An Innovative Thinking on the Concepts of Ex-Ante Value, Ex-Post Value and the Realized Value (Price)

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1 RISUS - Journal on Innovaion and Susainabiliy Volume 6, número ISSN: Edior Cienífico: Arnoldo José de Hoyos Guevara Ediora Assisene: Leícia Sueli de Almeida Avaliação: Melhores práicas edioriais da ANPAD An Innovaive Thinking on he Conceps of Ex-Ane Value, Ex-Pos Value and he Realized Value (Price) Dr. Mohammad Ali Tareq Senior Lecurer, Malaysia-Japan Inernaional Insiue of Technology (MJIIT), Universii Teknologi Malaysia Address: Johor Bahru, Johor, Malaysia areq@ic.um.my ; areq@canab.ne Absrac: Researchers have been considering he realized value as he ex-pos realizaion of he ex-ane value. They have argued ha ha he realized values have failed o esimae he expeced value in asse-pricing models. We provide a new definiion of he ex-pos measuremen and we show ha considering realized value as he ex-pos realizaion of he ex-ane value is misleading and his has led o he failure in esimaing he expeced value. Keywords: ex-pos, ex-ane, realized value, expeced value Paper received: 15/09/2014 Paper acceped for Publicaion: 30/03/2015 RISUS Journal on Innovaion and Susainabiliy, São Paulo, v. 6, n.1, p , jan/abr ISSN

2 An Innovaive Thinking on he Conceps of Ex-Ane Value, Ex-Pos Value and he Realized Value (Price) INTRODUCTION In heory, researchers can esablish risk-reurn relaionship; empirically, he unobservable naure of he ex-ane expeced reurn hinders he esimaion of he relaionship beween he risk and reurn. In general, i is believed ha invesors know heir expeced value and he variance-covariance marix, and based on hese, hey form he price of an asse in he marke. Thus in he ex books, researchers have been using realized reurns assuming ha realized reurns are normally disribued wih mean μ i and a variance of σ i 2 ; R i ~N(μ i, σ i 2 ), and over he sample period, he average of hese realized reurns will mach he ex-ane expeced reurn. This implici belief has led he researchers o assume realized reurns as a sample of reurns in esimaing he expeced reurn, i.e., hey consider realized value as he ex-pos realizaion of he ex-ane expecaions. We argue ha his belief on he convergence of realized reurn on ex-pos reurn is misleading. The dispariy beween ex-pos realizaion and ex-ane predicion is well addressed by Sharpe (1978); and Campello e al. (2008) also believe ha he disribuion of expeced reurns does differ from he disribuion of realized (ex-pos) reurns. Elon (1999) concluded ha he realized (ex-pos) reurn is a poor proxy for he expeced reurn because of is deficiency in reflecing he naure of ex-ane expecaions. None of he researchers have idenified he reason behind his dispariy beween hese wo values, however. rom he poin of view of asse-pricing model, we inroduce new definiion of he ex-pos value and show clear disincion beween he realized value (price) and he ex-pos value, and in urn, i proves ha realized price canno be he ex-pos realizaion of he ex-ane values. How he ex-pos value and he realized value differ from each oher? THE EX-ANTE RETURN, EX-POST RETURN AND THE REALIZED RETURNS The main focus of he asse-pricing model is o explain he risk-reurn relaionship. Theoreically, we can esablish risk-reurn relaionship (for example, CAPM). However, unobservable naure of he ex-ane expeced reurn hinders esimaing he empirical risk-reurn relaionship. As a resul, in empirical analysis, mos of he researchers consider realized reurn as he ex-pos realizaion of he (ex-ane) reurn, i.e., hey assume realized reurn as a sample of reurn. or example, hey assume ha (ex-ane) reurn 1 are normally disribued wih mean μ i and variance of σ i 2, R i, ~N(μ i, σ i 2 ). They have used he average realized reurn and sample 1 In general, expeced reurn has been considered as he ex-ane reurn by he researchers. As we have discussed laer in he paper ha he ex-ane lierally means he random fuure values. If we define ex-ane reurn as he expeced reurn, we are disregarding he randomness of he fuure values. Therefore, we have defied he reurns as (ex-ane) reurns in his paper insead of he expeced reurns as has been considered by he oher researchers. 49 RISUS Journal on Innovaion and Susainabiliy, São Paulo, v. 6, n.1, p , jan/abr ISSN

3 Mohammad Ali Tareq variance as esimaors of he ex-ane expeced reurn and he ex-ane variance. Neverheless resuls of he empirical analysis were almos inconclusive. Some researchers inuiively believe ha he realized reurn canno be he ex-pos realizaion of he (ex-ane) reurn and consequenly empirical esimaion differs from he exane expecaion. In his secion we depic he inabiliy of he realized reurn as he ex-pos realizaion of he ex-ane and presen ha ex-pos value is differen from he realized value. We porray our argumen from he pricing poin of view and in doing so we show ha he informaion se in he price is differen from he informaion se in he ex-pos value. Our argumen is based on he following simplified assumpions: (i). In an one-period seing, price is he discouned value of he nex period s expeced price, p i, = E(p i,+1) d i whered i > 1. In addiion, we assume ha E(p i,+1 ) incorporaes all fuure informaion available a. (ii). The sae of fuure economy changes wih ime. Assumpion (i) saes ha for any risky asse he invesors are assumed o expec posiive payoffs in fuure and can be considered as one of he basic assumpions in valuaion. Assumpion (ii) can be considered as he base of our argumen. Mos of he researchers assume a seady sae of he economy where here is no change in he fundamenal economic variables. Raher hey consider any change in he informaion se (surprises) as a change in variables oher han he fundamenals. And for a sample, hese surprises are expeced o be cancelled ou. We assume ha any change in he economy is a resul of he changes in he economic variables, boh fundamenals as well as firm specific ones. This may lead us o assume ha invesors forecass abou he asse s expeced price would increase (decrease) wih forecased posiive (negaive) changes in he economic variables. Besides, researchers have been using he realized reurn in empirical ess, and in realiy he economy is changing also. Thus our second assumpion is much closer o he realiy. Mos of he researchers have been using realized reurn in esablishing he empirical riskreurn relaionship; we inroduce 2 scenarios and argue on he inabiliy of he realized reurn o explain he risk-reurn relaionship. As we proceed, we discussed on he differen informaion ses in he asse-pricing, and gradually, we presen he difference beween he realized value and he ex-pos value. We conclude ha realized reurn canno be a sample of reurn. An example: We begin wih a simple example for beer undersanding of our argumen. We show ha when he assumpions (i) and (ii) hold, average realized reurns canno esimae he expeced reurn. Invesor s expeced price would rise (fall) wih he favorable (unfavorable) fuure economic forecass. We sar our argumen wih a series of unfavorable fuure economy in scenario 1. Under one-period model seings, we assume ha price in every period is formed RISUS Journal on Innovaion and Susainabiliy, São Paulo, v. 6, n.1, p , jan/abr ISSN

4 An Innovaive Thinking on he Conceps of Ex-Ane Value, Ex-Pos Value and he Realized Value (Price) based on he expeced price of he nex period. Le us assume he expeced prices for ( + 1) o ( + 4) a, ( + 1), ( + 2) and ( + 3) are 105, 95, 89 and 83 respecively. If we assume 5% expeced reurn 2 for he invesors, we would ge he price for o ( + 3) as (105/1.05), (95/1.05), (89/1.05) and (83/1.05) respecively. or his series he average realized reurn would be negaive. Noe ha our expeced reurn is 5% in scenario 1. The sample average realized reurn for hese ypes of series canno esimae he expeced reurn of 5%. Why average realized reurn fails o esimae he expeced reurn? Scenario 1: Realized reurn and he risk-reurn relaionship in downward Marke This able forecass he fuure values from (+1) o (+4) in a down-ward marke. The expeced reurn (cos of capial) is 5% (i.e., discoun rae, d i = 1.05). or simpliciy of he argumen we assume expeced reurn as consan E(p +τ ) p 105 +τ r +τ+1 = p +τ+1 p +τ or ( + 1) in scenario 1, researchers would consider (95/1.05) as he ex-pos realizaion of ex-ane price for, i.e., (95/1.05) is reaed as a realized value of he ex-ane disribuion of fuure price of p i,+1 for ( + 1) a. Can (95/1.05) a ( + 1) be a ex-pos value of he fuure price of ( + 1) for? The price a ( + 1) is he discouned expeced price of ( + 2). In his example, he price (95/1.05) a ( + 1) is derived from he informaion on he fuure price for ( + 2) which is available a ( + 1). In general, he ex-pos value a ( + 1) is he observed value from he informaion on o ( + 1). (95/1.05) canno be he ex-pos value a ( + 1) as his value is derived from he informaion of ( + 2) insead of he informaion se of o ( + 1). Under assumpion (ii), he expeced price of ( + 2), E(p i,+2 ), has no relaion o he disribuion of p i,+1 a ( + 1). So he realized reurn can neiher be he ex-pos reurn nor he sample of reurn. 2 Alhough he expeced rae of reurn (he discoun rae) migh change wih he changes in he economic forecass, for simpliciy, we consider consan discoun rae in his paper. Noe ha, he argumen of his paper can suppor a model wih changing discoun rae scenario also. 51 RISUS Journal on Innovaion and Susainabiliy, São Paulo, v. 6, n.1, p , jan/abr ISSN

5 Mohammad Ali Tareq In scenario 2, wih favorable economic forecass, he expeced values increase from 105 in ( + 1) o 150 in ( + 4). We can consider scenario 2 as an illusraion of he Japanese bubble during Wih his increase, he prices also increase from 100 a o 143 in ( + 3). The average realized reurn for his ype of upward series will be much greaer han he expeced reurn of he asse (5% in his case). Besides, as we have argued before, (120/1.05) canno be considered as he ex-pos value a ( + 1) because (120/1.05) is derived from he informaion se on he expeced price of ( + 2) available a ( + 1). Scenario 2: Realized reurn and he risk-reurn relaionship in upward Marke This able forecass he fuure values from (+1) o (+4) in an upward marke. The expeced reurn (cos of capial) is 5% (i.e., discoun rae, d i = 1.05). or simpliciy of he argumen we assume expeced reurn as consan E(p i,+τ ) p i,+τ r i,+τ+1 = p i,+τ+1 p i,+τ None of he researchers have argued on he informaion ses in he price as well as in he ex-pos reurn. In his secion, wih simple illusraive examples under assumpion (i) and (ii), we have shown ha he informaion ses in price and in ex-pos reurn are differen, and price canno be considered as he ex-pos realizaion of he ex-ane expecaion. REALIZED RETURN AND THE EX-POST RETURN In his secion, we provide a general discussion on he difference beween realized reurn and ex-pos reurn. We have divided informaion a ino wo pars for beer undersanding, and we define informaion as: Φ = Φ H + Φ +1 (1) where, Φ is he oal informaion se available a, Φ H is he pas informaion se on ( 1) o available a, and Φ +1 is he fuure informaion se on ( + 1) ha is incorporaed a. Pas informaion se is assumed o be comprised of he resuls of he operaing aciviies beween ( 1) o. In conras, he economic informaion as well as he firm s fuure policies is RISUS Journal on Innovaion and Susainabiliy, São Paulo, v. 6, n.1, p , jan/abr ISSN

6 An Innovaive Thinking on he Conceps of Ex-Ane Value, Ex-Pos Value and he Realized Value (Price) incorporaed in he fuure informaion se. Under assumpion (i), price p i is he discouned value of E(p i,+1 Φ +1 ). Similarly, price p i,+1, is he discouned value of E(p i,+2 Φ ). A H ( + 1), p i,+1 does no incorporae pas informaion se Φ ; 3 i is derived from he fuure informaion se of Φ The figure 1 explains he difference beween informaion ses in price and he ex-pos value. igure 1: Price and ex-pos value In he following discussion, we provide furher explanaion o confirm ha he realized reurn canno be he ex-pos reurn. We define (ex-ane) reurn a ime, r i,+1 as, r i,+1 = p i,+1 Φ +1 and, he realized reurn, r i+1, is defined as, p i, Φ +1 (2) +2 r i,+1 = p i,+1 Φ +1 p i, Φ +1 (3) The researchers consider (p i,+1 Φ ) as he ex-pos realizaion of (p i,+1 Φ +1 ), i.e.; hey have been assuming (p i,+1 Φ ) as he sample of he disribuion of fuure random price of (p i,+1 Φ +1 ). The reurn a in equaion (4.2) incorporaes he informaion abou he ime period ( + 1), available a. In equaion (3), (p i,+1 Φ ) has no relaion wih (p i,+1 Φ +1 ) under assumpion (ii), however. Insead, (p i,+1 Φ ) is he discouned expeced value of +2 ). The realized reurn of (+1) in equaion (3), includes informaion abou periods (p i,+2 Φ +1 3 Pas informaion, for example as cied by Elon (1999), high earnings announcemens of MacDonald, has lile or no role in forming fuure expecaion of he invesors. Does high earnings announcemen really lead o higher fuure price? In TSE, he annual earnings for Ninendo was he highes in March of 2009 a JPY 279 billon (approx); he price dropped from JPY 71,900 in 2007:10 o JPY 23,180 in 2009:10 following he earnings informaion, however. If posiive (negaive) pas informaion has an impac on he following price, he price would have increased (decreased) following he informaion. The drif in Ninendo s price, even wih he highes earnings informaion, can be an example of he absence of he effec of pas informaion on he price. 53 RISUS Journal on Innovaion and Susainabiliy, São Paulo, v. 6, n.1, p , jan/abr ISSN

7 Mohammad Ali Tareq (+1) and (+2) for (p i, Φ +1 ) and (p i,+1 Φ ) respecively. The informaion ses in r i,+1 and r i,+1 are differen. These wo values are derived from differen informaion ses of differen ime periods. As a resul realized reurn can neiher be ex-pos reurn nor a sample of reurn. ALTERNATIVE DEINITION O EX-POST VALUE In secion 3, we have shown ha he presen believe on he ex-pos reurn is misleading. How can we measure he ex-pos reurn? A, we consider, he ex-ane predicion follows 4, p i,+1 = (p i, Φ +1 ) + (x i,+1 Φ +1 ) (4) where, (x i,+1 Φ +1 ) is random operaing value for o ( + 1) based on available informaion se Φ +1 a. We assume earnings, x i,+1, as he random operaional oucome from o ( + 1) realized a ( + 1). We observe earnings for o ( + 1), i.e.; (x i,+1 he ex-pos value a (+1), (v i,+1 where, x i,+1 v i,+1 H Φ +1 ) as, H +1 ). Thus, we define Φ +1 = (p i, Φ +1 H ) + (x i,+1 Φ ) (5) is he observed earnings a ( + 1). The value in equaion (5) is he realized value of ex-ane random price of (p i,+1 Φ +1 ) for ( + 1) made a. The realized price, p i,+1, is no he ex-pos realizaion of (p i,+1 Φ +1 ); whereas, x i,+1 a ( + 1). The ex-pos reurn, r i,+1, can be wrien as: r i,+1 = v H i,+1 Φ (p i, Φ +1 ) is he observed earnings for o ( + 1) (6) 4 Readers migh quesion ha: How he invesors will forecas he fuure when he fuure looks gloomy and nex period s expeced value is hough o be negaive? ha is, when x i,+1 Φ +1 < 0, how he invesors will make heir fuure forecass? ollowing equaion (4.4), when x i,+1 Φ +1 < 0, invesors will consider anoher variable θ i,+1 Φ +2 which includes he informaion on he periods ( + 2) onwards. p i,+1 = (p i, Φ +1 ) (x i,+1 Φ +1 ) + (θ i,+1 Φ +2 ) The idea behind his is ha, even hough he nex period s values are negaive, he following periods informaion makes he forecass posiive in he sense ha: (x i,+1 Φ +1 ) + (θ i,+1 Φ +2 ) > 0 In he laer secion we conduc an empirical es on he ex-pos measure of he above relaionship. And we have provided proof supporing our assumpion. RISUS Journal on Innovaion and Susainabiliy, São Paulo, v. 6, n.1, p , jan/abr ISSN

8 An Innovaive Thinking on he Conceps of Ex-Ane Value, Ex-Pos Value and he Realized Value (Price) As a concluding remark of secion 4, he ex-ane value a is he expeced value of E(p i,+1 Φ +1 ) for (+1). E(p i,+1 Φ +1 ) is discouned o derive p i, a. The ex-pos value a ( + 1) is he realized (observed) value of ime s anicipaion of (p i,+1 Φ +1 ) ha we would observe as we move o ( + 1). In conras, price p i,+1 is derived from E(p i,+2 Φ ) a ( + 1). In secion 3 we have argued ha he informaion ses in hese values are differen. A ( + 1), price p i,+1 incorporaes he informaion se Φ on ( + 2), whereas he ex-pos H value a ( + 1), v i,+1 Φ +1 +1, is observed from he operaional aciviies of for ( + 1). A NUMERICAL EXAMPLE ON THE EX-ANTE RETURN, EX-POST RETURN AND THE REALIZED RETURN The Lain word ex-ane means beforehand. In models where here is uncerainy, ha is resolved during he course of evens, he ex-ane values are hose ha are calculaed in advance of he resoluion of uncerainy. In finance, for example, ex-ane is he fuure random values. As a numerical example, le E (p i,+1 ) and R i be $110 and 10% respecively. The price a, p i,, would be $100. Now, as we move o (+1), he price p i,+1 will be he discouned expeced value of (+2), i.e., E +1 (p i,+2 ). If for some reasons, a (+1) he invesors predic a macro-economic downurn in (+2). A (+1) invesors predic E +1 (p i,+2 ) o be $99. 5 Thus he price a (+1), p i,+1, will be $90. This downward movemen of he price is he resul of he unfavourable forecas of E +1 (p i,+2 +2 ). Under hese circumsances, he ex-ane reurns for boh and (+1) are 10% whereas he realized reurn a (+1) is -10%. The price $90 a (+1) is no observed from p i+1, raher p i,+1 is he discouned value of E +1 (p i,+2 ). Thus, realized value (or he price) canno be he sample of he ex-ane expecaions. Can realized reurn be he ex-pos realizaion of he ex-ane expecaions? Coninuing wih he numerical example, le he operaing earnings for o (+1), i.e.; x i,+1, be $8. Thus he ex-pos value v i+1 becomes $108 and he ex-pos earnings is 8%. The realized reurn a (+1) is -10% whereas he ex-pos reurn is 8%. How raional will subsiuing 8% wih -10% be? In oher words, we canno subsiue he ex-pos reurn of 8% wih he realized reurn of -10%.The ex-pos earnings is posiive whereas he realized earnings is negaive. The realized reurn canno be he ex-pos realizaion of he ex-ane reurns. In he empirical ess of CAPM, researchers are assuming ha he realized reurn, r i+1, is a sample of reurn r i, and he average of he realized reurn, r i, will be he bes esimae of he expeced reurn, E(r i). We have argued in his secion ha, he realized reurn canno be a sample of reurn. We believe, he inuiion ha realized reurn as he ex-pos realizaion of he 5 or simpliciy, we have assumed ha he variance-covariance marix for he invesor is consan and any change in he macroeconomic variable will be refleced in he changing expeced value. 55 RISUS Journal on Innovaion and Susainabiliy, São Paulo, v. 6, n.1, p , jan/abr ISSN

9 Mohammad Ali Tareq reurn is misleading. Redefining he ex-pos value will porrayed he disincion beween he expos value and he realized value. When he realized value is no he ex-pos realizaion of he ex-ane value, i concludes ha realized reurn is no he sample of reurn. The erm ex-pos lierally means "afer he fac". or any ex-ane value, he ex-pos value will be observed as we moved o ha paricular period and when all he uncerainy has been resolved. or example in figure 6.2, a, he fuure value of asse i for (+1) can be considered as he ex-ane value, i.e.; p i+1 would be he ex-ane value for. In conras, he ex-pos value is he observed value a (+1) from p i+1, and his value is he resul of operaions from o (+1). In oher words, he ex-pos value can be defined as he observed value of p i+1, forecased a, ha is realized a (+1). However, he realized price a (+1), p i+1, is he discouned fuure value of (+2). Alhough we ge wo values a (+1), he ex-pos value and he realized price, hese values are differen as hey are derived from differen ime periods. The price a, is derived from he value of (+1) E (p i,+1 ) The price a (+1), is derived from he value of (+2) p i,+1 v i,+1 p i, The ex-pos value a, is he resul of he operaing aciviies from o (+1) ( + 1) ig 2: Derivaion of Prices and ex-pos value We defined x i,+1 as he ex-pos earnings of he disribuion of random values of p i,+1 in (+1), which is observed as we move o (+1). The researchers believe ha he ex-pos average reurn provides a good esimae of he ex-ane expeced reurn. As he ex-ane expecaion is unobservable, empirical ess of CAPM assumes ha he probabiliy disribuion generaing he ex-pos oucomes is saionary over ime and realized reurn could be subsiued as he sample of ex-pos realizaion of he ex-ane expecaions. Are boh of hese values, he ex-pos and he realized value (price), generaed from he same disribuion of fuure values? Are boh v i,+1 and p i,+1 derived values of he disribuion of p i,+1? igure 2 clearly explains ha hese values are no he same. Insead, hese values provide differen informaion o he researchers. Alhough hese wo values rarely coincide, none of he values can be reaed as subsiuing he oher value. RISUS Journal on Innovaion and Susainabiliy, São Paulo, v. 6, n.1, p , jan/abr ISSN

10 An Innovaive Thinking on he Conceps of Ex-Ane Value, Ex-Pos Value and he Realized Value (Price) CONCLUSION In his paper, we focused on he belief of considering realized reurn as a sample of reurn. Under assumpions (i) and (ii), we have shown ha realized reurn canno be he ex-pos realizaion of he ex-ane expecaion. The researchers can esablish he risk-reurn relaionship in heory. The unobservable naure of he expeced reurn has led he empirical researchers o use realized reurn as a sample of reurn. And he measuremen of he empirical risk-reurn relaionship has been inconclusive and conroversial. As a resul, a number of researchers have inroduced new models o measure he empirical risk-reurn relaionship. or example, ama and rench (1992) have inroduced he 3-facor model in an aemp o explain he empirical risk-reurn relaionship. Their model gained populariy as hey focused on forming an empirical model ha would fi he realized reurn daa. The model is used o explain he ex-ane risk-reurn relaionship from he realized reurn daa. We have shown ha realized reurn can neiher be he ex-pos reurn nor he sample of reurn. Wha economic implicaion does he realized reurn daa conain? This hesis is he firs one o explicily define he ex-pos value, and we have shown ha realized value and he ex-pos value are differen because of he differences in he informaion ses. We conclude ha realized reurn canno be he ex-pos realizaion of he (ex-ane) reurn, i.e., realized reurn canno be a sample of reurn. REERENCES [1] Campello, Murillo, Long Chen, and Lu Zhang, Expeced reurns, yield spreads and asse pricing ess, The review of inancial Sudies [J], 2008, 21: [2] Elon, E. J., Expeced reurn, realized reurn and asse pricing ess, Journal of inance [J], 1999, 54: [3] ama, E.., and K. R. rench, The cross-secion of expeced sock reurns, Journal of inance [J], 1992, 47: [4] Sharpe, W.., New evidence on he capial asse pricing model: Discussion, Journal of inance [J], 1978, 33: RISUS Journal on Innovaion and Susainabiliy, São Paulo, v. 6, n.1, p , jan/abr ISSN