Bloomberg data.)
THEORY and TOOLS
1WK
CHAPTER 4 MONEY MARKETS
193
coupons are on actual/360 basis. For simpli®cation purposes weconsider that 30/360 is equivalent to actual/365 and use the latter.. The semiannual swap rate is converted into an annual rate usingthe approximation introduced above. For example, the six-monthoffered rate of 6.72 percent is considered equivalent to 6.8329 percent annually.. n is computed step by step for each maturity. For example:ask2 
bid3 
raskSW 2Y1  rbidZC 1Y
366  6:3889%365 6:04%366  5:9225%1360
rbidrbidSW 3YSW 3Y1rask1  raskZC 2YZC 1Y733

366  6:5024%365366  6:0475%1360
366  6:5024%365 11:91%733  6:5165%1360
Note that in each of these equations, rSW-nY denotes a differentinterim ¯ow. In 2 this ¯ow occurs on September 15, 2000, andinterest accrues from September 15, 1999Ðthat is, 366 days onactual/365 basis. In 3 , the ®rst rSW-nY occurs on September 15,2000, for 366 days as well, but the second appears on September17, 2001, and interest accrues from September 15, 2000Ðthat is,733 366  367 days. As for rZC-iY , it accrues from the very beginning of the swapÐthat is, today's settlement dateÐto each interim¯ow date.. The rates rZC-nY are computed using n according to the formula
given earlier. For example:733  rask1  raskZC 2YSW 2Y3601 ask2
1
1
733
366  6:3889%3651 6:04%
1
 13:26% ) raskZC 2Y  6:5165%
1,097  rbid1  rbidZC 3YSW 3Y3601 bid3
1
1  1,097 7336:5024%3651 11:91%
 20:89% ) rbidZC 3Y  6:8548%
1
194
THEORY and TOOLS
Here rSW-nY describes the last ¯ow of the swap, from September 15,2000, to September 17, 2001 (733 366  367 days) in rZC-2Y andfrom September 17, 2001, to September 16, 2002 (1,097 733  364days) in rZC-3Y .If we plot the market rates on one side and zero coupons on theother, we get a better picture of the relative yield curves and actual costof money, as shown in Figures 4-12 and 4-13. Note in these charts thatthe zero-coupon yield curve is remarkably linear, which is not the caseof the market curve.
FORWARD RATE AGREEMENTSConsider a situation of relatively low interest rates in which, as acorporation, you know that you will need a signi®cant amount ofmoney in six months. You would like to secure a rate now, but borrowing now is inappropriate because you have no need for the cash. Thesolution is to enter an agreement in which you borrow money at a rate®xed today, but settlement is postponed for six months. This type of
Figure 4-12 Zero-Coupon Curve, USD Chart
CHAPTER 4 MONEY MARKETS
Figure 4-13
195
Zero-Coupon Curve, USD Chart
agreement is called a forward rate agreement (FRA). It is exactly a forward contract on interest rates.Because the deal is actually done today, the negotiated rate has tobe a function of the current term structure. Indeed, consider the diagram in Figure 4-14. Assume we borrow a $1 zero coupon up to thematurity needed, at a rate rM (these ¯ows are represented in boldlines). We lend the same amount $1 up to the settlement desired forthe FRA, in our case six months from now at a rate rS (these ¯ows arerepresented in dashed lines). The net effect of these two operations isa wash: There is no ¯ow today. On the date of the settlement of theFRA, we receive a payment equal to $1  1  rS , including the interest on the dollar lent. At maturity we still have to pay back$1  1  rM .The net operation is exactly the FRA we are looking for. We receive$1  (1 + rS) on settlement and pay back $1  (1 + rM) on maturity. Byde®nition, the rate for this composite operation is the FRA rF :1  rS   1  rF   1  rM 
)
rF 
1  rM1  rS
rrS1 M1  rS
If we introduce the bid-ask spread, things get slightly morecomplex:
196
THEORY and TOOLSrM$1rS
rF$1 * (1 + rs)$1 * (1 + rM)
$1
Figure 4-14
FRA
askask1  rbidS   1  rF   1  rM 
)
raskrbidMSraskF bid1  rS
These equations are extremely useful as we will see shortly.Beyond the algebra, it is interesting to investigate the relationshipbetween the rate for a settlement now or postponed. Consider, forexample, an identical operation settled today, in three and in sixmonths. The different rates are shown in Table 4-8. Clearly the deferredsettlement has an impact on the cost of ®nancing, even more visible inFigure 4-15.The effect visible here is a manifestation of a more general rulebecause forward rates depend essentially on the slope of the yieldcurve. In an upward-sloping environment, which is the case above,FRAs are more expensive than spot rates, and vice versa. If we took anhypothetical downward-sloping curve such as shown in Table 4-9, theFRA prices would be cheaper than the spot rates, as shown in Figure 4-16.A ®nal word about terminology. FRAs are quoted using a convention ``n  m'': n indicates the settlement in months, m the expiry.A 2  6 FRA is a rate for a transaction of four months in two months.Table 4-10 is an example page of quotes from Bloomberg onSeptember 13, 1999, U.S. dollars. Note that the quotes are rankedby maturity: From 1  4 to 9  12, all deposits are 3M; from 1  7to 18  24, they are 6M; and so on, and that the bid-ask spread is 0.04percent systematically.TABLE 4-8Maturity
End Date
Nb. Days Regular Settlement, Mid
1MO
Oct. 15, 1999
30
5.32%
7.06%
5.74%
2MO
Nov. 15, 1999
61
5.38%
6.44%
5.75%
3MO
Dec. 15, 1999
91
5.45%
6.22%
5.76%
4MO
Jan. 17, 2000
124
5.90%
6.12%
5.81%
5MO
Feb. 15, 2000
153
5.89%
6.07%
5.87%
6MO
Mar. 15, 2000
182
5.88%
6.03%
5.91%
Note: For simplicity, the rates indicated are midmarket.
Settlement in 3M, Mid
Settlement in 6M, Mid
CHAPTER 4 MONEY MARKETS
197
Comparison Regular vs. Forward Rate(upward-sloping yield curve)
Figure 4-15
FRA versus Spot
SWAP PRICINGVariable Flows
The formula for forward rates will lead us to another interesting result,extremely useful to derive a general swap formula.52 Consider a variable payment at some point in the future life of a swap. By de®nition,this ¯ow is unknown because the rate has not been ®xed as of today. ItTABLE 4-9Maturity
End Date
1MO
Oct. 15, 1999
30
5.99%
4.09%
5.25%
2MO
Nov. 15, 1999
61
5.93%
4.70%
5.24%
3MO
Dec. 15, 1999
91
5.86%
4.92%
5.23%
4MO
Jan. 17, 2000
124
5.40%
5.02%
5.17%
5MO
Feb. 15, 2000
153
5.42%
5.07%
5.12%
6MO
Mar. 15, 2000
182
5.42%
5.11%
5.07%
52
Nb. Days Regular Settlement, Mid
Settlement in 3M, Mid
Settlement in 6M, Mid
We ignored the variable leg in the calculation of zero coupons, because we assumedswaps gave us access to money market rates. From now on we are more focused onswaps as stand-alone instruments, and we need to reinstate the variable leg in ouranalysis to price it accurately.
198
THEORY and TOOLSComparison Regular vs. Forward Rate(downward-sloping yield curve)
Figure 4-16 FRA versus Spot
is possible, however, to compute a forward rate based on today's termstructure and to use it as an estimate of the unknown rate.Figure 4-17 shows a variable payment on a notional of $1 at rF ,today's forward rate, used as a proxy for the unknown variable rate.The term rM stands for the zero coupon from today to the paymentdate of the ¯ow. The present value PV1 of this variable ¯ow is:PV1  rF = 1  rM . Now consider the situation in Figure 4-18, inTABLE 4-10Maturity
Bid-Ask
Maturity
Bid-Ask
1425364758697  108  119  121728
5.88/925.86/905.88/925.70/745.80/845.85/895.92/965.99/036.08/125.89/935.91/95
394  105  116  1212  1818  241  102  113  126  1812  24
5.93/975.86/905.91/955.99/036.40/446.55/595.89/935.95/996.02/066.30/346.58/62
(Source:
Compiled by author from Bloomberg data.)
CHAPTER 4 MONEY MARKETS
199
rMrFVariable Flow$1 * rF
Figure 4-17
Swap Variable
which two equal ¯ows of $1 are set at the beginning and end of theperiod on which the same variable ¯ow accrues.The present value PV2 of these ¯ows is PV2  $1= 1  rS $1= 1  rM , where rS stands for the zero coupon up to the beginningof the period. Considering the de®nition of rF , it follows that:rFrM rS1  rM 1  rS 1  rS   1  rM 1  rM1  rS   1  rM 11 PV21  rS 1  rM
PV1 
This result can be stated in the following manner: An unknown variablerate payment can be estimated as of today by a set of two ¯ows of notional ofopposite direction respectively at the beginning and end of the variable accrualperiod.Consider now two successive variable payments, as shown inFigure 4-19.And operate the above transformation, as shown in Figure 4-20.The $1 interim ¯ows cancel out, and we are left with only the ®rst andlast of these ¯ows to consider. In other words, whatever the variableschedule is, we can accurately price it if we include two ¯ows ofnotional of opposite directions at the beginning and at the end of thesequence.
Pricing
From this point a generic pricing formula will be straightforward. Aswap is a succession of ¯ows, some of them ®xed, some of them variable. Its value today is therefore the net present value of all these ¯ows.rMrS
rF$1$1
Figure 4-18
Swap Variable Nominal
200
THEORY and TOOLSrF2
rF1
rF2
rF1
Figure 4-19
Swap Variable Nominal
Consider as an illustration the two-year annual swap with a sixmonth variable schedule on a $1 notional, as shown in Figure 4-21.Let rvi stand for theÐunknownÐvariable rates, r1 and r2 for thezero coupon up to the payment dates of ®xed ¯ows, and rSW for the®xed rate of the swap.53 Next we operate the transformation of thevariable leg, and we get rid of interim variable ¯ows, as shown inFigure 4-22. We are left with four ¯ows in total, two annual resultingfrom the ®xed leg, two at each end resulting from the collapse of thevariable leg. The value V of this swap is therefore:   rSWrSW1rSWrSW1 V1  r21  r1 1  r21  r11  r2r21  r2The ®rst two terms in parentheses is the valuation of ®xed payments,and the third represents the two notional ¯ows resulting from theaggregation of the variable schedule.From there the generic formula for a swap of notional N, ®xedannual rate rSW , and duration n years is one step away:"#nX1rnV  N  rSW :1  ri 1  rni1where ri stands for the zero-coupon rates at each ®xed payment date.rF2
rF1
$1
Figure 4-2053
$1
$1 - $1
Swap Variable Nominal
We implicitly assume here that r1 and r2 are midmarket rates; therefore, we get amidmarket swap rate. It is relatively straightforward to price independently the bidand ask sides of the swap, so we don't detail all calculations and focus on midmarketfor simplicity.
CHAPTER 4 MONEY MARKETS1Y
201
1Yrv3
rv1
rv2
r1
rSW
rv4
rSW
r2
Figure 4-21
Swap Pricing
Several remarks can be made regarding this formula:. Semiannual swaps are slightly more complex, but the principle
remains the same. Indeed, the formula remains the same if westipulate that ri stands for the zero coupon up to the ith ®xedpayment, whenever that payment occurs. There is no need to consider even regular payments on either the ®xed or variable side.The formula above applies to any type of schedule, regardless of itsfrequency or regularity, as long as zero coupons and date calculations are handled properly.54. The term 1/(1 + ri ) is trickier than it looks: Because of our choice tosuppress time fractions, we did not care about date calculations.The numerator 1 is not equal to the numeral 1; it is in fact a succession of ni =360 where ni stands for the number of days between twosuccessive ®xed payments. By the same token, the denominator 1 ri is in fact 1  di =360  ri , where di there stands for the number ofdays between today and the successive ®xed payment dates.. This result can be used in a different way: Given a schedule of ®xedand variable payments, we can price the structure and extract avalue V. Conversely, if we force V to be zero, we get the fair value of1Y
1Y
$1
r1
$1+rSW
rSWr2
Figure 4-2254
Swap Pricing
The same exact formula can be used for a forward swapÐthat is, a swap negotiatedtoday but whose accrual period starts at some point in the future.
202
THEORY and TOOLS
this structureÐthat is, the swap rate that best re¯ects current market conditions:nX1rn1:where n rSW n 1  rn1rni1At this point we know how to extract zero coupons from swapsand conversely price standard and nonstandard swaps from zero coupons. As a consistency check, it is always possible to reprice a swapquoted in the market with the prevalent zero coupons, which we mightdo just to verify that the price we compute is in line with the market. Ifit is not, it may very well be there is an arbitrage opportunity, althougha calculation error is much more likely because in general, swap andcash markets are very closely integrated.
SHORT-TERM INTEREST RATE FUTURESCONTRACTSDe®nitions
A short-term interest rate futures contractÐcommonly labeled a eurofutureÐis a listed forward rate agreement.55 By trading a contract, eachparty enters a forward money market transaction at a rate set by theprice of the futures contract. We know from the previous chapter thede®ning characteristics of a listed futures contract:. As ®nancial futures, eurofutures are cash settled. No security is
delivered on expiry.
. As listed products, they are standardized with respect to the terms
of the underlying money market operation. Most eurofutures arethree-month FRAs, but some are one-month or one-year.56 A threemonth FRA is always 90 days in duration, regardless of the actualnumber of calendar days in a full three-month period. The timefraction is therefore always 90=360  1/ 4. The notional amount of
55
The market terminology with regard to individual contracts depends on the underlying currency. Eurodollar contracts are futures on the U.S. dollar, whereas eurosterling and euroyen are respectively contracts on the British pound and Japanese yen.The exception is the euro, for which the contract has been named euribor.56Eurodollar contracts are mostly three-month FRAs, but a one-month contract is alsolisted on the Chicago Mercantile Exchange. Similarly for the euroyen in Tokyo, themost liquid futures contract is a three-month FRA, but a one-year FRA is also listed.
CHAPTER 4 MONEY MARKETS
203
the FRA is ®xed for each contractÐfor example, $1,000,000 for eurodollars, ¥100,000,000 for euroyen, and £500,000 for eurosterling.. Eurofutures are quoted as 100 minus the rate of the FRA. Forexample, if the FRA is 5 percent, the price of the futures contractis 100 5  95.. The tick value is usually 0.005, but 0.01 and 0.0025 also exist.. The expiry cycle is usually quarterlyÐMarch, June, September, andDecember. Most exchanges list in addition interim maturitiesÐforexample, monthly contractsÐuntil the second quarterly expiry.57. As listed products, they are subject to margin requirements andmargin calls.. The ®nal settlement, often referred to as the SQ by analogy withindex futures, is often the current LIBOR.58. Eurofutures in general are wildly and aggressively cross listed,which was not the case for index futures, for example. In particular,the most liquid contracts, such as eurodollar, euroyen, and euribor,are listed on multiple exchanges in the United States, Europe, andAsia.The only characteristic we have not included is the point value, whichis here a consequence of the de®nition of the contracts, not an arbitrary®gure. In order to illustrate this point, consider the diagram in Figure4-23, which represents the FRA underlying a eurofuture. By de®nition,the forward cash transaction starts upon expiry of the futures contractand accrues typically for three months. Its price is rF , which is 100minus the price at which the future has been traded. If the FRA transaction were actually taking place, two ¯ows equal to the notional of theUnderlying FRA, rFFutures Expiry
N * (1 + rF)Toda y
N
Figure 4-2357
Eurofuture
If we are today in January 2000, the ®rst two quarterly futures are March and June,but January, February, April, and May are also listed. When March expires, two moremonthly contracts are listedÐJuly and AugustÐto complete the monthly cycle untilthe second quarterly expiry again. Note that the interim contracts are also threemonth FRAs.58Therefore, eurofutures, like swaps, are not symmetrical as well with respect to LIBIDLIBOR spreads.
204
THEORY and TOOLS
contract, N, would be exchanged at the beginning and at the end of theFRA. However, because the futures is cash settled at expiry, thenotional is never exchanged, and instead rF is compared to theLIBOR on expiry, and the difference is settled in cash.Back to the point value: Consider a euroyen contract quoted at99.70. If the tick size is 0.01, each move in price equal to the tick sizemodi®es the FRA rate by 0.01 percent. The FRA is a three-monthtransaction on ¥100,000,000 at 100 99:70  0:30 percent by de®nition,thus a move of 0.01 percent results in a change in value equal to¥100,000,000  0.01 percent  90=360  ¥2,500. Each move of 0.01 inthe quoted price of the futures contract results in a variation of ¥2,500in the value of the underlying position. If upon expiry the three-monthLIBOR is 0.35 percent, the difference 0.35 percent ± 0.30 percent = 5ticks results in a total variation of 5  ¥2,500  ¥12,500 in the value ofthe position. ¥12,500 are therefore exchanged between the two transacting parties, and there is no additional ¯ow afterwards. As a result ofthis de®nition, a buyer of the future is exposed on interest rates goingup because a rise in the rates implies a drop in the future price.Conversely, sellers bene®t from an increase in the short-term rates.
Fed Fund Futures
Fed fund futures are an interesting variant of eurofutures. Their periodicity is monthly, and they are settled on the last day of eachmonth on the simple arithmetic average of the effective fed fundrate for that month. For example, on the ®fteenth of a givenmonth, because all fed fund rates are already known from the beginning of the month, 50 percent of the ®nal settlement rate of thefutures contract is also known. Obviously the futures contract losesits liquidity toward the end of each month because the averagingmakes it very unlikely that a move just before expiry will signi®cantly change the settlement value.
Markets
Despite the seemingly complex mechanics, eurofutures are extremelysimple and powerful instruments. They have been designed with thesame inverse relationship with respect to interest rates that bonds have,but they are much simpler to trade, and indeed they have become overthe years the most liquid futures contracts across all categories. Figure4-24, for example, shows the open interest of the ®ve biggest currencymarkets from 1994 to 1999.
CHAPTER 4 MONEY MARKETS
205
Figure 4-24 Eurofuture Open Interest(Source: Compiled by author from Datastream data.)Note: Data unavailable for EURIBOR prior to January 1999. Data include all majorexchanges for a given contract. For euroyen, the LIBOR and TIBOR contracts have beenaggregated to re¯ect the overall amount of activity on Japanese interest rates.
As one would expect, eurodollars are clearly ahead in terms ofmarket activity, with euroyen and euribor contracts hovering muchlower. Note the spikes visible for each quarterly expiry, a phenomenonthat was present already with index futures and to which we will comeback later.
Cross Listings
As we mentioned earlier, cross listing is widespread with eurofutures.Considering the typical turnover in all these contracts, competition is®erce to attract volume. Table 4-11 presents an overview of the crosslisting network.Interestingly, despite identical characteristics, different exchangeshave adopted signi®cantly different margin requirements. Often thesediscrepancies have resulted in liquidity ¯owing to the cheapest market.5959
With respect to euroyen contracts, the LIFFE and TIFFE have implemented a clearingagreement in which all trades on the LIFFE are automatically cleared on the TIFFE.Therefore, historical data for the LIFFE euroyen show a trading volume and openinterest at zero because both are automatically aggregated in the statistics reported bythe TIFFE.
206
TABLE 4-11
Fed fund
CME
$3,000,000
0.005
Monthly
$25
Simple average
Last day of month
$135
Fed fund
CBOT
$5,000,000
0.005
Monthly
$41.66b
Simple average
Last day of month
$270
CME
$1,000,000
0.0025 (CME)
Quarterly
$25
3M LIBOR
Third Wednesday
$675
Quarterly
¥2,500
3M TIBOR
Third Wednesday
SIMEX
Tick
Cycle
Special Quotation
Final Settlement
Initial Margin
0.005 (SIMEX)¥100,000,000
0.005
¥16,200
TIBOR
TIFFE
Euroyen
SIMEX
(CME)
LIFFE
¥23,625
CME
(TIFFE)¥18,900(SIMEX)¥100,000,000
0.005 (LIFFE,
Quarterly
¥2,500
3M LIBOR
Third Wednesday
¥37,800
LIBOR
TIFFE
Euroyen
SIMEX
CME, SIMEX)
(CME)
LIFFE
0.01 (TIFFE)
¥80,000(TIFFE)
CME
¥10,800(SIMEX)Eurosterling
TIFFE
EURIBOR
EUREX
£500,0001,000,000
0.005
Quarterly
£12.5
0.002 (MATIF)
Quarterly
25
3M LIBOR
Third Wednesday
3M EURIBOR
Third Wednesday
£350500
MATIF
0.005 (EUREX,
(MATIF,
LIFFE
LIFFE)
LIFFE)750(EUREX)
Euro LIBOR
LIFFE
1,000,000
0.005
Quarterly
25
3M LIBOR
Third Wednesday
500
Note: Chicago Mercantile Exchange; CBOT, Chicago Board of Trade; SIMEX, Singapore International Monetary Exchange; LIFFE London International Financial Futures Exchange; TIFFE, TokyoInternational Financial Futures Exchange; MATIF, MarcheÂ aÁ Terme International de France; and EUREX, European Exchange.For a move of 0.01 point in the quoted price.bThe notional FRA is always 30 days long; therefore, the point value is $5,000,000  30/360  0.01 percent  $41.66.a
(Source: Compiled by author from Bloomberg data.)
THEORY and TOOLS
Listed
Eurodollar
Notional
Point Valuea
Contract
CHAPTER 4 MONEY MARKETS
207
As we did earlier with the Nikkei 225 futures contract, it is quiteinstructive to look at the market share of SIMEX versus TIFFE withrespect to euroyen open interest, and at the difference in price betweenthe two markets,60 as shown in Figures 4-25 and 4-26. The SIMEXseems to have captured over the years a reasonable market share,around 25 percent similar to that of the Nikkei contract. At the sametime, both euroyen contracts appear to be trading relatively closely,although differences are clearly apparent from time to time. Naturallyit remains to be seen if these differences will create pro®table arbitrageopportunities.
Yield Curves and Eurofutures
Besides swaps, eurofutures are the second-best source of informationto extract a zero-coupon yield curve. Their liquidity makes them
Figure 4-25 TIBOR Market Share(Source: Compiled by author from Datastream data.)60
Eurod