Preliminary. Comments welcome. Equity Valuation Using Multiples

Preliminary. Commens welcome. Equy Valuaion Using Muliples Jing Liu Anderson Graduae School of Managemen Universy of California a Los Angeles (310) 206-5861 jing.liu@anderson.ucla.edu Doron Nissim Columbia Universy Graduae School of Business (212) 854-4249 dn75@columbia.edu and Jacob Thomas Columbia Universy Graduae School of Business (212) 854-3492 jk1@columbia.edu January, 2000 We received helpful commens from David Aboody, Jack Hughes, Jim Ohlson, Sephen Penman, Michael Williams, and seminar paricipans a Columbia, Copenhagen Business School, Ohio Sae, and UCLA.

Equy Valuaion Using Muliples Absrac In his sudy we examine he valuaion performance of a comprehensive lis of commonly used price muliples. Our analysis indicaes he following ranking: forward earnings muliples perform he bes, followed by hisorical earnings measures, cash flow measures and book value of equy are ied for hird, and sales performs he wors. Conrary o he popular view ha differen indusries have differen bes muliples, we find ha hese overall rankings are observed consisenly for all indusries examined. Performance is improved by allowing for an inercep in he linear relaion beween price and value drivers, relaive o he raio formulaion ypically assumed in pracice. Performance is no improved, however, by he use of more complex value drivers, such as he shor cu value measures based on generic paerns for residual income growh pas he forecas horizon.

1. Inroducion Equy Valuaion Using Muliples In his sudy we examine he valuaion performance of a comprehensive lis of price muliples. The muliples we consider include hree measures of accrual flows (sales, COMPUSTAT earnings and IBES earnings), one accrual sock measure (book value), four measures of cash flows (cash flow from operaions, free cash flow, mainenance cash flow, and earnings before ineres, axes, depreciaion, and amorizaion (EBITDA)), and hree measures of forward earnings (EPS1, EPS2, and EPS3: 1, 2, and 3-year ou consensus analyss earnings forecass). We also consider more complex ways o incorporae value-relevan informaion, including varians of he popular shor cu value measures based on he residual income model. The muliple approach assumes firm value is direcly proporional o some value driver, such as earnings or book value. The approach is ypically applied as follows: firs, idenify a se of comparable firms; nex, generae a muliple equal o he mean (or median) raio of marke price o he value driver for ha se; and finally, generae firm value by applying ha muliple o he firm s value driver. Comprehensive equy valuaions, which require deailed pro forma analyses and presen value calculaions, should in heory perform beer han simple muliples. In addion o bringing less informaion o bear on he valuaion process, he muliple approach resuls in relaive no absolue valuaion, since firm value is esimaed relaive o he pricing of comparable firms. 1 There are, however, some concerns associaed wh implemening comprehensive valuaions. Firs, he quesion of how bes o conrol for risk remains largely unresolved. Alhough riskadjused discoun raes are used heurisically in pracice, here are concerns ha errors in 1 There is an elemen of relaive pricing even in he case of comprehensive valuaion, since socks are valued relaive o risk-free bonds. If here are concerns abou he risk-free rae (he so-called risk-free rae puzzle), hose concerns remain in sock valuaions based on discoun raes derived from risk-free raes. 1

assumed raes disor valuaions. Second, comprehensive valuaions require projecions o infiny. Raher han make specific projecions for all fuure years, simplifying assumpions (such as consan growh in free cash flows or a muliple of erminal earnings) are normally adoped o capure a erminal value, represening value beyond a horizon dae. Since a large fracion of oal value ypically resides in he erminal value, esimaes of firm value hinge subsanially on he simplifying assumpions. Given hese concerns abou comprehensive valuaions, muliples are used ofen in day-oday valuaion, eher as a subsue for or as a complemen o comprehensive valuaions. Analys repors, regulaory filings, valuaions for esae and gif ax purposes, and he financial press frequenly use muliples o value firms. When complemening comprehensive valuaions, muliples are ypically used o obain erminal values and o calibrae he comprehensive valuaion. The advanages of muliples, relaive o comprehensive valuaions, include exraordinary simplicy and he use of conemporaneous marke informaion. While his simplicy reduces informaion conen, also reduces poenial noise. I is no obvious a priori wheher he benefs of reduced noise exceed he coss of reduced informaion conen. Alhough he muliple approach bypasses explic projecions and presen value calculaions, relies on he same principles underlying he more comprehensive approach: value is an increasing funcion of fuure payoffs and a decreasing funcion of risk. Therefore, he muliple approach should perform reasonably well if he value driver reflecs fuure firm profabily, and he comparable group is similar o he firm being valued along various value aribues, such as growh and risk. To sudy he impac of selecing comparable firms from he same indusry, we conras our resuls obained by using indusry (as defined by IBES) comparables wh resuls obained when all firms in he cross-secion are used as comparables. 2

Regardless of he role of muliples vis-a-vis comprehensive valuaions, here is limed descripive evidence on he absolue and relaive performance of differen muliples, and he variaion across indusries in ha performance (e.g., Boasman and Baskin [1981], LeClair [1990], and Alford [1992]). Recenly, a number of sudies have examined he role of muliples for firm valuaion in specific conexs, such as ax and bankrupcy cour cases and inial public offerings (e.g., Beay, Riffe, and Thompson [1999], Gilson, Huchkiss and Ruback [2000], Kim and Rer [1999], and Tasker [1998]). Our sudy coninues in he same vein, bu is more comprehensive. As in mos prior research, we evaluae muliples by examining he disribuion of percen pricing errors: acual price less price prediced by he muliple, scaled by acual price. To eliminae in-sample bias and conrol for differences in he degrees of freedom across ess, we evaluae all muliples based on ou of sample predicion. Tha is, when calculaing muliples we always exclude he firm being valued. Our analysis consiss of wo sages. In he firs sage, we use he convenional raio represenaion (i.e., price doubles when he value driver doubles). In he second sage, we relax he requiremen ha value is direcly proporional o value drivers, while reaining he assumpion ha he relaion is linear. In essence, he second sage analysis allows for an inercep, whereas he firs sage does no. In he firs sage, muliples are calculaed using he harmonic mean of he raio of price o value driver (he reciprocal of he mean of he value driver-o-price raio) for comparable firms. Alhough his esimaor is rarely used (see Beay, Riffe, and Thompson [1999]), offers he desirable propery ha he percen pricing error is zero, on average. I is also recommended by Baker and Ruback [1999], based on deailed economeric analyses of alernaive esimaors. While he harmonic mean esimaor resuls in lower pricing errors han he simple mean or 3

median, our ranking of he relaive performance of differen muliples remains unchanged when he mean or median is used insead of he harmonic mean. The following is an overview of he relaive performance of differen muliples: forecased earnings perform he bes, even beer han more complex shor cu valuaions based on generic residual income growh paerns pas he erminal dae; among drivers derived from hisorical daa, earnings perform beer han book value; and IBES earnings (which exclude some one-ime ems) perform beer han COMPUSTAT earnings; cash flow measures, defined in various forms, perform poorly; and sales performs he wors. When comparable firms are resriced o be from he same indusry, performance improves for all muliples. We also find ha he relaive performance of he muliples we consider does no vary much across indusries. Tha is, conrary o general percepion, we do no find ha differen indusries are associaed wh differen bes muliples. This finding suggess ha our resul is driven by he inrinsic informaion conen of he differen value drivers, raher han heir abily o capure indusry-specific value-relevan facors. Turning from relaive performance o absolue performance, he forward earnings muliples describe acual sock prices reasonably well. For example, for 3 year ou forecased earnings or EPS3, he sandard deviaion of pricing error is abou 29%, and approximaely half he firms have absolue pricing errors less han 15%. While here are some firms wh very large pricing errors, sock prices for a subsanial majory of he firms are explained relaively well by simple muliples based on wo or hree year ou forecased earnings. The dispersion of pricing errors increases subsanially for muliples based on hisorical drivers, such as earnings and cash 4

flows, and is especially large for sales muliples. For example, approximaely half he firms have absolue pricing errors less han 21%, 25%, and 36% for IBES acual earnings, EBITDA, and sales, respecively. For he second sage, we esimae he inercep and slope of he price/value driver relaion by minimizing he sample variance of percen valuaion errors, subjec o he consrain ha he valuaion is on average unbiased. The procedure we follow is relaed o ha proposed by Beay, Riffe, and Thompson [1999]. As migh be expeced, allowing for an inercep reduces he dispersion of valuaion errors for all muliples, and he improvemen observed is inversely relaed o he performance of ha muliple in he firs sage (no inercep). 2 As in he firs sage, we find ha moving from a cross-secional comparison group o using comparable firms whin each indusry furher reduces pricing errors. These resuls sugges ha he radional raio formulaion should be replaced by a relaion ha allows for an inercep, especially for muliples ha perform poorly in he radional raio formulaion. We recognize, however, ha if simplicy is he primary moivaion o use muliples, he reducion in pricing errors may no be sufficien o compensae for he addional complexy inroduced by adding an inercep. To conras muliples wh comprehensive valuaions, we consruc inrinsic value measures based on he residual income model, assuming generic paerns for residual income pas he forecas horizon. Surprisingly, hese more complex value measures perform worse han simple muliples based on forecased earnings. We examine hree alernaive paerns for poshorizon residual income: (1) consan abnormal earnings pas year 5, (2) zero abnormal earnings pas year 5, and (3) ROE rending oward an indusry median (beween year 3 and year 12). 2 For example, for muliples based on IBES acual earnings, EBITDA, and Sales, approximaely half he firms have pricing errors less han 19%, 22%, and 29%, relaive o 21%, 25%, and 36%, respecively, in he raio formulaion. The improvemen is much smaller for muliples ha perform well in he firs sage; e.g. for EPS3, approximaely half he firms have pricing errors less han 14.6%, relaive o 15% in he raio formulaion. 5

These inrinsic value measures uilize informaion abou forward earnings a differen horizons, equy book values, firm-specific discoun raes, and indusry profabily. Furher, a srucure derived from valuaion heory is imposed o aggregae ha informaion. Despe hese advanages of inrinsic value measures over simple forward earnings muliples, hey do no perform beer han simple muliples based on forward earnings. 3 Preliminary invesigaions designed o uncover possible causes for his resul sugges ha errors in erminal value proxies and esimaed discoun raes are parially responsible. We find ha simply aggregaing earnings forecass for years 1 o 5 produces he lowes valuaion errors of all muliples. We also considered wo oher exensions o he muliple approach (resuls no repored in his version). 4 Firs, we combined wo or more value drivers (e.g., Cheng and McNamara [1996]). Our resuls, based on a regression approach (e.g., Beay, Riffe, and Thompson [1999]) indicae only small improvemens in performance over ha obained for forward earnings. Second, we invesigaed condional earnings and book value muliples. Tha is, raher han use he harmonic mean P/E and P/B values of comparable firms, we use a P/E (P/B) ha is appropriae given he forecas earnings growh (forecas book profabily) for ha firm. We firs esimae he relaion beween forward P/E raios and forecas earnings growh (P/B raios and forecas reurn on common equy) for each indusry-year, and hen read off from ha relaion he P/E (P/B) corresponding o he earnings growh forecas (forecas ROCE) for he firm being valued. Despe he inuive appeal of condioning he muliple on relevan informaion, we were unable o documen any improvemen in performance. Bradshaw [1999a and 1999b] is able 3 4 Bradshaw [1999a and 1999b] observes resuls ha are relaed o ours. He finds ha PEG, a consruc based on forward P/E raios and forecas long-erm earnings growh raes (g), explains more variaion in arge prices and recommendaions han more rigorous valuaion models. Deails of hose resuls are available from he auhors upon reques. 6

o find, however, ha a more resricive form of condioning (P/E equals forecas growh) improves performance for his sample of firms. Our findings have a number of implicaions for valuaion research. Firs, we confirm he validy of wo preceps underlying he valuaion role of accouning numbers: a) accruals improve he valuaion properies of cash flows, and b) despe he imporance of op-line revenues, s value relevance is limed unil is mached wh expenses. Second, we confirm ha forward-looking daa (specifically, near-erm forecased earnings) conain considerably more value-relevan informaion han hisorical daa. Third, we provide evidence on he signal/noise radeoff associaed wh developing more complex valuaion drivers. Finally, our resuls sugges ha forward earnings muliples should be used as long as earnings forecass are available, since hey ouperform he oher muliples in all 68 indusries we examine. The res of he paper is organized as follows: secion 2 conains a leraure review; secion 3 describes he mehodology; secion 4 describes our sample selecion process; secion 5 repors resuls and discusses implicaions; and secion 6 concludes he paper. 2. Leraure Review While mos of he popular exbooks on valuaion (e.g., Copeland, Koller, and Murrin [1994], Damodaran [1996]) devoe considerable space o discussing muliples, here is lle empirical research published on he valuaion properies of muliples. Mos exising papers ha sudy muliples use a limed daa se and consider only a subse of muliples, such as earnings and EBITDA. The mehodology used also varies from one sudy o anoher, making difficul o compare resuls from differen sudies. Among commonly used value drivers, earnings and cash flows have received mos of he aenion. Boaman and Baskin [1981] compare he valuaion accuracy of P/E muliples based on 7

wo ses of comparable firms from he same indusry. They find ha valuaion errors are smaller when comparable firms are chosen based on similar hisorical earnings growh, relaive o when hey are chosen randomly. Alford [1992] invesigaes he effecs of choosing comparables based on indusry, size (risk), and earnings growh on he precision of valuaion using P/E muliples. He finds ha valuaion errors decline when he indusry definion used o selec comparable firms is narrowed from a broad, single dig SIC code o classificaions based on wo and hree digs, bu here is no addional improvemen when he four-dig classificaion is considered. He also finds ha conrolling for size and earnings growh, over and above indusry conrols, does no reduce valuaion errors. Kaplan and Ruback [1995] examine he valuaion properies of he discouned cash flow (DCF) approach in he conex of highly leveraged ransacions. While hey conclude ha DCF performs well in valuaion, hey find ha simple EBITDA muliples resul in similar valuaion accuracy. Beay, Riffe, and Thompson [1999] examine differen linear combinaions of value drivers derived from earnings, book value, dividends, and oal asses. They derive and documen he benefs of using he harmonic mean, and inroduce he price-scaled regressions we use. They find he bes performance is achieved by using a) weighs derived from harmonic mean book and earnings muliples and b) coefficiens from price-scaled regressions on earnings and book value. In a recen sudy, Baker and Ruback [1999] examine economeric problems in idenifying indusry muliples, and compare he relaive performance of muliples based on EBITDA, EBIT and revenue. They provide heoreical and empirical evidence ha absolue valuaion errors are proporional o value. They furher show ha indusry muliples esimaed using he harmonic mean are close o minimum-variance esimaes based on Mone Carlo simulaions. Using he minimum-variance esimaor as a benchmark, hey find ha he harmonic mean dominaes 8

alernaive simple esimaors such as he simple mean, median, and value-weighed mean. Finally, hey use he harmonic mean esimaor o calculae muliples based on EBITDA, EBIT and revenue, and find ha indusry-adjused EBITDA performs beer han EBIT and revenue. Insead of focusing only on hisorical accouning numbers, Kim and Rer [1999] add forecased earnings o he convenional lis of value drivers, which includes book value, earnings, cash flows, and sales. They invesigae how inial public offering prices are se using muliples. Consisen wh our resuls, hey find ha forward P/E muliples (based on forecased earnings) dominae all oher muliples in valuaion accuracy, and ha he nex year EPS forecas (EPS2) dominaes he curren year EPS forecas (EPS1). I has been recognized ha he use of large daa ses could diminish he performance of muliples, since he researcher selecs comparable firms in a mechanical way. In conras, marke paricipans may selec comparable firms more carefully and ake ino accoun suaion-specific facors no considered by researchers. Tasker [1998] examines paerns in he selecion of comparable firms across indusries in acquision ransacions by invesmen bankers and analyss. She finds he sysemaic use of indusry-specific muliples, which is consisen wh differen muliples being more appropriae in differen indusries. 5 3. Mehodology In his secion we describe he differen value drivers considered, and he mehodology used in he wo sages of our analyses: esimaing he price/value driver relaion whou and wh an inercep. 5 Since is no clear wheher he objecive of invesmen bankers/analyss is o achieve he mos accurae valuaion in erms of smalles dispersion in percen pricing errors, our resuls may no be direcly comparable wh hose in Tasker [1998]. 9

3.1 Value Drivers The following is a lis of value drivers examined in his paper (deails of all variables are provided in he appendix): 6 Accrual sock: curren book value (BV). Accrual flows: sales, COMPUSTAT earnings (CACT) and IBES earnings (IACT). Cash flows: cash flow from operaions (CFO), free cash flow o deb and equy holders (FCF), mainenance cash flow (MCF, equal o free cash flows for he case when capal expendures equal depreciaion expense), and earnings before ineres, axes, depreciaion and amorizaion (EBITDA). Forward looking informaion: consensus one year ou, wo year ou and hree year ou earnings forecass (EPS1, EPS2 and EPS3), where eps *(1 ) 3 = eps2 + g, and g is he long erm eps growh forecas provided by analyss. Inrinsic pricing measure (P1*): This measure, which is based on he residual income (or abnormal earnings) valuaion approach, is considered since appears in a number of recen papers and s pricing properies are relaively beer undersood. 7 In essence, inrinsic value equals he book value plus he presen value of fuure abnormal earnings. For fuure years (beyond year +5) wh no available earnings forecass, abnormal earnings are esimaed by assuming ha hey do no grow. Deails of he implemenaion of P1* are discussed in he nex secion. All he variables lised above have been linked o value before. Accouning book value and earnings are used exensively for valuaion purposes. Ohlson [1995] and Felham and Ohlson 6 Some value drivers are no easily classified. For example, Sales, which is caegorized as an accrual flow, could conain less accruals han EBITDA, which is caegorized as a cash flow measure. 10

[1995] build valuaion models in which earnings and book value play insrumenal roles. In some marke inefficiency sudies (e.g., Basu [1977] and Saman [1980]), earnings and book value are assumed o represen fundamenals, and are even shown o conain value relevan informaion no refleced in marke prices. Accruals disinguish accouning numbers from cash flows. Accouning earnings could be more value-relevan han curren cash flows for a leas wo reasons: a) cash flows do no reflec value creaion in some cases (e.g., asse purchases), and b) accruals allow managers o reflec heir judgmen abou fuure prospecs. However, he flexibily allowed whin GAAP creaes he poenial for accouning numbers o be disored, hereby reducing heir value relevance. This poenial for earnings managemen, in combinaion wh he ruism ha price reflecs he presen value of fuure cash flows, has caused some o prefer cash flow muliples over muliples based on accouning numbers. To provide some empirical evidence on his debae, we consider four cash flow measures, and conras heir value-relevance wh wo muliples based on accouning earnings. The four cash flow measures considered are he mos popular ones used in pracice. Each measure removes he impac of accruals o a differen exen. EBITDA adjuss pre-ax earnings o deb and equy holders for he effecs of depreciaion and amorizaion only. CFO deducs ineres and ax expense from EBITDA and also deducs he ne invesmen in working capal. FCF deducs from CFO ne invesmens in all long-erm asses, whereas MCF only deducs from CFO an invesmen equal o he depreciaion expense for ha year. For earnings-based muliples, we consider repored earnings excluding exraordinary ems and disconinued operaion from COMPUSTAT, and acual earnings as defined by IBES. 7 Exising leraure gauges valuaion properies by comparing R 2 from cross-secional regressions. We use a differen meric, which we believe correcs some biases in he popular mehod. 11

The second measure is derived from he firs earnings measure by deleing some one-ime ems, such as wre-offs and resrucuring charges. To he exen ha he IBES measure is a beer proxy for permanen or core earnings (earnings ha are expeced o persis in he fuure), will be linked more direcly o price. Alhough he use of sales as a value driver has less heoreical basis, relaive o earnings and cash flows, we consider because of s wide use in cerain emerging indusries where earnings and cash flow are perceived o be uninformaive. The poenial mismach beween hisorical daa, such as repored earnings and cash flows, and he forward-looking informaion capured by prices has long been recognized in he leraure. Analyss forecass of fuure period earnings offer a possible soluion o his mismach. Liu and Thomas [1999] find ha revisions in analyss earnings forecass and changes in ineres raes explain a large porion of conemporaneous sock reurns. We include EPS1 and EPS2 because hese wo forecass are usually available for mos firms. To incorporae he informaion conained in he long-erm EPS growh forecas, we consruc EPS3 by adding he amoun implied by ha growh rae o EPS2. The discouned residual income model has been widely used as a way o calculae inrinsic values. Several recen sudies provide evidence ha he model explains sock prices (e.g., Frankel and Lee [1998], Abarbanell and Bernard [1997], Claus and Thomas [1999]) and reurns (e.g., Liu and Thomas [1999], Liu [1999]). Consisen wh many prior sudies, we assume zero growh in abnormal earnings pas a horizon dae. Alhough incorporaes more informaion han any of he simple muliples, his approach is no as deailed as a comprehensive valuaion based on pro forma projecions ha allow for firm-specific growh in abnormal earnings beyond he horizon dae. 12

3.2 Tradional Muliple Valuaion In he firs sage of our analysis, we follow he radional raio represenaion and require ha he price of firm i in year (p ) is direcly proporional o he value driver: p = b x + e (1) where x is he value driver of firm i in year, bis he muliple on he value driver and eis he pricing error. Since our focus is on percen pricing errors (ε /p ), no pricing errors, we divide equaion (1) by price, o obain he following. x e = +. (2) 1 b p p Baker and Ruback [1999] and Beay, Riffe, and Thompson [1999] discuss he problems associaed wh esimaing he slope using equaion (1), because he residual in ha equaion is approximaely proporional o price. When esimaing β, we eleced o impose he resricion ha expeced percen pricing errors (ε/p) be zero, even hough an unresriced OLS esimae for β from equaion (2) offers a lower value of mean squared percen pricing error. 8 Empirically, we find ha our approach generaes lower pricing errors for mos firms, relaive o an unresriced esimae, bu generaes subsanially higher errors in he ails of he disribuion. By resricing ourselves o unbiased pricing errors, we are in effec assigning lower weigh o exreme pricing errors, relaive o he 8 To invesigae he radeoff beween bias and dispersion of pricing errors associaed wh our choice of a resriced regression, we invesigaed he disribuion of pricing errors for he unresriced case. We esimaed equaion (2) for comparable firms from he cross-secion. (When using comparable firms from he same indusry, he esimaed muliples generaed subsanial pricing errors.) We find ha he disribuions of percen pricing errors for all muliples are shifed o he righ subsanially, relaive o he disribuions for he resriced case repored in he paper (our disribuions end o peak around zero pricing error). This shif o he righ indicaes ha he muliples and prediced valuaions for he unresriced case are on average lower han ours. We find ha he bias creaed by his shif causes greaer pricing errors for he bulk of he firms no in he ails of he disribuion, relaive o our resriced case. 13

unresriced approach. We are also mainaining consisency wh he radion in economerics ha appears o exhib a lexicographic preference for reduced bias over reduced dispersion. Our approach is o minimize mean squared percen pricing errors when esimaing æe ö equaion (2), subjec o he consrain ha hose errors be zero on average, i.e., E ç = 0. çè ø Since β is he only parameer o be esimaed in equaion (2), he unbiasedness resricion alone is sufficien o deermine ha parameer. Applying he expecaion operaor o equaion (2) and using he resricion, he esimae for b is he harmonic mean of he price-value driver raio. p 1 b = æ x ö E ç p çè ø (3) To eliminae in-sample bias, we esimae b for each firm using all relevan comparable firms excluding he firm ha is being valued. We predic he value of he firm by muliplying he b esimae by he firm s value driver, and hen calculae he percen pricing error as follows: 9 e p p -bˆ x =. (4) p The performance of muliples is evaluaed by examining he dispersion of he pooled disribuion of e / (lower dispersion indicaes beer performance). p 3.3 Inercep Adjused Muliples For he second sage of our analysis, we relax he direc proporionaly requiremen and allow for an inercep: p = a + b x + e. (5) 9 Noe ha some sudies measure he pricing error as he difference beween he prediced value and price (e.g, Alford [1992]) while we measure he pricing error as he difference beween price and he prediced value. 14

There are many facors, besides he value driver under invesigaion, ha affec price. The average effec on price of such omed facors is no likely o be zero. The inercep in equaion (5) capures he average effec of omed facors and misspecificaions and hus s inclusion may improve he precision of ou of sample predicions. pricing errors. As wh he simple muliple approach, we divide equaion (5) by price o focus on percen 1 x e = a + b +, (6) p p p 1 OLS esimaion of equaion (6), wh no resricions, minimizes he sum of he squares of percen pricing errors, bu he expeced value of hose errors is non-zero. 10 For he reasons menioned in secion 3.2, and o mainain consisency wh our esimaes from he no-inercep approach, we impose he resricion ha percen pricing errors be unbiased. 11 Tha is, we seek o esimae he parameers α and β ha minimize he mean squared error ( e / he expeced value of e / p is zero: p ), subjec o he resricion ha min var( ε / p ) = var[( p α β x ) / p ] = var[1 ( α α, β p æe ö s.. E ç = 0. p çè ø 1 p + β x )] (7a) (7b) To obain esimaes for α and β, we resae resricion (7b) as follows ε 1 E( ) = E 1 α β p p F HG x p I = KJ 0 (8) 10 11 In general, his bias could be removed by allowing for an inercep. Tha avenue is no available, however, when he dependen variable is a consan (=1), since he inercep capures all he variaion in he dependen variable, hereby making he independen variables redundan. As wh equaion (2), pricing errors from he unresriced approach for equaion (6) were higher for mos firms (in he middle of he disribuion) bu were smaller in he ails. 15

Solve (8) for α, and subsue ino (7a) o resae he minimizaion problem in erms of he following regression wh no inercep: F HG 1 I L M F H G I O KJ P KJ = N HG KJ Q E 1 1 J x x p β F 1 M I (9) p E p ( ) p E p p P where he differen E (.) represen he mean values of hose expressions based on he comparable group. The esimae for β is hen subsued ino equaion (8) o obain an esimae for α. Those esimaes are hen used along wh he value driver for he firm being valued o generae a valuaion. 4. Sample and Daa To consruc he sample, we merge daa from hree sources: accouning numbers from COMPUSTAT; price, analys forecass, and acual earnings per share from IBES; and sock reurns from CRSP. As of April of each year, we selec a cross-secion of firms based on he following creria: (1) all COMPUSTAT value drivers for he previous year are available; (2) he fiscal year ends in December; (3) price, acual EPS, forecased EPS for years +1 and +2, and a long erm growh forecas are available in he IBES summary file; and (4) none of he price raios is an oulier (defined as lying ouside he 1% o 99% of he pooled disribuion). The resuling sample includes 17,505 observaions beween 1981 and 1996. This sample is used for he descripive saisics repored in Table 1. For he resuls repored afer Table 1, we impose four addional requiremens: (5) share price on he day IBES publishes summary forecass in April is greaer han or equal o $2; 12 (6) monhly sock reurns are available in he CRSP files 12 Since our valuaion model has an inercep, valuaion error would be abnormally large for socks wh very low share prices. 16

for a leas 30 of he 60 monhs prior o April; (7) all muliples are posive; and (8) each indusry-year se has a leas five observaions (indusry as defined by IBES). These requiremens reduce he sample o 9,658 observaions. We adjus all per share numbers for sock spls and sock dividends using IBES adjusmen facors. If IBES indicaes ha he majory of forecass for ha firm-year are on a fully dilued basis, we use IBES diluion facors o conver hose numbers o a primary basis. We summarize all variable definions in he appendix. The P1* variable is calculaed using he discouned residual income model, assuming zero growh in abnormal earnings afer year five: P1 å æe ( eps -kbv ) ö E ( eps -kbv ) (8) 5 * + s +- s 1 + 5 + 4 = bv+ + s 5 ç s= 1 (1 k) è + ø k(1 + k) where bv = book value per share a ime (he end of year ), eps = earnings per share in year, k = he discoun rae for equy a ime. The discoun rae (k ) is calculaed as he risk-free rae plus bea imes he equy risk premium. We use he 10-year Treasury bond yield on April 1 of year +1 as he risk-free rae and assume a consan 5% equy risk premium. We measure bea as he median bea of all firms in he same bea decile in year. We esimae beas using monhly sock reurns and valueweighed CRSP reurns for he five years ha end in March of year +1 (a leas 30 observaions are required). 13 For a subgroup of firm-years (less han 5 percen), we were able o obain mean IBES forecass for all years in he five-year horizon. For all oher firms, wh less han complee 17

forecass available beween years 3 and 5, we generaed forecass by applying he mean longerm growh forecas (g) o he mean forecas for he prior year in he horizon; i.e., eps + s = eps+ s 1 *(1 + g). The book values for fuure years, corresponding o he earnings forecass, are deermined by assuming he ex-ane clean surplus relaion (ending book value in each fuure period equals beginning book value plus forecased earnings less forecased dividends). Since analys forecass of fuure dividends are no available on IBES, we assume ha he curren dividend payou raio will be mainained in he fuure. We measure he curren dividend payou as he raio of he indicaed annual cash dividends o he earnings forecas for year +1 (boh obained from he IBES summary file). 14 To minimize biases ha could be induced by exreme dividend payou raios (caused by forecas +1 earnings ha are close o zero), we Winsorize payou raios a 10% and 50%. 15 We also calculae four varians of P1* (P2* hrough P5*) ha we use o invesigae he informaion/noise radeoff among he componens of P1*. Definions for hese addional variables are provided in he appendix and he resuls are discussed in Secion 5. 5. Resuls 5.1 Descripive Saisics Table 1 repors he pooled disribuion of raios of value drivers o price. The able indicaes ha cash flow muliples are likely o perform poorly. Free cash flow and mainenance cash flow are ofen negaive (approximaely 30% and 20% of he sample, respecively). 13 14 15 We use decile median beas, since firm-specific beas are esimaed wh considerable error. Indicaed annual dividends are four imes he mos recen quarer s declared dividends. We use EPS1 as he deflaor because varies less han curren year's earnings and is less likely o be close o zero or negaive. The impac of alering he dividend payou assumpions on he resuls is negligible, because has a very small impac on fuure book value and an even smaller impac on he compued abnormal earnings. 18

Moreover, he mean of FCF/P is negaive, and he mean of MCF/P is close o zero, despe he deleion of observaions wh exreme values (op and boom 1%). Given he difficuly of mapping negaive value drivers o posive share values, we conclude ha hese wo value drivers are no suable for muliple valuaion purposes and drop hem from he remainder of he analysis. Table 2 repors he Pearson and Spearman correlaions among he raios of value drivers o price. Mos of he raios are highly correlaed, which suggess ha hey share a large porion of common informaion. The correlaions among differen earnings forecas raios are especially high, generally around 90%. Ineresingly, he correlaion beween earnings forecass raios and P1*/P is only abou 50%, which suggess ha book value and discoun rae adjusmens have a significan impac on he informaion conained in P1*. 5.2 Tradional Muliples The resuls of he firs sage analysis, based on he radional raio represenaion (no inercep), are repored in Table 3. The resuls repored in Panel A use he enire cross-secion of firms as comparables for compuing muliples, and he resuls in Panel B are based on comparables seleced from he same IBES indusry group. Ou-of-sample value predicions are made each year, and percenage valuaion errors are pooled across firm-years. We repor he following saisics ha describe he disribuion of he percen pricing errors: wo measures of cenral endency (mean and median) and five measures of dispersion (he sandard deviaion and four non-parameric dispersion measures: (i) 75%-25%, (ii) 90%-10%, (iii) 95%-5% and (iv) 99%-1%). Since we resric he muliples o yield unbiased valuaion on average, all he means are close o zero. 16 16 Since he valuaions are done ou of sample, is naural o expec some means o deviae from zero by chance. 19

The valuaion errors in Panel A exhib sligh negaive skewness, suggesed by he fac ha medians are higher han means. This implies ha he muliple approach undervalues mos firms by a small amoun and overvalues some firms by large amouns. This occurs because he prediced values are bounded from below a zero, while hey are no bounded above. A poenial way o make he error disribuion symmerical is o ake he log of P ˆ/ P(Kaplan and Ruback [1995]). However, we choose no o follow his approach because he percen pricing errors we consider are easier o inerpre. Examinaion of he sandard deviaion and he four non-parameric dispersion measures in Panel A suggess he following ranking of muliples. Forecased earnings, as a group, exhib he lowes dispersion of percen pricing errors. This resul makes inuive sense because earnings forecass reflec fuure profabily beer han hisorical measures. Consisen wh his reasoning, performance increases wh forecas horizon. The dispersion measures for EPS2 are lower han hose for EPS1 (sandard deviaion decreases from 0.348 o 0.311, iner-quarile range decreases from 0.440 o 0.368). The improvemen from EPS2 o EPS3 is less dramaic, consisen wh measuremen error in he long-erm growh forecas used o consruc EPS3. In addion o ranking he relaive performance of differen muliples, he resuls in Table 3 can also be used o infer absolue pricing errors. Specifically, halving he four non-parameric dispersion measures provides an esimae of he range of absolue pricing error whin which a cerain fracion of he sample lies. For example, he iner-quarile range of 0.347 for EPS3 in Panel A, indicaes ha approximaely half he sample has an absolue pricing error less han 17%. 17 17 This saemen assumes he disribuion is symmeric around zero. Because he disribuion is no precisely symmeric around zero, he numbers we provide are approximae. 20

Forecased earnings are followed by P1*, earnings, book value, cash flows, and sales, in decreasing order of performance. I is perhaps surprising ha P1* does no perform as well as forecased earnings, even hough he informaion in each of he forecased earnings is a subse of ha conained in P1*. The valuaion error of P1* has a sandard deviaion of 0.403 and inerquarile range of 0.504. This resul suggess ha alhough P1* incorporaes addional informaion such as firm specific bea, marke ineres raes, book value, and growh; hese explic adjusmens in combinaion wh he assumpion ha all firms abnormal earnings sop growing afer year 5 resul in a noisy valuaion measure. In secion 5.5, we invesigae furher he likely causes for he poor relaive performance of P1*. Comparing he wo summary accouning numbers, book value and earnings, we find ha earnings clearly ouperforms book value, which is consisen wh sree inuion. The valuaion error for book value (BV) has a sandard deviaion of 0.536 and iner-quarile range of 0.697, compared o a sandard deviaion of 0.477 and iner-quarile range of 0.579 for COMPUSTAT earnings (CACT). The performance of hisorical earnings is furher enhanced by he removal of one-ime ransory componens. Consisen wh he resuls in Liu and Thomas [1999], IBES earnings (IACT) have an even lower sandard deviaion of 0.448 and iner-quarile range of 0.549. Conrary o he belief ha Cash is King in valuaion, our resuls show cash flows perform significanly worse han accouning earnings. For example, he valuaion error of EBITDA has a sandard deviaion of 0.611 (28% higher han earnings) and iner-quarile range of 0.687 (19% higher han earnings). Beween he wo cash flow measures, CFO and EBITDA, here is lle difference in performance. 18 18 The free cash flow and mainenance free cash flow measures, which are excluded from his analysis because of he large proporion of negaive values, exhib even worse performance. 21

The sales muliple performs he wors. Is valuaion error has a sandard deviaion of 0.948, and iner-quarile range of 0.761, implying ha approximaely 50% of he firms have valuaion errors larger han 38%. This resul suggess ha sales do no reflec profabily unil expenses have been considered. A frequen reason for using sales as a value driver is when earnings and cash flows are negaive. Since we resric our sample o firms wh posive earnings and cash flows, our sample is less likely o conain firms for which he sales muliple is more likely o be used in pracice. In paricular, our sample is unlikely o conain Inerne socks (e.g. Hand [1999] and Trueman, Wong, and Zhang [2000]), and here are reasons o believe our resuls canno be generalized o ha group. To conduc he analysis using comparable firms from he same indusry, we searched for a reasonable indusry classificaion scheme. Because of he evidence ha SIC codes frequenly misclassify firms (Kim and Rer [1999]), we use he indusry classificaion provided by IBES. IBES indicae ha heir classificaion is based loosely on SIC codes, bu is also subjec o deailed adjusmens. 19 The IBES indusry classificaion has hree levels (in increasing fineness): secor, indusry, and group. We use he inermediae (indusry) classificaion level because secors are oo broad o allow he selecion of homogenous firms, and groups are oo narrow o allow he inclusion of sufficien comparable firms (given he loss of observaions due o our daa requiremens). The resuls repored in Panel B, which are based on comparable firms from he same IBES indusry classificaion, exhib improved performance over hose repored in Panel A. The improvemen is consisen wh he join hypohesis ha (1) increased homogeney in he valuerelevan facors omed from he muliples resuls in beer valuaion, and (2) IBES indusry 19 The IBES classificaion resembles he indusry groupings suggesed by Morgan Sanley. 22

classificaion idenifies relaively homogeneous groups of firms. 20 Generally, he improvemen is larger a he cener of he disribuions; ha is, small valuaion errors became much smaller while large valuaion errors do no change much. The muliples used in calculaing he percen pricing errors in Panels A and B were esimaed using he harmonic mean. To make our resuls comparable o hose in previous sudies (e.g., Alford [1992]) as well as o examine heir robusness, we replicae Panel B using he median insead of he harmonic mean. Those resuls are repored in Panel C. Consisen wh he evidence in Baker and Ruback [1999] and Beay, Riffe and Thompson [1999], we find ha median muliples perform worse han harmonic mean muliples. The relaive performance (i.e., ranking) of he differen muliples, however, remains he same. To offer a visual picure of he relaive and absolue performance of differen caegories of muliples, we provide in Figure 1 he hisograms for percen pricing errors for he following seleced muliples: EPS3, P1*, IACT, EBITDA, BV and Sales. The hisograms repor he fracion of he sample ha lies whin ranges of percen pricing error ha are of widh equal o 10% (e.g. 0.1 o 0, 0 o 0.1, and so on). To reduce cluer, we simply draw a smooh line hrough he middle of he op of each hisogram column, raher han provide he hisograms for each of he muliples. A muliple is considered beer if has a more peaked disribuion. The differences in performance across he differen caegories are clearly visible in Figure 1. The figure also offers a beer view of he shapes of he differen disribuions and enables readers o find he fracion of firms whin differen pricing error ranges for each disribuion. 20 Even if hese condions are saisfied, is no clear ha here should be an improvemen. Moving from he cross-secion o each indusry resuls in a subsanial decrease in sample size, and consequenly he esimaion is less precise. This fac is also refleced in he increase in he deviaion of he sample mean of he valuaion errors from zero. 23

5.3 Inercep Adjused Muliples In his subsecion, we repor resuls based on he second sage analysis, where we allow an inercep in he relaion beween price and value drivers. The opimizaion problem in equaion (7) is solved ou of sample o obain parameer esimaes, and valuaion errors are hen calculaed using hese parameers. Again, he analysis is conduced for comparable firms from he enire cross-secion (Table 4, Panel A) and he same indusry (Panel B). As prediced, relaxing he no-inercep resricion improves he performance of all muliples. The degree of improvemen is no uniform, however. Muliples ha perform poorly in Panel A of Table 3 improve more han hose ha do well. For example, EPS3 s valuaion errors exhib a small decrease in sandard deviaion (iner-quarile range): from 0.313 (0.347) o 0.306 (0.333), while sales valuaion errors decrease from 0.948 (0.761) o 0.676 (0.615). Alhough he improvemen in absolue performance of he muliples is no uniform, he rank order of muliples remains unchanged from Table 3 o Table 4. The improvemen generaed by allowing for an inercep can also be seen by comparing he resuls in Panel A of Table 4, based on comparable firms from he enire cross-secion, wh hose in Panel B of Table 3, based on comparable firms from he same indusry. Alhough simple indusry muliples are beer han simple cross-secional muliples, he inercep-adjused crosssecional muliples are beer han he simple indusry muliples for hisorical value drivers and are only slighly worse for forecased value drivers. The bes performance is achieved when we allow for an inercep and selec comparable firms from he same indusry (Table 4, Panel B). Comparing hese resuls wh hose in Panel A of Table 3 illusraes he join benefs of allowing for an inercep and resricing comparable firms o hose in he same IBES indusry. For example, he sandard deviaion (iner-quarile 24

range) for sales, he wors performer, improves from 0.948 (0.761) o 0.668 (0.574); and for EPS3, he bes performer, he improvemen is from 0.313 (0.347) o 0.289 (0.293). Consisen wh he resuls in Table 3, he improvemen in valuaion by allowing for an inercep (i.e., from Panel A o Panel B) is relaively uniform across muliples. 5.4 Adjusmen for Leverage Since Sales and EBITDA perain o he value of he whole firm (enerprise value) raher han equy alone, muliples compued on he marke value of equy are poenially in error. To correc for his mismach, we repea he analyses repored in Tables 3 and 4 for hese wo value drivers using enerprise value (marke value of equy plus book value of deb) insead of equy value. To facilae comparabily wh he resuls for oher muliples in Tables 3 and 4, we compue percenage errors in erms of equy value. In effec, we consruc muliples based on enerprise value for he wo value drivers, use he comparable firm muliples o esimae each firm s enerprise value, and hen subrac he book value of deb o esimae equy value. Table 5 repors he resuls of his analysis. The firs wo rows in each panel provide he resuls whou leverage adjusmen and are he same as he corresponding rows in ables 3 and 4. The nex wo rows are based on he leverage adjusmen. To our surprise, leverage adjusmen does no improve he f. Leverage-adjused Sales performs worse in all four panels of Table 5. For EBITDA, he leverage adjusmen reduces slighly he valuaion errors in Panel A, increases he valuaion errors in Panels B and C, and has only a marginal effec in Panel D. Alhough puzzling a firs glance, our resuls are consisen wh hose of Alford [1992], who finds ha adjusing P/EBIT muliples for differences in leverage across comparable firms decreases accuracy. 25

5.5. Poenial Errors in P1* We conduc an invesigaion ino possible reasons why he inrinsic value measure P1* does no ouperform forward earnings muliples, even hough incorporaes more informaion and imposes a srucure on ha informaion ha is prescribed by heory. One possibily is ha he assumpion ha abnormal earnings remain consan pas year +5 induces errors in he erminal value. To undersand beer his poenial source of error, we consider wo alernaive assumpions regarding erminal values: (1) zero abnormal earnings pas year 5 (i.e., erminal value equals zero for all firms), æe ( eps - kbv ) ö P2 = + ç, and ø 5 * + s +- s 1 bv å s ç s= 1è (1 + k) (2) ROE forecass from year +3 rends linearly oward he indusry median by year +12, and abnormal earnings exhib zero growh hereafer, P3 2 11 Ε ( 1) [ ( s) ] s [ ( ) eps s kbv ROE k bv ROE k s + + + ] bv + Ε Ε + +, 11 = bv + + + * 1 12 11 s s s= 1 (1 + k) s= 3 (1 + k) k(1 + k) E ROE + s for s = 4, 5,, 12 is esimaed using a linear inerpolaion o he indusry where ( ) median ROE. The indusry median ROE is calculaed as a moving median of he pas en years ROE of all firms in he indusry. To eliminae ouliers, indusry median ROEs are Winsorized o lie beween he risk free rae and 20%. 21 Resuls are repored in Table 6. As wh Table 5, we repor he resuls for all four combinaions: wo ses of comparable firms (enire cross secion and indusry) and wo value relaions (wh and whou an inercep). Because he resuls in all four Panels of Tale 6 are similar, we discuss only he resuls in Panel A. P2* produces essenially he same dispersion in 21 This measure has been proposed by Gebhard, Lee and Swaminahan [1999]. 26

valuaion errors as ha produced by P1*. This suggess ha he erminal value proxy in P1* conains considerable error, since dropping he erminal value alogeher in P2* does no affec he f adversely. We urn nex o he improvemen offered by he more complex erminal value proxy incorporaed in P3*. This inrinsic value measure allows for firm-specific paerns of profabily beween years +3 and +12, and indusry-specific erminal profabily afer year +12. Despe he inuive appeal of he adjusmen proposed in P3*, our resuls indicae ha he percen pricing errors are acually higher han hose observed for P1* and P2*. Again, he addional informaion ha is incorporaed in P3*, regarding he endency for firms in differen indusries o rever o indusry means, appears o be negaed by increased measuremen error. 5.6 Simple Aggregaion of Earnings Forecas Informaion Since he inrinsic value approach for incorporaing he informaion in he differen EPS forecass fails o improve on simple muliples based on specific EPS forecass, we examine wo alernaive, simpler, ways of incorporaing he informaion in he differen earnings forecass. The firs measure is based on he sum of he EPS forecass for he nex five years, 5 * 4 = å E( s) s= 1 P eps +. 22 The second measure aemps o conrol heurisically for he iming and risk of he differen earnings numbers by discouning he EPS forecass for he nex five years, 5 æ * E ( eps s) ö + P5 = å s ç (1 + k ) è ø. s= 1 The sandard deviaion (iner-quarile range) of valuaion errors for P4* is 0.309 (0.338), which is lower han ha for P1*. In fac, P4* performs beer han any of he oher muliples considered so far, including he hree forward earnings muliples (EPS1-EPS3). This 27

improvemen suggess ha a simple aggregaion of he earnings forecass a differen horizons allows us o incorporae he informaion in hose forecass, whereas he srucure imposed by compuing P* adds measuremen error. The similary of he valuaion error dispersions in he fourh row (P4*) and fifh row (P5*) indicaes ha our simple conrol for he iming and risk of fuure earnings does no improve he valuaion. 5.7 Ranking Muliples in Each Indusry Given our focus on undersanding he underlying informaion conen of he differen muliples, our focus has been on overall paerns, wh firms pooled across indusries. I has been suggesed, however, ha differen muliples work bes in differen indusries. For example, Tasker [1998] repors ha invesmen bankers and analyss appear o use preferred muliples in each indusry. Therefore, we deermine he exen o which he relaive rank of differen muliples, based on he dispersion of valuaion errors whin ha indusry, varies across differen indusries. Alhough we recognize ha our search is unlikely o offer conclusive resuls, since we do no pick comparable firms wh he same skill and aenion as ohers do in differen conexs, we wish o offer some general resuls. Since invesmen professionals use simple muliples (no inercep) and selec comparable firms from he same indusry, we use he same approach here. Then we pool he valuaion resuls over years for each indusry and rank muliples by he sandard deviaion of valuaion errors whin each indusry. Table 7 repors he resuls for he 68 indusries we analyze. The ranking goes from 0 (bes) o 11 (wors). We also repor summary saisics of he rankings a he boom of he able. 22 We hank Jim Ohlson for suggesing his value driver. 28

The overall resul shows remarkable consisency across all indusries. In almos all indusries, forecased earnings perform he bes, while Sales performs he wors. This resul, which is consisen wh he resuls in Kim and Rer [1999], suggess ha he informaion conained in forward looking value drivers capures a considerable fracion of value, and his feaure is common o all indusries. Turning o he oher value drivers, earnings perform beer han book value and cash flows in mos indusries. Book value performs well in cerain indusries in he finance secor, he energy secor (oil and gas), fores producs and gas uilies. Perhaps, accouning pracices in hese indusries cause book values o be relaed o marke values whin hese indusries in a more consisen manner. 6. Conclusions In his sudy we have examined he valuaion properies of a comprehensive lis of muliples. We consider boh he commonly used muliple approach, which assumes direc proporionaly beween price and value driver, and a less resricive approach ha allows for an inercep. To idenify he imporance of selecing comparable firms from he same indusry, we also repor resuls based on he comparable group including all firms in he cross-secion. Our resuls show he following rank ordering of muliples (from more accurae o less accurae): forecased earnings, earnings, cash flows ied wh book value, and sales. The ranking is robus o he use of differen saisical mehods, and similar resuls are obained whin individual indusries. We show ha boh he indusry adjusmen (selecing comparables from he same indusry) and he inercep adjusmen (allowing for an inercep in he price/value driver relaion) improves he valuaion properies of all muliples. While he indusry adjusmen is commonly used, he inercep adjusmen is no. We speculae ha muliples are used primarily 29

because hey are simple o comprehend and communicae and he addional complexy associaed wh including an inercep exceeds he benefs of improved f. Our resuls are consisen wh inuion regarding he informaion in differen value drivers. For example, fuure informaion reflecs value beer han hisorical informaion, accouning accruals add value-relevan informaion o cash flows, and profabily can be beer measured when revenue is mached wh expenses. Some resuls in his paper are surprising, however. For example, a discouned residual income analysis which explicly forecass erminal value and adjuss for risk performs worse han simple muliples based on earnings forecass. And adjusing for leverage does no improve he valuaion properies of EBITDA and Sales. We invesigae hese resuls furher and feel ha hese resuls indicae he rade-off ha exiss beween signal and noise when more complex bu heoreically correc srucures are imposed. We recognize ha our sudy is designed o provide an overview of aggregae paerns, and hus we may have missed more suble relaionships ha are only apparen in small sample sudies. We noe in conclusion ha our analysis assumes ha marke prices are efficien, and we evaluae muliples based on heir abily o mimic marke valuaions. If marke prices vary sysemaically from fundamenal or inrinsic value, we may need o revise our conclusions abou he relaive and absolue performance of he differen muliples considered here. To examine his possibily, we are currenly invesigaing he abily of hese muliples o predic fuure abnormal reurns. The resuls in his paper are valid if no relaion is observed beween fuure abnormal reurns and pricing errors from differen muliples. 30

APPENDIX This appendix describes how he variables are consruced. All he value drivers are adjused for changes in number of shares. BV: book value of equy, COMPUSTAT em #60 SALES: em #12 CACT: COMPUSTAT earnings (EPS excluding exraordinary ems), em #58 IACT: IBES acual earnings EBITDA: earnings before ineres, axes, depreciaion and amorizaion, em #13 CFO: cash flow from operaions, measured as EBITDA minus he oal of ineres expense (#15), ax expense (#16) and he ne change in working capal (#236) FCF: free cash flow, measured as CFO minus ne invesmen (#107 - #128 - #113 + #109) MCF: mainenance cash flow, measured as CFO minus depreciaion expense (#125) EPS1: EPS2: EPS3: IBES one year ou earnings forecas IBES wo year ou earnings forecas IBES hree year ou earnings forecas, measured as EPS2*(1+g), where g is IBES long erm growh forecas The P* measures: P1 æe ( eps -kbv ) ö E ( eps -kbv ) 5 * + s +- s 1 + 5 + 4 = bv+ å + s 5 ç s= 1 (1 k) è + ø k(1 + k) æe ( eps - kbv ) ö P2 = + ç ø 5 * + s +- s 1 bv å s ç s= 1è (1 + k) 31

P3 Ε ( eps kbv ) [ Ε ( ) ] [ Ε ( ) ] 11 ROE k bv ROE k bv 2 11 * + s + s 1 + s + s 1 + 12 + 11 = bv + s + + s s= 1 (1 + k) s= 3 (1 + k) k(1 + k) 5 * 4 = åe( s) s= 1 P eps + 5 æ * E ( eps s) ö + P5 = å s ç (1 + k ) è ø s= 1 The variables used in he P* calculaions are obained in he following way: The discoun rae (k ) is calculaed as he risk-free rae plus bea imes he equy risk premium. We use he 10-year Treasury bond yield on April 1 of year +1 as he risk-free rae and assume a consan 5% equy risk premium. We measure bea as he median bea of all firms in he same bea decile in year. We esimae beas using monhly sock reurns and valueweighed CRSP reurns for he five years ha end in March of year +1 (a leas 30 observaions are required). For a subgroup of firm-years (less han 5 percen), we were able o obain mean IBES forecass for all years in he five-year horizon. For all oher firms, wh less han complee forecass available beween years 3 and 5, we generaed forecass by applying he mean longerm growh forecas (g) o he mean forecas for he prior year in he horizon; i.e., eps + s = eps+ s 1 *(1 + g). The book values for fuure years, corresponding o he earnings forecass, are deermined by assuming he ex-ane clean surplus relaion (ending book value in each fuure period equals beginning book value plus forecased earnings less forecased dividends). Since analys forecass of fuure dividends are no available on IBES, we assume ha he curren dividend payou raio will be mainained in he fuure. We measure he curren dividend payou as he raio of he indicaed annual cash dividends o he earnings forecas for year +1 (boh obained 32

from he IBES summary file). To minimize biases ha could be induced by exreme dividend payou raios (caused by forecas +1 earnings ha are close o zero), we Winsorize payou raios a 10% and 50%. In he calculaion of P, ( ) * 3 E ROE + for s = 4, 5,, 12 are forecased using a linear s inerpolaion o he indusry median ROE. The indusry median ROE is calculaed as a moving median of he pas en years ROE of all firms in he indusry. To eliminae ouliers, indusry median ROEs are Winsorized a he risk free rae and 20%. 33

Table 1 Disribuion of Value Driver o Price Raios The variables are defined as follows: P is sock price; BV is book value of equy; MCF is mainenance cash flow (equivalen o free cash flow when depreciaion expense equals capal expendure); FCF is free cash flow o deb and equy holders; CFO is cash flow from operaions; EBITDA is earnings before ineres, axes, depreciaion and amorizaion; CACT is COMPUSTAT earnings before exraordinary ems; IACT is IBES acual earnings; EPS1 and EPS2 are one year ou and wo year ou earnings forecass; EPS3=EPS2*(1+g), where g is he growh forecas; and TP is enerprise value (price + deb). All oals are deflaed by he number of shares ousanding a he end of he year. 5 * Ε( eps+ s kbv 1) + s Ε( eps+ 5 kbv + 4) P1 = bv + s + 5 s= 1 (1 + k) k(1 + k), 5 æ * E( eps+ s- kbv s 1) ö +- P2 = bv + å s ç (1 + k ) è ø, s= 1 2 11 Ε ( 1) [ ( s) ] s [ ( ) eps s kbv ROE k bv ROE k s + + + ] bv + Ε Ε + +, 11 * 1 12 11 P3 = bv + + + s s s= 1 (1 + k) s= 3 (1 + k) k(1 + k) where E ( ROE + s) for s = 4, 5,, 12 is forecased using a linear inerpolaion o he indusry median ROE. The indusry median ROE is calculaed as a moving median of he pas en years ROE of all firms in he indusry. To eliminae ouliers, indusry median ROEs are Winsorized a he risk free rae and 20%. 5 5 æ * * E ( eps+ s) ö å P4 = E ( eps + ), and s s= 1 P5 = å s ç (1 + k ) è ø. s= 1 Sample is rimmed a 1% and 99% for each raio (excluding he P* raio) using he pooled disribuion. Years covered are 1981 hrough 1996. Sample size is 17,505. Mean SD 1% 5% 10% 25% 50% 75% 90% 95% 99% BV/P 0.549 0.354 0.014 0.121 0.178 0.300 0.483 0.729 1.001 1.205 1.653 MCF/P 0.034 0.079-0.281-0.092-0.035 0.016 0.043 0.069 0.097 0.123 0.193 FCF/P -0.016 0.161-0.675-0.274-0.154-0.041 0.019 0.054 0.092 0.127 0.252 CFO/P 0.102 0.094-0.151-0.021 0.017 0.053 0.093 0.146 0.207 0.254 0.391 Ebda/P 0.179 0.142-0.090 0.015 0.044 0.091 0.152 0.239 0.347 0.437 0.672 CACT/P 0.044 0.084-0.302-0.077-0.015 0.029 0.055 0.080 0.110 0.133 0.182 IACT/P 0.051 0.067-0.212-0.041 0.006 0.034 0.057 0.081 0.108 0.131 0.176 Sales/P 1.387 1.475 0.054 0.159 0.254 0.509 0.953 1.731 2.920 4.050 7.450 EPS1/P 0.072 0.041-0.055 0.016 0.030 0.050 0.070 0.093 0.119 0.140 0.180 EPS2/P 0.090 0.039 0.010 0.036 0.048 0.066 0.086 0.110 0.141 0.162 0.211 EPS3/P 0.106 0.043 0.025 0.049 0.060 0.078 0.099 0.127 0.164 0.188 0.247 P1*/P 0.693 0.308 0.176 0.299 0.370 0.492 0.647 0.847 1.048 1.207 1.727 P2*/P 0.607 0.251 0.161 0.258 0.318 0.424 0.571 0.762 0.935 1.053 1.308 P3*/P 0.790 0.466 0.161 0.273 0.350 0.494 0.691 0.987 1.305 1.570 2.477 P4*/P 0.536 0.217 0.124 0.248 0.308 0.395 0.500 0.640 0.822 0.942 1.236 P5*/P 0.354 0.131 0.076 0.169 0.210 0.270 0.339 0.421 0.525 0.594 0.746 Ebda/TP 0.123 0.075-0.081 0.014 0.041 0.081 0.122 0.163 0.207 0.241 0.325 Sales/TP 0.973 0.891 0.052 0.141 0.222 0.408 0.728 1.271 1.971 2.547 4.313 34

Table 2 Pearson (Upper Triangle) and Spearman (Lower Triangle) Correlaion Marices The variables are defined in able 1. Sample is rimmed a 1% and 99% for each raio (excluding he P* raio) using he pooled disribuion. Also, observaions for which any of he raios is negaive are deleed. Finally, observaions ha do no belong o an indusry-year group wh a leas five members are deleed. Years covered are 1981 hrough 1996. Sample size is 9,658. BV CFO Ebda CACT IACT Sales EPS1 EPS2 EPS3 P1* P2* P3* P4* P5* Ebda Sales /P /P /P /P /P /P /P /P /P /P /P /P /P /P /TP /TP BV/P 1.00 0.58 0.65 0.53 0.53 0.52 0.58 0.61 0.56 0.32 0.91 0.41 0.55 0.58 0.54 0.39 CFO/P 0.63 1.00 0.82 0.50 0.52 0.45 0.53 0.51 0.46 0.31 0.60 0.30 0.45 0.50 0.76 0.31 Ebda/P 0.70 0.87 1.00 0.59 0.61 0.49 0.62 0.59 0.52 0.30 0.65 0.30 0.51 0.55 0.79 0.27 CACT/P 0.52 0.55 0.65 1.00 0.93 0.26 0.82 0.72 0.65 0.35 0.62 0.33 0.66 0.70 0.63 0.22 IACT/P 0.54 0.59 0.67 0.93 1.00 0.27 0.85 0.75 0.67 0.38 0.64 0.35 0.68 0.73 0.65 0.22 Sales/P 0.60 0.55 0.61 0.37 0.39 1.00 0.37 0.44 0.43 0.17 0.46 0.20 0.42 0.40 0.45 0.91 EPS1/P 0.61 0.59 0.68 0.82 0.85 0.49 1.00 0.93 0.85 0.44 0.71 0.38 0.87 0.90 0.64 0.31 EPS2/P 0.63 0.56 0.65 0.72 0.75 0.55 0.93 1.00 0.96 0.44 0.71 0.35 0.96 0.95 0.61 0.38 EPS3/P 0.58 0.50 0.57 0.65 0.68 0.53 0.86 0.96 1.00 0.46 0.67 0.30 0.99 0.95 0.56 0.38 P1*/P 0.42 0.43 0.45 0.45 0.49 0.27 0.53 0.51 0.49 1.00 0.66 0.80 0.46 0.66 0.28 0.13 P2*/P 0.92 0.66 0.73 0.62 0.65 0.58 0.73 0.74 0.69 0.72 1.00 0.65 0.65 0.77 0.53 0.24 P3*/P 0.47 0.40 0.43 0.44 0.47 0.29 0.48 0.43 0.36 0.81 0.70 1.00 0.30 0.50 0.24 0.08 P4*/P 0.57 0.48 0.56 0.66 0.69 0.52 0.87 0.96 0.99 0.49 0.67 0.35 1.00 0.96 0.54 0.33 P5*/P 0.62 0.55 0.63 0.71 0.75 0.52 0.91 0.95 0.95 0.70 0.79 0.56 0.95 1.00 0.55 0.28 Ebda/TP 0.60 0.83 0.89 0.68 0.70 0.59 0.69 0.65 0.58 0.39 0.53 0.24 0.54 0.55 1.00 0.43 Sales/TP 0.43 0.38 0.38 0.27 0.28 0.93 0.38 0.44 0.45 0.16 0.24 0.08 0.33 0.28 0.50 1.00 35

Table 3 Disribuion of Percenage Valuaion errors for Simple Muliples Value and value drivers are assumed o be proporional: p = bx + e. Muliple is esimaed excluding he firm æx ö under valuaion, using harmonic means: b = 1/ E p. Percen pricing error is calculaed as follows, ç çè ø e ˆ p-bx = ; s pooled disribuion is repored. p p The variables are defined as follows: P is sock price; BV is book value of equy; CFO is cash flow from operaions; EBITDA is earnings before ineres, axes, depreciaion and amorizaion; CACT is COMPUSTAT earnings before exraordinary ems; IACT is IBES acual earnings; EPS1, EPS2 are one year ou and wo year ou earnings forecass; EPS3=EPS2*(1+g), where g is growh forecas. All oals are deflaed by he number of shares ousanding a he end of he year. 5 * Ε( eps+ s kbv 1) + s Ε( eps+ 5 kbv + 4) P1 = bv + s + 5 s= 1 (1 + k) k(1 + k) Panel A uses he whole cross-secion of firms as comparable firms, Panel B uses comparable firms whin each indusry (based on IBES indusry classificaion). Panel C also uses comparable firms whin each indusry, bu he muliple is calculaed using he median insead of he harmonic mean. Years covered are 1981 hrough 1996. Sample size is 9,658. Panel A: Valuaion using mean cross-secional muliples Mean Median SD 75%-25% 90%-10% 95%-5% 99%-1% BV 0.000 0.063 0.536 0.697 1.253 1.620 2.564 CFO -0.001 0.104 0.589 0.689 1.304 1.748 2.998 Ebda -0.001 0.129 0.611 0.687 1.296 1.674 3.019 CACT 0.000 0.047 0.477 0.579 1.120 1.510 2.371 IACT 0.000 0.041 0.448 0.549 1.060 1.418 2.242 Sales -0.001 0.259 0.948 0.761 1.694 2.394 4.797 EPS1 0.000 0.029 0.348 0.440 0.835 1.118 1.721 EPS2 0.000 0.026 0.311 0.368 0.743 1.008 1.570 EPS3 0.000 0.038 0.313 0.347 0.725 0.992 1.653 P1* 0.000 0.056 0.403 0.504 0.918 1.203 1.981 Panel B: Valuaion using mean indusry muliples Mean Median SD 75%-25% 90%-10% 95%-5% 99%-1% BV -0.021 0.064 0.530 0.543 1.152 1.586 2.668 CFO -0.023 0.056 0.554 0.559 1.171 1.665 2.938 Ebda -0.023 0.064 0.572 0.502 1.084 1.555 2.776 CACT -0.014 0.013 0.448 0.462 1.001 1.400 2.322 IACT -0.012 0.015 0.412 0.429 0.923 1.288 2.177 Sales -0.049 0.157 0.934 0.729 1.610 2.304 4.571 EPS1-0.007 0.018 0.312 0.333 0.711 0.981 1.658 EPS2-0.006 0.021 0.289 0.303 0.657 0.919 1.550 EPS3-0.006 0.026 0.293 0.301 0.658 0.927 1.561 P1* -0.010 0.038 0.377 0.369 0.799 1.136 1.991 36

Panel C: Valuaion using median indusry muliples Mean Median SD 75%-25% 90%-10% 95%-5% 99%-1% BV -0.109 0.002 0.601 0.571 1.284 1.770 3.054 CFO -0.102 0.000 0.628 0.587 1.273 1.816 3.334 Ebda -0.114 0.000 0.659 0.519 1.160 1.712 3.198 CACT -0.039-0.001 0.470 0.465 1.037 1.470 2.469 IACT -0.038-0.001 0.435 0.434 0.956 1.353 2.256 Sales -0.307 0.001 1.312 0.861 1.936 2.905 6.311 EPS1-0.028-0.001 0.323 0.339 0.729 1.023 1.774 EPS2-0.032 0.000 0.301 0.305 0.676 0.958 1.633 EPS3-0.039 0.000 0.307 0.305 0.685 0.968 1.646 P1* -0.063 0.000 0.411 0.377 0.838 1.228 2.199 37

Table 4 Disribuion of Percenage Valuaion Errors for Inercep Adjused Muliples Value and value drivers are assumed o be linear: p = a + b x + e. Muliple is esimaed excluding he firm under valuaion, by solving a consrain minimizaion problem: min var( e / p ) = var ( p -a - b x )/ p a, b ( ) æe ö s.. E ç = 0 p çè ø e ˆ p-aˆ -bx Percenage valuaion error is calculaed as follows, = ; s pooled disribuion is repored. p p The variables are defined as follows: P is sock price; BV is book value of equy; CFO is cash flow from operaions; EBITDA is earnings before ineres, axes, depreciaion and amorizaion; CACT is COMPUSTAT earnings before exraordinary ems; IACT is IBES acual earnings; EPS1, EPS2 are one year ou and wo year ou earnings forecass; EPS3=EPS2*(1+g), where g is growh forecas. All oals are deflaed by he number of shares ousanding a he end of he year. 5 * Ε( eps+ s kbv 1) + s Ε( eps+ 5 kbv + 4) P1 = bv + s + 5 s= 1 (1 + k) k(1 + k) Panel A uses he whole cross-secion of firms as comparable firms, Panel B uses comparable firms whin each indusry (based on IBES indusry classificaion). Years covered are 1981 hrough 1996. Sample size is 9,658. Panel A: Valuaion using inercep adjused cross-secional muliples Mean Median SD 75%-25% 90%-10% 95%-5% 99%-1% BV 0.026 0.084 0.458 0.541 1.038 1.408 2.264 CFO 0.032 0.108 0.465 0.520 1.013 1.386 2.427 Ebda 0.027 0.112 0.480 0.527 1.011 1.413 2.394 CACT 0.012 0.053 0.401 0.477 0.931 1.272 2.028 IACT 0.015 0.054 0.380 0.462 0.893 1.209 1.887 Sales -0.031 0.160 0.676 0.615 1.370 1.908 3.420 EPS1 0.013 0.039 0.321 0.392 0.767 1.030 1.610 EPS2 0.008 0.035 0.300 0.344 0.705 0.966 1.533 EPS3 0.000 0.042 0.306 0.333 0.706 0.976 1.629 P1* 0.016 0.068 0.365 0.435 0.809 1.089 1.830 Panel B: Valuaion using inercep adjused indusry muliples Mean Median SD 75%-25% 90%-10% 95%-5% 99%-1% BV -0.016 0.076 0.477 0.476 1.032 1.442 2.403 CFO -0.016 0.072 0.477 0.474 1.023 1.420 2.496 Ebda -0.020 0.082 0.496 0.443 0.976 1.384 2.440 CACT -0.012 0.034 0.396 0.406 0.884 1.250 2.068 IACT -0.011 0.035 0.372 0.385 0.826 1.175 1.917 Sales -0.035 0.146 0.668 0.574 1.291 1.852 3.287 EPS1-0.006 0.026 0.300 0.317 0.689 0.956 1.594 EPS2-0.004 0.028 0.284 0.296 0.642 0.900 1.511 EPS3-0.004 0.033 0.289 0.293 0.650 0.912 1.526 P1* -0.008 0.049 0.357 0.342 0.754 1.077 1.884 38

Table 5 Leverage Adjusmens for EBITDA and Sales Muliples The variables are defined as follows: P is sock price; EBITDA is earnings before ineres, axes, depreciaion and amorizaion; TP is enerprise value (marke value of equy plus book value of deb). Valuaions using simple and inercep adjused muliples are conduced using cross-secional and indusry comparable firms. When TP muliples are used, equy value is calculaed as he prediced enerprise value minus book value of deb. Years covered are 1981 hrough 1996. Sample size is 9,658..Panel A: Valuaion using mean cross-secional muliples Mean Median SD 75%-25% 90%-10% 95%-5% 99%-1% Ebda/P -0.001 0.129 0.611 0.687 1.296 1.674 3.019 Sales/P -0.001 0.259 0.948 0.761 1.694 2.394 4.797 Ebda/TP -0.045 0.039 0.601 0.630 1.231 1.671 2.994 Sales/TP 0.000 0.269 1.259 1.031 2.189 3.164 6.602 Panel B: Valuaion using mean indusry muliples Mean Median SD 75%-25% 90%-10% 95%-5% 99%-1% Ebda/P -0.023 0.064 0.572 0.502 1.084 1.555 2.776 Sales/P -0.049 0.157 0.934 0.729 1.610 2.304 4.571 Ebda/TP -0.035 0.026 0.581 0.535 1.088 1.518 2.696 Sales/TP -0.075 0.150 1.113 0.859 1.853 2.698 5.501 Panel C: Valuaion using inercep adjused cross-secional muliples Mean Median SD 75%-25% 90%-10% 95%-5% 99%-1% Ebda/P 0.027 0.112 0.480 0.527 1.011 1.413 2.394 Sales/P -0.031 0.160 0.676 0.615 1.370 1.908 3.420 Ebda/TP -0.017 0.048 0.513 0.521 1.040 1.429 2.602 Sales/TP 0.050 0.214 0.978 0.880 1.904 2.721 5.407 Panel D: Valuaion using inercep adjused indusry muliples Mean Median SD 75%-25% 90%-10% 95%-5% 99%-1% Ebda/P -0.020 0.082 0.496 0.443 0.976 1.384 2.440 Sales/P -0.035 0.146 0.668 0.574 1.291 1.852 3.287 Ebda/TP -0.002 0.078 0.492 0.447 0.976 1.352 2.432 Sales/TP -0.003 0.179 0.702 0.620 1.382 1.985 3.524 39

Table 6 Sources of Measuremen Error in P1* Analyzed Valuaions using simple and inercep adjused muliples are conduced using cross-secional and indusry 5 * Ε( eps+ s kbv 1) + s Ε( eps+ 5 kbv + 4) comparable firms. Variables are defined as: P1 = bv + s + 5 s= 1 (1 + k) k(1 + k), 5 * Ε( eps+ s kbv s 1) + P2 = bv + s s= 1 (1 + k), 2 11 * Ε [ ( ) 1 ] 1 [ ( 12) ( eps ) ROE s k bv s ROE k] bv + s kbv Ε s + + Ε + + + 11 P3 = bv + s + + s 11 s= 1 (1 + k) s= 3 (1 + k) k(1 + k), where E ( ROE + s) for s = 4, 5,, 12 is forecased using a linear inerpolaion o he indusry median ROE. The indusry median ROE is calculaed as a moving median of he pas en years ROE of all firms in he indusry. To eliminae ouliers, indusry median ROEs are Winsorized a he risk free rae and 20%. 5 5 æ * * E ( eps ) ö + s P4 = å E( eps + s), and P5 = å s s= 1 ç s= 1 (1 + k) è ø. Years covered are 1981 hrough 1996. Sample size is 9,658. Panel A: Valuaion using mean cross-secional muliples Mean Median SD 75%-25% 90%-10% 95%-5% 99%-1% P1* /P 0.000 0.056 0.403 0.504 0.918 1.203 1.981 P2* /P 0.000 0.053 0.395 0.548 0.957 1.226 1.846 P3* /P 0.000 0.108 0.551 0.640 1.185 1.562 2.693 P4* /P 0.000 0.041 0.309 0.338 0.711 0.980 1.631 P5* /P 0.000 0.029 0.309 0.366 0.733 0.982 1.592 Panel B: Valuaion using mean indusry muliples Mean Median SD 75%-25% 90%-10% 95%-5% 99%-1% P1* /P -0.010 0.038 0.377 0.369 0.799 1.136 1.991 P2* /P -0.009 0.036 0.346 0.375 0.803 1.097 1.756 P3* /P -0.016 0.062 0.491 0.444 0.993 1.400 2.488 P4* /P -0.006 0.028 0.291 0.294 0.646 0.915 1.528 P5* /P -0.006 0.024 0.288 0.297 0.655 0.920 1.507 Panel C: Valuaion using inercep adjused cross-secional muliples Mean Median SD 75%-25% 90%-10% 95%-5% 99%-1% P1* /P 0.016 0.068 0.365 0.435 0.809 1.089 1.830 P2* /P -0.001 0.045 0.356 0.463 0.846 1.110 1.733 P3* /P -0.011 0.079 0.482 0.532 0.988 1.371 2.381 P4* /P -0.002 0.039 0.305 0.326 0.700 0.971 1.611 P5* /P 0.000 0.031 0.299 0.342 0.698 0.956 1.545 Panel D: Valuaion using inercep adjused indusry muliples Mean Median SD 75%-25% 90%-10% 95%-5% 99%-1% P1* /P -0.008 0.049 0.357 0.342 0.754 1.077 1.884 P2* /P 0.000 0.051 0.336 0.356 0.777 1.067 1.720 P3* /P -0.013 0.074 0.446 0.397 0.899 1.287 2.359 P4* /P -0.004 0.034 0.286 0.288 0.636 0.894 1.509 P5* /P 0.000 0.034 0.284 0.287 0.647 0.903 1.495 40

Table 7 Indusry Rankings of Muliples Valuaions using he simple muliple approach are performed in each indusry. Muliples are ranked according o he variance of percen pricing errors using he pooled disribuion. Low rank numbers indicae low variance. Indusry classificaion is from IBES. Code is he firs four digs of he IBES indusry classificaion code. Years covered are 1981 hrough 1996. Sample size is 9,658. Secor Name Indusry Name Code BV CFO Ebda CACT IACT Sales EPS1 EPS2 EPS3 P1* P2* P3* finance finance & loan 101 7 8 11 5 4 10 3 2 1 9 0 6 finance financial services 102 9 5 11 8 7 10 4 3 2 0 1 6 finance Insurance 105 9 8 10 6 5 11 3 0 2 7 1 4 finance Invesmens 106 6 8 10 9 5 11 3 0 1 7 2 4 finance undesignaed finance 109 2 5 10 9 8 11 7 6 4 0 3 1 healh care drugs 201 10 9 8 7 6 11 3 2 1 5 0 4 healh care hospal supplies 202 9 10 8 6 4 11 3 2 1 7 0 5 healh care hospals 203 9 10 11 7 6 8 3 1 2 4 0 5 healh care bioechnology 204 6 9 10 8 7 11 5 2 0 1 4 3 healh care medical supplies 205 8 9 10 7 6 11 3 2 1 5 0 4 healh care services o medical prof 206 10 7 6 8 5 11 3 0 1 9 2 4 consumer non-durables clohing 301 10 9 8 4 5 11 0 3 2 7 1 6 consumer non-durables consumer conainers 302 7 9 10 8 6 11 1 3 4 0 2 5 consumer non-durables cosmeics 303 11 8 9 7 2 10 1 3 4 0 5 6 consumer non-durables food processors 304 10 9 8 5 6 11 3 1 2 4 0 7 consumer non-durables beverages 305 11 9 7 6 5 10 3 2 1 4 0 8 consumer non-durables home producs 306 7 10 9 8 4 11 0 2 3 6 1 5 consumer non-durables leisure imes 307 9 7 8 10 5 11 0 3 2 6 1 4 consumer non-durables obacco 309 11 9 8 7 0 10 1 2 4 5 3 6 consumer services communicaions 401 9 11 8 7 6 10 3 2 1 5 0 4 consumer services leisure 402 9 11 7 8 6 10 4 1 0 5 3 2 consumer services reailing foods 403 10 9 8 5 4 11 3 2 0 7 1 6 consumer services reailing goods 404 10 9 8 5 4 11 2 3 1 7 0 6 consumer services indusrial services 405 6 11 10 9 8 5 2 1 4 7 3 0 consumer services undesignaed conr svc 407 10 9 8 1 0 11 2 5 4 6 3 7 consumer durables auomoive mfg 501 6 11 10 7 8 9 4 1 3 5 2 0 consumer durables auo par mfg 502 9 10 8 6 5 11 0 1 3 7 2 4 consumer durables home furnishings 504 8 10 9 7 5 11 3 0 2 6 1 4 consumer durables leisure producs 505 10 4 5 8 6 11 1 0 2 7 3 9 consumer durables recreaional vehicles 506 10 5 7 9 6 11 2 0 3 8 4 1 consumer durables rubber 507 10 9 5 7 8 11 3 1 0 4 2 6 energy oil 601 6 8 7 10 9 11 5 1 2 4 3 0 energy coal 602 7 10 9 3 6 11 2 0 8 4 5 1 energy gas 607 2 7 6 10 8 11 3 1 5 9 4 0 ransporaion airlines 701 7 11 9 8 6 10 3 2 1 5 0 4 ransporaion railroads 702 8 9 11 7 4 10 3 1 2 5 0 6 ransporaion rucking 703 8 10 9 4 5 11 1 0 2 7 3 6 ransporaion marime 705 7 9 10 8 5 11 0 1 3 4 2 6 Coninued... 41

Table 7 Coninued Secor Name Indusry Name Code BV CFO Ebda CACT IACT Sales EPS1 EPS2 EPS3 P1* P2* P3* echnology compuers 801 8 10 9 7 6 11 3 2 1 4 0 5 echnology elecronics 803 9 11 5 7 6 10 3 1 2 8 0 4 echnology sofware & edp services 804 10 11 6 7 9 8 0 3 2 5 1 4 echnology undesignaed ech 805 6 11 9 8 7 10 0 1 3 5 4 2 echnology oher compuers 807 10 7 8 9 6 11 1 0 3 5 2 4 echnology semiconducors/componen 808 10 9 8 7 6 11 3 2 1 4 0 5 echnology elecronic sys/devices 810 8 10 9 6 7 11 3 2 1 5 0 4 echnology office/comm equip 811 10 9 6 8 4 11 3 2 1 7 0 5 basic indusries building & relaed 901 8 10 11 7 6 9 4 1 3 2 0 5 basic indusries chemicals 902 10 9 8 7 6 11 4 2 1 3 0 5 basic indusries conainers 903 6 9 10 8 7 11 5 3 1 4 0 2 basic indusries meal fabricaors & dis 904 8 6 9 7 3 11 5 2 1 10 0 4 basic indusries fores producs 906 4 8 7 10 9 11 5 1 3 6 2 0 basic indusries paper 907 9 10 7 6 8 11 3 0 1 5 2 4 basic indusries seel 908 6 9 5 11 8 10 4 2 3 7 0 1 basic indusries nonferrous base meals 910 6 10 7 8 9 11 5 0 4 3 1 2 basic indusries precious meals 911 9 8 10 6 7 11 1 2 3 5 4 0 basic indusries muli-ind basic 912 11 6 10 8 7 9 0 1 3 5 2 4 capal goods defense 1001 7 10 9 8 6 11 3 0 2 5 1 4 capal goods auo oems 1002 8 11 6 5 7 10 0 1 3 9 2 4 capal goods elecrical 1003 8 10 9 7 6 11 1 2 5 4 3 0 capal goods machinery 1004 10 9 8 7 5 11 3 1 0 6 2 4 capal goods building maerials 1007 8 10 7 9 6 11 1 0 2 5 3 4 capal goods office producs 1008 10 7 9 8 6 11 3 0 2 1 4 5 capal goods muli-ind cap good 1010 10 9 8 7 4 11 3 1 0 5 2 6 public uilies elecrical uilies 1101 8 10 9 7 6 11 2 0 3 4 1 5 public uilies gas uilies 1102 5 10 8 9 7 11 3 0 2 6 1 4 public uilies elephone uilies 1103 10 9 6 8 5 11 4 0 2 7 1 3 public uilies waer uilies 1105 7 9 8 5 3 11 0 4 2 10 1 6 miscellaneous/undesigna unclassified 9900 10 9 8 1 2 11 6 7 5 0 3 4 Mean 8.26 8.93 8.31 7.09 5.72 10.54 2.59 1.54 2.19 5.19 1.60 4.03 Med 9.00 9.00 8.00 7.00 6.00 11.00 3.00 1.00 2.00 5.00 1.00 4.00 SD 1.98 1.60 1.59 1.87 1.87 0.98 1.62 1.42 1.47 2.36 1.45 2.07 42

Figure 1 Disribuion of Percenage Valuaion Errors Using Simple Indusry Muliples Value and value drivers are assumed o be proporional: p = x + β, is esimaed using he æx ö b 1/ ˆ = E ç p çè ø, and he disribuion of percen pricing error, e p-bx =, is p p ploed below. The variables are defined as follows (all amouns are on a per share basis): P is sock price; BV is book value of equy; EBITDA is earnings before ineres, axes, depreciaion and amorizaion; IACT is IBES acual earnings; EPS3=EPS2*(1+g), where EPS2 is wo year ou earnings forecas and g is growh forecas, and 5 * Ε( eps+ s kbv 1) + s Ε( eps+ 5 kbv + 4) P1 = bv + s + 5 (1 + k ) k (1 + k ) s= 1 All muliples are calculaed using comparable firms whin each indusry (based on IBES indusry classificaion), and he firm being valued is excluded when compuing indusry muliples. Years covered are 1981 hrough 1996. Sample size is 9,658. The char below is derived from a hisogram wh columns of widh=0.1 (or 10% of price). For example, for EPS3, he fracion of he sample wh pricing error beween 0 and 0.1 is jus over 18%. 20 18 EPS3 16 P1* 14 IACT frequency in % 12 10 8 BV Ebda Sales 6 4 2 0 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0-0.1-0.2-0.3-0.4-0.5-0.6-0.7-0.8-0.9-1 -1.1-1.2-1.3-1.4-1.5-1.6-1.7-1.8-1.9-2 percenage valuaion error 43

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