
. Sell Q units of S @ FSQ .
71
Rigorously speaking, rZC is the relevant rate from today's stock settlement date to thestock settlement date on the futures expiry.72Note that rd is from the payment date, not the ex-date, to expiry. In Japan, forexample, dividends are typically paid three to four months after ex; therefore, thepayment date may very well be after expiry.
CHAPTER 4 MONEY MARKETS. Pay equity performance P  Q 
FSQ
223
FH 
. Receive accumulated capitalized dividends Q  d.*73
The resulting cash ¯ow is:C1 
N  1  rZC   Q  FSQ
Q  FSQ
FH   Q  d
N  1  rZC   Q  FH  Q  d
For the seller and the buyer to be willing to enter the contract, weneed to have C1  074 :FH  S  1  rZC 
d
This simple formula is consistent with our expectations: FH is proportional to S, ascending with rZC and descending with d:75Each term of this expression needs to be examined carefully. Thespot price S is the price at which the underlying is trading today. If S isa single stock and if we price during market hours, it is necessary toaccount for the market spread in calculating S. Selling the futurerequires buying the underlying security, which has to be done at themarket offered price. rZC is a zero-coupon rate, but the direction of thetrade dictates if we need to include a bid or ask price. The seller of thefuture is borrowing cash to cover the purchase of the underlying; therefore, an offered price here is necessary again. d* represents an amount73
The symbol * is often used to denote the capitalization of a series of ¯ows until acertain date. In the case of dividends, we do not know exactly when they are paid, butpresumably they are spread out all across the life of the future. d represents all thedividends that have been paid between t0 and t1 for one unit of underlying and forwhich we have received interest by lending the cash until t1 . For example, if there is asingle dividend d, then d  d  1  rd . Rigorously speaking, we should enter todayinto an FRA between the dividend payment date and the expiry of the future to lockrd . If we do not, the actual rate of ®nancing of dividends between payment date andexpiry may be different from rd , resulting in a pro®t or a loss.74What we did is essentially construct a replicating portfolio for the futures contractand apply absolute convergence reasoning. The price at which both parties are indifferent between the futures contract and its replicating portfolio is the fair price of thetransaction.75The formula we obtain for FH is derived from a cost-of-carry model. In the frameworkwe consider this approach to be absolutely satisfactory because in a real arbitragesituation we would actually replicate the underlying portfolio and carry it. There are,however, alternative models derived from speci®c hypotheses, notably on interestrates. See, for example, Eduardo Schwartz, ``The Stochastic Behavior of CommodityPrices: Implications for Valuation and Hedging,'' Journal of Finance, 52, 1997, pp. 923973 or Michael Hemler and Francis Longstaff, ``General Equilibrium Stock IndexFutures Prices: Theoretical and Empirical Evidence,'' Journal of Financial andQuantitative Analysis, 26, 1991, pp. 287±308.
224
THEORY and TOOLS
of accumulated dividends. In contrast to S and rZC , which are knownfairly precisely at the time of pricing, d* is not known for sure, onlyestimated. If after the trade the amount of dividends paid is not equalto what was expected, the difference is a pro®t or a loss to each party.Speci®cally, if dividends increase, the seller of the futures contract,being also the holder of the underlying, receives more cash thanexpected and realizes a pro®t. In contrast, the buyer does not realizea loss directly but paid too high a price for the futures contract in the®rst place.As a numerical illustration, consider the S&P 500 index at $1,400.Assume a three-month expiry, rZC = 6 percent, and d  $4.5. FH hasthe following value:6%  90$4:5  $1,416:5FH  $1,400  1 360The difference $1,416:5 $1,400  $16.5 is referred to as the theoreticalbasis of the futures contract, or as its fair value. It indicates the amountby which the future price differs from the price of its underlying.Because of the ®nancing bracket (1  rZC ) in FH , the fair value is generally positive unless the amount of dividends is such that d > S  rZC .Note also that because d* is usually fairly constant, the term (1  rZC )decreases with time as we get closer to expiry.We will revisit many of these issues when dealing with indexarbitrage, but it is useful to keep in mind that a future position createsa side exposure to interest rates and dividends.
ADR Futures
An ADR futures contract is a bicurrency futures contract in which theperformance is paid in a currency different from that of the underlying.76 Speci®cally, the payoff formula is modi®ed and becomes:¥=$
P  Q  f1
 FSQ
¥=$
f0
 FH 
¥=$
where f0 and f1 ¥/$ represent the exchange rate between two genericcurrencies yen and dollars, respectively, on t0 and t1 . The operatingmode to hedge this type of futures is quite similar to that of the previous case:76
In reality, there is no listed ADR or quanto futures contract. These particular structures are almost always traded OTC, which means that they are forwards rather thanfutures.
CHAPTER 4 MONEY MARKETS
225
On t0 :. Borrow an amount N ¥ from t0 to t1 @ rZC ¥.. Change N ¥ into N $ @ f0¥ =$ .
. Buy Q unit of S @ S0, for a total nominal N $ .. Sell one future @ FH .
The resulting cash ¯ows are:C¥0  N ¥
N¥  0
C$0  N $
QS0
On t1 :. Pay back loan N ¥ 
1  r¥ZC ).. Sell Q units of S @ FSQ.. Change Q  FSQ in yen to Q  f1¥ =$  FSQ .¥=$. Pay equity performance P  Q  f ¥=$  FSQf0  FH :1. Receive accumulated capitalized dividends in dollars and changethem to Q  f1¥ =$  d .The resulting cash ¯ows are:C¥1 
N ¥  1  r¥ZC   Q  f1¥ =$  FSQ
 Q  f1¥ =$  d C$1  Q  FSQ
Q  f1¥ =$  FSQ
f0¥ =$  FH 
N ¥  1  r¥ZC   Q  f0¥ =$  FH  Q  f1¥ =$  d
Q  FSQ  Q  d
Q  d  0
because every ¯ow cancels out on the dollar side.Again both parties are indifferent if C¥1  0. Therefore:!¥=$f1¥ dFH  S  1  rZC¥=$f0The ®nancing cost has been transferred to the second currency, and thedividend payment is modi®ed to account for the change in exchangerates between initiation and expiry. This last point, however, poses aproblem because f1¥ =$ is not known on t0 . We can easily solve this¥=$dif®culty by exchanging d today at the forward exchange rate fd , asshown in Figure 4-40.The foreign exchange risk on d¥ disappears and C¥1 then becomes:C¥1 It follows that:
N ¥  1  r¥ZC   Q  f0¥ =$  FH  Q  d¥
226
THEORY and TOOLSt0, S
t1, FSQ
P
rZC
d¥ = fd¥/$ * d
rd¥
Figure 4-40 ADR Futures Contract
FH  S  1  r¥ZC
d¥
!
¥=$
f0
in which all parameters are known on t0 .The ADR futures contract keeps the same risks as its standardcounterpart in terms of interest rate and dividends. There are twodifferences, however. First, the relevant interest rate is that of the currency in which the payoff is paid, not that of the underlying. Second,there is an additional risk on the exchange rate on the dividend side. Ifthe amount of actual dividends is different from expectations, the differenceÐalready a pro®t or lossÐis in addition exposed to the difference in exchange rate, from the initial forward rate fd¥ =$ to the spot rateon payment date.
Quanto Futures
A quanto futures contract is bicurrency for which the payoff is lockedwith respect to the exchange rate77:¥=$
P  f0
 Q  FSQ
FH 
It is relatively easy to build again a ¯ow diagram based on this payoff,and the fair value of the futures contract has to be equal to the following:!!¥=$¥=$f1f0d¥¥ FSQFH  S  1  rZC¥=$¥=$f0f0This formula, however, is not as simple as those we obtainedbefore. In particular, the middle term is problematic because it dependson the exchange rate as well as FSQ . Although f0¥ =$ is known at t0 ,neither f1 ¥/$ nor FSQ are. We could potentially use the forward rate77
The alternative formula P  f1¥ =$  Q  FSQ FH  is of no particular interest because itis equivalent to a standard futures contract for which the equity payoff is exchangedon expiry at the spot exchange rate.
CHAPTER 4 MONEY MARKETS
227
again to get rid of the uncertainty about f1 ¥/$, except that this time weneed to hedge FSQ , which is not known today. In short, a quantofutures contract cannot be priced easily with a ¯ow diagram, and weneed more sophisticated tools.We will not develop the pricing further, because we want to focuson relatively simple products, and quanto futures are indeed not suchsimple products.78 This exercise is essentially useful as a demonstrationof the limitations of the pricing procedure we have employed so far.
CONCLUSIONAs it happens, the money market is a massive component of modern®nancial markets, and therefore a signi®cant part of our attentionneeded to be devoted to the different instruments and practices. Thismay seem to be a contradiction to the original stated focus on equitymarkets, but in fact it is not.Except for extremely specialized strategies like mutual fundsÐbyand large limited to one single class of assetsÐevery trader needs tohave a clear and comprehensive knowledge of a multitude of productsbeyond his or her natural domain of expertise because rigorous riskmanagement demands it. This is all the more true for a professionalarbitrageur who may have a very broad mandate to focus loosely onequities and needs to be able to hedge other risks away.For example, even a simple product like an index futures contractincludes a signi®cant risk to interest rates and dividends.Understanding these risks allows a trader to manage them activelyÐfor example, in hedging them at a fair price. Out of all the risks likely toaffect the trading environment, many are directly related to the moneymarket, be it interest rate or foreign exchange, short term or long term.Therefore, the knowledge we built in this long and arguably painfulexamination is a prerequisite for all derivative traders alike.
78
In fact, quanto futures can be priced very easily with a Black-Scholes model includinga second currency.
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5A GGREGATE I NDICATORS:S TOCK I NDICES
INTRODUCTIONEquity indices have been around for as long as equities have beenaround. They are designed to serve the purpose of aggregating information in a legible and meaningful way. There is a multitude of different indices nowadays, broad based as well as industry speci®c, andsome have become milestones in the modern ®nancial landscapeÐtheDow Jones Industrial Average or the Nasdaq Composite, for example.Indices were originally developed to capture a relative appreciation, and in practice an index is really nothing more than a weightedarithmetic average of the prices of its components. As we will see, thereare mainly two ways to consider the question of the weightings, but forall practical purposes computing an index is a very simple exercise.Dif®culties arise only from corporate events affecting component companies. On one side, index sponsors have to monitor carefully thegeneral market trends to identify what the best candidates for inclusionare. An illustration of this point was made very clear when Dow Jonesdecided to include Nasdaq stocks in its industrial average for the ®rsttime in mid-1999. The move was long overdue for some, but in anycase it re¯ected the reality of today's marketsÐtechnology stocks arehere to stay. On the other hand, individual corporate events must betracked in a meaningful and timely manner if the index is to remainrepresentative of the activity taking place in its portfolio base.From our perspective, indices represent important instrumentsbecause a signi®cant fraction of the world's most liquid equity ®nancialproducts is index related, be it futures or options. This concentration is229
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230
THEORY and TOOLS
largely explained by fundamental corporate ®nance theory, notably thecapital asset pricing model (CAPM), but also by more down-to-earthconsiderations. The CAPM states that investors are not rewarded forunsystematic riskÐthat is, risk that can be diversi®ed away with alarger portfolio. The ultimate diversi®ed portfolio is the market, andbroad-based indices in general capture a sizable fraction of the overallcapitalization, making them ideal vehicles for investors trying to diversify their assets. On the other hand, because index futures are extremely simple and cheap to trade, many investors look no further to takespeculative positions. Even emerging markets have fairly liquid indexfutures; therefore, investors unable to ®nd liquidity on individualissues always have the option to use these.
DEFINITIONSRigorously speaking, an index is a measure of relative value. It captures the appreciation of a reference set of securities from a base period:Pw PI  P i i Oii wi  Piwhere wi stands for the individual weight of each component today, Pifor its price, and POi for its original base price. This de®nition impliesthat I is a dimensionless ®gure, being the ratio of two capitalizations.There may be variations of this simple formulationÐfor example,the Laspeyres or PaaÈsche formulas, each one named after the economist that ®rst proposed it in the mid-nineteenth century. Differencesreside notablyP in the weights used in the denominator. The aboveconvention, i wi  POi , means that I captures a change in overall capitalization due exclusively to a change in price because the same currentweights arePused in both numerator and denominator. An alternativeOOwould bei wi  Pi , which means that I now re¯ects a change inoverall capitalization, due either to wi or Pi .In any case the above formula is not extremely practical and needsto be re®ned slightly because nobody wants to keep track of base pricesof 50 years ago, for example. We transform the numerator into a moregeneric quantity, which we call the divisor of the index, and which wedenote as D:Pw PI i i iDThe divisor plays a fundamental role in the life and continuity of theindex, as we will see shortly. Depending on the exact de®nition and
CHAPTER 5 AGGREGATE INDICATORS: STOCK INDICES
231
properties of wi , the index can be capitalization weighted, priceweighted, or total return.
Capitalization-Weighted Indices
In a capitalization-weighted index, wi is the outstanding number ofshares. We denote this weight Wi , and the index becomes:PW PI i i iDThe reason that the index is called ``capitalization-weighted'' is relatedto the variation in I with respect to a variation in one of the prices, Pi .Speci®cally the de®nition of I implies that:IWW I iP iPiDi Wi  Piif Pi denotes a variation in the market price of the ith component.This result can be written as:IWi  PiPiPIWPPiiiiIn other words, the relative variation of I following a move of one of itscomponents is equal to the relative variation of that component timesits relative capitalization in the index.
Price-Weighted Indices
In a price-weighted index, wi is a ®xed weight, usually identical for allcomponents. The characterization ``price weighted'' comes from thedependence of I on each component's price. The above formula expressing a relative variation in I is still arithmetically valid:Iwi  PiPi PIPii wi  Pibut all wi are equal; therefore, this expression simpli®es to:IPiPi PIPii Piwhich shows that I=I is proportional to Pi =Pi again, but the ratio isnow the relative price of the ith component in the index.In practice, there are only a few price-weighted indices in theworld, and only two we want to considerÐthe Dow Jones IndustrialAverage and the Nikkei 225. The weight of Dow's constituents are 1,and its divisor is 0.20145 as of April 2000. For the Nikkei 225 the
232
THEORY and TOOLS
situation is slightly different: 222 components have a weight equal to1,000, 2 have a weight of 100, 1 has a weight of 1, and the divisor is10,179.8. The reason behind the difference in weight is the disproportionate price of some stocks. Nippon Telephone & Telegraph (NTT), forexample, has a price around ¥1,600,000, whereas most other stocks arebelow ¥10,000, which essentially means that without any adjustment,NTT would drive the index entirely.Even if the Dow Jones is considered a worldwide reference, the factthat until recently no derivative was listed on it is clearly indicative ofthe relative lack of interest price-weighted indices have received ingeneral. The calculation method accounts for a large part of this relative indifference because small stocks have a disproportionate impact.The Nikkei 225 is the only exception, and even if the Tokyo StockExchange also lists two capitalization-weighted indicesÐthe TokyoPrice Index (TOPIX) and Nikkei 300Ðthe Nikkei 225 remains the ¯agship market aggregate.1
Total Return Indices: Dividend Reinvestments
A total return index lists stocks for which dividends are systematicallyreinvested. Because the weight of a stock going ex is affected by thereinvestment, total return indices are necessarily capitalizationweighted.Consider a stock with a weight W and a price P that is going exdividend today by an amount d. If Y and T respectively stand foryesterday and today, the ex-dividend effect implies:PT  PY
d
The ``fair'' price of the stock today is its price yesterday adjusted for thedividend.2 For the capitalization of the stock to re¯ect the decrease inprice, we need to increase the weight W by an equivalent amount,which we denote   W. If we consider this increase to occur as ofyesterday's close, the following relationship holds:1
The reason for this is probably the size of the TOPIX, more than 1,600 stocks. When theNikkei 300 was listed, it should have corrected this situation; however, its listingoccurred in the middle of the Japanese ®nancial and economic crisis, which mayexplain the lack of interest it received.2Naturally, this relationship is valid only before the opening, the difference between theactual opening price and PY d being unpredictable. Note that d here is a grossamount and that the tax status of investors transacting the stock at open will effectively set the value of the opening discount. Cf. the discussion on dividend arbitrage inthe ®rst part.
CHAPTER 5 AGGREGATE INDICATORS: STOCK INDICES
233
PY    W  W  1    dThis equation states that the incremental capitalization captured by  W as of yesterday is equal to the total amount of dividend receivedtoday on W  1   shares.  is then simple to extract:
dPY
d
Therefore, for a stock going ex, the weight in a total return index isincreased to W 0 , where:PYW0  W PY dIn that case, although the index is still capitalization weighted, eachweight is not equal to the outstanding number of shares anymore butadjusted every ex date. The DAX 30 is the only major total return indexin our list.3
Dividend Yields
Let's extend the above de®nitions and denote:Pw ddI  i i iDwhere di stands for all the individual dividends paid by the indexconstituents over an identical period of time, typically a year.Regardless of whether the index is price or capitalization weighted,dI represents the total amount of dividends paid by the underlyinguniverse of stocks expressed in index points. The dividend yield yI isthen simply de®ned as:yI 
dII
dI and yI are particularly important with respect to the fair value ofindex derivative products, especially futures. When applied to anindex, the futures pricing formula becomes:FH  I  1  rZC 3
dI
The fact that all dividends are reinvested has a signi®cant effect on the DAX futurescalculation. Recall the pricing formulas for a standard futures contract, which explicitly included capitalized dividend payments on expiry. In the case of a total returnindex, these payments do not occur anymore because they are reinvested before;therefore, the fair value of the future is only the ®nancing cost until expiry.
234
THEORY and TOOLS
where rZC is the relevant zero-coupon rate to expiry. dI is here anextension of d to account for the capitalization of all dividends toexpiry:Pw  ddI  i i iDAlternatively, FH can be expressed as:FH  I  1  rZC
yI 
where yI is also an extension of yI to account for the ®nancing ondividend payments.The choice of one particular formula for FH depends on the repartition of dividends through time. If dividends are relatively homogeneously distributed until expiry, then the use of yI is acceptable andindeed simpli®es practical calculations. On the other hand, if dividendstend to cluster around particular periods, using yI leads to a possiblemispricing in FH .4
MAJOR INDICESIntroduction
Table 5-1 gives an overview of the major stock indices in the UnitedStates, Europe, and Japan. Interestingly, many indices appeared duringthe eighties and therefore are quite young by many standards.
Concentration
Indices are usually constructed in order of descending capitalizationand ``chained'' to cover different segments of the stock universe. Forexample, in Germany, the DAX 100 is constituted from the stocks of theDAX 30Ðthe 30 biggest capitalizationsÐand those of the MDAXÐthenext 70. (See Figures 5-1 through 5-5, for diagrams of the German,French, U.K., Japanese, and U.S. indices.)4
As a general rule, yI is used relatively rarely on equities, except for back-of-the-envelope approximations. In Japan, for example, dividends are highly concentrated aroundthe end of the ®scal yearÐMarch 31Ðor interim periodÐSeptember 30 for a largenumber of corporations. In the United States, although dividends are relatively homogeneous, most traders still use dI . In the United Kingdom dividends usually go ex onMondays, but their amount is not steadily distributed; therefore yI is not really applicable either. In France, dividends are also concentrated, around the second quarter ofeach calendar year.
TABLE 5-1Sponsor
FTSE 100FTSE 250FTSE 350FTSE All ShareFTSE Small CapDAXDAX Mid-CapDAX 100Dow IndustrialS&P 100S&P 500S&P 400S&P 600S&P CompositeRussell 3,0001
LSELSELSELSELSEFSEFSEFSENYSENYSENYSENYSENYSENYSENYSE
Nasdaq 100Nasdaq CompositeNikkei 50Nikkei 225Nikkei 300TOPIXTSE2CAC
NasdaqNasdaqTSETSETSETSETSESBF
Financial TimesFinancial TimesFinancial TimesFinancial TimesFinancial TimesFSEFSEFSEDow JonesS&PS&PS&PS&PS&PF. RussellCompanyNasdaqNasdaqNikkeiTSETSETSETSESBF
Number ofComponentsb
Value onDec, 31, 1999
10-YearCAGRc
Methodd
Institutede
Base Value
1002503507974443070100301005004006001,5003,000
£6,930.20£6,444.90£3,327.00£3,242.06£3,097.806,958.144,103.823,293.54$11,497.10$792.83$1,469.50$444.67$197.79$308.89$1,457.72
11.08%9.31%10.68%13.56%f12.43%14.54%6.77%13.40%15.36%17.02%15.31%18.03%12.04%25.30%14.93%
CWCWCWCWCWTRgTRgTRgPWCWCWCWCWCWCW
Jan. 3, 1984Dec. 31, 1985Dec. 21, 1985Apr. 10, 1962Dec. 31, 1992Dec. 31, 1987Dec. 31, 1987Dec. 31, 1987Oct. 1, 1928Jan. 2, 19761941±1943Dec. 31, 1990Dec. 31, 1993Dec. 31, 1994Dec. 31, 1986
£1,000£1,412.60£682.94£100£1,363.791,0001,000500Ð$50$10$100$100$100$140
1004,739502253001,35850640
$3,707.83$4,069.31¥2,008.78¥18,934.34¥324.26¥1,722.20¥2,609.735,958.32
32.41%24.50%0:69%6:95%4.64%5:01%0.85%h11.53%
CWCWPWPWCWCWCWCW
Feb. 1, 1985Feb. 5, 1971Jul. 4, 1988May 16, 1949Oct. 1, 1982Jan. 4, 1968Jan. 4, 1968Dec. 31, 1987
$250$100¥1,000¥176.21¥100¥100¥1001,000
235
(continued)
CHAPTER 5 AGGREGATE INDICATORS: STOCK INDICES
Exchangea
Name
236
TABLE 5-1 (continued)Exchangea
Sponsor
Number ofComponentsb
Value onDec, 31, 1999
10-YearCAGRc
Methodd
Institutede
Base Value
SBF 80SBF 120SBF 250
SBFSBFSBF
SBFSBFSBF
80120250
4,217.584,052.583,810.86
17.34%16.82%16.03%
CWCWCW
Dec. 31, 1990Dec. 28, 1990Dec. 28, 1990
1,0001,0001,000
a
LSE, London Stock Exchange; FSE, Frankfurt Stock Exchange; NYSE, New York Stock Exchange; Nasdaq, National Association of Securities Dealers AutomatedQuotation; TSE, Tokyo Stock Exchange; and SBF, SocieÂteÂ des Bourses FrancËaises.bAs of third quarter 1999.cTen-year compounded annual growth rate as of December 31, 1999. For indices instituted after December 31, 1989, the rate is from inception.dCW, capitalization weighted; TR, total return; and PW, price weighted.e``Instituted'' may be different from ``listed.'' For example, the Nikkei 300 was instituted as of 1982 but listed in 1995. The date of institution indicates the date upon whichthe index is based. On that particular date, its value is equal to the base value.fFrom December 31, 1990.gNote that the total return feature of the DAX makes a direct comparison of returns impossible with other indices.hFrom