December 31, 1991.Source: Compiled by author from exchanges' respective Web sites and Bloomberg.
THEORY and TOOLS
Name
CHAPTER 5 AGGREGATE INDICATORS: STOCK INDICES
237
DAX 30(€ 800)
DAX 100(€ 915)
MDAX(€ 115)
Figure 5-1 DAX(Note: All ®gures are the total capitalization of the index as of August 31, 1999, inbillions.)(Source: Author's estimates and Bloomberg except Russell data, Frank RussellCompany, reprinted with permission.)
CAC 40(€ 728)SBF 120(€ 900)
SBF 80(€ 172)
SBF 250(€ 1,010)
Figure 5-2 CAC(Note: All ®gures are the total capitalization of the index as of August 31, 1999, inbillions.)(Source: Author's estimates and Bloomberg except Russell data, Frank RussellCompany, reprinted with permission.)
FTSE 100(£1,260)FTSE 350
FTSE 250
(£1,496)
(£236)
FTSE All Cap(£1,570)
FTSESmall Cap(£74)
Figure 5-3 FTSE(Note: All ®gures are the total capitalization of the index as of August 31, 1999, inbillions.)(Source: Author's estimates and Bloomberg except Russell data, Frank RussellCompany, reprinted with permission.)
238
THEORY and TOOLS
Nikkei 50Overlap
(≈70 stocks)
Nikkei 225
Nikkei 300(¥255,913)TOPIX(¥371,548)
TSE2(¥14,719)
Figure 5-4 NIK(Note: All ®gures are the total capitalization of the index as of August 31, 1999, inbillions.)(Source: Author's estimates and Bloomberg except Russell data, Frank RussellCompany, reprinted with permission.)
Based on these diagrams, it is particularly interesting to push a littlefurther and examine the relative concentration of each index. With 500constituents, for example, the S&P 500 encompasses $10,414 billion incapitalization whereas the S&P 1500 Composite has three times as manyconstituents with only $1,190 billion in additional capitalization.Presented otherwise, with 33.3 percent of the stock base of the S&P1500, the S&P 500 captures 89.7 percent of the overall capitalization.Table 5-2 reproduces the same calculation for most of the above indices.If we plot these ®gures, we obtain the chart in Figure 5-6. The dashedline represents a logarithmic regression and gives an idea of a possibleshape of the underlying relationship between the two variables.The regression is ®tting relatively well and can be interpreted asshowing that the concentration of capitalization tends to be similar inevery major index. This is, however, true only for large indices, asillustrated in Figure 5-7. It represents the percentage of the overallcapitalization of each market captured by the 5 percent largest companies. When only so few companies are included, there are wide disparities in the total capitalization they capture, which indicates thatsimilar concentrations between the different markets appear indeedonly for large indices. Note that the trend is unambiguously towardmore concentration across the board.
CHAPTER 5 AGGREGATE INDICATORS: STOCK INDICES
S&P 100($6,873)
239
DowIndustrial($2,566)
S&P 500($10,414)S&PComposite($11,604)
S&P 400($853)
S&P 600($337)Russell 3000®($11,939)
Figure 5-5 S&P(Note: All ®gures are the total capitalization of the index as of August 31, 1999, inbillions.)(Source: Author's estimates and Bloomberg except Russell data, Frank RussellCompany, reprinted with permission.)
The overall issue of concentration is particularly important whendesigning or managing an index because too much capitalization in asmall number of stocks makes the index relatively skewed and lessrepresentative. For example, France Telecom and TotalFina in Franceor Deutsche Telekom and Allianz in Germany account for more than 25percent of the CAC 40 and DAX 30 indices, a proportion that is arguably too high to consider both indices as true broad based indicators.
SPONSORSHIPA sponsor is an institution responsible for the constitution and thecalculation of an index. In some cases these roles are assumed by theexchange or an institution af®liated with itÐFrance, Germany, or theTOPIX in TokyoÐbut most often a third (private) party is undertaking
240
THEORY and TOOLS
TABLE 5-2Index
Stock
Capitalization
Reference
Dow JonesS&P 100S&P 500FTSE 100FTSE 350DAXCACSBF 120
2.0%6.7%33.3%12.6%43.9%30.0%16.0%48.0%
22.1%59.2%89.7%80.2%95.3%87.4%72.1%89.1%
S&P 1500S&P 1500S&P 1500FTSE All ShareFTSE All ShareDAX 100SBF 250SBF 250
these tasksÐFinancial Times, Nikkei, or Standard & Poor's. The calculation is a fairly straightforward responsibility, but managing the constitution, in contrast, is a much more sensitive issue.There are numerous aspects to thisÐnotably concentration as weindicated, or industry representationÐbut the most important is transparency, which essentially means that clear standards and rigorousdisclosure procedures should be set with respect to inclusions anddeletions. The standard is necessary so that investors at large knowwhat to expect throughout the life of the index in terms of composition
Figure 5-6
Concentration
CHAPTER 5 AGGREGATE INDICATORS: STOCK INDICES
241
Figure 5-7 Capitalization(Source: Compiled by author from FIBV data.)
and may react accordingly.5 Failure to maintain such a standard resultsin a lot of volatility when announcements are actually made, when infact most of this volatility could be avoided.6Disclosure is also critical because of the importance of maintaining an index on which it is possible to trade continuously. In manycases sponsors have established explicit rules; for example, disclosure happens a certain number of days before the inclusion or dele5
For descriptions of addition and deletion standards, see, for example, www.nikkei.co.jp for Nikkei 225 and 300, www.dj.com for Dow Jones, www.spglobal.com for thefamily of S&P indices, www.exchange.de for the family of DAX indices, andwww.ftse.com for the family of FTSE indices.6In general, many broad-based indices effectively serve as benchmarks to a whole set ofstrategies, and inclusion of a new stock usually creates a lot of demandÐthat is,buying pressureÐwhereas exclusion has the opposite effect. Obviously index arbitrageurs are ®rst in line to act on these events because the nature of their position requiresthem to track the index rigorously. ``Index funds''Ðthat is, mutual funds tracking agiven indexÐare probably next in line, and they have more and more impact considering their size has grown explosively over the past decade. Beyond these, manyinvestors are also more or less directly benchmarking against speci®c indices. For anexamination of inclusions and deletions with respect to institutional ownership, see,for example, Stephen W. Pruitt and K. C. John Wei, ``Institutional Ownership and(continued)
242
THEORY and TOOLS
tion is effective.7 It is necessary to realize here that early announcement, although necessary, is very costly in terms of index return.Consider, for example, an index replacing one of its components A,by a stock B. When the switch is announced, assume that A loses 5percent and B rises by 10 percent. If the announcement is madebefore the effective date, the rise of B is not included in the index,whereas the decline of A is. Therefore, the index of®cially captures adecline and misses the corresponding increase. In fact, it is evenworse because in practice many stocks tend to ``overshoot'' on inclusion or deletion. Traders will overbuy the entrant and oversell thestock replaced, often resulting in a correction on the morning afterthe switch is effective. Not only is the index return recording theoriginal loss and missing the original gain, it is also recording thecorrection, during which A is included and potentially declines andB is not included anymore and potentially recovers. Naturally, it isdif®cult to generalize from these observations, but there cannot beany doubt that indices with recurrent inclusion and deletion events,like the S&P 500, suffer signi®cantly from the price movementsbetween an announcement and its effective date.The alternative is to disclose the inclusion or deletion event afterthe factÐfor example, just after the close on effective date. As ithappens, Standard and Poor's used this procedure commonly before1989, when it decided to revise it toward a ®xed delay of ®ve days.The case also presented itself in Japan with the Nikkatsu bankruptcy,and we will detail the exact situation in the chapter dedicated to(Footnote continued)Changes in the S&P 500, '' Journal of Finance, 44, 1989, pp. 509±513. See also AndreiShleifer, ``Do Demand Curves for Stocks Slope Down?'' Journal of Finance, 41, 1986, pp.579±590. Stocks recently included in the index are found to present an abnormal returnof 2.79 percent.The very existence of an ``inclusion premium'' is directly dependent upon the levelof index arbitrage activity. For example, the Dow Jones until recently did not have anyfutures contracts, and newly included stocks were not under price pressure. See, forexample, Messod Beneish and John Gardner, ``Information Costs and Liquidity Effectsfrom Changes in the Dow Jones Industrial Average List,'' Journal of Financial andQuantitative Analysis, 30, 1995, pp. 135±157.7See, for example, Messod Beneish and Robert Whaley, ``An Anatomy of the `S&PGame': The Effects of Changing the Rules,'' Journal of Finance, 51, 1996, pp. 1909±1930, for a complete discussion of the S&P 500 and its disclosure policy.In addition to disclosing inclusion and deletion, transparency also refers the availability of weights and divisor on a regular basis. Many exchanges have set upautomatic systems through which market participants can query constituents' characteristics when needed.
CHAPTER 5 AGGREGATE INDICATORS: STOCK INDICES
243
index arbitrage because it constitutes undoubtedly a risk to arbitrageurs. To set the stage, let's say that Nikkatsu was a Japanese movieproduction corporation that went bankrupt very suddenly in 1993. Ithad been in a very poor ®nancial condition for some time, but itsbankruptcy came as a surprise. Even more surprising, however, wasthe fact that Nikkei decided to remove the stock from the Nikkei 225immediately but announced it after the close.8 The stock went from¥229 to ¥25 ( 89 percent) the next day, while the entrant company,Shionogi & Co., rose from ¥985 to ¥1,050 (6:6 percent). Both movesconstituted a pure loss to index arbitrageurs and more generally toholders of benchmarked positions. To make matters worse, therewere rumors at the time of inside trading, notably on Nikkatsu,because the chart re¯ected a selling pressure shortly before theannouncement.Interestingly, what was common practice in the United Statesuntil 1989 was extremely ill perceived in 1993 in Japan. Nikkei'sdecision proved disastrous in terms of public con®dence and starteda wave of protests against its irresponsibility and total disregard ofinvestors' interests. Clearly, the development of index arbitrage, practiced pro®tably by the largestÐand most in¯uentialÐsecuritieshouses, and the aggressive growth of index funds have made a lateannouncement an unambiguous liability to the sponsor, even thoughit had been commonly accepted.9 Therefore, it is unlikely that aNikkatsu situation will occur ever again, at least on a major marketplace.10
8
The major criterion for inclusion in the Nikkei 225 is liquidity, measured broadlyspeaking by the traded volume. This selection mechanism, coupled to very rare alterations of the index constitution, leads to the awkward situation in which many companies with a fairly decent capitalization are excluded from the Nikkei 225 while somestocks with declining, and eventually very poor liquidity, are not deleted for quitesome time.9Index arbitrage was for a long time the biggest loser in a situation of late announcement. Today the total amounts invested in index funds dwarf those allocated toarbitrage, and the biggest loss is undoubtedly transferred to index funds holders,at least in the United States. A Nikkatsu-like decision today would prove primarilyunpopular to individual investors, not arbitrageurs, and would consequently not bevery smart from a commercial standpoint.10Interestingly, faced with a similar situation in September 1990 with a failure of one ofits constituentsÐPrime Motors Corp.ÐStandard and Poor's removed it on a shortnotice from the S&P 500 but left one full trading day between announcement andeffective date.
244
THEORY and TOOLS
INDEX TRADING AND FUTURES HEDGINGIndex Trading
Consider a portfolio set with the explicit strategy of tracking an index I,de®ned by its weights Wi and divisor D. It is very clear that we need toinclude all the index components in the portfolio, so let's denote Qi thequantity for each of them. The value V of the portfolio is therefore:XVQ  Pii iand I is still de®ned by:
PI
i Wi  P i
D
The analogy between the two formulas suggests that we considersetting Qi  Wi =D. The quantity Wi =D can then be interpreted as theimplicit number of shares of the ith component of the index. The convention generally accepted is to refer to a portfolio constituted withWi =D shares of each stock as a portfolio equal to 1 index. A portfolioequal to 10 indices would then be de®ned by 10  Wi =D) shares of eachstock.A simple numerical application will illustrate the point. Consider asimple index constituted by two stocks A and B, such that WA  1,000,WB  2,000, PA  $15, PB  $25, and D  100. I is then equal to:I
1,000  15  2,000  25 650100
Consider now a portfolio P of 10 shares of A and 20 shares of B. Thevalue V of this portfolio is:V  10  $15  20  $25  $150  $500  $650It is very easy to verify also that the capitalization of A in I  1,000 $15=$65,000 is equal to that of A in the portfolio  10  $15=$650. Weconclude that P is rigorously tracking I, and by convention we refer toP as 1 index:Now say that we have $100,000 to invest in P, and we would like to®nd what is the appropriate number of shares for A and B. Knowingthat 1 index is worth $650, $100,000 represents 153.8 indices $100,000=$650), which in turn translates into 1,538 shares of A 10  153:8 and 3,076 shares of B  20  153:8. Therefore, given anominal N, we know that we need N=I units of the elementary portfolio P, or (N=I)  WI =D shares of each component i. By extension, this
CHAPTER 5 AGGREGATE INDICATORS: STOCK INDICES
245
convention gives a meaning to the expression ``trading the index''because we can reconstruct fairly easily an equivalent portfolio.If we take the real example of General Electric (GE) within the S&P500, for example, W  3,295,312,000 and D  8,565,281,030; thereforeone index contains 0.385 shares of GE. The second biggest capitalization in the S&P 500 is Cisco (CSCO), which has W  6,937,630,000.Therefore, one index contains 0.81 shares of CSCO. Within the DowJones, W  1 and D  0:20145; therefore, one Dow Jones index contains 4.964 shares of GE.
Futures Hedging
Let's take again the typical arbitrage situation in which we buy anindex today and sell its futures contract at the same time.Speci®cally, we buy a portfolio P and Qi shares for each stock andsell it back on expiry. On the futures side, we want to know what isthe appropriate quantity of futures to sell, and we denote it . With thesame notations we used earlier for a standard futures, we arrive at thefollowing.On t0 :. Borrow an amount N 
P
Qi  Pi from t0 to t1 @ rZC .
. Buy 1 index @ I0 , for a total nominal N  I0 .. Sell  future @ FH .
The resulting cash ¯ow is:
X
C0  N
Qi  Pi  0
On t1 :. Pay back loan N . Sell 1 index @ ISQ .
1  rZC .
. Sell futures at expiry   PVF 
FSQ FH , where PVF is the future'spoint value.. Receive accumulated capitalized dividends d .The resulting cash ¯ow is:XQi  PSQC1  N  1  rZC  i
  PVF  FSQ    PVF  FH  d
We know by de®nition that:PFSQ  ISQ 
SQi Wi  P i
D

X Wi i
D
 PSQi 
Xi
Qi  PSQi
246
THEORY and TOOLS
which leads to:C1 
N  1  rZC  
X
Qi  PSQi  1
  PVF     PVF  FH  d
Setting   1=PVF is a convenient way to set C1 exactly equal to what itwas for a single security, with the same expression for FH . Therefore,the appropriate number of futures to hedge a portfolio equal to 1 indexis exactly 1=PVF where PVF is the point value of the future. Formulateddifferently, 1 future hedges PVF indices.11Taking the numerical example of the S&P 500, PVF  $250; therefore, 1 future hedges 250 indices. In other words, because 1 indexcontains 0.385 shares of GE and 0.81 shares of CSCO, 1 future hedgesexactly 96.25 shares of GE and 202.5 shares of CSCO. For the DowJones, PVF  $10; therefore, 1 future hedges 10 indices, or 49.64 sharesof GE.
INDEX CONTINUITYIntroduction
Continuity refers to the chaining of the index value from today totomorrow. In practice, it means that today's closing price as of todayshould be equal to yesterday's closing price as of tomorrow. The reasonfor such continuity is the importance of maintaining a tradable index,which goes back to the functions it is meant to serve.As we said, indices serve the purpose of aggregating market information in a convenient and meaningful way, and this has numerousimplications. On one side, the informational role of such an aggregation is clear because it serves as a lighthouse and revelator of marketactivity. Continuity is then a logical requirement because over a periodduring which the market is closed, an index simply cannot change invalue. At the same time this informational content is insuf®cient inpractice if it is not supported by the possibility to invest in it.12
11
This result is obviously dependent upon the set of conventions used to de®ne thenotion of ``1 index.'' In many cases 1 index is alternatively de®ned as a portfolio equalto Qi  PVF  Wi =D with the stated purpose of instituting a ``1 index , 1 futurescorrespondence.''12Index sponsors in general generate revenues by licensing their intellectual propertyÐnamely, their indices. A nontradable index has less licensing value; therefore, continuity is also an economical requirement.
CHAPTER 5 AGGREGATE INDICATORS: STOCK INDICES
247
Continuity is synonymous here with traceable changes, changes that canbe reproduced at no cost in a portfolio tracking the index.13It is therefore necessary for us to consider the question of continuity in detail from the dual perspective of numerical value and tradability, with respect to the different types of events likely to affect the indexcomposition.
Inclusions and/or DeletionsArithmeticConsider a very simple index with two components, A and B. WA , WB ,PA , and PB denote respectively A and B's weight and price today. I isthen given by:I
WA  PA  WB  PBD
Suppose now that B is replaced by C, effective on today's close. As oftomorrow morning, I becomes:I
WA  PA  WC  PCD0
Clearly D0 6 D because otherwise I would not keep the same numerical value. This very simple example shows the exact role of D: Thedivisor is the critical adjustment parameter to ensure numerical continuity ofthe index. In addition, D changes in value only if Wc  Pc 6 WB  PB Ðthat is, if the total capitalization included in the index is not constantthroughout the substitution.TradabilityThe above reasoning can be extended to a more general case:PP00i6j Wi  Pi  Wj  Pji6j Wi  Pi  Wj  PjIDD0where Wj , Pj , Wj0 , and P0j represent the characteristics of the stockdeleted and included respectively. This formula is generally used tocompute D0 from D.Now let's consider a replicating portfolio P, constituted of nindices. If we denote Qi the quantity held for each individual stock,the value V of the portfolio on the close before substitution is:13
In the example of Nikkatsu, the nondisclosure did not jeopardize the numericalcontinuity on the Nikkei 225, but its tradability. The change could not be reproducedat no cost.
248
THEORY and TOOLS
V
X
Q  Pi  Qj  Pji6j i
where Qi  n 
WiD

After the substitution, the number of shares to be held in a perfectlyreplicating portfolio is changed to: Wi0Qi  n D0for every single constituent. Because D0 6 D ) Q0i 6 Qi , we needto actually adjust every single position in the portfolio. There is, however, a way around that, by considering that the number of units n isnot constant. Let's suppose that n is changed to n0 after the substitution,and let's make n0 such that:  WiWi00 QinQi  n 0DDwhich implies:

D0n nD0

This relationship is convenient because we do not need to adjust all theQi anymore; therefore, the major part of the portfolio is unchangedthrough the substitution. In that case the only thing we need to do isget rid of Qj and acquire Q0j , given by:! 0WWjjQj  n Q0j  n0 DD0In other words, we can ``trade'' the substitution by simply selling the entirepositionÐQj sharesÐof the deleted stock, and buying Q0j shares of the newlyincluded one. As a consequence of the substitution, the number of index unitsin the portfolio is changed to n0 .This result shows that the substitution is tradable provided that itis announced prior to being effective. It does not say anything, however, about the cost of this trading. Consider, for example, that we sellQj at pj 6 Pj , and buy Q0j at p0j 6 P0j .If the substitution is effective tonight and we trade it, the value ofthe portfolio after the close becomes:XQ  Pi  Q0j  P0jVi6j iThe term Qj  Pj has disappeared, having been replaced by Q0j  P0j .Meanwhile by selling Qj , we received Qj  pj , and buying Q0j costQ0j  p0j . It is very clear that if pj 6 Pj , and p0j 6 P0j , the change in value
CHAPTER 5 AGGREGATE INDICATORS: STOCK INDICES
249
V is not compensated by the amount of cash required to settle thetrades. Therefore, we need to trade the substitution at the exact priceused in the index calculationÐthat is, the closing price.This is exactly what proved impossible in the case of Nikkatsubecause the announcement was made too late, after the close. Thenext day the substitution was effective, and investors realized a lossbecause prices had moved signi®cantly from Pj and P0j .14 Note that thede®ning parameter here is the divisor. Because D0 is computed with Pjand P0j , these prices are literally set in stone for the substitution.As a numerical application, consider the case of the substitution offour constituents of the Dow Jones operated in November 1999.Chevron, Goodyear Tire, Sears, and Union Carbide were removed, tobe replaced by Home Depot, Intel, Microsoft, and SBCCommunications, effective on November 1, 1999. Table 5-3 presentsall relevant prices as of October 29, 1999.The ®rst 26 stocks remain unchanged, and the weighted average ofthis ``core'' is simply the sum of all prices, $1,896:75. The group of fourstocks being deleted accounts for $221:81, and those about to be added,$296:63. The divisor before the change was 0.19740; therefore, computing the new divisor is relatively simple:I
$1,896:75  $221:81 $1,896:75  $296:630:19740D0
)
D0  0:20437
In reality, the new divisor given by Dow Jones was 0.20436, the difference being a rounding error. If we had a replicating portfolio of 100indices before, we have now n0  n  D0 =D  103:53 units, but thenumber of shares for each stock remains identical at 506.58.15
Corporate Events
As we did earlier with the examination of the impact of corporateevents on the outstanding number of shares, we need to considerhere the impact of these same events with respect to index tradability.The formalism we used before will be entirely relevant, and in general14
In a ``normal'' addition and deletion process, newly included stocks usually appreciate until the effective date and then correct slightly. Therefore, deciding not to buy Q0juntil after effective date can also be a very conscious trading decision. It constitutes apossible source of pro®t or loss, but it can be actively managed.15The role of the number of units may appear unclear in this example. It is in fact easyto show that it is proportional to the number of futures necessary to hedge theposition. Therefore, an increased number of units simply indicates that the positionwould have to be hedged by more futures because its total nominal increased.
250
THEORY and TOOLS
TABLE 5-3Name
Ticker
Closing Price
AlcoaAT&T Corp.AlliedSignal Inc.American Express Co.Boeing Co.Caterpillar Inc.CitigroupCoca-Cola Co.E.I. Du Pont de Nemours & Co.Eastman Kodak Co.Exxon Corp.General Electric Co.General Motors Corp.Hewlett-Packard Co.International Business Machines Corp.International Paper Co.J. P. Morgan & Co.Johnson & JohnsonMcDonald's Corp.Merck & Co.Minnesota Mining & Manufacturing Co.Philip Morris Cos.Procter & Gamble Co.United Technologies Corp.Wal-Mart Stores Inc.Walt Disney Co.
AATALDAXPBACATCKODDEKXONGEGMHWPIBMIPJPMJNJMCDMRKMMMMOPGUTXWMTDIS
Core:
60.750046.750056.9375154.000046.062555.312554.250059.000064.437568.937574.0625135.500070.437574.187598.250052.6250130.8750104.750041.250079.562595.062525.1875104.875060.500056.687526.50001,896.75
Goodyear Tire & Rubber Co.Chevron Corp.Sears, Roebuck & Co.Union Carbide Corp.
GTCHVSUK
Out:
41.312591.312528.187561.0000221.81
Home Depot Inc.Intel Corp.Microsoft Corp.SBC Communications
HDINTCMSFTSBC
In:
75.750077.43759