to handle only certain situations would requirecustom development, possibly useless in nonstandard situations.Finally, and this is probably the most powerful argument, a pricecorrection forces consistency as opposed to a valuation in which eachportfolio would be considered an independent entity. Consider, forexample, the case of two strategies trading the same stock, with theresult that the net position in this stock is nil. If each strategy uses aspeci®c valuation technique, the same stock will receive two differentimplicit prices, possibly different. This is obviously an absurd situation,even more so considering that the stock is not even part of the inventory. In contrast, if the question were to assign a single theoretical pricefor the stock, the structure of the valuation would at least ensure that aconsistent answer is provided.15
14
Note that S has been de®ned such that S  0 indicates an opportunity to buy B, whichis in practice the most common case.15This example raises a whole set of interesting issues. We said earlier that a ®nancialvaluation should be meaningful with respect to the external environment, but itshould naturally also be consistent with the internal environment. This problem iscritical in large organizations in which traders on different desks may be trading thesame instruments with different parameters.The implicit recommendation here is that one stock should have a unique marketprice, and in some circumstances a unique theoretical price.(continued)
282
FINANCIAL VALUATION AND RISK ANALYSIS
Therefore, we will ignore the speci®cs of the trading strategy goingforward, and we stick to a valuation performed instrument by instrument. When an adjustment is necessary to re¯ect a known singularity,it will be performed through a price revision and the introduction of atheoretical valuation.
Provisions on Anticipated Expenses
The question of transaction costs is particularly important when dealing with a valuation procedure. In general, these costs are relativelylow and typically masked in most cases behind the expected pro®t ofeach transaction. If we have to liquidate the whole position, however,they take a new dimension.Consider, for example, a $1 billion index arbitrage position on theS&P 500. With an index at $1,400, this position holds 2,857 futures,which represents a liquidation cost of $14,285 if we pay $5 per contractin execution commissions. Naturally, $14,285 may represent a ridiculous amount compared to the overall valuation, but in and of itself it iscertainly not negligible. On the cash side, if we take the average price ofa stock equal to that of the NYSE in 1999Ðthat is, $43.88Ðwe have22,789,425 shares in position, and if execution costs amount to 1 centper shareÐa reasonable assumptionÐthe overall liquidation cost isclose to $227,894.This very simple example shows clearly that transaction costs cannot be neglected when dealing with the overall liquidation process. Aswe will see, there are different ways to account for them, notably interms of realized versus expected value. As a general principle, theyshould be included as precisely as possible.
(Footnote continued)In some circumstances, it is possible to have different front-of®ce and back-of®ceprices. This is the case, for example, in Japan, where stocks under a heavy unbalance ofunexecuted orders move slowly up or down. If the unbalance persists at the time ofclosing, the last traded price may be far away from the current reference price thatre¯ects the most current market conditions but for which no transaction has taken placeyet because of the unbalance. The index calculation uses the reference price for thosestocks that are included in an index. There is no doubt that the reference price is a betterrepresentation of the market, but the back of®ce is under speci®c requirements to usethe last price at which a transaction occurred.
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283
INSTRUMENTSStocksIntroductionStocks are the easiest instruments to evaluate. If the valuation is performed after the close of the main market, the fair price of the stock isits closing price. The cash ¯ow generated by liquidating a position isthen exactly the position held times its price minus the establishmentprice:!PT PHTVT  QT 1  rTwhere QT denotes the number of shares held today, PHT the historicalprice paid to acquire today's position, and rT the relevant zero-couponrate between stock and future settlement dates.16 If we consider thesame position carried over between two days indexed as Y and T:!PT PHPY PHYYYPVY!T  QY 1  rT1  rYand again this ¯ow occurs on today's future settlement.17The term V TD represents the liquidating cash ¯ow resulting fromthe incremental position we traded today, which can be expressed as:PQT QY   PT  CFTTDV 1  rTwhere  CFT denotes the algebraic sum of all settlement ¯ows due totoday's deals. Therefore, it follows that:!PT PHPY PHYYYPTDVY!T  VY!T  V  QY 1  rT1  rYPQQY :PT  CFT T1  rT16
In general, futures are settled before stocks, which explains the form of the discountfactor. The extreme example in that respect was France, where many stocks settledonce a month but value date for the future is D  1.Note that the expression of VT , correct here, is not consistent with other formulasused in prior chapters. These were simpli®ed to make a particular point.17Under the simplifying assumption rT  rY ; PH disappears from the formula. In otherwords, the valuation does not have any memory beyond the day of acquisition. Thisassumption is often used in practice because it is a second-order approximation.
284
FINANCIAL VALUATION AND RISK ANALYSIS
If PHT denotes the average price as of todayÐthat is, including today'stransactionsÐwe have the additional relation:XHCFTQT :PHT  QY :PYwhich leads to:VY!T  QT 
PT PHT1  rt
!QY 
PY PHY1  rY
!
consistent with the initial de®nition of V. If we were to adopt theassumption rT  rY ; VY!T would simplify into:VY!T 
QT  PT
QY  PY   CFT1  rT
which is much simpler because it does not require any historical dataabout the position.18Regarding the overnight ®nancing charge, we have the following:. CT represents the balance of settlement ¯ows: CT  CY   CFT :O. IT is given by IT  IY  CY  IY   rOY , if rY stands for the appro-
priate overnight rate from yesterday to today.
Therefore, the ®nal valuation looks like this:!!PT PHPY PHTYQY  CY  IY   rOVY!T  QT Y1  rT1  rYThis de®nition satis®es all the principles we highlighted: It is cashdriven, continuous,19 parametric, inclusive of overnight ®nancing,and equal to a liquidation value. Above all, it is universal, which isof paramount importance. It applies to any type of situation, and it isthe only tool we need to value a stock position.DividendsThe inclusion of a dividend is fairly simple when valuing stocks.Because the closing price is what it is regardless of the dividend, a18
Note that the accuracy of rT  rY depends also on the number of days implied in therate calculation. For example, consider that today is Wednesday. Yesterday's futuresettlement is today, and yesterday's stock settlement is Friday. Today's future settlement is tomorrow, and today's stock settlement has been pushed to Monday becauseof the weekend. Therefore, a factor (1  rY ) would include a time fraction of 2/360whereas the fraction in (1  rT ) is 4/360 due to the weekend.19Note, for example, that VT1 !T3  VT1 !T2  VT2 !T3 , which indicates that we canbreak a given time period into as many intervals as we want.
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VALUATION
285
dividend going ex appears only as an additional expected cash ¯ow.Assuming the ex-date is today, the term VY!T is modi®ed andbecomes20:!HHPPPPTYYYYP QY  dVY!T1  rT1  rYif d denotes the dividend per share. The asterisk indicates that thedividend needs to be adjusted to account for its payment date, whichis after the ex-date. Therefore, d  d= 1  r, where r is the zerocoupon rate between the ex-date and the payment date. This formulation is potentially incomplete, however, because it does not re¯ect theexistence of taxes on dividends. If we call t the effective tax rate, 1t  d is the cash on payment date. The term QY  d must then bereplaced by QY  1 t  d .ApplicationsConsider that we are trading a stock S over three days, and we executethe following transactions, settling at D  321:. Day 1: Purchase of 100 shares at $93/4, closing price $10.
. Day 2: Purchase of 60 shares at $93/4 and 35 more at $101/2, sale of 25
shares at $9 and 20 more at $111/4. Closing price $11.. Day 3: No trading, closing price $101/2.
Table 6-1 summarizes the different variables in the day-to-day valuation of this position.The results in Table 6-1 are obtained from the following calculations:. CFT : On Day 1 we buy 100 shares
at $93/4, for a total amountof $975:  CF1  $975. The next day we buy 60 shares at thesame price, and 35 more at $101/2. The purchase amount is therefore60  $93/4 + 35  $101/2  $952:5. If we perform the same calculationon sales, we get in total  CF2  $502:5. These ¯ows areaccounted for on the day we trade but will actually settle on thesettlement dateÐthat is, Days 4 and 5, respectively.
YPThe dividend is included in VY!Tas opposed to V TD because it is a consequenceof the existence of a position QY as of yesterday.21For simplicity, we neglect the discounting at (1  rT ) in this example.20
286
FINANCIAL VALUATION AND RISK ANALYSIS
TABLE 6-1Variable
Day 1
Day 2
Day 3
QYQTPYPTCFT ÐpurchaseCFT ÐsaleCFTVYYP!TV TDVY !T without overnightCY !TIY !TVY !T with overnight
Ð100Ð$10±$975Ð±$975Ð$25$25±$975Ð$25
100150$10$11±$952.5$450±$502.5$100$47.5$147:5±$502.5±$0.13$147:37
150150$11$101/ 2ÐÐÐ±$75Ð$75Ð±$0.20±$75:20
YPYP. VY!T: On Day 2, applying the formula VY!T QY  PT PY YPYP= 100  ($11 $10  $100. Similarly, V2!3leads to V1!2
150  $101/2 $11 
$75.
PQT QY   PT  CFT givesV  100  $10 $975  $25. Similarly, V 2  150 100  $11$502:5  $47:5; and V 3  0 because Q3  Q2 .. CY!T : On Day 1 we settle a ¯ow equal to $975 for the purchaseof 100 shares: C0!1  $975. Similarly, on Day 2 we settle$502:5, C1!2  $502:5, and C2!3  0. C0!1 is physicallysettled on Day 4, C1!2 on Day 5.. IY!T : The overnight cost is computed on the actual treasury balance C  I. Therefore, on Day 2, we owe $975  5%=360  13:5cents for a 5 percent charge (actual/360). Similarly, on Day 3 weowe ($975:13  $502:5  5 percent=360  20:5 cents. Note againthat these ¯ows are accounted for today but occur physically onDays 5 and 6 respectively.. V TD : On Day 1, applying V TD 1
This simple example provides a perfect illustration of the differentprinciples we developed earlier. We have now a rigorous and systematic way to value any type of stock activity over a given period oftime. Pertinent parameters driving the valuation are unambiguouslyidenti®ed. The total P&L can be precisely allocated between the recurrent trading activity and the carry of preexisting positions, and wehave a clear vision of the treasury balance to monitor settlementdetails.
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287
Theoretical PricesLet's consider again a dividend distribution on a stock. If P denotes theprice of the stock on the last trading day, it becomes (P d) the nextday before the open. This modi®cation is not arbitrary because it issimply the translation of price continuity between yesterday andtoday. Therefore, it is legitimate to de®ne (P d) as the theoreticalprice of the stock on the ex-date, before the open. This price has avery short life span, however, because it disappears immediatelywhen the market opens.Similarly, there are other situations in which the introduction of atheoretical price may be legitimate. We mentioned before a mergersituation, for which the price of the acquired company stock can bemodi®ed to account for the expected convergence on maturity. Moregenerally any type of corporate event is likely to include a theoreticalvaluation, and Table 6-2 presents a summary of those we developed ina previous chapter. In most cases the theoretical substitution is relatively minor because it applies only to a short time span. The theoretical price is then an instantaneous representation for the purpose ofperforming a valuation.In some casesÐfor example, cross listings or mergers or acquisitionsÐthe theoretical price has a longer period of existence, typicallyfrom announcement to the effective date or even unlimited. This substitution, however, applies only to forward-looking prices, not to executed transactions. It has no effect on C or I.Back-Of®ce ValuationsBeing much more conservative than its front-of®ce counterpart, theback-of®ce valuation ignores entirely the possibility of a theoreticalTABLE 6-2Eventa
Mark to Market
Dividend
P
P0  P
Cross listing, ADRbSplit
P¥P
Merger or acquisitionc
PB
P$PP0 rP0PB  Ar
a
Theoreticald
ValidityBefore the open onthe ex-dateAny timeBefore the open onthe ex-dateUntil the effective dateof the merger
r represents the relevant ratio for each particular event.¥ and $ denote a generic pair of currencies, with $ being the principal exchange.cA is the acquirer, B the target company.b
288
FINANCIAL VALUATION AND RISK ANALYSIS
price. Along the same lines, the discount factor (1  rT ) is removedbecause there is no need to account for the difference in settlementbetween stocks and futures. Therefore, a back-of®ce valuation can beformulated simply as:XBO QT  PT QY  PY CFTVY!TIn the case of dividends, the difference is subtler, however. The backof®ce usually does not recognize a ¯ow until it is actually paid; therefore the dividend does not appear on the ex-date but on the paymentdate. In between it necessarily does not match the front-of®ce ®gure.
Futures and ForwardsFuturesThe mark-to-market valuation of a futures contract is arithmeticallystrictly similar to that of a stock:!FT FHTVT  QT  PV 1  rTwhere FT and FHT stand respectively for the market futures price andthe historical cost of the position, and PV is the point value of thefutures contract.However, the margining system is such that in reality we aremarked to market every day, and we would like this fact to be re¯ectedin the valuation. This is very simple to do, by setting the future settlement date as the effective date on which we perform the valuation.Under this assumption the discounting disappears, and V becomes:VT  QT  PV  FT FHTFrom there the differential valuation of a constant position is easy toexpress:YPFY FH QY  PV  FT FY VY!T QY  PV  FT FHYYAs for today's activity, the expression of V TD is exactly identical tothat of a stock position22:XV TD  QT QY   PV  FT CFT22
Note that CFT does not represent actual ¯ows because there is no ¯ow associated totrading a futures contract. This is in fact a convenient notation used to describe thealgebraic nominal sum of today's transactions.
CHAPTER 6
which leads to:VT!T  QT  FT
QY  FY   PV 
VALUATION
X
289
CFT
This result is very simple and intuitive: The mark-to-market valuation of afutures position from one day to the next is exactly equal to its margin call.This expression, however, seems to neglect the initial marginrequirement. The initial margin is important only from a treasury perspective because we get it back in full when we close the position,which is the underlying process behind the valuation. It does nothave to be included in V and can be handled by modifying C:. CT  CY  MCT  ICT
where MCT  QT  FT QY  FY   PV  CFT is the maintenance margin, and ICT  ICY  QT QY   IMis the initial margin.23. IT still given by IT  IY  CY  IY   rOY:It follows:VY!T  QT  FT
QY  FY   PV   CFT  CY  IY   rOY
ApplicationsConsider that we are trading an S&P 500 futures contract over threedays, and we execute the following transactions, settled at D  1:. Day 1: Purchase of 10 futures at $1,400, closing price $1,408.
. Day 2: Purchase of 60 futures at $1,415 and 35 more at $1,397, sale
of 25 futures at $1,420 and 20 more at $1,382. Closing price $1,402.
. Day 3: No trading, closing price $1,416.
Table 6-3 summarizes the different variables in the day-to-dayvaluation of this position. These results are obtained from the followingcalculations:. CFT : On Day 1 we buy 10 futures at $1,400, for a total notional
amount of 10  $250  $1,400  $3,500,000:  CF1  $3,500,000.The next day we buy 60 futures at $1,415, and 35 more at $1,397.The notional purchase amount is therefore 60  $250  $1,415 + 35 $250  $1,397  $33,448,750. If we perform the same calculationon sales and add up the two, we get  CF2  $17,663,750. Notethat despite the label ``cash ¯ow,'' these ®gures do not representany payment of any sort. They are only a set of conventions used tofacilitate the calculation process.
23
IM is the initial margin requirement per contractÐfor example, $23,438 for the S&P500 futures.
290
FINANCIAL VALUATION AND RISK ANALYSIS
TABLE 6-3VariableQYQTPYPTCFT ÐpurchaseCFT ÐsaleCFTVYYP!TV TDVY !T without overnightCY !TC Y  IYIY !TVY !T with overnight
YP. VY!T: On
Day 1
Day 2
Day 3
Ð10Ð$1,408±$3,500,000Ð±$3,500,000Ð$20,000$20,000±$214,380ÐÐ$20,000
1060$1,408$1,402±$33,448,750$15,785,000±$17,663,750±$15,000±$138,750±$153,750±$1,325,650±214,380±$29:78-$153,779:78
6060$1,402$1,416ÐÐÐ$210,000Ð$210,000$210,000±$1,325,679.78±$213:90$209,786:10
YPDay 2, applying the formula VY!T QY  PV YPleads to V1!2  10  $250  $1,402 $1,408) YP 60  $250  $1,416 $1,402) Similarly, V2!3
FT FY $15,000.$210,000.. V TD : On Day 1, applying V TD  QTQY   PV  PT   CFTgives: V 1  10  $250  $1,408 $3,500,000  $20,000. Similarly,on Day 2, V 2  60 10  $250  $1,402 $17,663,750 $138,750, and V 3  0 because Q3  Q2 .. CY!T : On Day 1 the margin call is equal to $20,000±$234,380 $214,380 because we have to include the initial margin on 10futures. Similarly, on Days 2 and 3 the margin calls are, respectively, $1,325,650 and $210,000. These three ¯ows are physicallysettled on D  1Ðrespectively, Days 2, 3, and 4.. CY  IY : The overnight cost is computed on the actual treasurybalance (C  1). On Day 2, we owe $214,380  5 percent/360 $29.78 for a 5 percent charge (actual/360). Similarly, on Day3 we owe: ( $214; 380 $153; 750 $1; 171; 900 $29.78)  5 percent/360  $213.90. These ®nancing charges physically appear onDays 3 and 4, respectively.As it appears, the valuation of a futures contract is even simpler thanthat of a stock, and the principles are rigorously identical. We havenow to consider to what extent a theoretical valuation changes theresults.
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291
Theoretical ValuesThe introduction of a theoretical valuation for a futures position isbased on the cost-of-carry model FTH  S  1  r d . Given theunderlying S, if we carry a position until maturity, we pay exactlyr  S (®nancing) and receive d (dividends). It would appear that, rigorously speaking, this formulation is valid only in the context of anindex arbitrage situation because in any other setting the underlyingstock position, if any, is not a rigorous index-replicating portfolio. Infact, all we need to use for the cost-of-carry model is the existence ofarbitrageurs with suf®cient capital to make the theoretical price a reality. Naturally we cannot be sure about the capital reserve available toarbitrageurs at any given time, but out of all arbitrage strategies, indexarbitrage is probably one of the safest, which essentially validates thetheoretical valuation.Algebraically speaking, the theoretical valuation consists only in asubstitution of prices, but this substitution must be done consistently.Because the cost-of-carry price implies the existence of a carry, the ®nalsettlement of a theoretical futures contract occurs necessarily on expiry.THTherefore, if FTHY and FT denote the theoretical prices yesterday andtoday:!THHTHHFFFFTYYYYP QY :PV :VY!T1  rT1  rYV TD 
QT
QY   PV  FTHT 1  rT
P
CFT
and thus:VY!T  QY  PV 

QT
FTHFHTY1  rT
QY   PV  FT 1  rT
FTHFHYY1  rYP
!
CFT
 CFT remains naturally unchanged because it corresponds to thetransactions of the day. As a result, IT is also unchanged.Consider, for example, a forward position on the S&P 500, expiringTHH60 days from today. Assume FTHT  $1,405, FY  $1,400, FY  $1,370,QY  QT  1, PV  $250, and rY  rT  5 percent. VY!T is then givenby:
292
FINANCIAL VALUATION AND RISK ANALYSIS
1
0B$1,405 $1; 370VY!T  $250  B@60  5%1360 $250  $34:71
$1,400 $1,370CC61  5% A1360
$29:75  $1,240:51
Without the discount factor, the difference in price is exactly $5 betweenTHFTHT and FY . This difference is reduced to $4.96 because of the 1-dayshift between yesterday and today (61 versus 60 days to maturity).Naturally if today is such that we bridge over a weekend, this differenceis even larger because the time fraction loses 3 days at once.This overall analysis is important because it con®rms two particularly important points. First, a theoretical valuation is always linked toa particular situation and to prevailing assumptions about that situation. On stocks the situation may be a corporate event, and on futures itis the existence of an active index arbitrage ``industry.'' In both caseswe do not necessarily need to be actively involved in any particularstrategy, but we need to be certain that somebody else is.Second, the valuation itself is very dependent on the rigorous analysis of the instrument, and it requires a close examination of issuesrelated to treasury management.Explanatory ValuationsConsidering that we can express FTHT as a function of other parameters,YPone step further:let's develop the expression for VY!T!THHTHHFFFFTYYYYP QY  PV VY!T1  rT1  rY!ST  1  r0T  dT FHSY  1  r0Y  dY FHYY QY  PV 1  rT1  rYr0T and dT denote respectively the relevant zero-coupon rate to expiryand capitalized dividend payment until then.24 The expression of
24 0
rT 6 rT because r0T accrues from today's stock settlement to the stock settlement onSQ, whereas rT accrues from today's future settlement to the future ®nal settlementon SQ. If the settlement cycle is different on stocks and futures, these rates are notnecessarily equal, although they are probably very close. It would be tempting to usean approximation and simplify the whole expression, but rigorously speaking, r0T andrT do not necessarily have the same number of days, notably around weekends.
CHAPTER 6
VALUATION
293
YPVY!Tis then very clearly dependent on S, r; and d. To separate thesecomponents, we write:!ST  1  r0T  dT  FHSY  1  r0Y  dY  FHYYYPVY!T  QY  PV 1  rT1  rY1  rT1  rY
 QY  PV  S0Y!Twhere:. S0Y!T  ST :
1  r0T1  rT
dY!T
FHY!T 
1  r0YSY :1  rY
is the movement in the underlying price,