Estimating and Testing Fundamental Stock Prices: Evidence from Simulated Economies

Transcription

1 Esimaing and Tesing Fundamenal Sock Prices: Evidence from Simulaed Economies R. Glen Donaldson Finning Ld. Associae Professor of Finance Faculy of Commerce and Business Adminisraion Universiy of Briish Columbia 2053 Main Mall, Vancouver, BC, Canada, V6T 1Z2 Phone: Fax: and 1 Mark Kamsra Associae Professor of Economics Deparmen of Economics Simon Fraser Universiy Burnaby BC, Canada V5A 1S6 Phone: Fax: kamsra@sfu.ca. Version: December 12, We hank Lisa Kramer for helpful commens and he Social Sciences and Humaniies Research Council of Canada for financial suppor.

2 A core ene of financial economics saes ha, in a marke populaed by raional invesors, he fundamenal price of an asse equals he expeced discouned presen value of is fuure cashflows. This implies ha in a raional and efficien marke, and in he absence of price bubbles, 2 sock price movemens are driven by forecased changes in dividends and discoun raes and no by he irraional exuberance of raders. Ineres in he exen o which sock prices are fundamenally driven has been paricularly high of lae, wih everyone from academics o he Chairman of he U.S. Federal Reserve Board offering opinions. Research ino fundamenal valuaion is herefore paricularly imely and, a leas for he ype of compuer-inensive research we underake in his chaper, is made possible by recen increases in compuing power available o financial economericians. Several differen procedures have been developed o esimae fundamenal sock prices and o es wheher marke prices deviae in significan ways from he esimaed fundamenal price. Many sudies have repored wha appear o be imporan deviaions from fundamenals, especially around marke crashes such as hose which occurred in 1929 and The resuling belief ha financial markes may be excessively volaile, and may poenially conain price bubbles ha push marke prices way from fundamenal valuaions, has conribued o he insiuion of policies such as rading hals in he face of large price moves. Wheher such policies help or hur financial markes depends in imporan ways on he exen o which marke prices are driven by fundamenal facors as opposed o irraional exuberance, and he answer o his quesion crucially depends on he accuracy of he fundamenal price esimaes and ess used in he analysis. To he bes of our knowledge, no one has ye underaken a horough and sysemaic invesigaion of he accuracy of various fundamenal price esimaing and esing procedures. 3 Thus, while a fundamenals-esimaion exercise migh value a share of sock a 2 The erm bubble is formally defined below, bu for now can be hough of as a siuaion in which some force oher han a fundamenal facor, such as dividends and discoun raes, drives sock prices. 3 Bollerslev and Hodrick (1995) provide a survey of much of his lieraure, and hemselves use simulaion echniques o invesigae ess of marke efficiency. Bollerslev and Hodrick (1995) focus much of heir simulaion sudy on he consan discoun rae case and consider only one ad hoc model for ime-varying discoun raes. This ad hoc model, like he consan discoun rae case, allows an analyic soluion for he 1

3 $X, we canno be sure how confiden o be in he esimae. For example, he esimae could be biased up or down and/or be very imprecise (i.e., have a large variance). Wih a small bias and low variance he esimae migh be very accurae and herefore of considerable use. Conversely, a significanly biased esimae wih very low precision migh be wildly inaccurae and hus of lile value. Esablishing he properies of various fundamenals esimaion and esing procedures would herefore be of significan benefi o academics, invesors and policy makers who ofen employ fundamenal price esimaes in heir work. The accuracy of fundamenal price esimaes can be obained analyically under only very resricive and special circumsances (such as assuming ha dividend growh raes and ineres raes will never change in he fuure, which is of course highly unrealisic). To circumven his analyical inracabiliy, we develop and es in his chaper a simulaion-based mehod for calculaing he properies of various fundamenal price esimaing and esing procedures. We are paricularly ineresed in deermining he saisical properies of fundamenal price and reurn esimaes commonly used in boh indusry and academia and in invesigaing he effecs ha esimaion inaccuracies may have on he variey of volailiy ess commonly employed in he lieraure. Our sraegy in his chaper is as follows: (a) use financial marke daa o esimae ime-series models for dividend growh and discoun raes, (b) use hese models o simulae dividend growh and discoun rae pahs for a variey of possible economies ha do no conain bubbles, (c) calculae fundamenal prices for hese bubble-free economies based on he simulaed dividend growh and discoun raes which prices we call marke prices since hey are fair-value prices for he simulaed marke economies, (d) use various fundamenal valuaion models o esimae fundamenal prices for each of he simulaed economies, and (e) compare marke prices versus fundamenal prices and invesigae saisical properies of common ess for excess volailiy and bubbles in sock prices using marke price. We consider formal discoun rae processes ha admi serial dependence and hence rule ou a general analyic soluion. Allowing serial dependence in discoun raes considerably complicaes he analysis, bu also considerably enriches he range of possible dynamics for prices and reurns. 2

4 he daa we simulaed daa under he (rue) null hypohesis ha here are no price bubbles. We apply his procedure o S&P 500 sock price daa. Resuls produced using our compuer-aided echniques sugges ha, while sock prices are indeed volaile, hey are no more volaile han one would reasonably find in an economy driven by fundamenals, hereby suggesing ha marke prices are no excessively volaile. We also find ha radiional ess for excess volailiy and bubbles over-rejec a rue null of no-bubbles in samples of he size radiionally employed in he lieraure. Indeed, mos ess we invesigae find overwhelming evidence of bubbles in daa series we consruc under he condiions ha here are no bubbles. In oher words, we demonsrae ha radiional ess for price bubbles frequenly find bubbles in daa ha do no in fac conain bubbles. In Secion I below we describe a variey of common fundamenal pricing models and ess for excess volailiy and bubbles. In Secion II we presen our Mone Carlo procedure for esimaing fundamenal prices/reurns and invesigaing es performance. We discuss he resuls of our effors in Secion III. Secion IV concludes. I. Fundamenal Pricing Mehods and Tess Define P M as a sock s beginning-of-period- marke price, r as he rae used o discoun paymens received during period, and E { } as he condiional expecaion operaor, wih he condiioning informaion being he se of informaion available o invesors a he beginning of period. Invesor raionaliy requires ha he curren marke price of a sock, which will pay a dividend D +1 a he beginning of period + 1 and hen immediaely sell for he ex-dividend marke price P M +1, saisfy Equaion 1. P M = E { } P M +1 + D r (1) 3

5 We can solve Equaion 1 forward o period T (where T > ) and subsiue realized dividends and discoun raes in for heir expeced values o produce Equaion 2. P X = T i=1 ( Π i k=1 [ r +k 1 ]) D +i + ( Π T k=1 [ r +k 1 ]) P M T (2) The superscrip X on he lef-hand-side price indicaes ha his is he ex pos raional price; i.e., he price ha an invesor would have raionally paid for he sock had she known ha he marke price of he sock in period T was going o be P M T and ha he sock was going o pay he sequence of dividends ha i acually paid beween periods and T. The ex pos raional price is no a fundamenal price, nor is i a price ha would ever be observed in he markeplace. However, for reasons explained below, i is noneheless useful in analyzing price behavior. I.A. Fundamenal Pricing Models To calculae he fundamenal price of sock, we need o solve Equaion 1 forward ino infinie ime under he ransversaliy condiion ha he expeced presen value of he marke price P M +i falls o zero as i goes o infiniy; i.e., here are no price bubbles. This produces he familiar resul ha oday s fundamenal sock price, P F, equals he expeced presen value of fuure dividends; i.e., P F { ( [ ]) } = E Π i 1 k=1 D +i. (3) i=1 1 + r +k 1 Noe from Equaion 3 ha he fundamenal price is a funcion only of dividends, discoun raes and ime, and no of he marke price. Defining he growh rae of dividends during period as g (D +1 D )/D allows he preceding equaion o be rewrien as: 4

6 P F { ( [ ])} 1 + = D E Π i g+k 1 k=1, (4) i=1 1 + r +k 1 or, defining he discouned dividend growh rae as y (1 + g )/(1 + r ), P F { } = D E Π i k=1y +k 1. (5) i=1 This can be rewrien as shown in Equaion 6 below (noe ha y is no in he ime informaion se since i depends on D +1 hrough g ): P F { } = D E Π i k=1 y +k 1. (6) i=1 In pracice, fundamenal valuaion calls for forecasing fuure discouned dividend growh raes (i.e., dividends and discoun raes) o solve for he sum produc in Equaion 4 (or, equivalenly, Equaion 6). Since Equaion 4 canno be solved analyically in general, his usually requires ha some resricive assumpions be made concerning he ime series processes driving dividends and discoun raes. The simples approach, inroduced by Gordon (1962), is o assume ha discoun raes and dividend growh raes will be consan, a r and g respecively, for all fuure ime so ha Equaion 4 reduces o Equaion 7, in which he superscrip G denoes he Gordon-model fundamenal price esimae: P G = D [ ] 1 + g. (7) r g The Gordon model is cerainly convenien, bu is exremely resricive assumpions of consan r and g do no end o produce he mos accurae valuaions possible. Several 5

7 aemps have herefore been made o relax Gordon s original resricions and ye sill reain an analyical soluion o Equaion 4. The lieraure in his area is raher large and includes papers by Malkiel (1963), Fuller and Hsia (1984), Brooks and Helms (1990), Hurley and Johnson (1994, 1998) and Yao (1997), among ohers. Early lieraure in his area broke fuure ime ino several chunks, wih dividend growh and discoun raes consan wihin each ime-chunk, bu differen beween chunks. For example, dividends migh be forecased o grow a a high rae for he firs hree years, and hen grow a some normal rae for he res of ime, so ha sum in Equaion 4 would be broken ino wo pars, each par solved separaely, and hen added ogeher (see, for example, Brooks and Helms (1990)). Recen work on exending he Gordon model has largely focused on allowing for more complicaed ime-series behavior of dividends and discoun raes, while reaining he abiliy o solve Equaion 4 analyically. Two paricularly good examples, found in Yao (1997), are he addiive Markov model (Equaion 1 of Yao (1997)) and he geomeric Markov model (Equaion 2 of Yao (1997)). These appear below as Equaion 8 and Equaion 9, respecively, P ADD = D 1 /r + [ 1/r + (1/r) 2] ( q u q d) (8) P GEO [ 1 + (q u q d ) % ] = D 1 r (q u q d ) % (9) in which q u is he proporion of he ime he dividend increases, q d is he proporion of he ime he dividend decreases, = T =2 D D 1 /(T 1) is he average absolue value of he level change in he dividend paymen, and % = T =2 (D D 1 )/D 1 /(T 1) is he average absolue value of he percenage rae of change in he dividend paymen. Donaldson and Kamsra (1996, 2000) develop an alernaive approach (hereafer referred o as he DK procedure) which does no require ha Equaion 4 be solved analyically. 6

8 The DK procedure insead uses Mone Carlo mehods o solve Equaion 4 numerically. The DK procedure eases he condiional consancy resricions on dividend growh raes and discoun raes of he Gordon procedure by modeling he discouned dividend growh rae y in Equaion 6 explicily as condiionally ime-varying. The basic idea of he DK procedure is o esimae an economeric model for he ime series behavior of y +i (he discouned dividend growh rae) in Equaion 6 (which is equivalen o Equaion 4), use his model and randomly-drawn innovaions o simulae ime-pahs for he possible fuure evoluion of y +i, hen ake he presen value of he forecased ime-pahs o find a fundamenal price. This is done housands (even millions) of imes, wih a differen sequence of randomly-drawn innovaions each ime, so as o inegrae ou he expecaion in Equaion 6 and hus produce a numerical (as opposed o analyical) esimae of he fundamenal price. Exhibi 1 goes here - The sequence of seps employed o produce a DK fundamenal price esimae are described below and depiced in Exhibi 1. Sep A: Use in-sample daa o specify and esimae an economeric model for condiionally ime-varying y. Sep B: Use he esimaed model from Sep A and daa up o period o simulae, ou-of-sample, possible realizaions of y +i ; i = 1,..., I, where I is chosen o be very large (infiniy, were i pracical), 4 for a cross secion of J differen possible economies, so ha we follow he evoluion of y across a panel of J simulaed economies over I periods of ime. We accomplish his by using he esimaed model from Sep A o make a condiional mean forecas of y, noed ŷ, hen simulae a populaion of J possible shocks around he mean 4 Noe ha as I increases, he produc Π I i=1 y +i converges o zero since y is less han uniy in seady sae. We found ha in pracice I = 400 was easily sufficien o have our simulaions converge o he poin where increasing I had no impac on he fundamenal price. In oher words, a dividend received 400 years in he fuure has essenially no impac on oday s price and hus 400 years in he fuure is equivalen o infiniy for pracical purposes in our simulaions. 7

9 and add hese shocks o ŷ o produce y j ; j = 1,..., J for he cross secion of j economies a ime. We hen repea his procedure hrough ime for each of he J economies, condiioning on y j, o form y j +i i = 1,..., I. We herefore produce an ou-of-sample panel of values of y for J economies sreching ou I periods, all of which have been simulaed based on he in-sample daa. Sep C: Calculae he fundamenal DK price based on his panel of simulaed y using Equaion 6 as follows: P DK = D J j=1 ( I Π i k=1y j +k i=1 ) /J. (10) Noe ha he preceding equaion can be rearranged by bringing he dividend level D inside he parenheses so ha P DK = ( ) J I D Π i k=1y j +k /J. j=1 i=1 I is ineresing o noe here ha he parenhesized erm in he preceding equaion i.e., D Ii=1 Π i k=1y j +k is for economy j he ex pos raional price of he sock from Equaion 2, wih he erminal period pushed ou o infiniy as I goes o infiniy. In oher words, if an invesor sood a he end of ime and looked back over hisory o see ha he sock had paid a sream of dividends ha had grown a rae g j +i and which were discouned a rae r j +i such ha he discouned dividend growh rae had been y j +i in his j h economy, he invesor would raionally feel ha she should have been willing o pay P X,j D Ii=1 Π i k=1y j +k o have purchased he sock back in ime, where P X,j commonly referred o by Shiller (1981) and ohers as he ex pos raional price (alhough when applied o daa i is runcaed a he end of he daa period, as shown in Equaion 2 above). 8 is

10 The represenaion of a single fan of he DK simulaion as an ex pos raional price is ineresing because i highlighs he relaionship beween he ex pos raional price, he Gordon price and he DK echnique, and herefore provides some insigh ino he likely properies of he price esimaes produced by various mehods. For example, i is clearly seen ha all Gordon-based mehods are resriced versions of he DK echnique. The basic Gordon model, for example, imposes condiionally consan dividend growh raes and discoun raes such ha each of he j economies in he simulaion are idenical and based on a consan y. If hese resricions are invalid and here is good reason o believe ha hey are indeed invalid hen he Gordon esimaor will be be biased and inefficien relaive o DK. The ex pos raional price calculaions performed by Shiller (1981) and ohers are also problemaic because hey esimae a fundamenal price based only on one realizaion of dividends and discoun raes: he realizaion acually observed in he rue marke. In oher words, Shiller s mehod sands a he end of ime and asks wha an invesor wih perfec foresigh would have paid back a dae for a share of sock, had she known he dividends and discoun raes ha were o occur in he fuure. 5 Conversely, a real-life invesor looks ino an uncerain fuure when making purchasing decisions and herefore considers a universe of possible economies ha migh unfold when valuing a share of sock his is he DK mehod. The ex pos raional price, based on only one ime-pah, will herefore provide an imprecise picure of he housands of possible economies, and heir associaed dividend ime-pahs, ha were considered when forming he marke price a which real-life raders buy and sell sock. The exen o which his imprecision migh affec our view on imporan asse-pricing quesions, such as wheher marke prices are excessively volaile or no, is sudied below. 5 Of course, if he invesor had perfec foresigh, and herefore faced no uncerainy, she would no require an equiy risk premium so he discoun rae she would use would be lower han he rae r used in he ex-pos price calculaion. This subley is ypically ignored in he lieraure ha uses ex-pos price calculaions. For consisency we will also follow he lieraure here. 9

11 I.B. Some Tess Using Fundamenal Prices One of he key feaures of analyically-based fundamenal pricing procedures is ha he fundamenal prices hey produce behave differenly han observed marke prices in imporan ways. For example, he ime pah of such fundamenal prices is ypically much smoher han ha of observed marke prices, which has led many researchers o conclude ha marke prices are excessively volaile. Indeed, here is an enire lieraure devoed o he sudy of excess volailiy in financial markes (reviewed by Camerer (1989) and Cochrane (1992)). Conversely, Donaldson and Kamsra (1996, 2000) find ha he DK procedure produces fundamenal sock prices ha behave subsanially he same as marke prices in erms of reurn volailiy. Of course, he evidence presened in all of hese papers is based on he comparison of one esimae of he fundamenal price (he esimae produced by he model in quesion) wih one realizaion of he marke price (he price series we see in he rue marke daa). Given ha he resuls come from only one price series, here is no way o be ruly sure how accurae (biased, precise, ec.) he fundamenals esimae migh be in general. The use of compuer-aided financial economeric echniques in an invesigaion of he accuracy of various fundamenals-esimaion procedures is herefore one imporan objecive of his chaper. Anoher objecive of his chaper is o invesigae he properies of various ess applied o fundamenal and acual sock price daa, in paricular ess for excess volailiy and price bubbles. Tess we invesigae in his chaper include he following sandard ess for price bubbles. Firs, Camerer (1989) observes ha marke prices would boom upward and crash downward wih bubbles expanding and collapsing, which would lead marke prices o be excessively volaile relaive o fundamenal prices if he marke price conained bubbles. This suggess a es wih a no-bubbles null hypohesis ha he variance of he percenage rae of change in marke prices and he percenage rae of change in fundamenal prices are equal, and an alernaive hypohesis ha he marke price conains bubbles and hus ha 10

12 he variance in he percenage rae of change in marke prices is greaer han he percenage rae of change in fundamenal prices. Second, a direc es of bubbles asks wheher he marke and fundamenal prices are coinegraed (e.g., Campbell and Shiller (1987)). Bubbles in he marke price will lead o a non-saionary difference beween he marke price and he fundamenal price esimae; a sandard uni roo es on his difference herefore ess he null of coinegraion. See Dickey and Fuller (1981) and Said and Dickey (1984) for a descripion of uni roo ess. Third, Mankiw, Romer and Shapiro (1985) (MRS) develop wo ess for bubbles. They exploi a decomposiion of he difference beween he ex pos raional price, P X (calculaed by insering realized dividends and discoun raes ino he presen value equaion as in Shiller (1981), as seen in Equaion 2 above), and he fundamenal price esimae P F, relaive o he observed marke price P M : ( P X ) ( P F = P X ) ( P M + P M ) P F. Since he oal volailiy of he wo sides of he preceding equaion are equal by definiion, he volailiy of each individual componen on he righ hand side should be less han or equal o he volailiy of he lef hand side. The es which compares he volailiy of he firs erm on he righ hand side wih he volailiy of he lef hand side will be labeled he MRS1 es, while he es which compares he volailiy of he second erm on he righ hand side wih he volailiy of he lef hand side will be labeled he MRS2 es. In boh cases he null hypohesis of no bubbles (i.e., no excess volailiy) is ha he erm on he lef hand side is more volaile han he righ hand side erm o which i is being compared. In considering hese ess i is worh asking wheher any of hem have proper size, or wheher hey over-rejec as some researchers argue. Noe ha hese ess involve a join null hypohesis ha he marke price does no conain a bubble and ha he fundamenal price esimae, produce by some fundamenal pricing model, shares similar properies wih 11

13 he marke s rue unobserved fundamenal price. A finding of sysemaic over-rejecion across simulaed marke economies (where he null of no bubbles is imposed) would herefore indicae ha he model used o produce fundamenal price esimaes is misspecified. In his chaper we employ Mone Carlo procedures o invesigae he performance of various fundamenals-esimaion mehodologies and of simulaed marke prices, and also of various procedures commonly used o es for excess volailiy and bubbles in sock prices. Alhough our paricular applicaion of Mone Carlo mehods is new, here is already a significan lieraure ha uses Mone Carlo mehods in asse pricing applicaions. Indeed, here is body of work ha (direcly or indirecly) simulaes sock prices and dividends, under various assumpions, o invesigae price and dividend behavior (e.g., Sco (1985), Kleidon (1986), Wes (1988a,b), Campbell (1991), Mankiw, Romer and Shapiro (1991), Hodrick (1992), Timmermann (1993,1995), and Campbell and Shiller (1998)). However, hese sudies ypically impose resricions on he dividend and discoun rae processes so as o obain fundamenal prices from some varian of he Gordon (1962) model discussed above and/or some log-linear approximaing framework. Raher han impose approximaions o solve Equaion 4 analyically, we will insead simulae he dividend growh and discoun rae processes direcly, and evaluae he expecaion hrough Mone Carlo inegraion echniques. This approach is compuaionally burdensome since i requires us o perform a Mone Carlo simulaion of a Mone Carlo simulaion, bu i is he only way o evaluae Equaion 4 wihou approximaion error. 6 We also ake care o calibrae our models o he ime series properies of he daa. Dividend growh, for insance, is srongly auocorrelaed in he S&P500 sock marke daa, in conras o he assumpion of a log random walk for dividends ofen imposed in his lieraure. 6 There is sill Mone Carlo simulaion error, bu ha is random, unlike mos ypes of approximaion error, and i can also be measured explicily. This simulaion error is analogous o simple cases such as he simulaion error associaed wih Mone Carlo experimens on he size of a es saisic. 12

14 II. Simulaing Fundamenal Sock Prices Consider again he pricing relaionship of Equaion 4 above, rewrien below for convenience, in which P is he price (for noaional convenience we drop he superscrip F ), D he dividend, g he dividend growh rae, and r he discoun rae: { ( P = D E i=1 Π i k=1 [ ])} 1 + g+k r +k 1 We now execue our sraegy of (a) calibraing models o marke dividends and ineres raes, (b) simulaing bubble-free economies using hese calibraed models, (c) calculaing fundamenal prices for hese bubble-free economies, (d) forming fundamenal price esimaes, including he Gordon models and he DK model, and (e) evaluaing hese fundamenal price esimaes and ess for bubbles. II.A. Dividends and Discoun Raes The firs sep is o esimae ime series models for dividend growh and ineres raes so ha he Mone Carlo simulaions generae dividends and discoun raes ha mach real-world dividends and discoun raes. This will allow us o generae non-bubble prices and reurns and compare hem o real world prices and reurns (e.g, a finding ha he real world prices and reurns look significanly differen han he simulaed (non-bubble) prices and reurns would provide evidence of non-fundamenal movemens in real world prices and reurns; possibly bubbles). This will also allow us o evaluae convenional fundamenal price esimaion mehods, including varians of he Gordon Growh model, and convenional ess for bubbles, such as he MRS ess, o explore he properies of he esimaors and es saisics when here are no bubbles in he price. Our dividend process is calibraed o he S&P 500 sock index annual dividend daa, 13

15 , colleced as described in Shiller (1989). The discoun rae is defined o be he risk free ineres rae plus a consan equiy premium, where he risk free rae is he ineres rae on a one-year U.S. T-bill as consruced by he U.S. Federal reserve, A consan equiy premium of 5.77% is added o he risk free ineres rae o produce a discoun rae consisen wih he sock price daa. 8 S&P 500 dividends grew a an average rae of 5.5% per year, over , wih a sandard deviaion of 3.7%. As dividend growh raes have a minimum value of -100% and no heoreical maximum, a naural choice for heir disribuion is he log normal in he S&P500 daa. The logarihm of 1 plus he dividend growh rae has mean and sandard deviaion over We esimaed simple auoregressive moving average (ARMA) ime series models for he logarihm of 1 plus he dividend growh rae and found he bes model by he Bayesian Informaion Crierion o be a moving average model of order 1 (MA(1)) wih he MA(1) coefficien equal o Sandard ess for normaliy of his error erm do no rejec he null of normaliy, 9 and sandard ess for auocorrelaion and auoregressive condiional heeroskedasiciy (ARCH) fail o rejec he null of homoskedasiciy and no serial correlaion As for ineres raes, since economic heory admis a wide range of possible ineres rae processes here are a variey of models possible, from consan o auoregressive and highly non-linear heeroskedasic forms. The auoregressive model of order 1 (AR(1)) of he logarihm of ineres raes, as described in Hull (1993) p.408, will be used here as i fis our daa well and resrics nominal raes o be posiive. Sandard specificaion ess for 7 We choose his measure because, as Cochrane (1992) noes, here is a long radiion in he volailiy es and invesmen or capial-budgeing lieraure ha measures ime-varying discoun raes from ineres raes plus risk premiums ha are consan over ime. 8 Finance heory requires ha E {(1 + R)/(1 + i + π)} = 1, where R is he log of one plus he sock reurn, i he riskfree rae and π he equiy premium (i.e., 1/[1 + i + π] is he pricing kernel ). See, for example, Campbell, Lo and MacKinlay (1997). Selecion of he equiy premium ha produces he requisie pricing kernel was accomplished wih a grid search, and his leads o a risk premium of 5.77% in he annual S&P 500 daa over These ess include he Shapiro-Wilk, Kolmogorov-Smirnov, Cramer-von Mises, and Anderson-Darling ess. See SAS Procedures Guide (1999). 10 See Engle (1982) for he seminal reamen of ARCH effecs. 11 These are based on auocorrelaions of he residuals and residuals squared. 14

16 normaliy, auocorrelaion and ARCH on he error erm from an AR(1) model of he logarihm of ineres raes do no rejec he null of no misspecificaion. The 1-year T-bill raes have mean and sandard deviaion 0.03 over The AR(1) coefficien esimae in he regression of log ineres raes on lagged log ineres raes equals Finally, he error erms from he MA(1) model of log dividend growh raes and log ineres raes are correlaed, wih a correlaion coefficien of Properies of fundamenal prices and reurns produced by Equaion 4 hinge delicaely on he modeling of he dynamics of he dividend growh and ineres rae processes. For insance, fundamenal prices will equal a consan imes he dividend level and fundamenal reurns will be very smooh over ime if dividend growh and ineres raes are equal o consans plus independen innovaions. However, modeling hese daa series o capure he serial dependence of dividend growh raes and ineres raes observed in he daa, as we have done, will ypically lead o ime-varying price-dividend raios and variable reurns of he sor we see in he S&P500 sock marke daa. II.B. The Mone Carlo Experimen We now deail he Mone Carlo experimen by which he price, P, is arrived a given condiioning informaion on he dividend level, D, dividend growh rae, g, and ineres rae, r. Tha is, we deail for he e h economy (where e = 1,..., E) he formaion of he price P e given D e, g e, and r e. P e is he marke price ha Equaion 4 saes would obain in economy e if he sock marke in economy e is raional, efficien, and bubble-free. In erms of iming and informaion, recall ha P e is he sock s beginning-of-period- marke price based on fundamenal facors, r e is he rae used o discoun paymens received during period and is known a he beginning of period, D e is paid a he beginning of period, g e (D+1 e D e )/D e and is no known a he beginning of period since i depends on D+1, e and E { } is he condiional expecaion operaor, wih he 15

17 condiioning informaion being he se of informaion available o invesors a he beginning of period. Finally, recall from Equaion 4 ha invesor raionaliy requires P e { ( [ ])} = D e 1 + g E Π i e +k 1 k=1. i=1 1 + r+k 1 e Based on he preceding equaion, we generae prices by generaing a muliude of possible sreams of dividends and discoun raes, presen-value discouning he dividends wih he ineres raes, and averaging he resuls; i.e., by conducing a Mone Carlo inegraion. Hence we produce P e, e = 1,..., E uilizing only dividend growh raes and discoun raes. The exac procedure is described below and summarized in Exhibi 2. Exhibi 2 goes here - II.B.1. Simulaing Sock Prices for a Raional, Efficien, Bubble-Free Marke Sep 1: When forming P e, he mos recen fundamenal informaion available o a marke rader would be g e 1, D e, and r e. The quaniies g e 1, D e, and r e mus herefore be generaed direcly in our Mone Carlo experimen, whereas P e mus be calculaed based on hese g, r and D since his is how prices would be deermined by a raional marke paricipan in a bubble-free economy. In he seps below he risk free T-bill rae is indicaed as r and he discoun rae (i.e., he risk free ineres rae plus he risk premium of 5.77%) as r. The objecive of Seps 1(a)-(c) is o produce dividend growh raes and ineres raes ha replicae he real world dividend growh and discoun rae daa. Tha is, he simulaed dividend growh raes and ineres raes mus have he same mean, variance, correlaion srucure and auocorrelaion srucure as he real world dividend growh raes and ineres raes. Sep 1(a): Noe ha since he logarihm of one plus he dividend growh rae is modeled 16

18 as a MA(1) process, log(1 + g e ) is a funcion of only innovaions, labeled ɛ e g. Noe also ha since he logarihm of he ineres rae is modeled as an AR(1) process, log(r e ) is a funcion of log(r e 1) and an innovaion labeled ɛ e r. Se he iniial dividend, D e 1, equal o he S&P500 s dividend value for 1951 (observed a he end of 1951), and he lagged innovaion of he logarihm of he dividend growh raes ɛ e g,0 o 0. To mach he real-world ineres rae daa, se log(r e 0) = 3.05 (he mean value of log ineres raes required o produce ineres raes maching he mean and variance of observed T-bill raes). Se he sandard deviaion of he innovaion o he log ineres rae process o 0.242, and he sandard deviaion of he innovaion o he log dividend growh rae process o Then generae wo independen sandard normal random numbers, ɛ e 1,1 and ɛ e 2,1, and form wo correlaed random variables, ɛ e r,1 = 0.242(0.21ɛ e 1,1 + ( ).5 ɛ e 2,1) and ɛ e g,1 = ɛ e 1,1. These are he simulaed innovaions o he ineres rae and dividend growh rae processes, formed o have sandard deviaions of and respecively o mach he daa, and o be correlaed wih correlaion coefficien 0.21 as we find in he S&P 500 and T-bill daa. Nex, form log(1 + g e 1) = ɛ e g,0 + ɛ e g,1 and log(r e 1) = log(r e 0) + ɛ e r,1. 12 Also form D e 2 = D e 1(1 + g e 1). Sep 1(b): Produce wo correlaed normal random variables, ɛ e r,2 and ɛ e g,2 as in Sep 1(a) above, and condiioning on ɛ e g,1 and log(r e 1) from Sep 1(a) produce log(1 + g e 2) = ɛ e g,1 + ɛ e g,2, log(r e 2) = log(r e 1) + ɛ e r,2 and D e 3 = D e 2(1 + g e 2). Sep 1(c): Repea Sep 1(b) o form log(1 + g e ), log(r e ) and D e for = 3, 4, 5,..., T and for each economy e = 1, 2, 3,..., E. Then calculae he dividend growh rae g e and he discoun rae r e, = r e Sep 2: For each ime period = 1, 2, 3,..., T and economy e = 1, 2, 3,..., E we mus 12 Noice ha he AR(1) parameer for he log ineres rae process is esimaed o be 0.83 bu we have se i o 0.94 in he simulaions. I is well known ha he coefficien esimae in an AR(1) OLS regression is biased downwards; see for insance Kennedy (1992) p.147. Mone Carlo experimens were employed o deermine he appropriae correcion for our daa, as in Orcu and Winokur (1969), and his led o he seing of The inercep erm had o be adjused as well o reflec his new seing. 17

19 calculae presen value prices, P e. In order o do his we mus solve for he expecaion of he infinie sum of discouned fuure dividends condiional on ime 1 informaion for economy e. Tha is, we mus produce a cross-secion of dividends and ineres raes ha migh be observed in periods, + 1, + 2,... given wha is known a period 1 and use hese o solve he expecaion of Equaion 4. The couner j below indexes he cross-secion of fuure economies ha could possibly evolve from he curren sae of he economy. Sep 2(a): Se ɛ j,e g, 1 = ɛ e g, 1 and log(r j,e 1) = log(r e 1) for j = 1, 2, 3,..., J. Generae wo independen sandard normal random numbers, ɛ j,e 1, and ɛ j,e 2, and form wo correlaed random variables ɛ j,e r, = 0.242(0.21ɛ j,e 1, + ( ).5 ɛ j,e 2,) and ɛ j,e g, = ɛ j,e 1, for j = 1, 2, 3,..., J. 13 These are he simulaed innovaions o he ineres rae and dividend growh rae processes, respecively. Form log(1 + g j,e ) = ɛ j,e g, 1 + ɛ j,e g, and log(r j,e ) = log(r 1) j,e + ɛ j,e r,. Sep 2(b): Produce wo correlaed normal random variables ɛ j,e r,+1 and ɛ j,e g,+1 as in Sep 2(a) above, and condiioning on ɛ j,e g, and log(r j,e ) from Sep 2(a) produce log(1 + g+1) j,e = ɛ j,e g, + ɛ j,e g,+1, and log(r+1) j,e = log(r j,e ) + ɛ j,e j = 1, 2, 3,..., J. Sep 2(c): Repea Sep 2(b) o form log(1 + g j,e +i) and log(r j,e +i) for i = 2, 3, 4,..., I, r,+1 for j = 1, 2, 3,..., J, and economies e = 1, 2, 3,..., E. Solve for he dividend growh rae g j,e +i, he dividends D j,e +i, and he discoun rae r j,e,+i = r j,e +i for i = 0, 1, 2,..., I. Sep 2(d): The presen discouned value of each of he individual J sreams of dividends is now aken in accordance wih Equaion 4. Noe ha for each of he j sreams he presen value price so calculaed for ha sream is he ex pos raional price for he j h economy; he price a raional invesor would pay for he sock if she knew for cerain ha he j h economy would obain. We can herefore call he j h presen value P X,j,e, where he superscrip X denoes ex pos raional. 13 For our random number generaion we made use of a variance reducion echnique, sraified sampling. This echnique has us drawing pseudo-random numbers ensuring ha q% of hese draws come from he q h percenile, so ha our sampling does no weigh any grouping of random draws oo heavily. 18

20 In considering hese prices, noe ha according o Equaion 4 he sream of discoun and dividend growh raes should be infiniely long, while in our simulaions we exend he sream only a finie number of periods, I. Since he raio of gross dividend growh raes o gross discoun raes i.e., he ys in Equaion 6 are less han one in seady sae, he individual produc elemens in he infinie sum in Equaion 6 (equivalenly Equaion 4) evenually converge o zero as I increases. Indeed, his convergence o zero is exacly wha is required for he absence of price bubbles! We herefore se I large enough in our simulaions so ha he runcaion does no maerially effec our resuls. As menioned previously, we found ha seing I = 400 (years) accomplishes his conservaively. Tha is, he discouned value oday of a dividend paymen received 400 years in he fuure is essenially zero. Also noe ha he seps above are required o produce P e, D e, g e 1, and r e, for e = 1,..., E; he inermediae erms superscriped wih a j are required only o perform he numerical inegraion ha yields P e. Sep 2(e): Perform Seps 1(a)-(c) and 2(a)-(d) for = 1,...T, rolling ou E independen economies for T periods. The lengh of he ime series T is chosen o be 47 o imiae he 47 years of annual daa we have available from he S&P 500 from This produces D e, r e, and P e, for = 1, 2, 3,..., T and e = 1, 2, 3,..., E. We can also consruc he marke reurns for hese economies, R e = (P+1 e + D+1 e P e )/P e and he equiy premium, π e, ha agens in he e h economy would observe. The equiy premium saisfies he pricing-kernel condiion: To avoid iniial condiions conaminaing he simulaions, he dividend growh raes and ineres raes are simulaed for over a hundred periods before dividends and prices are calculaed. 15 As wih deermining he equiy premium for he S&P 500 daa, his requires a nonlinear esimaion problem o be solved and was accomplished wih a grid search. The equiy premium is resriced o be posiive and hus is se o 0 if π e < 0. 19

21 E { } 1 + R e = r e + π e Exhibi 3 summarizes his procedure by which we calculae marke prices for he E economies in Sep 2(e). Exhibi 3 goes here - II.B.2. Calculaing Fundamenal Prices Now ha we have calculaed marke prices for each of he E economies, we nex move o calculae esimaed fundamenal prices for he same economies based on various fundamenals-esimaion procedures, such as he Gordon and DK models. Sep 3: I is useful for fuure reference o firs calculae ex pos raional prices for hese E economies; hese are he prices ha obain by subsiuing realized ineres raes and dividends for heir expeced values in Equaion 2. To do his, Define P X,e as he ex pos raional price for economy e a ime and rea P e T as he runcaion poin price; i.e., he las price observaion in he sample. 16 We will use he realized dividends D e, ineres raes r e and equiy premium π e o perform he calculaion for each economy e = 1, 2, 3,..., E. This calculaion of ex pos raional prices for each of he E economies and = 1,..., T ime periods produces E ime series of ex pos raional prices produced by our simulaions. Sep 4: We mus also form Gordon Model prices for each of hese E economies. Tha is, for each of he e = 1, 2, 3,..., E simulaed economies for which marke prices were calculaed in he previous secion, we now calculae a fundamenal price using he Gordon model. Sep 4(a): Se ḡ e = T =1 g e /T and r e = T =1 r e /T + π e, and form he Gordon price 16 In he lieraure (e.g., Shiller(1981)), when calculaing ex pos raional prices, a runcaion poin is chosen o be eiher fixed, a say he las observed marke price, or o be a moving poin 20 or more years ahead of he dae for which we are calculaing he price. Wih a fixed dae, he ex pos price converges o he marke price a he erminal dae. 20

22 . [ ] 1 + ḡ P G,e = D e e r e ḡ e Sep 4(b): In addiion o he classic Gordon model, here have been a variey of exensions o make he Gordon model more realisic by allowing dividend growh raes o vary over ime. We implemen he rinomial dividend models of Yao (1997): he addiive Markov model and he geomeric Markov model specified in Equaion 8 and Equaion 9 above. The addiive model has us esimaing for each e = 1, 2, 3,..., E economy he average absolue value of he change in he dividend paymen, e = T =2 D e D 1 /(T e 1), he proporion of he ime he dividend increases, q e,u, and he proporion of he ime he dividend decreases, q e,d, in order o form he price esimae P ADD,e = D e 1/ r e + [ 1/ r e + (1/ r e ) 2] ( q e,u q e,d) e. The geomeric model has us esimaing for each economy e = 1, 2, 3,..., E he average absolue value of he percenage change in he dividend paymen, e,% = T =2 (D e D e 1)/D e 1 /(T 1), o form he price esimae P GEO,e = D e 1 [ 1 + (q e,u q e,d ) e,% ] r e (q e,u q e,d ) e,% Sep 5: Finally, we also consider he Donaldson and Kamsra (1996) procedure, which focuses on he raio of dividend growh o discoun raes, log(y ) log((1 + g )/(1 + r + π)), raher han on dividend growh and ineres raes separaely. The DK procedure calls for evolving many possible sreams of ys ino he fuure wih a Mone Carlo simulaion and hen aking he presen value as in Equaion 10. Donaldson and Kamsra (1996) argue ha y is beer behaved han g or discoun raes 21

23 alone and ha i makes more sense o forecas y since his is he objec of invesor ineres in Equaion 6. In oher words, invesors care abou he raio of gross dividend growh o he gross discoun rae, no abou each variable individually, so i makes more sense o forecas he raio y. The DK procedure is described below and summarized in Exhibi 4. Exhibi 4 goes here - Sep 5(a): To implemen he DK procedure, we sar by esimaing a model for log(y) ha capures is dynamic evoluion over ime. For each of he e = 1, 2, 3,..., E simulaed economies for which marke prices were calculaed in he previous secion, we now use he Bayesian Informaion Crierion o selec he bes model from he se of ARMA(p,q) models, (p,q)=(1,0), (1,1) and (2,0) for log(y ). We find ha (p,q)=(1,0) is usually, bu no always, chosen. For each of he E economies we hen use he esimaed model o make condiional mean forecass log(y e ), = 1,..., T, condiional on only daa observed before period for ha economy. We also esimae he sample sandard deviaion (ˆσ e ) of he residual of he model wihin each economy. Wih E = 1000 economies we would do his sep 1,000 imes, once for each of he e = 1,..., E economies. Sep 5(b): Now simulae discouned dividend growh raes for each of he e = 1, 2, 3,..., E simulaed economies. Tha is, produce log(y e ) ha migh be observed in period in economy e given wha is known a period 1 in economy e. To do his for a given period and economy e, simulae a populaion of J independen possible shocks (draws from a normal disribuion, 17 mean 0 and sandard deviaion equal o ˆσ e ), which we label ɛ j,e, j = 1,..., J, and add hese shocks separaely o he condiional mean forecas log(y e ) from Sep 5(a) so as o produce log(y j,e ) = log(y e ) + ɛ j,e, j = 1,..., J. This is a simulaed cross-secion of J possible realizaions of log(y e ) considered a ime 1 for economy e, i.e. differen pahs ha economy e may ake nex period. Noe here ha we are performing a 17 The regression error from ime series models of log(y), formed wih he S&P 500 and T-bill daa, are normally disribued, leading us o his choice of disribuion for he shocks here. Limied experimens wih he Mone Carlo log(y) series indicaes he regression error invariably appears o be normally disribued. As y is a raio of log normal random variables, his is no surprising. 22

24 Mone Carlo simulaion on each of he e economies ha were hemselves generaed by a Mone Carlo simulaion, so ha if we generae E = 1000 economies in Seps 1-2 above, and J = 1000 economies in Sep 5, we are in oal generaing 1, 000, 000 economies for he DK simulaion (which is precisely wha we have done for he invesigaions in his chaper). Sep 5(c): Nex, for each economy e, use he esimaed model from Sep 5(a) o make he condiional mean forecas log(y j,e log(y j,e +1), condiional on only he j h realizaion for period, ) and ɛ j,e, and he daa known a period 1, and simulae a populaion of J independen shocks ɛ j,e +1, j = 1,..., J as in Sep 5(b) o form log(y j,e +1). Sep 5(d): Repea Sep 5(c) o form log(y j,e +2), log(y j,e +3),..., log(y j,e +I) for each of he J economies, where I is he number of periods ino he fuure he simulaion is run (I = 400 in our simulaions, as explained above). Then form he simulaed ex pos raional price, P s,j,e, where P s,j,e = D e ( y j,e + y j,e y j,e +1 + y j,e y+1y j,e j,e +2 + ) ; j = 1,..., J, corresponding o he J possible economies based on he previously simulaed economy e. Sep 5(e): Calculae he DK fundamenal price for each ime period = 1,..., T and each economy e = 1, 2, 3,..., E: P DK,e = J j=1 P s,j,e /J. The DK procedure oulined above is represened diagrammaically in Exhibi 4 above. For he experimens in his chaper we se E = 1000, J = 1000, and I = Less han 2% of he simulaions (economies) yielded DK models ha were no sable, or ha produced rollous ha were no sable, and were excluded from he analysis below. These economies were no remarkable oherwise, and he majoriy of he experimenal resuls we presen are qualiaively unchanged wheher hese simulaions are included or no. 23

25 II.B.3. Sensiiviy of he Mone Carlo Resuls Careful analysis of any Mone Carlo simulaion mus include a discussion of he simulaion error iself. In a world of unlimied resources, he simulaion error can be driven down o negligible scales by increasing he number of replicaions, in our case increasing he number of simulaed economies, E, from 1,000 o several million, and increasing he fans, J, used in he calculaion of each economy s marke price from 1,000 o several million. This chaper s Mone Carlo experimen involves a simulaion of a numerical inegraion (o calculae marke prices) as well as he mehod of Donaldson and Kamsra (1996) which is also a numerical inegraion (o calculae he DK prices). The scale of he simulaion quickly reaches frighening proporions as we increase he number of replicaions (economies), E, or fans, J, in he numerical inegraions. The choice of 1,000 economies, each sreching ou for 47 years, and hen he choice of 1,000 fans used o calculae he marke prices for each year of each economy, led o roughly one monh of CPU ime on an SUN UlraSparc II 400 Mhz machine, he pracical limi for his experimen given compeing demands for his resource. To deermine he simulaion error, we mus conduc a simulaion of he simulaions. Unlike some Mone Carlo experimens (such as hose esimaing he size of a es saisic under he null) he sandard error of he simulaion error for mos of our esimaes (reurns, prices, ec.) are hemselves analyically inracable, and mus be simulaed. In order o esimae he sandard error of he simulaion error in esimaing marke prices, we esimaed a single marke price 1,000 imes, each ime independen of he oher, and from his se of prices compued he mean and variance of he price esimae. If he experimen had no simulaion error, each of he housand price esimaes would be idenical. Wih he number of fans, J, equal o 1,000 we find ha he sandard deviaion of he simulaion error is only 0.28 % of he price, which is sufficienly small as o no be a source of concern for our sudy We furher invesigaed he sensiiviy of seleced resuls by increasing he number of simulaed 24

26 III. The Resuls of he Mone Carlo Experimens We now explore resuls based on our simulaed marke economies and he fundamenal prices calculaed from hem. The firs se of resuls focus on summary saisics for reurns, dividend-price (D/P) raios, dividend growh raes, ineres raes and risk premia produced by our simulaed economies as compared o daa from real-life financial markes. The second se of resuls focuses on ess of bubbles in he simulaed economies, as well as on some relaed saisics. Recall ha no experimen presened here is calibraed o acual S&P 500 prices. All experimens are calibraed only o dividend growh raes and discoun raes, and hen prices are calculaed based on he underlying fundamenal facors. Grea care was aken o ensure ha our experimens faihfully replicaed he mean, variance, cross-correlaion and auoregressive srucure of log dividend growh raes and log discoun raes. If S&P 500 prices conain bubbles, we would expec ha S&P 500 reurns and price/dividend raios would be excessively volaile relaive o our simulaed economies, which do no hemselves conain bubbles. Furhermore, since we have no bubbles in our simulaed economies, he properies of he fundamenal price esimaes and he ex pos raional price under he null of no bubbles can be obained from our simulaions. This allows us o invesigae he size properies of ess for bubbles based on hese bubble-free price esimaes. Table 1 Goes here Recall from he discussion above ha we have 47 years of daa from he S&P 500 and ha each of our 1000 simulaed economies herefore also produce 47 years of daa. For each of he 1001 economies under invesigaion (1 acual economy and 1000 simulaed economies) we calculae he mean, sandard deviaion, skewness and kurosis of a variey of ineresing economies, E, or he number of fans, J, o as large as 10,000 (resricing o wo years he number of periods each economy was followed ou), bu found no qualiaive changes o our experimenal resuls. Of course, his did reduce he simulaion error by a facor of square roo 10, bu was no of much pracical gain since he simulaion error was already exremely small. 25

27 variables, including dividend growh raes, ineres raes, sock reurns and dividend-price raios. For example, we calculae he mean reurn of he 47 years of he acual S&P 500 which appears in he op lef cell of he resuls shown in Table 1. We also calculae he mean reurn for each of our 1000 simulaed economies, which produces 1000 mean reurn esimaes, he disribuion of which is summarized in he remaining cells of he firs row of resuls shown in Table 1 (he X h percenile repors he reurn for he economy wih he X h larges reurn). We begin our analysis of he resuls in Table 1 by considering resuls for dividend growh raes and discoun raes. Given ha we calibraed he discoun raes and dividend growh raes o he rue daa, we would expec ha discoun and dividend growh raes from our simulaed economies would appear quie similar o he rue marke daa. And indeed, we see means, sandard deviaions, skewness and kurosis for our simulaed ineres raes and dividend growh raes very nearly idenical o he rue ineres rae and dividend growh rae daa. 20 Furhermore, each and every saisic for dividend growh raes and ineres raes lies near he cener of he disribuion of he simulaed series. Given ha we generaed dividend growh raes and ineres raes o mach he rue daa his is no surprising and is only presened as a verificaion check. The rue challenge, which may provide some insigh ino he exisence of bubbles in he S&P 500 daa, is o see if S&P 500 reurns and prices also look like reurns and prices from economies ha share dividend and ineres rae characerisics wih he rue economy, bu have been simulaed o have no bubbles. From he firs row of Table 1 we see ha, on average, our simulaed economies produce mean reurns very close o reurns from he acual S&P 500. Indeed, he S&P 500 s reurn lies near he middle of he disribuion of reurns from our simulaions. The skew and kurosis of S&P 500 reurns also fall well wihin he 90% confidence inerval formed by he simulaed economies. Only he sandard deviaion of marke reurns is (barely) ouside he 20 For insance, he S&P 500 dividend growh raes have a sligh posiive skew of 0.257, while he 5 and 95 perceniles of he simulaed series of dividend growh raes are -.45 and 0.66 respecively. 26