to be overwhelmingly borrowers to fund future industrial development, it was indeed a natural thing to consider an offered rate as theappropriate benchmark.With respect to zero coupons, the existence of this asymmetry isnot practical because it potentially complicates signi®cantly the algebra. There are three simple ways to solve this dif®culty. One is toignore it altogether and decide that we do not need to includeLIBOR-LIBID spreads on the variable leg in zero-coupon calculation.The second is algebraic and involves a simple approximation. In thisapproach, we neglect the compounding of the 0.125 percent cost. Ineffect, regardless of the frequency of variable resets, we assume that theoverall cost to lend regularly is 0.125 percent annually.45The third approach, more qualitative, is different in nature.Consider, for example, an index arbitrage activity, which is often structurally a borrower of cash. The ``working capital'' of the arbitrage ishighly negative, and in the vast majority of cases the re®nancing cycleis very short termÐtypically from futures expiry to futures expiry. Inthat environment, if we need periodically to lend money because of aswap, the cash is not actually lent away but rather deducted from theoutstanding balance that has to be borrowed periodically. In otherwords, the loan matching the variable leg of a swap is in fact a deduc-
44
As a general rule, the same asymmetry exists with all other interest rate derivativeproducts indexed on a reference rate, either a bid or an offer such as the LIBOR. Thisproblem does not surface with stocks because of the existence of a unique referenceprice at the close, which is neither a bid nor an offer.The fact that the swap is no substitute for cash also explains why spreads are muchlower than that of money market transactions, 1/16 instead of 1/8. There is no exchangeof liquidityÐusually the scarce resourceÐonly a contractual commitment, whichdoes not require the same margin.45This approximation is relatively benign. With a semiannual compounding, assumingmarket rates at 5 percent, the annual cost is 0.1265 percent versus our assumption of0.125 percent. With quarterly compounding, the cost becomes 0.1273 percent. Theseerrors represent respectively $15,000 and $23,000 annually on a $1 billion notional.
CHAPTER 4 MONEY MARKETS
187
tion from a borrowing operation that would be traded presumably atthe LIBOR: It can be considered as a virtual loan at the LIBOR. Thezero-coupon calculation is therefore very straightforward, we considerthat we can effectively lend at the LIBOR.46The important part in this reasoning is to understand that zerocoupons are context dependent. If we use swaps to manage an interestrate exposure, with no particular emphasis on treasury management,we do not even need to care about the money market spread in anyway. If, on the other hand, liquidity is an issue, then we need toincorporate the overall funding need of the trading activity. If it happens to be on the borrowing side, again the spread can be neglected. Ifit is on the other side or unclear, we still need to make sure we accountfor the situation properly.In the following we will essentially ignore the spread to keep calculations to their simplest expression. The zero-coupon rates we willobtain, therefore, will be a general approximation, suf®cient for ourpurpose.
Pricing
In our de®nition, zero coupons are money market transactions duringwhich no interim payment is made and for which interest accrueslinearly. Regardless of the maturity, all the interests and principal arerepaid in one lump sum at expiry.Let's denote rZC-6M ; rZC-1Y the zero-coupon rates for six months andone year. We know these rates from market quotes because the LIBORare for zero coupons and include maturities up to one year. If wedenote rSW-1Y as the one-year annual swap rate, we know that rSW-1Y rZC-1Y because there is no interim ®xed payment on the swap. Nowconsider a two-year term loan built from an annual swap priced atrSW-2Y , and the ¯ow diagram shown in Figure 4-9. (Note that the variable leg being ignored here, we indicate only the frequency of the ®xedleg in the denomination of swaps.)Assume we borrow $1 from this two-year structure: We receive $1today, and we pay the intermediate interest payment of $1  rSW-2Y byde®nition of the swap and the ®nal payment $1  (1+rSW-2Y). These¯ows are represented in bold lines. At the same time, we lend anamount equal to $1  rSW-2Y = 1  rZC-1Y  for one year, so that the46
The opposite situation in which a trading activity would have a positive workingcapital is naturally different. In that case we would still need to account for the cost ofthe spread.
188
THEORY and TOOLSToda y
Year 1
Year 2
rSW-2Y$1rZC-1Y
$1*rSW-2Y
$1*rSW-2Y /(1+rZC-1Y) $1*rSW-2Y
Figure 4-9
$1*(1+rSW-2Y)
Zero Coupon, Two Years
®nal payment of this ¯ow falls on the same date as the swap interiminterest payment. These ¯ows are represented in dashed lines.The net effect of this structure is given in Figure 4-10.We have eliminated the intermediate ¯ow, and the resulting rate istherefore a two-year zero coupon: The nominal transacted is$1 $1  rSW-2Y = 1  rZC-1Y , and the net interest at expiry is rSW-2Y .Therefore47:rSW 2Y1  rZC-2Y   1  rSW-2Y11  rZC 1YAs we mentioned, we removed the explicit basis for the date calculations.It is worth coming back to this issue for a second. Consider, for example,that today's settlement date is January 15, 1998Ðtherefore, we are twodays before, January 13. The swap is two years and therefore ends onJanuary 15, 2000. The interim payment is due on January 15, 1999. Theone-year zero coupon is 4.5 percent, the swap ®xed rate is 5 percent, andall dates are considered to be actual/360.In the diagram, the rSW-2Y on the left side of the above equationapplies to the period 1998 to 1999, while on the right side it applies tothe period 1999 to 2000. There are 365 days between January 15, 1998,rZC-2Y
$1 - $1 * rSW-2Y /(1 + rZC-1Y)$1 * (1 + rSW- 2Y )
Figure 4-1047
Net Zero Coupon, Two Years
In agreement with what was announced earlier, the term 1  rZC-2Y actually incorporates the time fraction. Alternatively, it could have been written 1  2  rZC-2Y becauseit is a two-year zero coupon.
CHAPTER 4 MONEY MARKETS
189
and January 15, 1999. January 15, 2000, being a Saturday, the ¯ow ispostponed to January 17, 2000Ðthat is, 367 days from January 15,1999.In other words, the above equation should be written as:01365  5%BC732r367  5%ZC2Y360B1C 11@A365  4:5%3603601360which gives rZC-2Y  5:14 percent. The numeric value has little importance, but two points are worth noticing in this example:. This relation rZC-2Y > rSW-2Y is expected considering that with no
interim payment, a zero coupon should pay more at the end.
. Rigorous analysis matters tremendously when manipulating inter-
est rate calculations, speci®cally in an environment in which nonexplicit conventions are frequent.
It follows that:rZC 2Y 
1
1  rSW 2YrSW 2Y1  rZC 1Y
1
To anchor the approach, consider now a three-year swap and similarinterim zero coupon lending to remove the swap ®xed payments, asshown in Figure 4-11. The same conclusion holds as to the structureresulting from our superposition. It is a three-year zero coupon, and itsprice is given by:rSW 3YrSW 3Y1  rZC 3Y   1  rSW 3Y11  rZC 1Y 1  rZC 2Y
$1
rSW-3YrZC-1Y
rZC-2Y
rSW-3Y /(1 + rZC-1Y) + rSW-3YrSW-3Y /(1 + rZC-2Y)
Figure 4-11
rSW-3Y
$1 * (1 + rSW-3Y )
Zero Coupon, Three Years
190
THEORY and TOOLS
And:rZC 3Y 
1
1  rSW 3Y 
rSW 3Y1  rZC 1Y
rSW 3Y1  rZC 2Y
1  rSW nY!n 1XrSW nY
rZC nY 1
i1

1
1
1  rZC iY
If we denote:n 
n 1XrSW nY1  rZC iYi1
the generic formula becomes48:rZC nY 
1  rSW nY1 n
1
Are we done with the computation of a zero coupon term structure?Recall that swaps in the United States are conventionally quoted assemiannual ®xed payments, and our formula has been derived fromannual ®xed swaps.Instead of manipulating algebra, let's introduce a small approximation frequently used. Knowing the price rS of a semiannual swap, theequivalent annual rate rA can be derived from49:48
It is easy from there to compute rZC-nYrZC nY
rSW nY :
rSW-nY 
X 1  rSW nYn
1
n
which shows unequivocally what we suspected: rZC-nY > rSW-nY . Furthermore in general n > m if n > m, which indicates that for n > m; rZC-nY rSW-nY > rZC mYrSW mY : the zero-coupon rates diverge from their swap equivalent. The furtherahead we look, the higher the zero-coupon yield compared to the swap rate ofsame maturity.49Note that the factor 1/ 2 in 1/ 2 rS should have been omitted because it represents thetime basis for the date calculation, and we stated earlier that we would not exteriorizethese fractions. We include it only to emphasize the semiannual to annual transformation. To be extremely rigorous, note also that 1/ 2 is not accurate because throughthe life of a swap there may be six-month periods longer or shorter 1/ 2 a year.Therefore, the formula is a simplifying approximation in many respects.
CHAPTER 4 MONEY MARKETS
191
r 21  S  1  rA2This formula is not rigorously correct because it supposes that theinterim payment is reinvested at the exact same rate rS for all interimsix-month periods. For our purposes, however, it is suf®ciently accurate, and we can use it as a small trick to convert semiannual swaps toan annual schedule and then perform the zero-coupon analysis presented above.The ®nal issue pending is the question of the spread.50 If we denote``ask'' as the rate at which we are likely to borrow, and ``bid'' as the oneat which we are likely to lend, the zero-coupon formula becomes:raskZC nY 
1  raskSW nY1 askn
1
askn 
n 1XraskSW nY1rbidZC iY1
The bid side is obtained by inverting the ask and bid rates in theseformulas.Table 4-7 is an illustration of those results and shows the zerocoupon yield curve for the U.S. dollar. For the purpose of these calculations, today is September 13, 1999. The settlement is September 15,1999. To make things clearer, here are the major steps in the process:. Gray-shaded areas indicate market rates as obtained in real time
from major information providers (Bloomberg, Reuters, Telerate,BRIDGE, and so on). LIBORs are contributed by the BritishBankers Association, whereas swap rates are available from specialized brokers.. Maturity dates are calculated by adding the relevant number ofmonths and years to the settlement date and adjusting for holidays.For example, September 15, 1999, + 4 months  January 15, 2000,which is a Saturday. The ¯ow is postponed to the followingMonday, January 17, 2000.51 All date calculations below one yearare on actual/360 basis. Over one year, swaps are 30/360 and zero
50
The spread here refers to the bid-ask spread on the ®xed leg of the swap. The spreadwe mentioned earlier in reference to swap asymmetry was due to the variable leg ofthe swap. It is being ignored here as we indicated.51For simpli®cation, we ignored banking holidays. If by any chance one of our maturities is a banking holiday, the ¯ow has to be postponed one more day.
192
TABLE 4-7Maturity
Date
Nb Days
USD
USD
Swap Semiannual
LIBID
LIBOR
Bid
Ask
Swap AnnualBid
Ask
Sigma nBid
Zero CouponAsk
Bid
Ask
Sep. 22, 1999
7
5.2313%
5.3563%
5.2313%
5.3563%
1MO
Oct. 15, 1999
30
5.2550%
5.3800%
5.2550%
5.3800%
2MO
Nov. 15, 1999
61
5.3150%
5.4400%
5.3150%
5.4400%
3MO
Dec. 15, 1999
91
5.3850%
5.5100%
5.3850%
5.5100%
4MO
Jan. 17, 2000
124
5.8388%
5.9638%
5.8388%
5.9638%
5MO
Feb. 15, 2000
153
5.8250%
5.9500%
5.8250%
5.9500%
6MO
Mar. 15, 2000
182
5.8163%
5.9413%
5.8163%
5.9513%
7MO
Apr. 17, 2000
215
5.8213%
5.9463%
5.8213%
5.9463%
8MO
May 15, 2000
243
5.8263%
5.9513%
5.8263%
5.9513%
9MO
Jun. 15, 2000
274
5.8325%
5.9575%
5.8325%
5.9575%
10MO
Jul. 17, 2000
306
5.8600%
5.9850%
5.8600%
5.9850%
11MO
Aug. 15, 2000
335
5.8900%
6.0150%
5.8900%
6.0150%
12MO
Sep. 15, 2000
366
5.9225%
6.0475%
5.9225%
6.0475%
2Y
Sep. 17, 2001
733
6.2500%
6.2900%
6.3477%
6.3889%
6.00%
6.04%
6.4674%
6.5165%
3Y
Sep. 16, 2002
1,097
6.4000%
6.4400%
6.5024%
6.5437%
11.91%
12.00%
6.8548%
6.9099%
4Y
Sep. 15, 2003
1,461
6.5100%
6.5500%
6.6160%
6.6573%
17.57%
17.70%
7.2256%
7.2883%
5Y
Sep. 15, 2004
1,827
6.6000%
6.6400%
6.7089%
6.7502%
23.01%
23.18%
7.6114%
7.6834%
6Y
Sep. 15, 2005
2,192
6.6800%
6,7200%
6.7916%
6.8329%
28.18%
28.40%
7.9975%
8.0806%
7Y
Sep. 15, 2006
2,557
6,7400%
6,7800%
6.8536%
6.8949%
33.03%
33.29%
8.3854%
8.4816%
8Y
Sep. 17, 2007
2,924
6,7900%
6.8300%
6.9053%
6.9466%
37.61%
37.92%
8.7934%
8.9053%
9Y
Sep. 15, 2008
3,288
6,8400%
6,8800%
6.9570%
6.9983%
41.92%
42.27%
9.2110%
9.3414%
10Y
Sep. 15, 2009
3,653
6.8800%
6.9100%
6.9983%
7.0294%
45.95%
46.28%
9.6531%
9.7787%
(Source: Compiled by author from