, and borrow periodically at the LIBOR. Theinterests paid on the LIBOR loans match exactly the variable paymentsfrom the swap, which means that the effective borrowing cost of the17
The same result can be obtained using the risk framework we introduced before. If Vis the valuation of a position of (n  ) futures, its delta is V=S  n    F=S  n    r because F  S  1  r d . Hedging V=S means setting  so thatV=S    0. This is true if (n    r   )   n  r= 1  r.18In addition, note that the hedge ratio changes with maturity. For example, if we take a®ve-month futures contract it becomes [ 1= 1  5 percent  5=12  0:97959.The process by which the number of hedging futures is changed is usually referredto as tailing the hedge. The tail denotes the small fraction of the hedge covering themargin calls. See, for example, Ira G. Kawaller, ``Tailing Futures Hedges/TailingSpreads,'' The Journal of Derivatives, October 1997.
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structure is the ®xed rate of the swap. We described this approachearlier.Now consider that we forget or intentionally forego entering one ofthe LIBOR loans. Assuming we need the cash, the shortfall will becovered day to day by an overnight carry. In contrast to the LIBOR,which would have been known for the entire three months, the overnight rate is known with certainty only from one day to the next, and itcan vary signi®cantly depending on the liquidity instantaneously available in the marketplace. As a result of this variability, the effective costof carrying the overnight for three months might end up being different from the original three months' LIBOR at which we should havetraded. Indeed, because of its particular nature, the overnight rate canbe affected by a number of factors including day-of-the-week or endof-month effects. Consider as an illustration the ®ve charts in Figures7-2 through 7-6 giving overnight and three months' LIBOR for USD,GBP, FRF, DEM, and JPY.These charts show the immediate difference between a three-monthLIBOR and the overnight. Clearly there are systematic differencesbetween the two rates, and their relative behavior is highly dependenton the market. What is not directly available, however, is the actualdifferential cost of carrying a position overnight instead of hedging itat a three-month LIBOR. Figure 7-7 presents exactly that information. Itis obtained by compounding the overnight over a three-month period,and comparing the compounded return to the three-month LIBOR atthe beginning of the period. It therefore represents a direct P&L forsomebody who would have made a choice to stay at the overnight.19Interestingly, the chart is most often under the x axis, which indicates that for a borrower it is often better to carry the overnight riskthan to hedge with the three-month LIBOR. Conversely, a lenderwould have a strong interest in entering a term loan. It appears thatthe shape of the chart changes with the country, but in general thesame conclusion holds: It is cheaper for borrowers to carry the overnight risk. This risk is not entirely free, however, because if the overnight shoots up suddenly for one or several days, the total costadvantage might disappear very quickly.Another example of spread risk can be found in the United Stateswith the fed fund futures contracts. As we have seen, the fed fundfutures are settled on the average effective fed fund rate everymonth. The effective fed fund is a weighted average of all the trades19
A negative ®gure indicates that the compounded overnight is below the three-monthrate at the beginning of the compounding period.
348
FINANCIAL VALUATION AND RISK ANALYSIS
Figure 7-2 Fed Fund(Source: Compiled by author from Datastream data.)
Figure 7-3 Overnight GBP(Source: Compiled by author from Datastream data.)
CHAPTER 7 RISK ANALYSIS
Figure 7-4 Overnight FRF(Source: Compiled by author from Datastream data.)
Figure 7-5 Overnight DEM(Source: Compiled by author from Datastream data.)
349
350
FINANCIAL VALUATION AND RISK ANALYSIS
Figure 7-6 Overnight JPY(Source: Compiled by author from Datastream data.)
Figure 7-7 Compounded Overnight USD(Source: Compiled by author from Datastream data.)
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351
of the day; therefore, it is different from the close which represents asnapshot of the price traded at the end of each day. The differencebetween these two rates can be signi®cant, both ways, as shown inFigure 7-8. Somebody hedging a treasury position with a fed fundfutures contract but executing overnight transactions toward the endof each day would effectively carry the spread risk between the two. Infact, even trading earlier during the day would not remove the riskbecause there cannot be any guarantee that the effective rate for theday will be equal to the rate traded at some particular time.
Dividends
The dividend risk arises essentially because of a difference betweenexpected and actual values for dividend payments. As we havedescribed, the procedure by which dividends are set implies that theactual amount is unknown until a few days before it is actually paid. Ifthis amount had to be estimated before, and there are numerous situations in which it is necessary to do so, any difference between expectations and actual values results in a pro®t or a loss.Consider, for example, a risk arbitrage in which the target companyB and acquirer A have similar historical nominal dividend payments,say, $0.10 per quarter. If we take a typical merger arbitrage situation, thetarget company B is trading at a discount compared to the terms of the
Figure 7-8 Fed Fund Closing versus Effective(Source: Compiled by author from Datastream data.)
352
FINANCIAL VALUATION AND RISK ANALYSIS
acquisition. Assuming for simplicity a merger ratio of 1, we are long Band short A for the same number of shares. In that scenario, when weentered the spread, dividend payments were expected to cancel out:+$0.10 every quarter from B, $0.10 with the same frequency on A.The ®rst possible problem is a sequencing issue. If dividend ex-datesdo not occur at the same time, we may have to pay more on A than wereceive on B, if, for example, more ex-dates of A are included in theperiod until maturity. This difference would naturally have to beincluded in the arbitrage pro®tability. More generally, assume thatthe ex-dates coincide, so that we are perfectly matched. The risk hereis an increase in A's dividends, or alternatively, a decrease in B's. If, forexample, A went up from $0.10 to $0.12 and B down from $0.10 to $0.05,we would realize a loss of $0.07 (= $0.02 + $0.05) per share of A and B. Inpractice, downward revisions of dividends are not well perceived in themarketplace, so a company involved in a merger would certainly avoidthat option. However, an increase is always possible, especially if Awants to show its ®nancial strength and emphasize what it perceivesto be a signi®cant positive event, namely, the merger.There is no easy way to hedge this risk because dividend paymentsare set by the board and the information related to the decision process ishighly proprietary. Many companies have adopted consistent and transparent policies, making their decisions relatively predictable. When alarge number of companies is involved, however, such as with anindex arbitrage position, on average it is not possible to consider thedividends really predictable. This situation makes any hedge problematic because the underlying variable does not ®t a traditional pattern.Interestingly, a dividend exposure may exist even without explicitequity exposure. If we take the example of a portfolio hedged withindex futures, the value of the futures is highly dependent on the indexdividend expectations, whereas the underlying portfolio may not be areplicating portfolio. In that case, if the futures are evaluated with acash-and-carry model, they create an exposure on the index dividendsdespite the fact that the index is not necessarily a concern in any wayfor the trading strategy. As a result, the overall portfolio carries thedividend risk of the actual stock position, plus an exposure on theindex dividends through the futures position.2020
Note that even if futures are held as a hedge, the dividend exposure on each leg is inthe same direction. Consider, for example, that the portfolio is long stock and shortfutures. If dividends go up on the stocks and on the index, the portfolio will show apro®t on both legs because the theoretical value of the futures goes down, whichresults in a pro®t on a short position.
CHAPTER 7 RISK ANALYSIS
353
Short Squeezes
We already gave one example of a short squeeze on the ®nancingside, although it carried another name: ``end-of-month effect.''Demand for liquidity is so high in some instances that the marketprice of this liquidity exhibits an abnormal spike to re¯ect the shortage of the scarce resource. Despite its abnormal nature and possiblemagnitude, this spike is in general relatively benign because centralbanks are available to attenuate its effect as part of their regularmarket interventions.More interesting and more costly is a short squeeze on stocks forshort sellers. We evoked this risk in several instances already, notablywith regard to corporate events. The extreme form is a squeeze inwhich the price of the scarce resourceÐthat is, the borrowing feeÐbecomes irrelevant because there are simply no more shares availablein the marketplace. The effect can be devastating because short sellershave no choice but to close their positions at the current market price.Consider, for example, the situation of an arbitrageur of closed-endfunds. Trying to capture a premium, the arbitrageur would short sellthe fund and hedge with a replicating portfolio of some sort. If the fundpremium starts widening, the arbitrageur is losing on the spread, andat some point if the fund becomes really expensive relative to its underlying assets, holders will be tempted to sell, and to do so they will haveto recall the shares they have lent out. Naturally, because the totalnumber of shares is ®xed, the arbitrageur should be able to ®ndanother counterpart ready to lend a replacement position. In practice,for a short time it is extremely possible that all holders will recall theirshares at once to sell them, and during that period the arbitrageur willbe unable to keep the short position. If sellers do rush to the market, thepremium will collapse again, at which time the arbitrageur will ®ndnew shares to borrow, having been forced out of the market at theworst time.A short squeeze occurs when a scarce resource becomes even morescarce, which essentially means that it becomes impossible to acquirewhen it is most needed. Due to its nature, it is also extremely dif®cultto hedge and, in the particular case of borrowed securities, virtuallyimpossible because lenders usually keep the ultimate privilege to recalltheir shares when needed.
Execution
We exposed earlier the fundamental risk of execution related to theexpiry of a futures contract on an average value. Similarly, when creat-
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FINANCIAL VALUATION AND RISK ANALYSIS
ing synthetic futures with American options, like those on the S&P 100,the risk of exercise on one of the legs creates an execution risk becausethe leg must be replaced, at a price likely to be different from its fairvalue when originally acquired.More generally, we de®ne execution risk as the need to execute inthe market a parameter that is already implicitly or explicitly priced inan existing position. This parameter may be a stock price or any othermarket price. For example, if a swap structure is used to borrow cashover a long period of time, the implicit assumption is that it will alwaysbe possible to borrow short-term liquidity at the reset price of thevariable legÐthat is, the LIBOR. This assumption creates precisely anexecution risk because any difference between the LIBOR and theactual loan rate at a reset date will result in a pro®t or a loss, unaccounted for previously.21In many instances, and arbitrage is a powerful example, executionis a risk but at the same time the primary driver of pro®tability. Indeed,arbitrageurs act as cross-security market makers and as such pro®tfrom instantaneous price discrepancies. Because the two legs of a position can never be really executed at the same time, the necessary timelag between them creates an execution risk, which should constitutethe only signi®cant risk in the strategy if it is truly an arbitrage.Therefore, execution, more than any other, is a risk we do not necessarily want to hedge away, even if we were able to.Other RisksDespite the fact that they are less visible, nonmarket risks are extremely important in many respects. Clearly, their magnitude and monitoring are much more dif®cult to assess with an adequate level ofcomfort, which makes them sometimes even dif®cult to de®ne. Moreimportantly, however, their main characteristic is their discretenature.Although a market price is relatively continuous, nonmarket risksare of a different nature because they essentially boil down to ``all ornone.'' In the case of credit risk, for example, if a counterpart failsbefore ®nal settlement, chances are that the claim on its assets will bedif®cult and painful to recover, to the point where it may be wiser towrite it off. On the other hand, if its ®nancial strength is adequate,payment will occur with no particular dif®culty. Therefore, except21
If we knew from the very beginning that we would have to pay more than the LIBOR,this difference would not be a risk because it would be entirely predictable.
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for somebody in the business of explicitly trading it, this risk is notwelcome in general because it is extremely dif®cult to encapsulate in arisk analysis framework.22The same is true of a larger set of risks, some of which are not eventradable in the market. Considering what our perspective has beenfrom the very beginning, the following presentation will be essentiallyan introduction.
Credit Risks
Credit risk denotes the risk that a counterpart fails to settle its liabilities when they come due. For example, if a corporation fails to meeta covenant attached to one of its obligations, bond holders may havethe option to force it into receivership or bankruptcy.23 In virtually allcases, however, default is de®ned with respect to ®nancial strength,not operating performance. This last point is critical because it meansthat power to trigger judicial action rests with a particular category ofinvestorsÐdebt holders.24 In that situation, stockholders' rights aresubordinated to those of debt holders and creditors, and this precedence is well known. In other words, the failure of a corporationcreates a credit risk to creditors but a market risk to shareholdersbecause the two groups have different rights and expectations fromtheir investment.Generally speaking, credit risk is probably the most pervasivetype of risk beyond that created directly by a market exposure.Virtually every single ¯ow expected at one particular moment ispotentially at risk, regardless of its nature and paying party.Naturally numerous safeguards have been implemented over theyears at every key juncture in the ®nancial system, such as the22
Naturally it is possible to follow and closely watch for degradation of ®nancialstrength in many institutions because a situation of default naturally does not appearovernight. Such ongoing monitoring requires skills and resources not necessarilyavailable to the population of traders we focus on, which is why it is necessary tomake a distinction between professionals explicitly focused on credit issues and othernonexperts.23Bankruptcy law differs widely in many countries. In some instances, it implies liquidation, while in others, like the United States, the corporation is temporarily protected from its debt holders and creditors and allowed to remain in business underappropriate judicial control.24Naturally shareholders have full latitude to judge the performance of the management team and accordingly take legal steps to protect their interests. Bond holdershave additional rights because of their status when the solvency of the corporation isunder scrutiny.
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FINANCIAL VALUATION AND RISK ANALYSIS
execution, clearing, and settlement infrastructure for publicly tradedsecurities. Even if the situation is not perfect, it is reasonable to consider that in many countries the appropriate controls and proceduresare in place to guarantee that investors carry credit risk as a choice,not as a necessity.In the trading world of an equity arbitrageur, credit risk is primarily arising from transactions entered over the counter. These can bemoney market or derivative, but in all cases a natural hedge againstcredit risk, more and more used, is a reciprocal margining agreementbetween counterparts that are used to trading with each other. Themodel is that of margin calls on futures and options, and it is a naturalstep to take even if it is administratively expensive because all partieshave a strong interest in protecting themselves.25
Legal and Compliance Risks
Legal and compliance departments are extremely important in securities dealing institutions, especially in the United States, where thesecurities law is complex and intricate. It requires ®nancial institutionsto monitor themselves on numerous dimensions, from training andregistration of their personnel to disclosure guidelines and ethicalrequirements when dealing with investors and clients.Broadly speaking, legal risk arises on two dimensions: compliancewith the prevalent securities law and enforcement of private contracts.Naturally both aspects are highly dependent on the country's legalstructure and maturity of the capital market, which makes it impossible to offer a generic description here. As an example of the intricaciesinvolved, we consider brie¯y the case of the United States.In the United States, the regulatory framework for ®nancial market participants establishes two major regulatory institutions. TheSecurities and Exchange Commission (SEC), on one side, regulatesthe market for securities, and options on securities. The CommodityFutures Trading Commission (CFTC), on the other hand, regulatesthe market for commodity futures and options on futures, wherethe term commodity includes a wide range of assets, physical as
25
The market for credit risk derivatives has grown tremendously in size and maturityover the years, and there are now sophisticated products to address some of the needsof market participants. Still the principle of preemptive margins is widespreadbecause of its simplicity and appeal. For example, some bond covenants force theissuing corporation to establish a ``sinking fund'' to progressively cover the principalredemption payment at maturity.
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357
well as ®nancial. The SEC was established in 1933 as a result ofthe debacle of 1929, whereas the CFTC was established in 1974 bythe Commodity Futures Trading Commission Act, as a signi®cantrevision to the previously existing Commodity Exchange Act (CEA).Despite the apparently well de®ned jurisdictional frontier betweenthe two institutions, the legal environment has been characterizedby ®erce battles between them, often considered as pure ``turf battles.'' The current state of affairs is certainly not written in stone asmore and more complex instruments present blended characteristics,making any de®nite classi®cation extremely dif®cult. In addition tofederal-level regulations, a number of other rules apply to marketparticipants, notably state laws, also known as blue sky laws. Otherself-regulated organizations (SROs), such as the National Associationof Securities Dealers (NASD) or New York Stock Exchange (NYSE)impose in addition their own requirements on their members. Theresult of this patchwork is a complex set of issues, making compliance a nontrivial challenge and an expensiveÐbut necessaryÐproposition.Given the context, enforcement of private contracts is not surprisingly also a complex issue. Depending on the ®nancial structureof a particular product, any one of the two agencies can legitimatelyclaim jurisdictional power, in which case the original legal framingof the transaction may prove inadequate. It is therefore possible torefute a liability on the grounds that it was not properly quali®ed tostart with and that the losing party did not receive appropriateinformation and/or advice. An example of such an attempt is thehighly publicized Procter and Gamble (P&G) swap with BankersTrust (BT) in 1993. P&G entered at that time into a speculativeand highly leveraged swap on interest rates, which ended up losingalmost $160 million. The loss prompted P&G to sue BT on severalaccounts, notably breach of ®duciary duty, negligent misrepresentation, and fraud. P&G also tried to have the transaction requali®ed asa security under SEC jurisdiction, which would have been muchmore sympathetic to end-user concerns. This requali®cation, however, was denied, and the rulings of the judge in 1996 also foundthat the swap was exempt from exchange-listing requirements underthe Swaps Exemption of the CEA. Denying that the swap wasactionable under the CEA or securities laws, the judge essentiallyupheld the legality of the original transaction in the context in whichit had been tradedÐthat is, a swap documented by a standard formISDA master agreement. Eventually P&G and BT reached a negotiated agreement ending the legal battle, but the case served in
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FINANCIAL VALUATION AND RISK ANALYSIS
many respects as a road map of the legal intricacies attached tocomplex OTC structures.26This short digression is highly illustrative of the complexity ofthe issues in OTC contracts and more generally of the prevalentlegal framework applicable to derivative products. In our context,the legal risk is mitigated by the fact that OTC transactions areusually entered between traders with similar and straightforwardobjectives, a situation with two practical consequences. First, thetraded structures are usually fairly simpleÐstandard forwards orswaps. More importantly, because both parties have arguably thesame high level of understanding in the technicalities of the products, it is dif®cult for either of them to sue the other for misrepresentation, for example.
Other Risks
Beyond credit and compliance, there are numerous other possiblesources of risk within a trading institution. Even if individual tradersdo not necessarily feel they are concerned, risk managers have to havea view broad enough to encompass as many uncertain or unexpectedevents as possible.For example, reconsider the case of the tax rebate. We detailedearlier how ®scal arbitrage was widespread in countries where taxauthorities implicitly or explicitly permit it. This arbitrage naturallyis not risk free, and here the risk can be quali®ed as ®scal riskÐthatis, the possibility that actual tax liabilities may end up being higherthan expected. Equivalently, the rebate effectively available may turnout to be lower than originally planned, resulting in a pure loss.Other areas of uncertainty and risk are the operational performanceof a given institution, or its IT infrastructure. The previous example of afailure in the order execution system just before an expiry is only asmall illustration of this particular category. Miscommunicationbetween internal departments may also result in wrong informationbeing passed along, which could in turn result in a disastrous tradingdecision or conversely in no decision when one would have beennecessary. Consider, for example, the impact of corporate events onstock portfolios, for which we mentioned several times that tradersshould at the very least follow the terms of the proposed transaction26
See, for example, Robert McLaughlin, ``Over-the-Counter Derivative Products,''(New York: McGraw-Hill, 1999) for a legal description of the case and of itsimplications.
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very closely. In many cases an internal department, typically the backof®ce, concentrates all the details and makes them broadly available toavoid a situation in which each individual trader would search for andtrack the same information independently. This aggregate internalsource has to be particularly reliable and timely in the way it processesexternal information because a number of critical decisions depend onit.It is fair to say then that establishing an exhaustive catalog of allpossible sources of risk is extremely dif®cult. Clearly, risk is a pervasive notion, and even if we considered primarily its ®nancial side, it isfar from being the entire story.
VALUE-AT-RISKIntroduction
The notion of value-at-risk (V@R)27 was ®rst introduced by J. P.Morgan as an internal risk reporting metric. The idea was to providean aggregate value of maximum loss under a given con®dence interval.Instead of compiling pages and pages of charts and ®gures, the aggregation allowed the senior management of the bank to assess the bankwide ®nancial risk for the next day in just a simple ®gure. In 1994, J. P.Morgan started distributing its framework for free, accompanied by allthe elements necessary to make it a turnkey solutionÐnotably thevariance-covariance matrices for different classes of assets. The framework was called RiskMetrics, and it has since been spun off from J. P.Morgan as an independent entity. Ultimately V@R has been adoptedby the Bank of International Settlement (BIS) as a required ®rm-widerisk management system, indicating that it has now become a worldwide standard in risk management.To understand what is behind a value-at-risk, consider a singleassetÐfor example, a share of General Electric (GE). If we held 1share of GE for a year until the beginning of May 2000, for example,the maximum daily pro®t or loss would simply be the highest changein the price of GE over the period. The histogram of daily returns lookslike that shown in Figure 7-9. The line shows the distribution of normalreturns, and it is apparent that there are signi®cant differences between27
For an overview of the different risk management techniques and particularly V@R,see, for example, VAR: Understanding and Applying Value-at-Risk (New York: RiskPublications, 1997), or Philippe Jorion, Value at Risk: The New Benchmark forControlling Market Risk (New York: McGraw-Hill, 1997.)
360FINANCIAL VALUATION AND RISK ANALYSIS
Figure 7-9 Histogram, GE(Source: Compiled by author from Datastream data.)
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the two curves although the shapes are similar, a point we will developlater. From this histogram, the value-at-risk is relatively easy to ®ndbecause it represents the maximum loss for a given probability level.Visually, it is easier to extract from a cumulative distribution, as shownin Figure 7-10. Considering that 95 percent of the time the return isabove 3:4 percent, at the 5 percent con®dence level the value-at-risk isclose to 3:4 percent.28 If GE closes today at $150, for example, thenominal V@R is then $5.1 (= $150  3.4 percent).Naturally even if the basic methodology holds in a wide range ofapplications, the actual calculation can be extremely complex if theportfolio is not held constant or if it contains several stocks or evenmore if it has sophisticated instruments such as bonds or options.Consider, for example, a situation in which we hold Microsoft(MSFT) in addition to GE. MSFT has its own return distribution andbehaves in a way signi®cantly different from GE. Therefore, the distribution of returns on the portfolio is a bivariate distribution where thecorrelation between MSFT and GE plays a particular role.In fact, the approach we have taken, based on historical data, is notthe only framework that has been proposed to answer the V@R question. Before we proceed to a more general discussion of V@R from arisk management perspective, let's examine the different alternatives.
Methodology
V@R calculations can be classi®ed broadly in three categories: analytical method, historical method, and stochastic simulation method. Theanalytical method, essentially that developed originally by J. P.Morgan and embedded in RiskMetrics, makes the assumption thatall assets have returns distributed normally. Under this umbrella, itis fairly easy to construct the distribution of returns of any portfoliobecause it is a linear combination of normal returns. Speci®cally, thevariance of a portfolio is given by:XXX22P i  i  2 i  j  ij  i  ji
i
j
where i is the weight of asset i, i is the standard deviation of itsreturn, and ij is the coef®cient of correlation between assets i and j.The return of the portfolio is then normally distributed, and its cumulative distribution is available from a simple calculation when Pis known, which requires only the individual terms i and ij .28
Rigorously speaking, the V@R is between 3:6 percent andbe obtained through a simple linear interpolation.
3:4 percent, and it can
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Cumulative Distribution, GEFigure 7-10
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Therefore, the RiskMetrics model conveniently requires only minimalinput, and it applies quickly to a wide range of assets.The historical method is the one we applied earlier to GE. It consists of assembling historical data over a relevant period for the portfolio under consideration and basing the cumulative histogram onthese observed data points. It has the advantage of using actual data,removing the need to simplify the assumptions about normality. At thesame time its principal drawback is that it does not properly incorporate sudden changes in the size of the portfolio or, in our example, inthe number of shares we hold. As a remedy to this weakness, a modelcalled historic simulation has been proposed. Instead of collecting onlydaily pro®t and losses for the portfolio, the historical simulation modelis based on collecting historical market parameters. The parameters arethen analyzed for their 95 percent con®dence value, and the V@R is thevaluation of the portfolio based on this new set of parameters.The stochastic simulation method, based on a Monte Carlo simulation, is designed to incorporate ``stress testing'' in the risk managementprocess. Traders know all too well that the past is usually not a reliablepredictor of the future; therefore computing a V@R on any other sort ofhistorical record is dangerous because it describes only ``knownrisks''Ðthat is, risks that have occurred already. A stochastic simulation is based on a set of randomly generated scenarios, for which avirtually unlimited number of hypotheses can be incorporated. Thisapproach guarantees that the V@R truly re¯ects a large fraction of``unknown'' behavior, possibly highly volatile.Naturally these three methods have their strengths and weaknesses, and the above descriptions cover only their most fundamentalprinciples. The most important of these is probably the tradeoff speedversus accuracy. The Monte Carlo simulation, for example, requires atremendous number of runs to provide adequate precision, whereas aV@R assessment based on historical analysis and RiskMetrics can becomputed in seconds. There is much more to say about each one; Table7-1 presents a useful overview.As it appears, there is no best answer here again for the ``V@Rproblem.'' Risk is highly dependent on time and context, which hasan impact on the choice of a particular method.
Value-at-Risk and Risk Management
We brie¯y mentioned earlier the key difference between a V@Rapproach and the one we chose based on partial derivatives. Inessence, V@R has two main drawbacks, making it a powerful tool
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TABLE 7-1 A Comparison of VAR Approaches
Assumption aboutasset returnsValuation model(s)Cash ¯ow mappingSpeed
Data requirement
Risk Metrics
Historical
HistoricalSimulation
StochasticSimulation
Assumes normality
No assumption
No assumption
User-de®ned/¯exible
RequiredNoMedium
RequiredNoSlow
Not requiredNot requiredYesNoDepends on