of circumstances. The arbitrageur must have been able to buy at some point inthe past a long-term roll, probably at a pro®t or at least without anysigni®cant loss. If that is the case, the market was pricing at that time along-term roll close or below parÐthat is, it was also implicitly pricingall the interim short-term rolls close to or below par. Therefore, buyinga long-term roll cheap gives an early indication that interim rolls willmost probably be cheap, in turn giving some substance to the risk ofbeing squeezed at a later date.To better understand the source of this risk, let's reconsider a standard roll position. Consider, for example, that we hold the exact opposite position, as shown in Figure 8-24. We can buy 20 rolls today, or 20indices at the SQ, or choose any combination of the two. There is noproblem whatsoever here because we can always buy stocks, at amaximum cost equal to the transaction costs.36 Therefore, the dif®cultyappears only because going short on stocks is dif®cult and/or costly.This is particularly true in Japan, for example, where the securities loanmarket is not mature yet compared to that of the United States orEurope.
Interest Rates
As an illustration of the interest rate exposure inherent to an indexarbitrage position, consider that we enter a trade today in which webuy a replicating portfolio at I and sell futures at their theoretical price+20
-10
Tomorrow, SQ
Toda yFigure 8-24
36
-10Short Squeeze
This assumes naturally that we have the capital to buy additional sharesÐthat is, thatthe current position has not reached a limit.
412
COMMON ARBITRAGE SITUATIONS
FTH . If r denotes the interest rate from today's stock settlement date tothe ®nal settlement date of the futures, and d denotes the accumulateddividend amount expected until the expiry, then we know thatFTH  I  1  r d . This formula implies that we borrow an amountequal to the cash outlay necessary to establish the positionÐthat is, I, ata rate r until the expiry. Consider now that we do not perform thismoney market transaction and that the interest rate changes to r0 as oftomorrow.37 If for the sake of simplicity we keep the stock marketunchanged from today to tomorrow, we have no pro®t or loss onthe stock side but possibly a P&L on the futures position. Along thesame lines, we consider d  d0 for simplicity although this wouldprobably not be true in practice. Today the short futures are valuedat FTH , and tomorrow at F0TH  I  1  r0  d .Speci®cally, if PV denotes the point value of the futures and Q theposition held, the P&L P is approximately equal to38:
FTH F0THP  Q  PV 1r


r r0 Q  PV :I 1r

To illustrate the magnitude of this risk, consider a $1 billion S&P 500positionÐthat is, 2,857 futures with an index at $1,400. We assume thatthe futures expiry is in two months and that r  6 percent and r0 = 6.10percent. Under these assumptions, P is equal to:232  6% 6:1%6712P  2,857  $250  $1,400  452  6%112
$165,008
If the position is optimistically expected to yield 0.73 percent over thecourse of a regular yearÐthat is, $7.3 millionÐa loss of $165,000 represents approximately one week of trading (5.6 days  $165,000/$7,300,000  250). Although it is probably nothing dramatic in absoluteterms, this loss can be easily avoided by entering the appropriate37
If we do not perform the money market transaction, the position will still be ®nancedbut at the overnight rate. The result of the discussion does not change because we arestill entirely exposed on a change in r.38P is an approximation because we use r  r0 in order to simplify the expression. Intheory, we should use:FTH FH F0TH FHP  Q  PV 01r1rwhere FH is the historical price at which the futures has originally been traded.
CHAPTER 8 INDEX ARBITRAGE
413
money market transaction because the $165,000 is then offset by apro®t from rising interest rates.39This example is relatively straightforward and intuitive because itis apparent that we need to ®nance the carry of the stock position at agiven rate. The same risk exists, however, in a situation in which we donot explicitly have to carry the position yet. Consider, for example, thatwe roll today part of our position through a one-year forward transaction. We buy a number of futures and sell an equivalent number ofone-year forward contracts, resulting in Figure 8-25. The futures position is exposed on the two-month rate, r2 M . The forward, on the otherhand, depends exclusively on the one-year rate, r1 Y . If the entire portfolio is properly hedged as of today before the forward is traded, itincludes a replicating stock position, an equivalent short futures position, and a money market hedging the interest rate risk until expiry.When the forward is ``substituted'' to the futures,40 the money marketis not appropriate anymore to hedge the risk associated with r1Y . Thissituation is another practical example of the distinction between treasury management and interest rate exposure. As of today the treasurybalance is nil because the money market covers the entire amount ofstock purchased, so there is no need for any adjustment on the treasury
2 Months
1 Year
*
r1Y, d1Y
r2M, d2M
*
Toda y10 MonthsFront-Month Future
Long-Te rmForward
Figure 8-25 Interest Rate Risk
39
Note that the magnitude of the loss depends as much on the absolute change ininterest rates, 0.10 percent in our example, as it depends on the remaining time toexpiry. If there were only one month left, the loss would be $82,915, still signi®cantbut much less so.40The forward is not substituted for the futures, but the net result of the rollÐthat is, afutures purchase and forward saleÐcan be seen as a substitution. Naturally, depending on the price at which each instrument is traded, the entire operation may result ina pro®t or a loss. We leave this particular element out here because we are concentrating on the interest rate component.
414
COMMON ARBITRAGE SITUATIONS
as a result of the forward trade. In contrast, the interest rate exposure ofthe entire portfolio has been signi®cantly modi®ed because r2M is notthe relevant variable anymore. To reestablish a satisfactory hedge, weneed to enter into a transaction that has no effect on the treasury whilesimultaneously hedging a rate exposure.In fact, the situation is slightly subtler than that because in 2months the existing money market disappears. What we really needthen is a transaction that allows us to access cash liquidity in 2 monthsat a rate set today, for a total maturity of 10 months. This is typicallywhat a FRA is designed for, although a swap of eurofutures could beused as well.The risk created by the forward transaction is not on the 2-monthor 1-year rate but rather on the 2  12 FRAÐthat is, 10-month rate in 2months. We can hedge it with a 2  12 FRA, or a swap starting on thefutures expiry date, or a eurofutures strip starting on the futures expirydate as well. The best choice depends on a number of factors, the mostimportant probably being the net cost to perform the hedge. FRAsbeing cash transactions, we know we can expect a wide bid-ask spread,and consequently they are unlikely to be the cheapest solution. Swapsare a very strong alternative, but for the particular case of a 10-monthmaturity, eurofutures are probably the most attractive consideringtheir high liquidity and very low bid-ask spread. If the forward hadbeen a longer-term transaction, for example, beyond 2 to 3 years, then aswap would have probably been the most likelyÐand in fact onlyÐdecision.Naturally it is always possible to neglect that risk and keep theposition unhedged. However, because the true maturity here is 10months, the effect of an adverse move on interest rate can be sizable.If we recall the example we used before, a 0.10 percent move would inthat case result in a loss of41:0110  0:1%BC12P  $1,000,000,000  @A  $786,16312  6%112These examples show clearly that interest rate risk can be signi®cant,although relatively easy to manage in general.41
The term 10  0.1 percent/12 corresponds to a loss of 0.10 percent in 10-months, whilethe discount factor (1 + 12  6 percent/12) is necessary because the loss occurs in 12months at the expiry of the futures. The formula is not exactly the forward valuationbut an estimate of the loss associated with a change in the 2  12 market FRA.
CHAPTER 8 INDEX ARBITRAGE
415
Recomposition
By de®nition of index arbitrage, an arbitrageur is supposed to carry areplicating portfolio of the underlying index, which is the only way toguarantee that the futures hedge is effective. This operating modeimplies a dynamic replication throughout the life of the positionÐthat is, adjustments whenever the index itself is adjusted by its sponsor.Naturally the most important question here is whether these changescan be implemented at no costÐthat is, if they are what we called``realizable.'' We argued in the chapter dedicated to indices that thisproperty was a fundamental requirement for sponsors, in terms ofconsistency in the index behavior as much as from a purely economicperspective.To assess the risk embedded in a recomposition event, let's consider again the case of Nikkatsu in 1993. Nikkatsu was a movieproducer and went bankrupt suddenly in July, 1993. Although thecompany was known to be in ®nancial dif®culties, its collapse surprised the market. Nihon Keizei Shinbun, the Nikkei 225 sponsor,decided to remove Nikkatsu from the index on July 1, 1993, effectiveimmediately, and replaced it with Shionogi Corporation. The changetook place at the close of the day, and the new divisor was computedaccordingly. At that point arbitrageurs faced a tough situationbecause they were naturally unable to realize the index substitutionat a time the market was closed. Furthermore, it was quite certainthat the price of Nikkatsu would fall sharply the next day, andindeed this is exactly what happened. Figure 8-26 shows closingprices for Nikkatsu and Shionogi over the period, scaled to 100 percent on the day of the announcement.Arbitrageurs were still required to implement the adjustment inthe index, so most of them tried to liquidate Nikkatsu and buyShionogi the next day. In doing so, they incurred a signi®cantloss, which we can estimate easily. Assuming a $1 billion positionÐthat is, ¥100 billionÐwe know that we would have heldclose to 555,500 shares of Nikkatsu.42 The price loss for Nikkatsuwas ¥204, from ¥229 to ¥25, while Shionogi gained ¥65 from ¥985 to¥1,050. In selling Nikkatsu, arbitrageurs thus lost ¥113.3 million, and¥36.1 million more in buying Shionogi. The total loss, close to $1.5million, represents 17.6 percent of the expected pro®t for the year if
42
Cf. footnore 29. The Nikkei 225 is price weighted so that all components have thesame weight.
416
COMMON ARBITRAGE SITUATIONS
Figure 8-26 Nikkatsu(Source: Compiled by author from Bloomberg data.)
we assume an overall pro®tability of 0.85 percent. As in the case ofSumitomo, this is not dramatic but it is certainly signi®cant.This is a very telling example of a situation that is virtuallyimpossible to hedge, even though it may have a sizable negativeimpact. Nowadays examples of such adverse events are relativelyrare, and the best way to prepare for them is to know the exactrules applied by a sponsor whenever an index member ®les forbankruptcy.
CorrelationPercentage of Execution RevisitedWe have so far mentioned several times the notion of replicating portfolio, assuming that we would have at all times a perfect portfolio withthe right number of shares for each constituent, based on the underlying index de®nition. It is now time to revisit this particular assumption and examine the correlation risk associated with an unperfectposition.Indeed, the constitution of a perfect portfolio is a relatively easytask theoretically, but it is more dif®cult in practice. As an illustration,let's revisit the execution risk and more particularly the measure of the
CHAPTER 8 INDEX ARBITRAGE
417
percentage of execution. If we denote eS this percentage on the cashside, eS can be de®ned as43:P
Qexec PlastiieS  iPQi  Plastii
where Qexecand Qunexecare the executed and unexecuted parts of theiiis the average price of execution of executedoriginal quantity Qi , Pexecilast44orders, and Pi is the last traded price of stocks for which thereremains an unexecuted part. This de®nition does not make any difference for the relative executed position of each individual stock, however. If, for example, a small number of stocks are entirely executed,while a large fraction of the remaining portfolio is almost entirelyunexecuted, eS may be relatively high although the position is notwhat we would like it to be.As a simple example, consider that we are buying an index withonly two stocks, A and B, each with a weight of 50 percent. If A isentirely executed while B is entirely unexecuted, the total percentage ofexecution is 50 percent, which means that we would trade 50 percent ofthe total size on the futures leg. Naturally this is radically differentfrom 50 percent executed on each stock, which would still require usto trade the same number of futures. To get a sense of the magnitude ofthe difference, let's call PA and PB the prices at which A and B tradewhen we execute A, and P0A and P0B the same prices when we execute B.If we assume for simplicity the index to be price weighted, the following relationships are true:I
43
PA  PB2
I0 
P0A  P0B2
Note that eS is homogeneous to a capitalization. Therefore, eS is really a priceweighted percentage of execution.There is an alternative de®nition of eS :P exec execQi  PieS  P exec i exec P unexec lastQi  Pi  Qi Pii
i
If this last formula is used, the de®nitions of L and S later in the text should bemodi®ed accordingly.eF , the percentage of execution on futures, does not appear here because we consider a situation in which we are hedged at all timesÐthat is, eF  eS .44Therefore, at all times, Qi = Qexec Qunexec:ii
418
COMMON ARBITRAGE SITUATIONS
if I and I 0 respectively denote the instantaneous value of the index inthe two instants when A and B are executed. If F and F0 denote theprice of the futures at these two moments, and bTH is the fair value forthe day, then we know that F > I  bTH because otherwise we wouldnot have entered the arbitrage in the ®rst place. Let's call bM the marketI  bTH . Inbasis when we originally enter the arbitrage, bM  Fessence, this situation represents an opportunityÐthat is, bM Ðwe aretrying to capture but for which only 50 percent of the portfolio isexecuted immediatelyÐthat is, A. To complete the picture, let's makethe ®nal assumption that the opportunity has disappeared when we®nally execute BÐthat is, F0  I 0  bTH :The purpose of the exercise is to show how the execution partitionbetween A and B affects the overall pro®tability of the trade. If P&Lstands for the expected pro®t for the trade, it is given by the differencebetween the realized value of the future and that of the index. If Fexecand Iexec respectively denote the executed prices on futures and index,then the P&L is given by:Iexec  bTH 
P&L  FexecIn addition, we know that:Iexec 
PA  P0B2
I  bTH  bM   I 0  bTHF  F0I  bM  I 0 bTHFexec 222Iexec is a result of the de®nition of PA and P0B , while Fexec results fromthe fact that we trade the total number of futures at two different pricesF and F0 . The total pro®t is then equal to: 0PA  PBPA  P0B bM PA  P0B22 bTH bTHP&L 22which can be simpli®ed into:  0PA  PBPA  P0BP&L 44and ®nally:P&L 
 0PA
4
PA

PB

4
PA  P0Bb M22
P0B

b M2
Interestingly, despite the fact that A is fully executed immediately, westill carry an exposure to its price through P0A . Along the same lines,
CHAPTER 8 INDEX ARBITRAGE
419
even if B is fully unexecuted when we initiate the trade, the total P&L isexposed on PB as much as it is on P0B .Consider now a situation in which we execute half the total position on A and B instead of executing A only and the remaining halvesinstead of executing B. The only change is Iexec , which becomes:  0PA  PBPA  P0B22Iexec 2The total pro®t is naturally affected and is now equal to: 001PA  PBPA  P0BPA  PB P0A  P0B bM B 2C222C bTH BbP&L TH@A22Therefore:P&L 
bM2
This formula is clearly much more attractive than the previous onebecause it depends only on the market basis, not on the prices of Aor B anymore. In fact, the pro®t in that case is the average of the marketbasis when we executed the ®rst 50 percent of the entire positionÐthatis, bM Ðand the market basis when we executed the remaining 50 percentÐthat is, 0: P&L = (bM + 0)/2. This result is very intuitive and canbe easily generalized: If execution within the position is homogeneousfor each index constituent, the total pro®t on the entire trade is theaverage of the market basis throughout the execution period.45 In contrast, if the execution is not homogeneous, then the overall pro®tabilitydepends not only on the current price of each constituent but also onthe price it was trading at if it is ``overexecuted'' or will be trading at ifit is ``underexecuted.'' This dependence is usually referred to as acorrelation risk between the portfolio and the index because despite thefact that the portfolio is perfectly hedged against futures, it may nottrack the index accurately, and it may lose value compared to thefutures due to its inexact composition.The level of risk associated with correlation can be measured bytwo aggregate indicators, which we denote L and S. L is the total45
Rigorously speaking, the execution is homogeneous if, for example, at 10 percent ofexecution, every single component in the position has been executed at 10 percent ofits entire size.
420
COMMON ARBITRAGE SITUATIONS
nominal created by overexecutionsÐthat is, stocks for which we executed more than the overall percentage of execution. S is the similarnominal for stocks that have been underexecuted. Algebraically L andS can be written as46:XXLQexeceS  Qi   PlastSeS  Qi Qexec  PlastiiiiQexeceS Qii
Qexec<eS Qii
Because we assume a position correctly hedged at all times, we knowthat we can expect L  S. However, what is important here is themagnitude of L and S compared to the original nominal, or the totalnominal executed up to this point.47 In fact, the situation is exactlyequivalent to a perfect replicating portfolio of size eS  Qi , hedgedwith the appropriate number of futures, and two additional oppositeeS  Qi and si  eS  Qi Qexec.portfolios (li ) and (si ) with li  QexeciiL and S are the current nominal of each of these portfolios. This spreadposition between L and S is unwanted and the result of the randomexecutions coming from the market. Ideally we want to have L and S assmall as possible relative to the total executed position, but managingthem in real time is not an easy task. In fact, the execution problem tominimize correlation risk can be formulated as trying to maximize eSwhile bM is advantageous while keeping L and S as low as possible. Inthe S&P 500, for example, it is dif®cult to follow 500 components atonce, and this is all the more dif®cult if we require a homogeneouspercentage of execution across the board. Therefore, in practice arbitrageurs try to achieve a homogeneous percentage within the largestcapitalizations as fast as possible, knowing that the smallest components will have only a marginal effect on the overall pro®tability andrisk. In other words, it is generally true that the overall pro®tability of aparticular operation is determined within a few minutes of the oppor46
47
Note that the alternative de®nition in which we would replace Plastby Pexechas noiiinterest because we want to follow the evolution of L and S in real time.L
X
S
Qexeces Qii
Qexeci
X
S
QexeceS Qii
Xi
Qexec Plastii
Qexec Plastii
by de®nition of es .
Qexec  Plastii
eS  Qi
Qexec <eS Qii
which eventually leads to:L
X
es  Qi   Plasti
eS 
Xi
X
Qexec Plastii
Qexec <eS Qii
Qi  Plast0i
eS 
Xi
Qi  Plasti
CHAPTER 8 INDEX ARBITRAGE
421
tunity, even if the execution can spread through a much longer periodof time.IllustrationAs a hypothetical numerical illustration of the discussion, consider aprice weighted-index with three components, A, B, and C. Assume thethree stocks have a weight of 100 in the index, and that the divisor is 1.If we want to buy one index we have to purchase 100 shares of each:QA  100, QB  100, and QC  100. In addition we assume the follow 20, Qexec 80, Qexec 50, PA  $10, PB  $15,ing parameters: QexecBACand PC  $30. eS , L, and S are easy to compute:eS 
20  $10  80  $15  50  $30$2,900 52:7%100  $10  100  $15  100  $30 $5,500
L  80
52:7  $15  $409:1
S  52:7
20  $10  52:7
50  $30  $409:1
In essence, we have here a portfolio of 52.7 shares of A, B, and Cperfectly hedged against the appropriate number of futures, and aspread position between 27.3 shares of A on one side and 32.7 sharesof B and 2.7 shares of C on the other side. Because the perfect portfolioof 52.7 shares is a perfect replication of the index, its pro®tabilityis preserved as long as the market basis remains advantageous.However, if we wait too long to complete the trade, regardless of bM ,the spread between L and S may evolve unfavorably and dilapidate theoriginal pro®t.To evaluate if that scenario is potentially threatening, consider thatwe expect to realize a pro®t of 0.15 percent on this trade. With an indexat $5,500, 0.15 percent gives a nominal pro®t of $8.25. In other words,the market basis is probably close to $8 when we enter the arbitrage.The spread position between L and S has a nominal of $409.1, whichmeans that an adverse move of 1 percent on each leg would generate aloss equal to the expected pro®t on the entire portfolio. Despite thehypothetical nature of the argument, it is clear that a 1 percent move isvery common, and the risk of losing the entire expected pro®t or evenmore is extremely real. The best way to diminish the risk here would beto buy A and C more aggressively to increase eS and decrease L and S.4848
Alternatively we could sell B, but this does not really make sense because we wouldhave to buy it back at some later point to complete the trade.
422
COMMON ARBITRAGE SITUATIONS
CONCLUSIONHere ends our discussion of index arbitrage. The reader should get asense of the diverse dimensions involved in managing a position ofthat type, in terms of risk and opportunities. The dif®culty naturally isto bring every aspect together in practice, incorporating all the valuation and risk analysis techniques to make the best decision at a giventime. As always, judgment and experience are powerful allies, and across-border experience is particularly valuable to gain a broad view ofissues such as execution, dividends, or correlation beyond one singlemarket. This is exactly what we have attempted to achieve here, aglobal perspective hopefully rich in rewarding insights.
REFERENCESThe following are valuable resources on the subject of index arbitragepro®tability:George So®anos, ``Index Arbitrage Pro®tability,'' The Journal ofDerivatives, fall 1993.Wolfgang Buhler and Alexander Kempf, ``DAX Index Futures:Mispricing and Arbitrage in German Markets,'' Journal of FuturesMarkets, 15, 1995.Kee-Hong Bae, Kalok Chan, and Yan-Leung Cheung, ``The Pro®tabilityof Index Futures Arbitrage: Evidence from Bid-Ask Quotes,''Journal of Futures Markets, 18, 1998.Michael Brennan and Eduardo Schwartz, ``Arbitrage in Stock IndexFutures,'' Journal of Business, 63, 1990.Peter Chung, ``A Transactions Data Test of Stock Index Futures MarketEf®ciency and Index Arbitrage Pro®tability,'' Journal of Finance, 46,1991.
9R ISK A RBITRAGE
INTRODUCTIONIn the most general sense, risk arbitrage describes trading strategies inwhich the entire pro®t is based on the assessment and trading of oneparticular class of risk. Over the years, however, the term has becomemore and more speci®c and now refers almost unequivocally to thesystematic arbitrage of corporate events, most notably mergers andacquisitions. The underlying concept is even more straightforwardthan index arbitrage. The terms of the corporate event specify exactlywhat investors will receive in exchange for their holdings when theoperation effectively closes: stocks, cash, or a combination of the two.Let's denote, for example, A the acquirer, T the target, and r theexchange ratioÐthat is, 1 share of T is exchanged for r shares of A atexpiry.1 An arbitrage situation can be represented as shown in Figure 9-1,where PA denotes the market price of A and PT that of T. In this diagramLastPT trades at a discount because r  PLastA > PT , although if we account forthe bid-ask spreads, the opportunity is not necessarily easy to capture.PLastorIt would appear that the total ``available'' pro®t is r  PLastTABidAskPT if we incorporate the bid-ask spreads, but in fact, this isr  PAnot rigorously true notably because of dividends and borrowing costs.To obtain a clear expression of the pro®tability, let's examine the situation more rigorously. Figure 9-2 shows the relevant parameters if weenter a position on t0 and close it on expiry t1 . The initial trades are1
We implicitly assume that the operation is not a straight tender offer; otherwise, thediscussion is trivial because investors just tender shares bought at a lower price. Alongthe same lines, the exchange might incorporate a cash part without any signi®cantchange in the reasoning.
423
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424
COMMON ARBITRAGE SITUATIONSASKr.PALASTr.PA
ASK
PT
Discount
BIDr.PA
LAST
PT
BID
PT
Figure 9-1
Arbitrage Situation
executed at PA and PT respectively. At expiry the closing prices2 arerespectively P0A and P0T . In addition, we call dA , dT , rA , and rT thedividends for A and T and the relevant zero coupons between thepayment date and expiry. Finally, rE , and bA denote the zero couponfrom t0 to t1 and the market borrowing cost for A:3If we want to take advantage of the arbitrage, we need to enter thefollowing transactions:On t0 :. Buy 1 share of T at PT .
. Secure a borrowing of r shares of A until expiry at bA :4t
, P
0
A
, P
t
T
, P'
1
A
, P'
T
Toda y
Expiry
d
T
r
T
d
r
E
Figure 9-2
r
A
A
, b
A
Arbitrage Cash Flow
Note that we do not necessarily have r  P0A  P0T . This is true only if the two stocksclose at par on the last trading day before the merger is effective.3This assumes that we sell AÐthat is, T trades at a discountÐwhich is generally thecase. However, if T were trading at a premium