even subtler because some costs areasymmetric. The ``selling tax'' in Japan, for example, applied before itdisappeared only to the sale, not purchase, of stocks. If we buy ¥100million worth of Japanese stocks today, we do not pay any tax, but we
320
FINANCIAL VALUATION AND RISK ANALYSIS
should still provision an amount equal to the selling tax and hedge itagain with the appropriate long position. If we were now to take ashort position, there would be no need to account for any transactiontax anymore because buying back the short will not create any taxliability. However, the borrowing fee, usually charged as a percentageof the position periodically, would then need to be provisioned andhedged.
Conclusion
This simple discussion of costs and fees is not meant to be an exhaustive description of the costs associated with maintaining a tradingposition, arbitrage related or not. In practice, each market has itsown particularities in the types and magnitude of trading fees. It isuseful, however, in presenting the issues associated with these costsfrom a valuation and risk perspective. It shows clearly that a carefulanalysis of the nature and behavior of all trading expenses is necessaryto manage the overall position rigorously.
COST OF CAPITALIntroduction: The 1988 Basle Accord
In 1988 after more than a year of negotiations, the Basle Committee onBanking Supervision of the Bank of International Settlements (BIS)reached an agreement among its members on an international regulatory framework for bank capital requirements. This agreement markedan important phase in the development of international regulatoryguidelines because it included the 12 leading industrial countries andbridged over numerous legal and accounting differences.The goal of the committee was twofold: on one side, protect theinternational ®nancial system by imposing an international norm forcapital requirement, and level the playing ®eld with respect to banklending.39 The ®rst point is relatively straightforward to understandbecause imposing a minimal capital requirement results in a betterprotection of creditorsÐthat is, depositors. The second argument isbased on the competitive advantage created by a lack of harmoniza39
In particular, because Japanese banks had been aggressively taking advantage of thelack of harmonization in banking regulation to underprice their competitors andacquire market share. See, for example, John Wagster, ``Impact of the 1988 BasleAccord on International Banks,'' Journal of Finance, 51, 4, 1996.
CHAPTER 6
Bank A
VALUATION
321
Bank B
Assets
Liabilities+ Equity
Assets
Liabilities+ Equity
$100
$90
$50
$40
$10
$10
Figure 6-14 Balance Sheet
tion. Consider, for example, two banks with the same amount of equitycapital, $10. Bank A has no minimum capital whereas bank B musthold 20 percent of its total assets in the form of equity. Without any setlimit bank A can lend and grow its balance sheet signi®cantly, whilebank B is limited to $50 in assets because of its 20 percent requirement.Figure 6-14 shows an example of the two balance sheets. If both bankswant to present a return on equity of 15 percent to their shareholders,they must realize a pro®t of $1.5. On an asset base of $100, bank Aneeds a return of 1.5 percent, while bank B needs for the same performance a return of 3 percent ( $1.5/$50). Therefore bank A has morepricing power than bank B, and can gain market share without damaging its reported pro®tability.The trick here is that A's equity is much more risky than B'sbecause the level of debt is much higher. In other words, any downturnin the market may force A out of business while B would have lessexposure and remain solvent. To compensate for this higher risk, A'sshareholders would not be satis®ed with a return on equity close tothat of B but would demand much higher. This simple example showsthat a minimum capital requirement has a signi®cant impact on theeconomic and competitive situation of a bank.The Basle agreement was designed to address these issues andimpose a minimum capital requirement for banking institutions. Theprinciple adopted was to divide the equity part of a balance sheet intotwo tiers and de®ne a maximum asset coverage for this capital. Tier 1capital represents ``pure'' equityÐthat is, paid-up share capital forcommon stocks and disclosed reserves. Tier 2 includes capital assimilated to equity under speci®c conditions: undisclosed reserves, assetreevaluation reserves, general provisions and general loan-lossreserves, hybrid (debt-equity) capital instruments, and subordinateddebt.40 In front of the total capital available in each of these groups,40
For speci®c details of these categories, see Basle Committee on Banking Supervision,International Convergence of Capital Measurement and Capital Standards, 1988.
322
FINANCIAL VALUATION AND RISK ANALYSIS
the Basle committee de®ned risk weightings to apply to each assetclass, with the recommendation that the weighted asset nominal bematched with at least 8 percent equity, out of which 4 percent shouldbe Tier 1. For example, in the case of bank A, assuming $50 of the $100assets are risk weighted at 100 percent and the $50 remaining are riskweighted at 50 percent, the minimum capital is 8 percent  ($50  100percent + $50  50 percent)  $6, out of which $3 should be Tier 1Ðthatis, pure equity.This framework has a signi®cant impact on the management of abanking institution because it transforms equity into a scarce resource.Every dollar of asset on the balance sheet must have an equity base,which in turn means that this asset has to have a minimum returncommensurate with the institution's cost of capital. For example, ifequity holders expect 15 percent from their investment, a low-riskasset weighted at 20 percent has to have a minimum return of 0.24percent ( 20 percent  8 percent  15 percent), while a risky assetweighted at 100 percent would have to have a minimum return of1.2 percent ( 100 percent  8 percent  15 percent). In effect, therule serves as an incentive for banks to preferably hold low-risk assetsunless they are compensated with an appropriate return.Despite its general nature, this discussion has far-reaching consequences in the day-to-day business of most professional proprietarytraders. As we indicated before, many securities houses are subsidiaries of large banks, and they are therefore fully consolidated in thebank's balance sheet. All the assets held by the trading subsidiary areincorporated in the minimum capital rule, which essentially means thattraders are also subject to a strict capital allocation monitoring. Thetotal capital consumption associated with a particular activity leadsto a cost that has to be subtracted from the activity's pro®ts. If thesepro®ts are insuf®cient to sustain pro®tability after the cost of equity,then the entire activity should be redesigned or interrupted.In practice, the minimum-capital rule is extremely complex toimplement, for many reasons. First, a bank always evolves in its owndomestic regulatory environment, which may impose further restrictions and disclosure than those set in the BIS recommendations.41 Inaddition, the allocation process is a ®rm-wide undertaking, whichmeans that every individual activity should theoretically review itsrequired return and consumption of capital regularly and base its
41
The Basle Accord imposes minimum standards but leaves entire latitude to domesticregulatory authorities to go further in strengthening capital requirements.
CHAPTER 6
VALUATION
323
investment decisions on the result of this examination.42 Finally, bynature, trading positions ¯uctuate wildly in size or composition, andit can be therefore extremely dif®cult to compute or estimate a reliablerisk-weighted asset nominal.43
Arbitrage and Capital
Being part of a larger trading ¯oor, arbitrageurs are naturally affectedby the BIS guidelines. However, arbitrage positions are by de®nitionalways perfectly hedged or hedgeable, and it is therefore tempting toargue that their intrinsic risk is lower, justifying a lower cost of capitalthan a regular equity position. Consider, for example, a fully hedged$100 stock position. Under a generic capital calculation, each leg isaccounted for separately, and if their risk weight is 50 percent, thetotal requirement is $8 [ ($100 + $100)  50 percent  8 percent].Alternatively, if the position is a real arbitrage, it might be alreadyvery conservative to charge capital on one leg only, or even less thanthat.Consider, for example, an index arbitrage position we denote N.Assume the risk weight for stocks is 15 percent and 7 percent onfutures.44 The total capital need C is given by C  8 percent  (15percent  N  7 percent  N) if we consider the position as a juxtaposition of stocks and futures. C is therefore equal to 1.76 percent of thetotal nominal. If we require a return of 15 percent on this capital, thetotal charge on the arbitrage activity is 0.264 percent (= 1.76 percent 15 percent). In other words, to carry a $1 billion position, the trader42
In many institutions the cost-of-capital allocation process is run periodically, fromonce a year to once a month. At this time a snapshot is taken of all the assets of thebanks, and each activity is internally ``charged'' its capital consumption.43There is a fourth component to that complexity in countries like Japan and the UnitedStates where securities houses are subject to speci®c regulation. Usually subsidiariesof large banks, these institutions must satisfy their own sets of solvency criteria and ®twithin those imposed to their parent companies because they are consolidated intheir balance sheet. We leave that layer of complexity aside. For more informationabout the regulation of ®nancial institutions in the United States, see, for example,Clifford Ball and Hans Stoll, ``Regulatory Capital of Financial Institutions: AComparative Analysis,'' Financial Markets, Institutions and Instruments, New YorkUniversity Salomon Center, 1998. For a comparative analysis of capital requirementsin the United Kingdom, Europe, and the United States, see Elroy Dimson and PaulMarsh, ``Capital Requirement for Securities Firms,'' Journal of Finance, 50, 3, 1995.44Fifteen percent is the actual weighting required by the Securities and ExchangeCommission for liquid stocks, and 7 percent corresponds to the margin deposit onthe S&P 500 assuming an index close to $1,400.
324
FINANCIAL VALUATION AND RISK ANALYSIS
should generate at least $2.64 million in pro®t, and his or her personalperformance will be evaluated only above that threshold. On the otherhand, if the position was recognized for an arbitrage and risk weightedaccordingly, the net-capital requirement would be signi®cantly smallerand the trader's variable compensation would kick in much faster.Indeed, this example re¯ects one of the criticisms of the Basleframework; it is very capital intensive because it does not recognizeany correlation between the assets in their risk weighting.45 An alternative could be to allow arbitrage positions to be recognized as logical``components'' of an entire portfolio and compute a capital consumption accordingly. Japan is an interesting example in that respectbecause it does allow securities houses to reduce their capital requirement by presenting index arbitrage positions as ``integrated.'' C is thencomputed with:C  rC  rM  M  rU  Uwhere U and M represent the nominal respectively unmatched andmatched within the position, while rM ; rU are their respective riskweightings. rC is the overall capital ratio, 8 percent. The principle ofmatching used to make a distinction between M and U based on thenumber of futures in the position, nF . For each futures contract, weshould hold a rigorous number of shares of each constituent of theindex if the portfolio is to be perfectly replicating. Let's denote QTHithis quantity, and Qi the actual quantity held. M and U are de®nedrespectively as:XX Pimin Qi ; nF  QTHUQi nF  QTHMi   Piii
i
where i covers the number of securities in the portfolio, and Pi is therelevant market price or the ith security. M captures the nominalentirely matched between the futures contract and stock position. U,on the other hand, captures the mismatch between the two, eitherbecause the portfolio is not perfectly replicating the index or becausewe do not have the appropriate number of futures to cover the stockposition. If Qi  nF  QTHi , which is supposed to be the case, then U  0and M  N.Naturally U is included in C to introduce a penalty for a positionthat is not perfectly maintained with respect to its underlying index.This is done by setting rU > rM . Indeed, typically rU  100 percent,45
This criticism is examined from the perspective of an arbitrageur, but it extendsnaturally way beyond this type of trading strategy.
CHAPTER 6
VALUATION
325
which is a re¯ection of the fact that the unmatched nominal isunhedged and entirely at risk. In contrast, rM can be quite lowÐforexample, 10 percentÐto incorporate the stability and low level of riskexpected from a true arbitrage position. Under these assumptions thepenalty due to U can be quite important. Consider, for example, asituation in which M  95 percent  N and U  5 percent  N. If theposition were entirely matched, the cost C would be C  8 percent  (10percent  M)  0:8 percent  N. If rE  15 percent is the expected returnon equity, the cost of capital KE is KE  rE  CÐthat is, KE =N  0:12percent. If we incorporate now the unmatched part U; C  8 percent (10 percent  M  100 percent  U)  8 percent  (10 percent  95percent  N  100 percent  5 percent  N)  1:16 percent  N. Inthat situation, KE =N  0:174 percent. The difference of 0.054 percentis an increase of 45 percent in the total cost of capital, amounting to$540,000 for a $1 billion total position. A slight deviation of 5 percent inthe management of the position has therefore a huge impact on the costcharged on the activity under capital consumption, providing thetrader with a signi®cant incentive to monitor the position correctly.The cost of capital is an important notion for a proprietary tradingactivity because of its implication in terms of expected and minimumpro®tability. Rigorously speaking, it should be monitored every day,but in reality, it is measured only infrequently. It does impact day-today trading decisions, however, because traders have to make sure thatthey pursue the most pro®table opportunity for a given amount ofcapital put at risk. Arbitrageurs should even be more focused whentheir strategy is highly capital intensive, such as index arbitrage.
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7R ISK A NALYSIS
INTRODUCTIONIn common nomenclature, risk is de®ned as the possible occurrence ofan unfavorable event. For example, a poker gambler may take the riskthat someone else at the table has a better hand. Alternatively a pedestrian crossing a street with heavy traf®c takes the risk of being hit by acar. In essence, the notion of risk is often associated with a negativeoutcome. In the world of ®nancial markets, risk is a symmetrical concept referring to the randomness embedded in markets and assetsprices. A risk can lead to positive as well as negative outcomes andstill be considered inappropriate.1 Indeed, even if there exist manyrisks leading primarily to negative events, the vast majority of themcan evolve both ways. The added value of a trader is then to chooseand manage those that he or she is in a position to understand.The notion of risk is the ultimate concern on a trading ¯oor, and itis arguably far more important than being able to evaluate precisely anopportunity, for example. This point is particularly clear in the of®cialmandate most traders receive from their employers: They are givenP&L objectives but risk limits. In other words, it is acceptable to losemoney or fail to meet a pro®t targetÐto a certain point at leastÐbut itis an absolute requirement to monitor the level of risk and maintain itunder a ®xed limit. In a period of troubled or volatile markets, tradersmay not know precisely how much they lose because the market1
For example, a trader taking a huge risk as a result of mismanagement will probablybe sanctioned even if the ®rm turns a pro®t.
327
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328
FINANCIAL VALUATION AND RISK ANALYSIS
moves too fast, but they will never get away with not knowing howmuch they can possibly lose if everything goes wrong. Incidentally,emphasizing maximum loss instead of expected pro®t is a sound riskmanagement practice, valid far beyond the world of ®nance.Risks come in many shapes and forms, and cataloguing them is adif®cult task because the perception of a particular event as a risk isvery subjective and highly dependent on the internal and externalcontext. It is trivial to say that the buyer and seller of a given securitydo not perceive the risks in the same way, but it should nevertheless bekept in mind. The same trivial idea leads to the recognition that arbitrageurs in general love volatility whereas investors at large tend toconsider it a risk and a nuisance, and this difference has had a longstanding in¯uence on regulating authorities.From our perspective we focus primarily on market risks, althoughwe shortly examine other risks as well. In contrast to a valuation procedure, we cannot base the discussion here on a distinction by instruments because many products carry several classes of risk. Therefore,we adopt a functional presentation, in which we classify and discussrisk type by type. Furthermore, a risk in itself is only accessible throughrisk measurement mechanics, the same way a distant star can be seenin detail only through a magnifying telescope. The quality of the telescope has a huge impact on the ability of scientists to draw conclusionsabout the star, and this is also true of the risk management system thatconditions the ability of a trader to make the right decision. In addition,it is necessary to note that the proposed framework is not meant toembody the ultimate truth, but rather to be one robust and relativelysimple alternative. If there can be little debate on the market price of astock, a risk analysis may be performed under different mathematicalformulations or models, with different results, in turn leading to adequate decisions along different paths. Before we start, however, a fewwords are necessary on the ability of human beings to take and handlerisks in general.
RISK AND HUMAN PSYCHOLOGYIt is a very clear and known fact that risk is not handled the same wayby different individuals. For example, assume that you play heads ortails with a friend and that you pay $6 if you lose and receive $10 if youwin. The probability of each outcome is 50 percent; therefore, the fairvalue of the game from your perspective is $2 ( $10  50 percent Ð$6  50 percent). In other words, you would be ready to pay your
CHAPTER 7 RISK ANALYSIS
329
friend $2 to enter the game, and he should consider the deal a fair one.In reality, three types of reactions are possible with respect to such agame. Risk-neutral individuals will ®gure out the math and accept thefair price, $2, to carry the risk of paying $6 for a total upside of $10.Risk-averse individuals will not be willing to enter the bet at fair price, sothey will either refuse or expect more than $2. Risk-seeking individualswill be willing to receive less than $2 for the opportunity to receive $6.If this simple example underlines the principle behind the notionof risk aversity, there are naturally many nuances and re®nements.Clearly the willingness to carry risk depends on a number of factors,including sociodemographic elements such as age or income, but alsoan innate inclination that pushes individuals one way or the other.Beyond acceptance, another dimension is the ability of individuals tohandle risk and uncertainty. In the above example the result of the betis available immediately, and the amount is small. In other instances,if the amount were signi®cant and the outcome uncertain for sometime, even somebody with an inclination to take the bet in its originalform would probably think twice to account for the persistentuncertainties.This discussion is naturally nothing more than an introduction tothe complex problems behind human psychology, but it is essential tokeep it in mind when examining the notion of risk.2 In contrast to aportfolio evaluation, which is by and large a mechanical process, riskmanagement calls upon intrinsic individual characteristics and behavior. A good trader is not necessarily someone with a Fields Medal orNobel Prize, but he or she must have the ability to evaluate, assume,and handle risks consistently.3 In practice, this often translates into theability to withstand heavy losses without losing sight of the best decision at a given time. Therefore, even the best risk management systemis no substitute for judgment and experience. We try to present thewhole process as highly rigorous and analytical, which it is, but morethan in any other part in this book, the mechanics we present are toolsand should be taken as the necessary ®rst step, not as an end in and ofthemselves.
2
For a short discussion about anomalies in ®nancial markets, see, for example, JayenduPatel, Richard Zeckhauser, and Darryl Hendricks, ``The Rationality Struggle,'' TheAmerican Economic Review, 81, 2, 1991.3This last characteristic is particularly important because most people tend to be riskaverse in bad times and risk seeking in good times, which more often than not leads toan inconsistent set of decisions and accumulating losses.
330
FINANCIAL VALUATION AND RISK ANALYSIS
PRINCIPLESPurpose
In our approach, a risk analysis is nothing more than a dynamic valuationÐthat is, a valuation carried out for an incremental variation intrading parameters. As such, a risk analysis shares the fundamentalcharacteristics of a valuation: It is essentially cash driven and presentsrelevant information as of today. There is, however, a signi®cant difference in principle. If the valuation assumes liquidation, a risk analysisassumes continuation in the position, even though it uses liquidationbecause it is based on a standard valuation. Its purpose is to identifyand quantify risks, risk being de®ned as any random occurrence ofevents with an impact on the overall P&L. Its results must allow atrader to make the appropriate decisions to carry the position withthe best chances of realizing a pro®t or avoiding a loss.This de®nition means that a risk analysis is a parametric procedurelike a regular valuation, and indeed the reason we imposed a valuationto be parametric in the ®rst place was to simplify the risk analysisprocess. In contrast to a valuation, however, a thorough analysis ofrisk must also account in some form for nonparametric sources ofrisks. Earlier we gave the example of a system failure, but there aremany other sources that could be included, and some will be exploredlater on.The problem with nonparametric random events is that they areextremely dif®cult to quantify accurately. For example, in the case of asystem failure, depending on the magnitude of the failure, the consequences may be devastating or relatively minor. In practice, there areno speci®c rules to deal with these risks but rather a few generalprinciples, and being paranoid is one of them. In the case of a systemfailure, a thorough evaluation of contingency plans and alternativeexecution routing systems would be a good place to start, makingsure that these plans are well known and regularly tested andupdated.4 ``Expect the unexpected'' and plan accordingly is indeedthe best way to address these potential problems, keeping in mindthat the cost of carrying some of these risks may be prohibitivelyexpensive.4
The system failure is indeed a very serious problem for ®nancial institutions. The 1995Kobe earthquake in Japan, or the ®re at the Credit Lyonnais headquarters in 1996 inFrance, for example, reminded market participants that natural disasters were a truepossibility. Virtually all trading ®rms maintain secret fully functional alternative trading ¯oors, to be used in case of emergency.
CHAPTER 7 RISK ANALYSIS
331
Isolation of Individual Risks and HedgingPrinciplesIn a risk analysis, we compute the change in the value of the positionbased on a unitary variation in each of its underlying parameters. Aunitary variation in our context is a variation small enough that a Taylorexpansion of the valuation V is acceptable to the ®rst order.The principle is identical to an explanatory valuation but implemented differently. An explanatory valuation is meant to break VY!Tinto distinct parts:VVVY!TY!T P1 P2 PY!T  VY!T nP1P2Pnwhere:VV PY1 ; . . . ; PTi ; . . . ; PYn  V PY1 ; . . . ; PYi ; . . . ; PYn PiPTiPYiA risk analysis is meant to access Ri , the ith risk factor, de®ned by:Ri 
VPi

V PT1 ; . . . ; PTi  i ; . . . ; PTn i
V PTi  i i
V PT1 ; . . . ; PTi ; . . . ; PTn 
V PTi 
where i is arbitrarily small. This process reduces V into a number ofindependent linear functions, each carrying the impact of the variationof one particular parameter:V PTi  i 
V PTi   Ri  i
This relationship is used in turn to evaluate the extent of the P&Lassociated with any value i . Graphically, the approach can be represented as shown in Figure 7-1. The actual dependency of V is approximated wherever necessary by a straight line, with a slope equal to Ri .Hedging a risk in that representation is equivalent to trading anadditional instrument in the position so that Ri is neutralized. If v is theincremental valuation due to this additional instrument, we wouldideally want to have:v PTi   0
and
v PTi  i 
v PTi  
Ri   i
so that the addition of v turns out to be both pro®table and risk reducing. Naturally, in general, these relations are dif®cult to combine, andhedging an unwanted risk may turn out to be extremely costly.Furthermore, the inclusion of one instrument to hedge a given class
332
FINANCIAL VALUATION AND RISK ANALYSISVActual VSlope = (∆V/∆Pi) = Ri
0
1
Pi = Pi Pi = P i
Figure 7-1
2
Pi = P i
Pi
Risk Factor
of risk may have the effect of creating risk of another type. Therefore, ifon paper the procedure is fairly simple, it is much less so to implementin reality. In any case, this approach illustrates the key bene®ts ofhandling risk through a partial derivative framework: Each type ofrisk is systematically quanti®able, allowing a rigorous and tailoredrisk management.Dif®cultiesNaturally this method has clear limitations, visible, for example, inFigure 7-1. The linear approximation is only truly valid in the neighborhood of the point around which it is calculated, which means that ifthe market moves signi®cantly from that point, the actual P&L may bedifferent from the one calculated. This dif®culty can be solved by sampling more points like P0i , P1i , and P2i in Figure 7-1, at the expense ofcomputing time.A more serious problem, however, comes from the assumed independence between variables. Varying a single parameter at a timewhile keeping all the others constant is inaccurate because it doesnot re¯ect the reality of the marketplace. For example, interest ratesexhibit a cross dependency throughout the yield curve; therefore, keeping each one of them independent can give inconsistent results. Evenfor instruments as simple as stocks, varying the price of one whilekeeping the others constant in a portfolio is a strange proposition.There can be two broad solutions to this problem. On one side, it ispossible to assume ``rigid'' relationships between parameters, implyingthat a cross correlation is always equal to 1. The second option is toincorporate correlations into the model and proceed with a more complex mathematical framework. The V@R was derived from the latterperspective, so we will develop it when examining V@R in details.
CHAPTER 7 RISK ANALYSIS
333
As an illustration of the ®rst approach, consider that we hold twostocks S1 and S2 . Assuming they are independent, we would de®ne R1and R2 as:V S1   1 
V S1   R1  1
V S2  2 
V S2   R2  2
with 1 and 2 de®ned arbitrarily small, possibly equal. In that case aglobal variation in V is simply a linear combination of these two terms,and we can hedge the position if necessary independently on S1 and S2 .If we were to adopt a rigid correlation between S1 and S2 , wewould assume that S1 and S2 move in parallelÐthat is, a move of percent on S1 gets exactly re¯ected on S2 because the entire market isaffected equally. We then de®ne an aggregate risk factor R as:R
VS1  1  ; S2  1  
V S1 ; S2 
In that case, if we want to hedge the position, we cannot consider S1and S2 individually because they are in fact part of a bigger market. Weneed to hedge against variability from the whole market, S1 and S2being only two components of this market.As an application, consider that we are long 5 shares of S1 andshort 6 shares of S2 . If we were to consider S1 and S2 independently, wewould have:R1 
V S1   1 1
V S1 

5  S1   1 1
5  S1
5
and similarly R2  6: R1 and R2 are here homogeneous to a numberof shares. If we wanted to hedge V against a move in S1 we wouldhave to sell 5 shares of S1 or buy 6 shares of S2 for a protection against amove of S2 .On the other hand, if S1 and S2 are rigidly linked, R becomes:R
5  S1  1  
5  S1

6  S2  1    6  S2
 5  S1
6  S2
Here R is homogeneous to a nominal, not to a number of shares anymore. This nominal could be hedged with S1 , S2 , another asset, or acombination of other assets. R is used here to hedge an entire class ofrisks, those related to equity prices, not those created by a particularinstrument in that class.55
In some cases, for example, index arbitrage in which the portfolio is supposed to be aperfect rep