avorableshift in the market. Assume, for example, that the total size is $15million on the S&P 500Ðthat is, 43 futures with an index at $1,400.For the ®rst seven minutes the difference in executed percentages isrelatively steady at around 30 percentÐthat is, 13 futures. If we take
Time Diagram: Percentage of Execution
100%90%80%70%60%50%40%30%20%10%0%
Stocks0
1
2
3
4
5
6
Time (arbitrary unit)
Figure 8-14
Percentage of Execution
7
Futures8
9
10
396
COMMON ARBITRAGE SITUATIONS
the arbitrage to be a buy tradeÐthat is, long cash and short futuresÐthe arbitrageur is essentially short 13 futures for a period of sevenminutes. If the market goes down during that interval, he or she cancomplete the cash side or buy back the extra futures at a pro®t. If themarket goes up, however, he or she may end up losing a signi®cantpart of the anticipated pro®t, if not all. If the expected pro®t is $2.5 on43 futuresÐthat is, $26,875, which we know is a very optimistic ®gureon averageÐa move of $4 on the futures is enough to dissipate 50percent of the original pro®t, and in fact $4 over a few minutes isextremely common. It is tempting to argue that if the futures wentup, the arbitrageur would still have an opportunity to complete thecash side because it is presumably going up more slowly. This is true toa large extent, but it does not really change the outcome: The trade ismuch less pro®table if not losing.We argued earlier that nonsimultaneous execution was not entirelynecessary and should even be avoided in a pure arbitrage situation.Indeed, if an arbitrageur built a short in his or her execution patternbecause of an anticipated move, he or she might as well have taken astraight short position without bothering with the complications of atwo-legged strategy. It is all too tempting to blend pure speculativepositionsÐhighly riskyÐto risk-free trades and try to make the bestout of the combination. In practice, the result is rarely satisfactorybecause these strategies do not require the same skills, and blendingthem does not enable a trader to make an assessment of his or herability to successfully trade one or the other.Execution on Expiry: The Case of Sumitomo in June 1996Beyond real-time execution another type of execution risk is related tothe expiry of the futures. We mentioned before the case of an expiry onan averageÐfor example, in France and the United KingdomÐasbeing more risky because of the nature of the averaging process. Infact, even futures expiring on outright prices such as opening can betricky to execute under some circumstances.An interesting case of this situation was illustrated by SumitomoCorp. in Japan. Sumitomo is a large Japanese corporation involved in anumber of commercial activities, and commodity trading is one ofthem. For years Sumitomo has been highly respected in the coppermarket because it employed one of the most renownedÐandfearedÐcopper traders, Yasuo Hamanaka. In June 1996, however, amajor scandal broke out, bringing to light years of fraudulent activityon Hamanaka's part. Sumitomo originally declared losses of $1.8billion, soon to grow to $3.5 billion by September.
CHAPTER 8 INDEX ARBITRAGE
397
On the day of the announcement, Friday, June 14, 1996, the TokyoStock Exchange suspended trading in Sumitomo (ticker: 8053) for theday.27 As it happened, this particular day was a quarterly Nikkei 225futures expiry, and Sumitomo was included in the index. Figure 8-15shows Sumitomo's closing price over the period and its daily tradingvolume. The stock eventually opened on the following Monday toclose at ¥1,010, losing 16.5 percent.28 This particular gap is in itselfan interesting example of a leptokurtic distribution, but more interesting to us, it had a signi®cant impact on index arbitrageurs. Manyarbitrageurs were planning to unwind their stock position at theopen by realizing the SQ value of the futures. Because liquidity atopen is usually plentiful, such an unwinding is easy to implementand risk free. The fact that Sumitomo was suspended, however, complicated the situation because the applicable rule stated that the priceattributed to Sumitomo for the SQ calculation would be the last traded
Figure 8-15 Sumitomo(Source: Compiled by author from Bloomberg data.)27
See, for example, ``Sumitomo Says Unauthorized Dealings in Copper Caused $1.8Billion in Losses,'' Wall Street Journal, June 14, 1996, p. A3.28Interestingly, it traded in London on June 14 at a price anywhere between 10 and20 percent compared to the last traded price in Tokyo, ¥1,210.
398
COMMON ARBITRAGE SITUATIONS
priceÐthat is, ¥1,210. In short, arbitrageurs were facing an SQ computed with Sumitomo at ¥1,210 but were unable to execute on thatprice. Furthermore, there was naturally a sense that Sumitomowould trade sharply lower, but in the absence of an active market,``sharply'' could have meant anything. Buyers were in a relativelycomfortable situation because any price lower than ¥1,210 was totheir advantage. Sellers, on the other hand, were desperate, andmany of them turned to London as soon as possible becauseSumitomo was listed on the London foreign market.To get a sense of the possible impact, assume a $1 billion positionÐthat is, ¥100 billion if we take a convenient exchange rate of ¥100 for $1.With a Nikkei around ¥18,000, a ¥100 billion position is equivalent to555,500 shares of Sumitomo.29 Therefore, a drop of ¥200 in price if theposition could not have been liquidated at ¥1,210 would have resultedin a net loss of ¥111 millionÐthat is, $1.1 million. Assuming the overallpro®tability of the position was close to 0.85 percent as we proposedearlier, $1.1 million represents 13 percent of the overall total, $8.5 million. In other words, this particular event does not necessarily compromise the pro®tability of the activity, but it is serious enough topotentially wipe out a nonnegligible part of the expected pro®t forthe entire year.There is really no way to hedge against this particular risk becauseit depends on a number of factors typically impossible to predict. It is,however, particularly important to be aware of the exact rules forhandling the SQ calculation under unusual circumstances. By andlarge, virtually all exchanges will use the latest traded price availablewhen one particular index constituent is suspended, but they maydiffer signi®cantly in the criteria applied to determine whether astock should be suspended or not.
Time Diagrams and RollsShort-Term RollsConsider a situation in which we enter an arbitrage position by buying a stock replicating portfolio and selling the appropriate number offutures. Assuming we carry the entire position or part of it until thefutures expiry, let's consider in detail what happens on that particularday.29
The weight of Sumitomo in the Nikkei 225 was 1,000, and the divisor ¥10,000.Therefore, ¥100 billion is equivalent to (1,000/¥10,000)  (¥100,000,000,000/¥18,000) 555,500 shares of Sumitomo.
CHAPTER 8 INDEX ARBITRAGE
399
After the close of the previous day, the futures do not trade anymore, but technically speaking, they still exist because the expiry hasnot occurred yet. In essence, the expiry occurs when the SQ price isknown with certainty because the futures contract is fully de®ned thenand all ¯ows can be settled. If we consider a market where the SQ isreferenced at the open, the position is still technically hedged by a shortposition until that moment, although the futures leg cannot be tradedanymore. Immediately after the open, when all stocks have opened, theSQ is known and the short leg simply disappears. The holder of anarbitrage position is then left with an outright stock position, suddenlynot hedged anymore. This is clearly a problem because a $1 billion riskfree arbitrage position is an entirely different proposition from a $1billion long stock portfolio. There are really only two solutions tothat potential problem: Sell the portfolio at the SQ so that the expiryof the futures is matched by the realization of the same price by liquidating the portfolio, or trade the futures position away so that it disappears before expiry.In the second case, to remain hedged, for every futures that we buyback, we need to sell an equivalent nominal of something, and if wedetermine that we want to keep the stock position intact through theSQ, we need to sell a futures of a later expiry. This type of calendarspread is commonly referred to as a roll, and the position has simplybeen rolled over to a further expiry. It is convenient and enlightening torepresent the whole operation as a time diagram, as shown in Figure8-16. When the position has been entirely rolled over, we do not holdthe nearest futures anymore, but the entire stock position is nowhedged with futures further away.Because this type of spread is traded extremely frequently, it hasbecome a listed instrument in and of itself. Typically two to three
2 Months
Toda y
3 Months
SQ
SQ
Far Future
Near Future
Calendar Spread = RollFigure 8-16
Roll
400
COMMON ARBITRAGE SITUATIONS
weeks before a futures expires, the next maturity becomes liquid andtraders can get very deep market on the roll. The quote convention is``far minus near''Ðthat is, a March to June roll quoted at $20 indicatesthat the June futures is quoted $20 above March. As a consequence,selling the roll means selling the far maturityÐJune, for example, in aMarch to June roll.Although two futures on the same index clearly have commoncharacteristics, they do not have the same price. The further awaythe futures, the larger the ®nancing component in its price, whichexplains why in general far maturities are more expensive than nearones. Let's consider as an illustration a March to June spread with thefollowing characteristics: rMar  6 percent, rJun  6:5 percent, tMar  5days, tJun  95 days, I  $1,400, dMar  $0:20, dJun  $7. The theoreticalprices of each futures are as follows:5  6%TH$0:20  $1,400:97FMar  $1,400  1 360
FTHJun  $1,400 
95  6:5%1360
$7  $1,417:01
The theoretical roll is then $1,417.01 - $1,400.97 = $16.05. Naturally theroll can trade at a price different from its theoretical price, the sameway a futures can trade above or below its theoretical price. The samecauses having the same effects, there is an arbitrage opportunity if theroll is trading too far from its ``fair'' price.Consider, for example, that the roll is too expensive compared to itstheoretical value. Under these circumstances, it is very intuitive thatarbitrageurs will be natural sellers. Imagine one of them, with no position, selling a March to June spread above its fair value.30 Until theexpiry of March, the spread position is exposed to the market throughthe quoted roll price. If the transaction was originally March to June at$16, and if the market quotes $17 on a given day, unwinding theposition would create a loss of $1. If the spread is kept until Marchexpires, we know that the occurrence of the SQ will require the arbitrageur to buy something to replace the expiry leg, or to sell the Junefutures in order to remain reasonably hedged.30
Naturally, the same applies to any investor in the market. An arbitrageur in that casemay pursue the opportunity more aggressively, but the market at large is very likelyto enter at least into a ``passive arbitrage'' mode.
CHAPTER 8 INDEX ARBITRAGE
401
The arbitrageur has two choices: Buy a replicating portfolio at theSQ price, or sell the June futures at a price as close as possible to the SQ.Buying a portfolio of stocks is a reasonably safe proposition in general;however, selling futures is highly uncertain because the market price ofthe contract depends on a number of factors and may be signi®cantlydifferent from the reference at which we need to trade.31 In general,thus, arbitrageurs tend to buy cash, and this is all the more true if thereis an additional advantage in holding stocks, like a tax rebate or incremental revenues from stock loans.Obviously, traders are very aware of the possible dif®culties associated with selling the roll in the ®rst place, which means that in general the purpose of the sale is indeed to buy a replicating portfolio. Thisis in fact a very interesting and safe way of doing that because theposition transforms itself into a fully hedged arbitrage portfolio onexpiry of the long futuresÐMarch in our example. Conversely, buyingthe roll requires selling stocks on expiry, which in turns implies borrowing costs if the stocks are not held in the ®rst place. Therefore, it isno surprise that the roll is in general used much more frequently tobuild a long stock position than the opposite.Consider now the consequences of this discussion. If the roll istrading above fair value, arbitrageurs will probably sell it at ®rst toroll their existing position. Once these are entirely rolled, they still havethe option to roll more and to buy stocks on expiry. On each roll sold,they make a pro®t equal to the difference between the market price andthe fair value of the roll. Alternatively, if the roll is trading below fairvalue, they will suffer a loss if they decide to roll existing positionsanyway, which means that they will have to liquidate on SQ. Theymight even consider selling more than what they hold if the roll issuf®ciently cheap. Therefore, an expensive roll usually creates a buyingpressure on the SQ while a cheap roll has the opposite effect. Naturallythese are not absolute rules, but they are fairly reasonable as far asarbitrageurs are concerned. With respect to the market at large, however, it is more dif®cult to draw conclusions, but we can safely assume
31
Naturally, if the SQ is an average, as in France or the United Kingdom, we know thatthe situation is not as simple as it is on an opening. The problem then is to weigh therisk associated with the realization of the SQ on cash versus executing on the futures.Again experience is a helpful guide in these situations.The price we need to realize on the cash is the SQ price of March contract.Therefore, the reference price on the June contract is the theoretical price of the Junefutures based on an index price equal to the SQ.
402
COMMON ARBITRAGE SITUATIONS
that ``passive arbitrageurs'' will have a similar analysis of the situationand similar conclusions.Therefore, if there were a way to observe the level of roll in themarket, we could then infer a number of conclusions for the next SQ,and possibly anticipate its direction. This type of focus on the SQ mightappear surprising, but quarterly events of such nature are generallywidely regarded as ``signals'' of things to come. In other words, apositive SQÐthat is, an expiry for which stocks move upÐis usuallyconsidered a bullish sign or at the very least a sign of con®dence in themarketplace. Accordingly there is some value in trying to get a sense ofSQ imbalances, and fortunately there is in fact a way to assess the levelof roll in the market through the futures open interest.Open InterestThe open interest indicates the number of open contracts in the marketfor a given expiry. If a buyer and a seller with no prior position trade afutures contract, the open interest of the contract is increased by 1because this is a ``new'' commitment. If, in contrast, a contract is tradedbetween two parties and closes a position that was opened before, theopen interest is reduced by 1.Consider now an investor with a long position in a particular contract, say, March. If the investor decides to roll the position into June,he or she buys the rollÐthat is, he or she sells his or her March contracts and buys an equal number of June contracts. In doing so, theaggregate open interest of March and June combined remainsunchanged because one contract literally replaces the previous one.Consider now an arbitrageur selling the roll with no preexisting position in futures. The transaction leaves the aggregate open interestunchanged if the buyer closes a position or increases its absolute sizebecause the arbitrageur is now long one March and short one Junewhen he or she had no position before.Therefore, we can draw conclusions from the aggregate level ofopen interest only if we have a sense of the preexisting positions. Inpractice, this is often the case because different classes of investors tendto have recurrent positions. Index arbitrageurs, for example, tend to beall on the same side. In the United States and Japan, they are long cashand short futures so they are natural sellers of the roll to keep theirpositions. In France and Germany, on the other hand, they are muchmore often short stocks and long futures; therefore, they would benatural buyers of the roll. Investors at large tend to be in general onthe long side, except hedgers using the futures as a protective device.This schematic presentation is very useful in understanding the impact
CHAPTER 8 INDEX ARBITRAGE
403
of the roll of the market in general and on the SQ in particular. If theroll is expensive and the aggregate open interest increases just beforethe SQ, chances are that arbitrageursÐand possibly other investorsÐare selling it aggressively, and on expiry they have to buy cash toreplace the dying futures. The result is most probably a buying pressure building in the market on the SQ. The opposite conclusion couldbe made from observing a cheap roll and an increase in open interest.This situation is much less frequent, however.As an illustration, Figures 8-17 through 8-21 present the aggregateopen interests of several futures over the past year.32 Beyond thenumerical values that are relatively unimportant here, the shape ofeach chart reveals again a number of details about the underlyingmarkets. Very regularly the aggregate open interest literally jumpsbefore the SQ, to fall back again immediately after. This is an effectthat is visible virtually everywhere, albeit with different magnitudes.
Figure 8-17 S&P 500 Open Interest(Source: Compiled by author from Datastream data.)
32
The y axis is the open interest in the number of contracts for each designated maturity. ``First expiry'' denotes the nearest futures contract, ``second expiry,'' the next one.Note that the chart for the S&P 500 was presented in an earlier section. France has amonthly cycle; therefore, the second expiry includes also the third one correspondingto a quarterly cycle when necessary.
404
COMMON ARBITRAGE SITUATIONS
Figure 8-18 FTSE 100 Open Interest(Source: Compiled by author from Datastream data.)
Figure 8-19 Nikkei 225 Open Interest(Source: Compiled by author from Datastream data.)
CHAPTER 8 INDEX ARBITRAGE
Figure 8-20 DAX 30 Open Interest(Source: Compiled by author from Datastream data.)
Figure 8-21 CAC 40 Open Interest(Source: Compiled by author from Datastream data.)
405
406
COMMON ARBITRAGE SITUATIONS
This is clearly the sign that an unusually large number of contracts hasbeen created just before the expiries.To better understand the phenomenon, consider the case of theUnited States. On average, over the period the total open interest isbetween 350,000 and 400,000 contracts. Far from an expiry, only thefront month is truly liquid; therefore, the open interest is heavily concentrated on this particular contract. As the expiry approaches, investors have four alternatives. They can unwind their front month positionbefore the expiry, which does not increase the open interest.33 They canroll their front month to the back month, which keeps the open interestexactly unchanged. They can decide to let their position expire on theSQ, with or without intervening in the cash market. This strategywould have no impact on the open interest before the SQ, but itwould decrease it signi®cantly on that particular day. And ®nally,they can overrollÐthat is, purchase or sell the roll beyond their originalposition. In both cases the open interest actually increases before theSQ and decreases again on the SQ when the front month expires. Table8-2 shows a summary of the diverse strategies and their respectiveimpact on the open interest.The only decision that increases open interest is the last one, andit corresponds to a situation in which arbitrageurs and investors atlarge take a position on the roll beyond their natural need. Thisposition may be speculative in the sense that arbitrageurs are tryingto take advantage of the mispricing of the roll, or it may be morefundamentalÐif it is associated, for example, with the tax advantagedue to dividends. In addition, we do not know just from the abovecharts if this position is built by selling or buying the roll becauseboth lead to the same effect. By and large, selling is much moreTABLE 8-2DecisionLiquidate before SQNo action, let it expireRoll to back monthOverroll
33
Impact on Total Open Interest Impact on Total Open(Front + Back Month) before SQInterest on SQor ééé+
éé
For example, if a long position is liquidated, if the buyer of the contracts is notliquidating a short position, the open interest is unchanged. If the buyer is liquidatinga short, then the interest is decreased. In any case, it is never increased.
CHAPTER 8 INDEX ARBITRAGE
407
frequent, however, because it is a very simple and safe way to build along stock position on the SQ.As far as local particularities are concerned, experience shows thatthe roll has been historically much more often expensive than cheap inthe United States and Japan, which unambiguously points towardarbitrageurs making pro®t every quarter on their existing positionand probably buying more cash through the roll.34 Incidentally, notethat the overall level of total open interest remains relatively constantin the United States while it decreases slightly in Japan, which may be asign of a general decrease in activity over the end of the year in thatparticular market.In the United Kingdom the same conclusion does not hold, and infact, the United Kingdom chart presents the only signi®cant case of a¯at aggregate open interest in September 1999. It is dif®cult to makeany de®nitive conclusion, but the basis at that time was edging negative, and this may have caused arbitrageurs and investors in general tobe less aggressive in their strategy on the SQ.Germany presents a very interesting chart, consistent with the conclusions we made earlier. Considering that most dividends are concentrated during the second quarter of each year, the roll at that time wasaggressively sold to allow arbitrageurs to buy stocks massively inanticipation of the dividend season. The total open interest droppeddramatically in June, which probably indicates that a sizable fraction ofpositions held after March were liquidated in June when the dividendgame was over.France also presents an interesting situation, primarily because thefutures cycle is monthly, which means that we have 12 expiries insteadof 4. In contrast to Germany, however, the dividend effect is relativelysilent, and certainly insigni®cant compared to the sharp increase inopen interest visible at the end of the year. This situation is dif®cultto interpret. Clearly, dividend advantages are pursued much lessaggressively than in Germany, but beyond that preliminary conclusion,a more exhaustive examination of the market would be necessary tomake better sense of the chart.In conclusion, open interest constitutes another source of valuableinformation on the SQ but much more generally on market behaviorin general. Naturally it should be examined in conjunction withthe market basis because both present essentially complementaryinformation.
34
We included these pro®ts when assessing the overall arbitrage pro®tability.
408
COMMON ARBITRAGE SITUATIONS
Hedge RatiosWe neglected in the above presentation an important factor about theroll, precisely the fact that different futures do not have the same hedgeratio. We mentioned in the chapter dedicated to risk analysis thathedging the ®nancing of futures margin calls required a modi®cationof the hedge ratio by a factor 1= 1  r where r denotes the relevant rateto the expiry.Different maturities have different r; therefore, they should not beadjusted by the same factor. Numerically, consider the two futuresmentioned earlier:5  6%TH$0:20  $1,400:97FMar  $1,400  1 36095  6:5%$7  $1,417:01FTHJun  $1,400  1 360Every replicating portfolio equivalent to one future should in fact behedged with:11r
1  0:9995  6%1360
March futures or:11r
1  0:98395  6:5%1360
June futures. Therefore, a March to June roll should be traded with aratio of 0.984 June for 1 March (= 0.983/0.999). If we had to roll a $1billion long positionÐthat is, short 2,857 futuresÐfrom March to June,we would have to buy 2,857 March contracts and sell only 2,811.3 Junecontracts ( 2; 857  0:984). In practice, the roll is always traded in themarket with the same number of contracts on each leg; therefore, wewould have to sell 2,857 rolls and buy back 45.7 June contracts( 2:857 2; 811:3) independently.Long-Term RollsWe've considered so far a short-term rollÐthat is, from one expiry tothe next. Naturally longer maturities are also commonly traded, forexample, March to September, or even June to September or June toDecember. However, it is sometimes interesting to roll the positionlonger term, typically beyond one year.
CHAPTER 8 INDEX ARBITRAGE
409
The reasons for doing that are multiple. First and foremost, rollingan existing position long term allows a trader to keep the bene®ts ofholding this position for a longer period. For example, if dividends oropportunities to lend stocks are signi®cant enough, an arbitrageur maydecide that he or she would like to enjoy this rent for as long aspossible. Alternatively, the price of a particular roll may be such thatan arbitrageur will decide to buy it, rolling a short only because itmakes economic sense. Overall, long-term rolls are interesting becausethey magnify the problems or opportunities associated with shorterones, typically with respect to interest rates, stock loans, and dividends. For example, even if it does not necessarily create any pro®timmediately, a long-term roll may prove bene®cial if it hedges orreduces an existing risk.Because long-term rolls are usually not listed, they must be tradedover the counter, as forwards, equity-linked swaps, or syntheticfuturesÐthat is, futures created by a call-put position. Depending ona number of factors, the market is more or less liquid, and credit riskreappears as a signi®cant issue again. This is all the more true becauselong-term instruments of that nature tend to be bought by institutionalinvestors rather than by traders.All the markets we consider have fairly liquid OTC markets forlong-term roll, typically up to two to three years. Beyond that term,trading depends on ®nding the right counterpart, which means thatquoted prices may be less favorable.Risks of Short SqueezesConsider an arbitrage position with the time diagram shown in Figure8-22. We are long 15 indices as of today, and against that short 25futures and long 10 long-dated forwards.35 Although it is not necessarily apparent immediately, this position carries the possibly signi®cantrisk of a short squeeze at some point in the future.Consider, for example, that we liquidate the ®rst 5 futures. At theclose of the last trading day of the next futures, the diagram looks as itis shown in Figure 8-23. The expiry of the 20 futures tomorrow requires35
By ``long 15 indices,'' we mean long a stock position replicating the index and suchthat the number of underlying indices is 15.Note that the diagram is simpli®ed and not rigorously accurate. The number offutures and forwards adds up to 15, which is the hedge against the stock position.However, the number of futures for each expiry should be adjusted by 1= 1  r tore¯ect the hedge of the margin call ®nancing. Forwards, however, are not subject tothis adjustment because they are not marked to market every day.
410
COMMON ARBITRAGE SITUATIONSStock Position
+15
+10
Toda y-5
Forward
-20Futures
Figure 8-22
Roll
that we sell 20 more futures of a later expiry, or 20 indices at the SQprice. Alternatively, we could sell 10 indices and roll only 10 futures.There is apparently no difference compared to a standard futuresexpiry, but this is only super®cially true. In general, an arbitrageurholding a long position against futures has entire latitude in decidingwhat should be rolled and what should be liquidated. In the particularcase we are considering however, this is not true anymore becauseeven if the stock position is liquidated, 10 more futures are expiringso the arbitrageur has to roll 10 futures, or sell 10 more indices short, tohedge the position adequately.This obligation removes any opt-out possibility from the arbitrageur if the market is unfavorable. Typically when the roll is below par,arbitrageurs tend to liquidate their position if they are long and if theydon't gain somewhere elseÐon dividends, for example. In the abovesituation, the arbitrageur would have no choice but to sell the roll, atany price; otherwise he or she would not be able to maintain a hedgedposition. If he or she wanted to sell stocks instead, he or she wouldface the risk that too many borrowers would appear at the same time,in effect making stock borrowings prohibitively expensive. This scenario is in fact extremely probable because, as we said many timesearlier, index arbitrageurs tend to react at the same time in the same
+10
Toda y
Tomorrow, SQ-20
Figure 8-23 Short Squeeze
+10
CHAPTER 8 INDEX ARBITRAGE
411
direction. Therefore, if one were facing a particularly unfavorable rollprice, most certainly all of them would carry out the same analysis andmake the same conclusion.To make matters more complicated, it is important to realize thatthis type of situation can develop only under a speci®c set