Variance Bounds Tests for the Hypothesis of Efficient Stock Market

Transcription

1 67 Variance Bounds Tess of Efficien Sock Marke Hypohesis Vol III(1) Variance Bounds Tess for he Hypohesis of Efficien Sock Marke Marco Maisenbacher * Inroducion The heory of efficien financial markes was regarded as inviolable in academic lieraure for a long ime. An efficien financial marke is characerized by he complee reflecion of all relevan informaion in he marke prices, which means ha permanen deviaions from he fundamenal jusified valuaion are impossible (Fama, 1970). Miller and Modigliani (1961) se up a firs model o capure he idea of efficien sock markes. In his model he value of a sock should equal he raional expeced, discouned value of all fuure dividends of he sock. In he mos recen financial crisis he validiy of efficien financial markes was brough ino quesion. Long ime before he financial crisis Shiller (1981) recognized he phenomenon ha especially sock price indices like he Sandard and Poor s 500 Composie Sock Price Index are much oo volaile o be explained by he radiional fundamenal value model of Miller and Modigliani. Based on his observaion Shiller develops a variance bound for efficien sock markes. He concludes he violaion of he fundamenal value model since he variance of * Marco Maisenbacher received his degree in Economics (B.Sc.) from he Universiy of Bonn in March The presen aricle refers o his bachelor hesis under he supervision of Prof. Dr. Jörg Breiung, which was submied in January 2013.

2 68 Variance Bounds Tess of Efficien Sock Marke Hypohesis Vol III(1) he S&P 500 sock price index is five imes higher as suggesed by his variance bound. Shiller (1981) iniiaes he discussion abou excess volailiy on financial markes. In he following years especially Flavin (1983), Marsh and Meron (1986) and Kleidon (1986) criicize Shiller s conclusion from his variance bound es due o undesired small finie sample properies in his es. In response o Shiller (1981) and he rising criic agains his variance bound es, Mankiw, Romer and Shapiro (1985) develop a modified variance bound relaion which holds in finie samples. Afer some furher refinemens of heir es, Mankiw e al. (1991) obained more differeniaed resuls wih respec o he validiy of he efficien marke hypohesis. According o heir resuls, here is a overall endency o rejec he hypohesis of efficien sock markes bu his finding is less pronounced compared o he earlier sudy of Shiller (1991). In his paper he framework of Mankiw e al. (1991) is applied o an updaed daa se in order o es he hypohesis of efficien sock markes. The Original Variance Bound for Efficien Sock Markes The idea of a variance bound in efficien financial sock markes bases upon he presen value model by Miller and Modigliani (1961). This model can be characerized by he following wo equaions: P = i=0 ( ) i+1 1 D +i (1) 1 + r P = E P, (2) where D denoes he dividend of a sock for period and r is he expeced reurn assumed o be consan. E ( ) denoes he expecaion condiional on informaion available a ime, especially he presen and all pas prices of he sock.

3 69 Variance Bounds Tess of Efficien Sock Marke Hypohesis Vol III(1) Therefore P is he unobservable ex pos raional price of he sock and P is he raional forecas of P a ime. Equaions (1) and (2) describe he fundamenal value model, where he price of a sock equals he expeced, discouned value of all fuure dividends. According o his model flucuaions of he marke price of a sock should be fully explained by he emergence of new informaion abou fuure dividends. Shiller (1981) challenged he validiy of he fundamenal value model. The (ex pos observable) series P appeared o be much smooher han he series of he acual marke prices P. Shiller (1991) quesioned ha he dispariy of he volailiies can be adequaely explained by he mere emergence of new informaion. In order o analyze his issue empirically, Shiller derives he firs variance bound for efficien sock markes. Equaion (1) can be reformulaed as P = P + ɛ, (3) where ɛ denoes he raional forecas error a ime wih zero mean. Using he fac ha under raional expecaions ɛ has o be uncorrelaed wih all known informaion a ime, he variance of P simplifies o V ar(p ) = V ar(p ) + V ar(ɛ ) (4) and since V ar(ɛ ) 0, i follows ha V ar(p ) V ar(p ). (5) Equaion (5) saes he simples form of a variance bound in efficien sock markes and indicaes ha he variance of he ex pos raional prices has o be a leas as large as he variance of he marke prices. Shiller (1991) compares he sample

4 70 Variance Bounds Tess of Efficien Sock Marke Hypohesis Vol III(1) variance of P and P compued from he Sandard and Poor s 500 Composie Sock Price Index for and finds he variance of he marke prices P o be five imes higher han he one obained from heir raional ex pos counerpar P. To ensure he variances of boh series o be finie, Shiller assumes boh series o be rend saionary. Obviously, his resul quesions he validiy of he fundamenal value model for sock markes. In response o Shiller s work many auhors criicize he assumpions and inerpreaion of his es. Flavin (1983) noes ha boh sample variances of P and P underesimae he rue populaion variances. However he negaive bias for he variance of P is larger, which means ha he undersimaion for V ar(p ) is sronger han for V ar(p ). This negaive bias resuls from replacing he unknown expecaion by sample means. Marsh and Meron (1986) challenge he enire inerpreaion of Shiller s resuls. According o hem he violaion of Shiller s variance bound in equaion (5) does no necessarily imply a violaion of he presen value model. In conras Shiller s mehod is a es of he join hypohesis of efficien financial sock markes wih consan reurn and a rend saionary dividend process. Following his argumen a violaion of he variance bound in (5) migh occur due o a violaion of he assumed consan reurn or he rend saionary dividend process even hough he assumpion of an efficien sock marke is fulfilled. Kleidon (1986) reveals a second weakness in Shiller s inerpreaion. He shows ha from a single ime series of he realized observaions of P and P nohing can be concluded in erms of he validiy of Shiller s variance bound. Kleidon explains his seemingly counerinuiive argumen as follows. The variance bound (5) is based on repeaed samples of he process {P1,..., PT } because differen realizaions of fuure dividends resul in differen sequences of {P1,..., PT }. Shiller s variance bound implies ha among all possible realizaions he variance of P is

5 71 Variance Bounds Tess of Efficien Sock Marke Hypohesis Vol III(1) expeced o be larger han he variance of P. Accordingly he variance bound represens a cross-secional relaion beween differen saes of an economy a ime and no relaion beween he ime series variances. The Modified Variance Bound Tes In response o Shiller s firs rial, Mankiw e al. (1991) develop a modified es ha is more accurae in finie samples. In order o derive heir es saisic, some new definiions need o be inroduced. Mankiw e al. (1991) define he ex pos presen value P h i for h periods as for he sraegy of buying a sock a ime and holding P h = h 1 j=0 ( ) j+1 ( ) h 1 1 D +j + P +h (6) 1 + r 1 + r where D +j denoes he dividend in period + j and P +h is he marke price in period + h. Under he assumpions of he presen value model i holds ha P = E P h. (7) The invesmen horizon h can be chosen as variable or consan. In he variable case, he invesmen horizon coincides wih he end of he sample such ha h = T. Alernaively h can be chosen as consan for every observaion such ha P h displays he ex pos presen value for he sraegy of holding he sock unil period + h and selling i for he marke price. This new definiion of P h is a basic componen for he modified es. The derivaion of his es is based on he ideas of Mankiw e al. (1985). Le P 0 be an arbirary ( naive ) forecas

6 72 Variance Bounds Tess of Efficien Sock Marke Hypohesis Vol III(1) of he fundamenal value of he sock: P 0 = ρ k+1 D+k (8) k=0 where D +k denoes naive forecas for he dividend D +k a period and ρ is he known (consan) discoun facor. The naive forecas does no have o be raional (i.e. he forecas may neglec available informaion). I is imporan however ha he marke paricipans may have access o he naive forecas a period, ha is, he naive forecas is enailed in he invesor s informaion se. In order o derive Mankiw e al. s (1985) modified es he following ideniy serves as saring poin: P h P 0 = (P h P ) + (P P 0 ). (9) From equaion (7) i follows ha P h P displays he raional forecas error ɛ which is independen of any available informaion a period. Therefore i holds ha: E [(P h P )(P P 0 )] = 0. (10) Squaring equaion (9), using expecaions and subsiuing equaion (10) yields: E (P h P 0 ) 2 = E (P h P ) 2 + E (P P 0 ) 2. (11) Equaion (11) will remain valid if he condiional expecaions are normalized wih any scaling variable W known a. Equaion (11) can be reformulaed as ( P h E P 0 W ) 2 = E ( P h P W ) 2 + E ( P P 0 W ) 2. (12)

7 73 Variance Bounds Tess of Efficien Sock Marke Hypohesis Vol III(1) In a las sep define q = ( P h P 0 W ) 2 [ (P h P W ) 2 ( P P 0 ) 2 ] +. (13) W Equaion (12) implies ha E (q ) = 0 holds. However, by he law of ieraive expecaion, his implies E s (q ) = 0 for all s 0. Therefore a es of he hypohesis of efficien sock markes implies ha H 0 : α = 0, (14) in he regression q = α + ε, where ε is an error erm wih expecaion zero. Since he expecaion of q is zero for all, he expecaion of he mean q is zero as well. In order o consruc a es saisic for he null hypohesis (14), asympoically valid sandard errors have o be consruced. Mankiw e al. (1991) sress he issue of auocorrelaion in he errors. For consan holding periods h and under he assumpions of efficien markes q and q j are correlaed for j < h bu uncorrelaed for j h. In he case of variable holding periods h he correlaion does no vanish afer a fixed lag. To accoun for he auocorrelaion in he error erms, Mankiw e al. (1991) use Newey-Wes sandard errors which are asympoically valid. In he case of a consan holding periods h, he runcaion lag is se o h 1 since he auocorrelaion vanishes afer his lag. In he case of variable holding periods he rule of humb of Newey-Wes is chosen for he runcaion lag. 1 The final es saisic is he square of he -saisic α 2 / V ar( α), which is a wo-sided Wald-saisic wih a χ 2 disribuion wih one degree of freedom. Noe ha he esimaed variance of α is calculaed wih he Newey- Wes sandard errors. 1 Newey-Wes s rule of humb for he runcaion lag for unknown auocorrelaion is P = in[4(t/100) 9 2 ].

8 74 Variance Bounds Tess of Efficien Sock Marke Hypohesis Vol III(1) Finally here are wo more useful implicaions of Mankiw e al. s (1985) es which can be derived from equaion (12): ( P h P 0 ) 2 ( P h E E W ) 2 P (15) W and ( P h P 0 ) 2 ( P P 0 ) 2 E E. (16) W W The firs upper bound in (15) claims ha he expeced squared error wih a naive forecas (P 0 ) should be a leas as large as he expeced squared error wih he opimal forecas (P ). The upper bound in (16) saes ha he volailiy of P around P 0 should be a leas as large as he volailiy of P around P 0. If he null hypohesis is rejeced boh upper bound relaions can be helpful o deec he source of rejecion. In conras o Shiller s es, Mankiw e al. (1985) show ha heir upper bound relaions in (15) and (16) are unbiased regardless of he sample size and he underlying dividend process. This is achieved by cenering he variances around a naive forecas and no around he sample mean. Empirical Analysis In his secion he modified es of Mankiw e al. (1991) is applied o real daa in order o es he hypohesis of efficien sock markes. All ime series are annual daa from 1871 o The sock price series consiss of daa of he Sandard and Poor s 500 Composie Price Index, where he price of a year is represened by he average of he daily closing prices for January. The dividend series consiss of dividends per sock, added over 12 monhs and adjused o he index for he fourh quarer of each year. Boh series are convered o real unis wih he

9 75 Variance Bounds Tess of Efficien Sock Marke Hypohesis Vol III(1) Consumer Price Index. In order o conduc he es of efficien sock markes he naive forecas P 0 has o be specified. In a firs version of he es Mankiw e al. (1991) specify P 0 derived from he no-change forecas of fuure dividends so ha he expeced dividends are idenical o he las observed value (D 1 ). Therefore P 0 can be expressed as: P 0 = 1 r D 1. (17) Table 1 presens he resuls of he es using he naive forecas in (17) and consan reurns of 5%, 6% and 7% for differen holding periods. The Columns (ii), (iii) and (iv) display he sample mean of [(P h P 0 )/P ], [(P h P )/P ] 2 and [P P 0 )/P ] 2. The scaling variable W is P in order o diminish he issue of heeroskedasiciy since he variables are growing over ime. Column (v) shows he resul of he es saisic, which is he squared difference of column (ii) wih he sum of column (iii) and (iv), divided by he variance of his difference. The variance is calculaed wih he Newey-Wes sandard errors. Under he null hypohesis (14) he es saisic is asympoically χ 2 disribued wih one degree of freedom, implying a criical value of 3.84 for a significance level of Column (vi) shows he respecive p-values. The heory of efficien sock markes predics ha he enries in column (ii) should equal he sum of he enries in column (iii) and (iv) which is esed by he saisic in column (v). Furhermore he enries in column (ii) should be larger han he enries in column (iii) and (iv), as presened in he upper bound relaions in (15) and (16). In erms of he validiy of inequaliy (15) Table 1 shows ha he relaion holds excep for he case of r = 5% wih variable holding periods h = T. Only in his case he naive forecas in (17) is a beer forecas han he marke price in erms of he forecas error variance. The inequaliy (16) is sable as well since column

10 76 Variance Bounds Tess of Efficien Sock Marke Hypohesis Vol III(1) (ii) almos always exceeds column (iv). The only excepion is he case of r = 7% and variable holding periods. Accordingly he volailiy of P h around he naive forecas in (17) is almos always larger han he volailiy of P around he naive forecas. Therefore, he marke prices are no excessively volaile around he naive forecas. The p-values of he es saisic for he hypohesis ha column (ii) equals he sum of column (iii) and (iv) show a endency for acceping he null hypohesis of efficien sock markes. Only he p-values of 5 and 10 year holding periods wih an expeced reurn of 5% yields a rejecion of he null hypohesis a he 10% significance level. However he picure for he variable holding periods is differen. The p-values imply a significan rejecion of he null hypohesis for every expeced reurn. The presen value model wih consan expeced reurn is no suppored for variable holding periods and he naive forecas defined in (17). Table 2 shows he resuls of a similar es as before bu wih a differen naive forecas. The alernaive naive forecas consiss of a hiry year moving average of he dividends and can be wrien as: P 0 [ = 1 1 r i=1 D i ]. (18) Mankiw e al. (1991) choose his paricular forecas o smooh he series P 0. The smoohed naive forecas should help o deec he excess volailiy of he marke prices. Furhermore, he scaling variable W is se o P 0, which is supposed o avoid a bias in he es resuls due o possible excess volailiy in he series P used above. The es resuls in Table 2 indeed display excess volailiy of he marke prices around he naive forecas defined in (18). The values in column (iv) exceed he values in column (ii) for every holding period and every expeced reurn. However he second upper bound relaion always holds since column (ii) always exceeds column (iii). This implies ha he marke price P is a beer

11 77 Variance Bounds Tess of Efficien Sock Marke Hypohesis Vol III(1) forecas for he ex pos raional price P h han he naive forecas in (18). The p- values imply accepance of he null hypohesis for consan holding periods below 10 years for every expeced reurn. However for he 10 year holding periods he null hypohesis has o be rejeced. In he case of variable holding periods he rejecions are much weaker compared o he ess presened in Table 1. In he case wih r = 5% he null can be acceped a he 5% level. In general he es wih he modified naive forecas displays a sronger endency o accep he hypohesis of efficien sock markes excep for he 10 year holding period. Nex we relax he assumpion of consan expeced reurns. We follow Mankiw e al. (1991) and consruc ime varying expeced reurn as he sum of a variable, riskless ineres rae (r ) and a consan risk premium (φ). Therefore he one period nominal discoun facor is given by ρ = 1/(1 + r + φ). Under he hypohesis of efficien sock markes i holds ha: P = h 1 E j=0 ρ j+1 D +j + ρ h P +h E P h. (19) For he naive forecas Mankiw e al. (1991) assume ha he dividends grow wih he riskless ineres rae. Therefore he naive forecas can be expressed as P 0 = 1 φ D 1. According o Mankiw e al. (1991) a es wih variable discoun facors is especially suiable if changes in he ineres raes are considered as imporan drivers for marke price volailiy. In his case one would expec a rejecion of he null hypohesis of efficien sock markes in ess wih consan expeced reurn. Table 3 on page 11 shows he es resuls for differen risk premiums (4%, 5%, 6%) and he same holding periods as in he ess before. The daa for he riskless ineres rae (r ) are annual commercial paper raes of riple A ranked companies

12 78 Variance Bounds Tess of Efficien Sock Marke Hypohesis Vol III(1) which are aken from he Federal Reserve Bank of S.Louis. Furhermore all daa are in nominal erms now since he dynamic of he inflaion is capured by he riskless ineres rae. The resuls of he new ess in Table 3 do no suppor he validiy of he presen value model wih variable discoun raes. Even hough he null hypohesis canno be rejeced for low holding periods of one and wo years, here is a srong endency owards rejecion for longer holding periods. Especially he variable holding periods lead o srong rejecions. The upper bound for he volailiy of he marke prices is, like in he case of consan reurns, almos always me. The upper bound of he forecas error variance holds for consan holding periods only. In general he es wih variable discoun raes exhibis a sronger endency for rejecing he null hypohesis compared o he es wih consan reurns. Table 4 shows he es resuls for variable discoun raes based on he smoohed [ naive forecas P 0 ] 30 i=1 D i. The scaling facor W is again P 0. The = 1 φ 1 30 es resuls end o be similar o hose of he analog es wih consan reurns. The upper bound of he volailiy is almos always violaed, whereas he upper bound for he forecas error variance is always me. The null hypohesis is acceped more ofen wih he smooher naive forecas compared o he es before. The resuls of he ess wih variable discoun rae show ha variaion in he ineres raes does no seem o play an imporan role for he volailiy of he sock marke prices. The ess do no yield more evidence in favor of he null hypohesis bu end o sronger rejecions. Conclusion In his paper wo differen variance bounds ess were considered. Shiller s (1981) approach leads o a srong rejecion of he hypohesis of efficien sock markes. However, due o he undesired small sample properies and he sensiiviy in

13 79 Variance Bounds Tess of Efficien Sock Marke Hypohesis Vol III(1) erms of he underlying dividend process, his resuls are no reliable. The resuls of he modified variance bound es of Mankiw e al. (1991) yields a more differeniaed picure. Using his es, he hypohesis of efficien sock markes canno be generally acceped, however i canno be definiely rejeced neiher. For low holding periods here is a endency for acceping he null hypohesis. In conras, especially variable holding periods lead o a clear rejecion. The degree of significance of he rejecions depends on he choice of he naive forecas. The smoohed naive forecas yields weaker rejecions of he null hypohesis. Furhermore i was shown ha variaion in he ineres raes is no a significan driver of price volailiy on sock markes since he es wih variable discoun raes leads even o sronger rejecions. To sum up, he modified es of Mankiw e al (1991) wih an updaed sample neiher delivers clear evidence agains he fundamenal value model nor does i generally suppor he hypohesis. References (2012): 6-Monh Commercial Paper Rae, hp://research.slouisfed. org/fred2/series/wcp6m?cid=120, Federal Reserve Bank of S. Louis, Economic Research. Fama, E. F. (1970): Efficien Capial Markes: A Review of Theory and Empirical Work, The Journal of Finance, 25(2), Flavin, M. A. (1983): Excess Volailiy in he Financial Markes: A Reassessmen of he Empirical Evidence, Journal of Poliical Economy, 91(6), Friedman, M., and A. J. Schwarz (1982): Moneary Trends in he Unied Saes and Unied Kingdom: The Relaion o Income, Prices, and Ineres Raes, , Universiy of Chicago Press, p. Chaper 4: The Basic Daa. Kleidon, A. W. (1986): Variance Bounds Tess and Sock Price Valuaion Models, Journal of Poliical Economy, 94(5),

14 80 Variance Bounds Tess of Efficien Sock Marke Hypohesis Vol III(1) Mankiw, G. N., D. Romer, and M. D. Shapiro (1985): An Unbiased Reexaminaion of Sock Marke Volailiy, The Journal of Finance, 40(3), (1991): Sock Marke Forecasabiliy and Volailiy: A Saisical Appraisal, Review of Economic Sudies, 58(3), Marsh, T. A., and R. C. Meron (1986): Dividend Variabiliy and Variance Bounds Tess for he Raionaliy of Sock Marke Prices, The American Economic Review, 76(3), Miller, M. H., and F. Modigliani (1961): Dividend Policy, Growh, and he Valua- ion of Shares, The Journal of Business, 34(4), Shiller, R. J. (1981): Do Sock Prices Move Too Much o Be Jusified by Subsequen Changes in Dividends?, The American Economic Review, 71(3), (2012): ONLINE DATA ROBERT SHILLER, ~shiller/daa.hm.

15 81 Variance Bounds Tess of Efficien Sock Marke Hypohesis Vol III(1) Tables Appendix ] 2 (i) (ii) (iii) iv) (v) (vi) h [ ] P h P E 0 2 [ P h P = E P P + E [ P P 0 P ] 2 χ 2 p-value r = 5% T r = 6% T r = 7% T Noe: column (i): holding periods; column (ii)-(iv): sample esimaor (sample means) of he expecaion from (12, weighed wih he marke price; column (v): χ 2 (1) es saisic for he hypohesis, ha column (ii) equals he sum of (iii) and (iv) ; column (vi): p-values of he es saisic. The naive forecas is defined in (17). Table 1: Tes wih a naive forecas, consan expeced reurn (r)

16 82 Variance Bounds Tess of Efficien Sock Marke Hypohesis Vol III(1) (i) (ii) (iii) (iv) (v) (vi) [ ] P h P h E 0 2 [ ] P h 2 [ ] P P 0 = E P P 0 2 P 0 + E P 0 χ 2 p-value r = 5% T r = 6% T r = 7% T Noe: See Table 1. The naive forecas is defined in (18). Table 2: Tes wih naive forecass based on smoohed dividends, consan expeced reurn ] 2 (i) (ii) (iii) (iv) (v) (vi) h [ ] P h P E 0 2 [ P h P = E P P + E [ P P 0 P ] 2 χ 2 p-value φ = 4% T φ = 5% T φ = 6% T Noe: See Table 1. The naive forecas is P 0 = 1 φ D 1. The daa are in nominal erms. Table 3: Tes using naive forecas wih recen dividends and variable ineres raes.

17 83 Variance Bounds Tess of Efficien Sock Marke Hypohesis Vol III(1) (i) (ii) (iii) (iv) (v) (vi) [ ] P h P h E 0 2 [ ] P h 2 [ ] P P 0 = E P P 0 2 P 0 + E P 0 χ 2 p-value φ = 4% T φ = 5% T φ = 6% T Noe: See Table 1. The naive forecas is P 0 erms. = 1 φ [ 1 30 ] D 30 i i. The daa are in nominal Table 4: Tes wih smoohed naive forecas and variable ineres raes