moreshould also accommodate different formats. Typically the back-of®cedepartment in charge of clearing and settling trades performs its ownvaluation every day, necessarily different from the front-of®ce ®gurebecause of distinct conventions. The best illustration of this situationis the difference between a mark-to-market and theoretical ®gure: Themark-to-market calculation re¯ects market conditions and arguably ismore valid than any sort of convergence assumption. At the sametime an absolute convergence is a unique characteristic that should beincorporated in the valuation, making the theoretical value morerepresentative. In fact, reliability and meaningfulness sometimes con¯ict, and when this is the case, it is usually better to produce twodocuments rather than try to impose one of them as the absolutetruth.Finally, it is necessary to mention that the framework we develophere goes far beyond the sphere of pure arbitrage. Indeed, we wouldargue that any type of ®nancial valuation should be based on the sameprinciples because traders are generally looking for similar informationregardless of the products they trade. For the same reasons we do notmake any distinction for geographic particularities because thereshould be few, if any.To build a valuation procedure on solid ground, we adopt a stagedapproach. First, we develop a certain number of general and fundamental principles with a clear emphasis on the underlying need theyserve. As a second step, we take a standard portfolio and apply theseprinciples systematically to every instrument. Before we do that, however, let's de®ne clearly what we call valuation.
1
A principle very well known in a variety of other settings as GIGO: garbage in,garbage out.
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DEFINITIONSThe valuation of a ®nancial position is equal to the cash ¯ow generated by afull liquidation of this position at its fair market value. The concept is verysimple and very straightforward: We liquidate every single instrumentin the portfolio at a price as close as possible to its fair market valueand derive the total amount of money brought in by this liquidation.This de®nition has two practical consequences. On one side, avaluation is a static ®gure. Although the position itself is constantlyevolving, performing a valuation means freezing its constitution andparameters at the same moment. Therefore, any possible variation ofany of these parameters is not re¯ected in the valuation ®gure itself.There is indeed a need for a dynamic analysis, which we call a riskanalysis and describe in the next chapter. The other side of the de®nition is that we do not know how to account for noncash items if theycannot be turned into cash. A typical example would be the possibilityof a system failure on a futures expiry. At this time the inability totransmit orders to the stock exchange could prove disastrous. We donot really know how to value this risk, and accordingly we cannotincorporate it in the total valuation although it is extremely real.
PRINCIPLESLiquidation, Cash, and Present Value
In performing a valuation, cash is the only object of interest. The purpose of the operation is to compute the net present value of all liquidating ¯ows, regardless of their effective date. The term liquidation hastherefore a dual meaning. On one side it means that we have to neutralize unknown ¯ows still dependent on the market, and on the otherside we must ensure that they are adequately offset by a money marketoperation. At the end of the liquidation, there is no more uncertain ¯ow inthe future, as every single one of them has been ``canceled out'' and discountedtoday.To better understand the implication of this procedure, consider along forward position. If we sell the forward today, the settlement ¯owof this transaction will take place at expiry, as opposed to a futurescontract for which the settlement would have taken place in the nextdays with the margin call. The sale does neutralize the forward becausethe ®nal ¯ow is thereafter known with certainty. The second steprequires us to borrow or lend today the amount associated with theforward ¯ow, so that its value is canceled out. If we were supposed to
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receive $1,000,000 as a ®nal settlement in two months, we would haveto borrow today an amount such that we have to reimburse exactly$1,000,000 in two months. The amount we borrow, which is exactly theamount of cash generated by the liquidation of the forward, is thevaluation we are looking for. After we secure both the sale and theborrowing, we still have to settle two ¯ows on expiryÐ$1,000,000 onthe forward side and the opposite amount on the money market.However, at this time the settlement is only a technical formality.For reasons that will become apparent later, it is often practical tochoose the date on which all ¯ows are discounted to be the regularsettlement date on futures; D  1, for example, in the United States. Itis relatively unimportant as long as it is consistent within a givenportfolio.
Continuity and Time Partition
Continuity refers to the consistency of the valuation through time.Typically professional traders have to run a valuation on their positionevery day, and the difference between two of these ®gures representsthe pro®t or loss generated over the period.2 This P&L is critical inmany respects because it determines the performance of the tradingactivity and needs to be examined thoroughly.There are many pieces of information we want to extract from theP&L statementÐnotably a decomposition by instrumentÐbut we areconcerned here with the chronological analysis. This layer tries todetermine what is the fraction of the P&L generated by new transactions as opposed to the carry of preexisting ones.Consider as an illustration that we hold a position of 1 share of S,priced at $100, $110, and $95 over a three-day period. The P&L, withobvious notations, is V1!2  $10; V2!3  $15. Now assume thatwe buy 1 additional share of S on Day 2 at $105. At the end of Day 2,we have now 2 shares, and the P&L becomes: V1!2  $15,V2!3  $30. V1!2 has increased by $5 due to the purchase of 1additional share, but this ®gure itself does not allow for any distinctionbetween preexisting and newly acquired positions. Therefore, in computing our P&L, we want to introduce the notion of a preexistingposition, and we write:preexisting
V1!2  V1!22
 V2new
More generally we de®ne a pro®t and loss as a difference of two valuations.
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which would translate here into: V1!2 preexisting  $10, V2new $5.In practice, this equation can be generalized and written as:YP V TDVY!T  VY!T
The total valuation from yesterday to today is equal to the valuation generatedby the position carried over plus the increment due to today's transactions.The denominations ``yesterday'' and ``today'' are commonly used todescribe any day such that yesterday < today.
Parametric Valuations
As we indicated, we want to consider only cash when evaluating aposition. Another way of specifying the same constraint is to imposeour valuation to be parametric, which essentially means that its mathematical formulation is exclusively built upon numerical parametersknown with certainty at the time of valuation. These parameters can bemarket parameters such as interest rates, closing prices, internal costssuch as that of capital, or external constants such as the tax rate ondistributed income. In any case, our valuation will take the form V f P1 ; P2 ; P3 ; . . . ; Pn  where P1 ; . . . ; Pn represents the parameters wedeem relevant.This formalism is a direct consequence of the fundamental characteristic we postulated. If we care about cash and cash only, we shouldbe able to associate each component in the valuation to a numericaldescriptor. It is also extremely convenient because we want the valuation process to re¯ect accurately current market conditions and moregenerally the speci®cs of the position as of today. Therefore, we imposean algebraic formulation such that all these characteristics are capturedinto a set of parameters. If at a future date we want to re-create themarket as we know it today, we know we can reuse today's parameters. Along the same lines, if we are trying to compare two differentdates, we know this comparison is reduced to a set of parameters andcan be automated relatively simply.
Overnight Financing
A valuation breaks the time line in two, before and after. In terms ofcash ¯ows, this separation has a very tangible consequence becausemost of the ¯ows ``after'' are unknown and in any case not received
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FINANCIAL VALUATION AND RISK ANALYSIS
yet, whereas all the ¯ows ``before'' are known and have actually beenreceived.If we look precisely at the amount of money on balance in theaccount associated with the trading activity, there are two differentsets of ¯ows. Certain ¯ows result from the life cycle of a traded instrument, whatever it is, and we call them settlement ¯ows. Others areinterest payments resulting from the carry of the entire balance fromone day to the next, and we refer to these as overnight ®nancing expenses.Suppose, for example, that today is a forward expiry and that we aredue to receive $1,000,000 from the counterpart as settlement of theforward tomorrow. This payment is clearly a direct result of a tradedposition and thus is a settlement ¯ow. Once the payment is received, itwill earn interest at the overnight rate, for example, 5 percent fromtomorrow to the day after tomorrow. At that time the balance amountsto $1,000,138.89, adding $138.89 of interest (= 5 percent/360 $1,000,000).3 This amount of interest is a ®nancing ¯ow because it isassociated to an overnight money market operation. Because of thedifference in nature between settlement and ®nancing, we need tomake a distinction and track them separately. Settlement ¯ows arecritically important because they must be exchanged with third partiesand thus they must be easily accessible. Financing ¯ows, which areexclusively the results of overnight transactions, must also beaccounted for because their incidence on the overall result of the activity is not negligible, and yet they are not necessarily the result of anyactive trading decision.If we denote C and I the overall balances due respectively tosettlement ¯ows and overnight interests, the following relationshipsformalize our discussion:IT  IY  IY  CY   rO
CT  CY   CFT
IY and IT represent respectively the accumulated amount of ®nancing¯ows for yesterday and today; CY and CT, on the other hand, are theaccumulated amount of settlement ¯ows.4  CFT is a generic termdescribing the algebraic sum of all settlement ¯ows receivable and3
If in contrast we had lent the same $1,000,000 for a week through a money marketoperation, the ®nal interest payment would have been a settlement, not a ®nancing¯ow, because it would have been the result of a particular transaction, as opposed tomoney left on the account after every claim had been settled.4The notion of an accumulated amount implies a time origin on which the accumulation started. In practice, this time is often the beginning of each ®scal year for theinstitution performing the valuation.
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payable today. rO is the relevant overnight rate from yesterday totoday.In short, the balance of ®nancing interests today is equal to yesterday'splus yesterday's total balance carried at the overnight rate. Along the sameline the total balance of settlement ¯ows today is yesterday's balance combined to all ¯ows expected on today's settlement date.5 In practice, C and Iare computed instrument by instrument and aggregated over the entireposition.Despite the somehow arti®cial nature of this formalism, it is essential because it corresponds to an internal reality. In many institutions,different departments are responsible for managing C and I independently. As we mentioned earlier, treasury departments are usually incharge of overnight ®nancing, which in practice means managing I forall sorts of activities within the institution.6 C, on the other hand, isusually monitored by the back of®ce because it includes ¯ows thathave to be paid and received in a timely manner.
Front Of®ce versus Back Of®ceDistinctionAs we mentioned already, the existence of distinct internal controlprocedures requires the presentation and reconciliation of differentperspectives from different parties. A very clear illustration of therisk created by a failure to design and implement such independentprocedures, was, for example, the collapse of Baring's in 1995. NickLeeson at that time was head of both the back and front of®ces,7 and hewas in a position to dissimulate losing trades on a large scale, a privilege that he used and abused over and over again.5
It is important to note that C and I are de®ned as of today's settlement date. In theexample of the $1,000,000 forward settlement tomorrow, IT is the overnight ®nancingcharge payable or receivable on the day after tomorrow, on tomorrow's balance, due to theexpiry today. Because it appears due to today's activity, we de®ned it as today's overnight charge although the overnight charge actually paid today is different and is theresult of the trading activity a few days ago.6Treasury management has in fact two layers. A trader is naturally involved in optimizing his or her own treasury, whereas the treasury department has a broadermandate with respect to optimizing cash requirements for the institution at large.7Front of®ce refers to the trading and personnel infrastructure dealing directly with themarket. Back of®ce, on the other hand, denotes the organizational structure dealingwith clearing, settlement, and accounting procedures. In some instances there exists anintermediate step not surprisingly called middle of®ce, for preprocessing informationbetween the front and back of®ces.
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FINANCIAL VALUATION AND RISK ANALYSIS
In the context of a closely monitored professional trading ¯oor,traders and back-of®ce risk managers produce their own independentP&L statements. Requirements for each of these may vary from oneinstitution to the next, but in general back-of®ce ®gures tend to bemuch more conservative, notably in their appreciation of cash ¯ows.Consider as an illustration the case of a one-month money markettransaction entered today at a rate of 5 percent. If the traded nominal is$1,000,000, we receive that amount today and pay back $1,004,167 onemonth from now. One week later, let's look at the valuation of thesame operation assuming that the three-week rate is still at 5 percent.According to our principles, we enter an offsetting money marketoperation to match the $1,004,167 ¯ow in three weeks, so there is nomore ¯ow on expiry. Therefore, the total P&L generated by the twoopposite transactions is8:P  $1,000,000
$1,004,1673  5%148
$1,038:42
This amount corresponds to the implicit interest charge incurred overthe week elapsed from the initial operation and is exactly the frontof®ce valuation.A back-of®ce valuation would look at the same situation from anaccounting perspective and associate the valuation to the current costof the operation.9 In other words, if we have to pay $4,167 over amonth to use $1,000,000, for a week, we should pay one fourth.Therefore, the P&L is here:P
$4,1674
$1,041:75
Both ®gures are very close, but they are not equal. The difference iseven accentuated if interest rates move. In a front-of®ce valuation,assuming the three-week interest rate changes to 6 percent, the P&Lbecomes:8
For simplicity, we assumed a year is 12 months, and 4  12 = 48 weeks. We alsoneglected overnight ®nancing for the $1,000,000 sitting on the account.9Readers familiar with accounting know that one of the most fundamental principles inaccounting is the going concern: An economic entity should be considered from theperspective of an ongoing business. Front-of®ce ®nancial valuation, on the other hand,is very clearly focused on immediate liquidation. Hence a lot of basic differencesbetween the two points of view.
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P  $1,000,000
$1,004,1673  6%148
VALUATION
275
$415:11
The loss has decreased because the present value of the ®nal ¯owis lower. The back-of®ce ®gure, however, is unchanged because thecurrent cost remains identical.There can be little argument about the necessity to include changesin interest rates to accurately value a money market transaction from afront-of®ce perspective. The fundamental principle of meaningfulnesssimply requires it. On the other hand, the back-of®ce and accounting®gures ful®ll different needs, and these needs are better served withdifferent conventions. Naturally there is no doubt that these differencesshould be reconciliated very precisely, and this situation opens the doorto long and intricate explanations if both parties do not recognize fromthe start their distinct nature. In general, ®nancial controllers and riskmanagers are highly in¯uential in this respect because it is their primaryresponsibility to ensure that reliable ®gures get reported every day.10
Mark to Market versus Theoretical
It is always possible to argue in any trading situation about the fairprice of a particular instrument, as opposed to the price at which it istrading in the marketplace, based on some sort of proprietary knowledge or expertise. After all, nobody would ever buy a stock without theconviction that it is priced too low.A valuation based on some esoteric justi®cation can be characterized as theoretical, in contrast to a valuation marked-to-market. In general,sound risk management requires regarding the whole theoretical argument with a lot of suspicion and basically denying it systematicallyexcept in a certain number of well-de®ned situations. In other words,the market is always right, and whatever its price is, there is no wayaround it.As it happens, many of the well-de®ned situations in which atheoretical valuation is warranted, if not necessary, are arbitragerelated. As we explained before, an absolute convergence has someunique features that make its theoretical valuation a requirement.Therefore, in the particular context in which we evolve, we need toprovide an alternative theoretical valuation when necessary.10
A sound risk management practice is to extract P&L ®gures from statements produced by ®nancial controllers, which forces traders to justify their own results, especially if they are higher.
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FINANCIAL VALUATION AND RISK ANALYSIS
However, even if a theoretical valuation has to be called upon, forexample, for futures and forwards, in principle it is, and will alwaysremain, a ®nancial monster because it relies on the assumption that themarket is ``wrong.'' For that reason, a back-of®ce P&L will not surprisingly always use exclusively mark-to-market prices, which is commonly one of the major differences with front-of®ce results. Problemshere are much more acute than pure methodology differences for tworeasons. On one side, a theoretical valuation substitutes nonobservable``arbitrary'' prices to observable immediate prices. And it also introduces a disconnection with reality that may have far-reaching consequences.To illustrate both points, consider a futures contract on the S&P500. Assume that the index closes at $1,400 today, with a basis of $15.Fair value for the futures is therefore $1,415, but assume it closes at$1,418 in the market. The theoretical futures in that situation are valuedat $1,415 whereas the mark-to-market price is $1,418. If we are long onefuture, the theoretical valuation will be lower than the mark-to-marketone by $3Ðthat is, $750 ( $3  $250). For a large position this difference might represent a sizable amount of money, with the consequenceof disconnecting risk managers' and traders' ®gures. In addition, themargin call of the futures contract is computed on $1,418 as opposed to$1,415. Therefore, valuing the position at $1,415 means that valuation isnot in line anymore with actual monetary ¯ows, which is disturbingbecause we speci®cally requested the valuation process to care onlyabout actual cash ¯ows. This apparent contradiction disappears onexpiry because the futures contract comes in line with cash and the®nal margin call matches the theoretical valuation of the position perfectly. Still, at any time before we can have no certainty about actualmargin calls versus theoretical valuation, and we have to be aware ofthat.Therefore, introducing theoretical valuations is a dangerous exercise, and it should be practiced recurrently as we said only in wellknown situations, such as absolute convergence situations. Whennecessary, we will present both formulations for each instrument.
Valuations and Random Variables
By and large, all market prices can be considered random variables,and rigorous option pricing, for example, is highly dependent on theassumed distributions of these variables. The different models usednowadays differ on the characteristics of each particular parameterbehavior, as well as on a number of simplifying assumptions. For
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example, the Black and Scholes option model treats stock prices asrandom walks but does not account for a stochastic interest rate, andit has been adapted since.11Therefore, it is a natural question to ask about the existence of astochastic valuation. By de®nition, such a calculation would accountfor the variability in all market parameters in a mathematically satisfactory way, producing a valuation ®gure potentially more accurate.Clearly, the choice we have made to freeze a position before valuing itscomponents is not an accurate re¯ection of the reality of the marketplace.Generally speaking, there can be two ways to deal with thisproblem. On one side, we can freeze the market, value our position,and account for variability in a second step, which we called a riskanalysis. Varying each parameter within a given range and measuringthe effect on the position then gives a simple picture of the underlying parametric risks. If we detect, for example, an increase in valuation as a result of an increase in the price of a given security, weconclude that we are long this security and may decide to hedge theposition by selling an appropriate amount of it. This is essentiallywhat we set out to do here. The second method would be toembed variability in the valuation, trying to come up with anexpected valuation and a distribution for this ®gure. This is what avalue-at-risk does, and we will consider this methodology in moredetail shortly.Each method has its strengths and weaknesses, but arguably the®rst one is the most useful from a front-of®ce perspective. A criticalaspect of managing a trading position is differentiating sources of riskin an intelligible way, so that each risk can be independently monitoredand managed. This may not be always possible because many risks areentangled and correlated, but the ultimate goal should be to carry onlythose risks that we really want to carry, leaving to the market those wedo not care about. In that perspective a two-staged approach explicitlybreaks the valuation into a series of elementary processes, each onefocusing exclusively on one category of risk. Incidentally, this operating procedure has a very well known mathematical counterpart, calleda partial derivative. If we denote f a function of two variables x and y,an elementary variation in f , denoted df , is a function of an elementaryvariation dx or dy:
11
The case of interest rates is indeed particularly meaningful because numerous modelshave been proposed over the years, probably more than for equities.
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FINANCIAL VALUATION AND RISK ANALYSIS
df x0 ; y0  
@f@x

x0 ;y0 
dx 
@fdy@y x0 ;y0 
The notation
@f@x
x0 ;y0 
describes an elementary variation in f due to a move in x, around thepoint x0 ; y0 , assuming y is constant. Respectively,
@f@y
x0 ;y0 
assumes x is constant and measures an elementary variation in f due toa move in y. Overall the total variation df is a linear combination ofboth terms. This formalism is extremely useful in our context because itessentially means that a risk analysis is nothing more than the numerical estimation of partial derivative, a point that we will develop in thenext section.In contrast a value-at-riskÐV@RÐusually provides one single®gure assorted of an expected probability. For example under a 95percent certainty, the maximum loss or maximum pro®t of a givenposition is such and such. This massive aggregation of information isdif®cult to interpret at the trader's level because it incorporates manydifferent types of uncertain parameters. For this exact reason, a V@R isin fact much more useful at the highest level of risk management,typically executive committees and eventually COO and CEO level.At the same time its methodology incorporates a certain number oftechnicalities clearly missing in a partial derivative analysisÐnotablythe observed correlation between parameters.12Therefore, it is fair to say that our choice of a frozen valuation isnecessary in many respects because we place ourselves on the tradingdesk, but we should de®nitely be aware of its limitations. In the context12
Consider, for example, that we hold a position exposed on the 5-year and 10-yearinterest rates. Should these be considered independent parameters? No straightanswer to this question is satisfactory because these rates are clearly closely correlated, but it is also very possible for one to change signi®cantly without the other'sbeing affected. A partial derivative approach would consider that they move independently, whereas a V@R would typically measure their correlation based on available historical data. The reality of the market is somewhere in between.
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of arbitrage, optionality is very low or nonexistent and instrumentsrelatively simple so these limitations are benign.
Partial Derivatives and Explanatory Valuations
Beside their fundamental application to risk management, partialderivatives have a lot to offer to traders with regard to the behaviorof their position. Typically even if markets are relatively quiet, a lot ofparameters change from one day to the next. A valuation processconsiders each class of assets distinctly, which can in turn be brokendown into smaller categories to facilitate some ®ne-grain checking ifnecessary.Consider, for example, a standard futures contract. Its theoreticalprice is a function of the price of its underlying, of the interest rate tomaturity, and of the capitalized expected dividend amount untilmaturity. If all three change from today to tomorrow, the aggregatevaluation as such does not provide a lot of details in and of itself. Still itis extremely useful, if not necessary, to isolate the impact of each oneindividually, as a safety check about what is really happening in theposition.An easy way to solve this dif®culty is to perform a valuation independently on each parameter, which is equivalent to a partial derivative formulation.13 If PY1 , PY2 ,. . ., PYn denotes the n valuation parametersas of yesterday, and PT1 , PT2 ,. . . ; PTn for their value today, we can write:VVVY!TY!T P1:P2 PY!T  VY!T nP1P2PnEach term

VPi

 PY!Ti
represents the partial valuation due to a change in Pi , and should beclose to 0 for a position perfectly hedged with respect to Pi . It is computed very simply by running two valuations with slightly differentparameters:VV PY1 ; . . . ; PTi ; . . . ; PYn  V PY1 ; . . . ; PYi ; . . . ; PYn PiPTiPYi13
Note that the implementation of this technique is possible because we imposed thevaluation to have a closed parametric form.
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FINANCIAL VALUATION AND RISK ANALYSIS
The only parameter changed is Pi , which is taken as of today. All otherPi are kept equal to what they were yesterday. In practice, the partialvaluation is performed only for the most important market variables,typically spot prices, interest rates, and time. By analogy to optionspricing, the quantities computed are respectively called delta, theta, andrho.
Instruments versus Portfolio Valuations
As we indicated, it is relatively natural to perform an independentvaluation for each speci®c class of instruments, but in many casesthis approach does not look optimal. Recall, for example, how theSPAN margining system, by looking at the complete portfolio, led toa cheaper margin cost than that computed by looking at each individual component, notably because of cross-hedging opportunities. Foran arbitrage valuation we can apply the same type of reasoning andcome up with a similar conclusion because looking at only one leg ofthe arbitrage negates the original intent, which was to set up a fullyhedged risk-free position.Therefore, it would be tempting to tailor a generic valuation procedure to particular situations. For example, when considering an indexarbitrage, we could measure the difference between cash and theoreticalfutures and base the overall valuation on this difference, instead ofvaluing each instrument independently. An extreme example of sucha process would be to simply postulate this valuation to be zero, on thegrounds that it is a risk-free situation by de®nition. Obviously, thisproposition would not be very popular with risk managers.Still it is absolutely legitimate to examine the issue of valuationbetween the whole and the parts. V@R techniques are indeed a partialanswer to that concern, because by de®nition they explore a givenportfolio as a whole, and they link distinct instruments through theirhistorical correlation. However, we argue that there should never be anyportfolio aggregation for the purpose of evaluating a position, because this isexactly what a theoretical valuation is meant to address.Consider, for example, a situation of risk arbitrage in which thetwo merging stocks should be held in proportion to their merging ratio.If we were to take a portfolio approach, we would consider the twolegs as one single position, and evaluate the spread compared to itsexpected valueÐthat is, 0. Alternatively, we can value one of thestocks, for example, the acquirer, at its market price, while in contrastthe acquisition target carries a theoretical price deducted from themerge ratio. Let's call A the acquirer, B the acquisition, and r the
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ratio, and assume that PA  $100, PB  $49, and r  1=2. Looking atthe portfolio as one single entity essentially means de®ning a newsecurity S such that S  PB r  PA  $1. In that situation we arelong 2 shares of B, and short 1 share of A, and the spread has atheoretical valuation of 0 and a mark to market of$1.14Alternatively, if we value A and B independently, we can attribute atheoretical value to B; PTHB , which would be in that case $50  r  PA .The mark-to-market value for the entire position would still be PB r PA and its theoretical value, PTHB Ð r  PA . It is apparent that bothmethods lead to the same numerical values, but the second option ismuch more satisfactory for several reasons.First, if only B carries the label ``theoretical,'' we avoid the dangerof a nongeneric valuation. Introducing particularities in ®nancialvaluations is always a dangerous exercise because it creates the riskof losing track of the market. The second reason is that it is simpler toimplement and monitor. A price correction on one single security isvery straightforward to perform, whereas a speci®c risk arbitrage software module designed