2.562550.8750296.63
(Source: Compiled by author from Datastream data.)
CHAPTER 5 AGGREGATE INDICATORS: STOCK INDICES
251
it is not surprising to ®nd that virtually all events are tradable quitesimply, provided that the information is available in a timely manner.The general idea is that corporate events are never price driven. Inother words, if the capitalization of a particular stock is affected by aspeci®c capital operation, the index should not be because it is meant torecord only price appreciation. Therefore, the divisor of an index isadjusted following a corporate event in such a way as to ensure continuity and make the event invisible in the index performance.DividendsDividend payments are recorded only in total return indices, and theweight is modi®ed in the following way:PY0W WPY dTherefore, for a total return index, a dividend ex-amount is recordedwith the following adjustment:!PjP PjPi6j Wi  Pi  Wj Pj di6j Wi  Pi  Wj  PjIDD0As we observed for an inclusion or deletion, D0 6 D means that wepotentially need to adjust the entire portfolio stock by stock.16 We solvethis dif®culty in the same manner by saying that the number of indexunits n has been changed to n0 due to the dividend ex-date. Therefore,the quantity Qj we should hold is modi®ed accordingly to Q0j :!!  0WWjWPjjjQ0j  n0  n0 Qj  n DD0D0Pj dBecause of the relationship between n and n0 , Q0j becomes:!! WjPjPj0 Qj Qj  n DPj dPj dThis very simple result means that the adjustment taking place in theindex is directly replicable within the position. We need to buy only asmall additional quantity Qj of the stock going ex, equal to:16
Note that in the case of an inclusion or deletion, we cannot be sure that D0 > D. In thecase of an ex-dividend event for a total return index, this is always true.
252
THEORY and TOOLS
Qj  Q0j
Qj  Qj 
!
PjPj
d
Qj  Qj 
dPj
!d
It would be very simple as well to show that Qi must be boughtat the close Pj or at the ``fair'' opening price Pj d. In practice, becausethe actual opening price may be signi®cantly different from Pj d, it isalways safer to buy Qj at the close of the last trading day.SplitsSplits are extremely straightforward events with regard to index behavior. If r denotes the split ratio, we know that Q0j  r  Qj and P0j  Pj =r.Therefore, the index is simply not affected in any way: PjPPi6j Wi  Pi  Wj  rWPWPiijjri6jID0DD0  D, and the replicating portfolio is unaffected. There is really nothing particular to do except passively record the changes in Wj and Pj .This development is true for capitalization-weighted indices butnot for price-weighted indices. In the case of a price-weighted index,the weight Wj remains unchanged through the split: PjPPi6j Wi  Pi  Wj WPWPiijjri6jID0DHere D0 6 D, and for this reason the number of units again changes ton0  n  D0 =D. The new theoretical quantity Q0j is equal to Qj :!  Wj0WjWj000 Qjnn Qj  n D0D0Dbut we still have a problem because we receive a distribution ofr 1  Qj free shares due to the split, and we need to sell these shareson the effective day of the split. They need to be sold at the price Pj =r,which is the fair price of the stock after the split but before the open.Again we do not know for sure the exact opening price; therefore, it ismore prudent to sell them the day before on the close, automaticallyrealizing the price Pj used in the divisor adjustment.If we denote  the proportion of shares we actually need to sell, thenew position held is Qj  1  on the close. The number of free shareswe receive is then Qj  1   r 1, and the total position isQj  1   r. This total position should be equal to Q0j for the portfolioto be a perfectly replicating portfolio again:
CHAPTER 5 AGGREGATE INDICATORS: STOCK INDICES

Qj  1   r  Q0jQ0j  Qj
)

r
253
1r
This result means that to remain perfectly hedged after a split in aprice-weighted index, we need to sell r 1=r  Qj shares of thestock to be split at the close of the last trading day of the split.Consider as an illustration a portfolio of 100 Dow Jones indices,and the split 3:2 of Home Depot (HD) on December 31, 1999. As ofDecember 30, the Dow divisor was 0.20436, and each constituent hadan equivalent quantity of 4.893 shares. A portfolio of 100 indices wouldhave held 489.33 shares of HD. HD closed at $100:1875, and the nextmorning after the split its fair price was $66:7917  $100:1875  2=3.At the same time, the Dow closed at $11,452:86, indicating a totalweighted average of all constituents of $2,340:50  $11,452:860:20436. The new divisor D0 is then given by:I
$2,340:50 $2,340:500:20436
$100:1875  $66:7917D0
) D0  0:20144
Therefore, the next day we hold n0  n  D0 =D  98:57 indices, andwe need to have 489.33  98:57=0:20144 shares of each constituent,which is the case already except for HD. For HD we have now 734shares  3=2  489:33 due to the distribution of free shares after thesplit. To remain hedged, we should hold 489.33 shares only, whichmeans 244.67 shares in excess. We can sell those shares at the openon the ex-date at the risk that the opening price is lower than $66:8125,or we can decide to sell them preemptively on the close of the lasttrading day.17 In that case, we sell   Qj  163:1  0:33  489:33shares, after which we have 326.22 left. These 326.22 generate 163.11free shares with the split, reconstituting the needed 389.33 position.Figure 5-8 is a quick diagram to help in visualizing the situation.Mergers and AcquisitionsMergers and acquisitions are complex events with respect to indexcomposition and behavior. On one side, the stocks involved may ormay not trade at par with respect to the terms of the transaction, whichpotentially creates an opportunityÐand a risk. The second dimensionof the event is which stock will remain in the index after the transaction. Not surprisingly, different con®gurations lead to different results,but more interestingly some of these may be pro®table.17
As it happens, the actual opening price on December 31 was $67.4375, 62.5 centshigher than its fair value. The value $66.8125 is different from the theoretical fair price$66.7917 because of the rounding to the nearest 16th.
254
THEORY and TOOLS
Figure 5-8
Substitution
As an illustration, consider a capitalization-weighted index constituted of three stocks, out of a universe set to A; B; C; or D. We considerthat A absorbs B with a ratio r, meaning that at maturity holders of Breceive r shares of A for each share they hold. In terms of index composition, four scenarios are possible:. A; B, and C in the index; no stock is added after the merger.. A; B, and C in the index; D is added.. A; C, and D in the index.. B; C, and D in the index.
Let's examine each situation in detail and draw some generalizationsfrom there.A and B Included, No Addition18If the index is A; B, and C, continuity is expressed by:WA  PA  WB  PB  WC  PC WA0  PA  WC  PCDD0WA  r  WB   PA  WC  PCD0
I
The number of units in our portfolio is changed to n0  n  D0 =D onlyif PB 6 r  PA Ðthat is, if both stocks are not trading at par. Its value isalso modi®ed just by virtue of the merger from:V  QA  PA  QB  PB  QC  PCto:V 0  QA  r  QB   PA  QC  PC18
This situation occurs, for example, when the index is all-inclusive, like the TOPIX inJapan. When two stocks merge, there is not necessarily a replacement immediately.
CHAPTER 5 AGGREGATE INDICATORS: STOCK INDICES
255
Because of the relationship between n0 and n, it is very simple to checkthat the portfolio remains a perfect replication of the index without anyfurther adjustment. In addition, there is a creation of value if A istrading above par with respect to B:V0
V  r  PA
PB   QB
In other words, the holders of a portfolio replicating an index passivelybene®t from the risk arbitrage opportunity created by the merger.Note, however, that the bene®t occurs only on the last trading day ofthe merger, when the difference with par is probably small.A and B Included, B ReplacedThis situation is very similar to the one before. Continuity is given by:WA  PA  WB  PB  WC  PCDWA  r  WB   PA  WC  PC  WD  PDD0
I
assuming B is replaced by D. Again we have n0  n  D0 =D, and thesame mechanical effect of the merger creates a passive pro®t or lossupon expiry. The only difference from the previous case is that D needsto be bought on the close of the last trading day of the merger, with:WDWDnQ0D  n0 D0DA Only IncludedContinuity here is given by:WA  PA  WC  PC  WD  PDDWA  r  WB   PA  WC  PC  WD  PDD0
I
This situation requires us to buy additional shares of A because itsweight increases in the index. Although before we held B as part ofour replicating portfolio, we do not here, which means that the difference with par does not automatically materialize in the position onexpiry. If we buy an additional quantity QA number of shares atthe close, the position is perfectly hedged with respect to the index,at no cost or pro®t:
256
THEORY and TOOLS
WA  r  WBQAQA  Q0A QA  n0 D0   WAWBWB0nQ n0 rnrAD0D0Dbased again on the fact that n0  n  D0 =D.B Only IncludedThis case is particularly interesting because although relatively infrequent. Continuity here is given by:WB  PB  WC  PC  WD  PDDWA  r  WB   PA  WC  PC  WD  PDD0
I
Because we hold B before the merger, we bene®t again automaticallyfrom a difference with par between PA and PB . We still need to buyshares of A to adjust the portfolio because the transformation of Bshares into A shares provides only a fraction of the capitalization weneed:  WA  r  WBWAWB0000n n rQA  n 00DDD0 WA r  QB nDr  QB is what we receive in A shares due to B's position, but n  WA =Dis missing and must be bought at the close of the last trading day.SummaryTable 5-4 presents a rapid summary of the different alternatives. The column represents the ``Automatic win'' pro®t or loss. In cases 1, 2,and 4, because of a native position in B, an index-replicating portfolioautomatically and passively bene®ts from a merger if the price relationship between A and B is such that r  PA PB > 0 on expiry. Inpractice, this relationship is almost always true because it simplyindicates that B is trading at a discount to its fair acquisition price.However, if for some exceptional reason PB were trading at a premium,  would be negative and a replicating portfolio would thenautomatically record a loss. The only way to hedge against this loss isto trade the risk arbitrage before at a favorable price. Case 3 is verysimilar to cases 1 and 2 and bene®ts from the merger, but the arbitrage has to be implemented manually because it is not automaticallyre¯ected in the divisor's adjustment.
Situation

Adjustment
1 A and B in, no addition
  r  PA
P B   QB
é
2 A and B in, D added
  r  PA
P B   QB
Buy QD0  n 
3 A in, remains in
0
4 B in, replaced by A
  r  PA
P B   QB


WDDWBBuy QA  n  r D WABuy QA  n D
Risk Arbitrage``Free'' Option
``AutomaticWin''
``DoubleChance''
Yes
Yes
Ð
Yes
Yes
Ð
Yes
Ð
Yes
Ð
Yes
Yes
CHAPTER 5 AGGREGATE INDICATORS: STOCK INDICES
TABLE 5-4
257
258
THEORY and TOOLS
The ``Risk arbitrage option'' describes the option embedded in theposition to bene®t from the merger before expiry. In three out of foursituations, we hold A as part of the replicating portfolio; therefore, ifthe arbitrage of the merger is pro®table, we can trade it very easily,without having to borrow A, which can be dif®cult and expensive.19In short, the ®rst generalization we can make is that it is alwayspossible for the holder of a replicating portfolio to bene®t from a mergersituation within an index. In cases 1 and 2, the arbitrage can be implemented any time before expiry, and in any case, accrues automaticallyanyway on the last trading day. In 3 the arbitrage is also possible at anytime, but it does not materialize automatically. In 4, the arbitrage isautomatic on expiry, but it is not necessarily possible before becausewe do not hold A.Now consider cases 3 and 4, and the ``Double chance'' feature. Inboth situations we have to buy A upon expiry of the merger, as part ofthe normal life cycle of the index. However, buying A is not necessarilyoptimal because we can trade B instead and tender the shares to themerger on the effective date. Therefore, we have an additional option tobene®t from the merger, in addition to the full extent of the position wealready hold. In 3, instead of buying n  r  WB =D shares of A, we canbuy n  r  WB =D=r  n  WB =D shares of B at a better price if Btrades at a discount. Similarly, in 4 we can buy n  WA =D=r sharesof B instead of n  WA =D shares of A.The difference between cases 3 and 4 is only a question of magnitude in the ``Double chance'' opportunity. For example, in 3, if wedecide to buy n  WB =D shares of B, the nominal of this trade is: nWB WB  PB  PB N3  n DDwhereas in 4, if we buy n  WA =D=r shares of B, it is:  n W n PBnA WA  WA  PA  PB N4 rDDDrif we assume for simplicity that r  PA  PB . It is apparent that N3 isproportional to the total capitalization of BÐWB  PB Ðwhereas N4 isproportional to that of AÐWA  PA . If the merger is a merger of equalentities, there is no signi®cant difference between N3 and N4 . In gen19
Note that the automatic win or risk arbitrage options are nonexclusive. In 1 and 2 it ispossible to enter arbitrage before expiry and still bene®t from the automatic win atexpiry.
CHAPTER 5 AGGREGATE INDICATORS: STOCK INDICES
259
eral, however, B tends to be smaller than A, which means that the``Double chance'' in 3 is usually smaller than in 4.This discussion leads to the second generalization: If one of themerging companies is not included in the index at the time of the merger,there is an additional opportunity to arbitrage the event, the most advantageous situation being that of a substitution operated in parallel to themergerÐthat is, event 4.Price-Weighted IndicesLet's examine the different cases again, this time from the perspectiveof a price-weighted index:A and B In, No Addition20:I
WA  PA  WB  PB  WC  PC WA  PA  WC  PCDD0
but we know that WA  WB  WC  W, so:  WW PA  PB  PC   PA  PC IDD0A and B In, D Added:I
WA  PA  WB  PB  WC  PC WA  PA  WC  PC  WD  PDDD0
which simpli®es to:  WW PA  PB  PC   PA  PC  PD IDD0A In, Remains In:I
WA  PA  WC  PC  WD  PD WA  PA  WC  PC  WD  PDDD0
and clearly D0  D.B In, Replaced by A:I
WB  PB  WC  PC  WD  PD WA  PA  WC  PC  WD  PDDD0
again simpli®es to:20
In the case of the Dow Jones or Nikkei 225, the merger would result in one lessconstituent; therefore, this case is practically impossible.
260
THEORY and TOOLS
I
  WW PB  PC  PD   PA  PC  PD DD0
If we draw a table similar to the one above, it would result in thesummary shown in Table 5-5.In cases 1 and 2, the weight of A is unchanged by the merger,and we need to sell B because it is not part of the index anymore.We still have the option to arbitrage the merger because of a nativeposition QA , but there is no automatic win anymore. On the surfaceit looks as though there is potential for a ``Double chance'' becausewe need to sell QB , and we could decide to sell A instead and enterthe arbitrage. In practice, however, if we take advantage of thearbitrage before expiry to the full extent of what the position allows,we do not hold A anymore on expiry. Therefore, if we want to sellmore shares of A instead of QB , we need to borrow them, whichmay be impossible or expensive. Therefore, the double chance is infact uncertain.Case 3 is really a nonevent because the index is absolutelyunchanged with D0  D, and so is the replicating portfolio.Case 4 is more interesting because there is a need for adjustment inthe position. If we do nothing, the position QB is automatically transferred to an equivalent r  QB shares of A, which creates an ``Automaticwin.'' However, this quantity is not necessarily equal to what we shouldhave: Q0A  n0  W=D0   n  W=D  QB . Therefore, we must sell  QB shares of B so that after the merger, r  1   QB is equal to Q0A 21:r 1
  QB  Q0A  QB
)

r
1r
At the time the merger is effective, the position in B is reduced to(1   QB  QB =r, and the pro®t generated by the automatic win isthen: QB  r  PA PB  rBecause we can sell r    QB shares of A instead of   QB shares of Bto adjust the position, the situation presents a possible doublechance feature. In fact, if r  1, we need to sell A, which we donot hold and have to borrow. Therefore, the double chance is uncertain because we are not assured that we can ®nd the shares. If r < 1,21
In general, r  1, which means  > 0. However, r < 1 also occurs, in which case weneed to buy   QB  shares.
Situation

Adjustment
Risk Arbitrage``Free'' Option
``AutomaticWin''
``DoubleChance''
1 A and B in, no addition
0
Sell QB
Yes
Ð
Uncertain
2 A and B in, D added
0
Yes
Ð
Uncertain
3 A in, remains in
0
Sell QBBuy QDé
Yes
Ð
Ð
4 B in, replaced by A
  r  PA
Sell n  r 1 WQB Dr
Ð
Yes
Uncertain
PB  
QBr

CHAPTER 5 AGGREGATE INDICATORS: STOCK INDICES
TABLE 5-5
261
262
THEORY and TOOLS
we need to buy B, which is always possible, and the double chanceis then certain.ConclusionThis detailed examination of corporate events explains in a large partwhy we focused heavily on understanding them in the ®rst place. Itcan be argued that only index arbitrageurs and index fund managersreally care about index-replicating portfolios, but the reality is muchbroader.Indeed, any arbitrage position for which one of the legs is anindex futures contract is potentially affected by a corporate event.Because it is possible to associate each futures contract to an elementary quantity of each constituent in the index, a futures contractis also a form of replicating a portfolio and needs to be managedaccordingly. The rigorous exercise of precisely tracking opportunitiesbased on a simple arithmetic de®nition is also extremely useful inmany other settings where the same type of analysis must beperformed.
CORRELATIONSAs was indicated before, many indices are chained to include andsegment as many constituents as possible within the underlying universe of listed stocks. Therefore, it is interesting to consider the relativebehavior of embedded indices more generally.Consider, for example, two indices I1 and I2 constituted of m1 andm2 securities, such that I1 is included in I2 . By de®nition:m1P
I1  i1
m2P
w1  Pi
I2  i1
and
D1
wi  PiD2
if I1 and I2 are computed according to the same averaging rule. If werank the stocks according to their inclusion in I1 , we can write:m2P
I2 
i1
D2
D1  I1 
m1P
wi  Pi
i1
wi  Pi 
m2Pim1 1
D2
m2Pim1 1
wi  Pi
D2wi  Pi

m2P
wi  PiD1im1 1 I1 D2D2
CHAPTER 5 AGGREGATE INDICATORS: STOCK INDICES
263
If we denote:m2P
C1=2 
wi  Pi
im1 1
D2
the above expression becomes: D1 I1  C1=2I2 D2This simple relationship indicates how we can transfer a positionon I1 to I2 and vice versa. To illustrate this point, consider we hold n1indices I1 . For each stock in I1 , the number of shares in the portfolio is: Wi1Qi  n D1This number of shares is equal to what we would hold if we had n2indices I2 in portfolio, with n2 such that n2  n1  D2 =D1 . However, tohave a complete I2 position, we should also hold stocks that are notincluded in I1 for an amount equal to: wifor m1  1  i  m2Q2i  n2 D2This is exactly the quantity associated with C1=2 , if C1=2 is considered anindependent index. Therefore, to transform n1 indices I1 into n2 indicesI2 , we need to trade n2 indices C1=2 as if it were itself a third distinctindex. Naturally the reverse transformation is exactly the opposite.This result is not extraordinarily surprising, but it is interesting inat least three aspects. First, it opens the door to cross-index arbitrage.We know how to transform I1 into I2 , which means that if both indiceshave listed futures, we can indifferently trade any of them and operatea transformation when needed. Second, we have implicitly created anew index, strictly equal to I2 I1 . If needed, we can use it to take aposition on the difference I2 I1 , for example, to bene®t from anexpected appreciation of small capitalization stocks compared to traditional larger companies. Finally, we even have a futures contract onD1 =D2   F2 behaves exactlythis new index because the difference F1like a futures contract on C1=2 . In the United States, for example, theexistence of a long series of embedded indices offers a certain numberof opportunities for investors trying to build nonconventional indexpositions.
264
THEORY and TOOLS
CONCLUSIONThis chapter covers many of the technicalities necessary to manage aportfolio replicating a given index. In fact, beyond that speci®c caseapplicable to index arbitrageurs, the day-to-day maintenance of a stockportfolio requires a precise understanding of the mechanics, and moreimportantly the opportunities, attached to corporate events. The formalism we went through should enable a rigorous analysis of any situation, from the most common to the least frequent, giving solid tools tounderstand the details and bene®t from them.
P ART IIFINANCIAL VALUATIONAND RISK ANALYSIS
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6V ALUATIONS
INTRODUCTIONAfter the exhaustive examination of all the instruments independently of each other, we must now turn to the issue of evaluatingan entire position. The underlying assumption in all prior analyticalresults was a static forward-looking perspective: All products wereabout to be traded, and we systematically examined how theyshould be priced in order to determine their fair value at the timeof the trade.The perspective we want to take now is that of a live position,established at various points in the past. It is a collection of variousinstruments of different nature, notably in terms of payment schedule. This situation corresponds to the reality of a trading deskwhere transactions are accumulated day after day and where wemust be able to value incoming trades as well as established positions. Furthermore, we must design a uni®ed process so that allinstruments can be systematically valued, in such a way that the®gure we obtain is reliable and meaningful. Reliability is here aninternal metric and points toward rigorous pricing procedure, withthe lowest possible level of uncertainty. Meaningfulness, on the otherhand, refers to the overall environment and implies that valuationparameters be chosen wisely to be as representative as possible. Forexample, there cannot be much debate about the value of a futurescontract based on a cost-of-carry model, but there may be more discussion about the relevant amount of dividends or the right interestrate to use. Clearly the futures contract can be priced rigorously withany choice of parameters, but its value is not meaningful if it is not267
Copyright 2001 The McGraw-Hill Companies, Inc. Click Here for Terms of Use.
268
FINANCIAL VALUATION AND RISK ANALYSIS
speci®cally tailored to the characteristics of the underlying positionand market environment.1Another dimension of a valuation statement is that it is distributed to a relatively broad audience. Traders have full responsibilityfor the day-to-day management of their position, but the valuation isalways an absolute reporting requirement to senior management andother departments such as internal control and audit. A practicalconsequence of this requirement is that the valuation documentshould be readily understandable by nontraders and further