Volatility of stock prices and market efficiency

MANAGERIAL AND DECISION ECONOMICS, VOL. 7, 119- 122 (1986) zyxwv Volatility of Stock Prices and Market Efficiency RAMAN KUMAR Department of Finance, Virginia Polytechnic Institute and State University, Blacksburg, Virginia, USA and ANIL K. MAKHIJA zyxwv Graduate School of Business, University of Pittsburgh, Pittsburgh, Pennsylvania, USA Recently, historical price series along with the dividend series have been used to severely question the Efficient Markets Hypothesis. The literature suggests that the stock prices vary too much to be explained by subsequent changes in dividends. It is argued in this paper that these results require the assumption of stationarity of the price process and that this assumption is not compatible with the random walk model of Efticient Markets. A non-stationary dividend process, which is compatible with the random walk model of Efficient Markets, results in a reversal of earlier results. The new results are shown to be consistent with the empirical findings. Simulations are run to verify the results. The Efficient Markets Hypothesis (EMH) has received a great deal of attention and support in finance. Substantial evidence supporting the weak form of EMH and the semi-strong form of EMH was documented over a decade ago by Fama (1970).’ Recently, however, historical price series along with the dividend series have been used to severely question the EMH. Shiller ( 1 98 1a, 1981b), Grossman and Shiller (1981) and LeRoy and Porter (1981), among a growing literature, suggest that stock prices vary too much to be explained by prevalent valuation models in an efficient market. Shiller (1981a) argues that in an eficient market the variance of stock prices should be less than the variance of the ex-post rational stock prices, where the ex-post rational stock prices are defined as the present value of actually realized subsequent dividends. Several other variance relationships are derived by Shiller and others. Observed data violate these conditions. Consequently, they have concluded that either markets are inefficient or that the valuation models used are inadequate. The purpose of this paper is to provide a concise demonstration of the role of an implicit assumption in this new literature. Although we focus on Shiller’s (1981 a) relationship of variance of stock prices with the variance of ex-post rational stock prices, the analysis is applicable to the other works in this area. In this paper, as well as in Shiller (1981a), the analysis assumes a zero growth rate and the empirical data is detrended for comparison.’ We show that in deriving the relationship between the variance of stock prices and variance of ex-post rational stock prices, it has been implicitly assumed that stock prices follow a stationary process over In deriving the subsequent relationships Shiller (1981a) uses a stationary dividend process which implies a stationary price process. We question the compatibility of the assumption of the stationary price process with random walk model of EMH. Using a non-stationary dividend process (implying a nonstationary price process) which is compatible with the random walk model of EMH, we arrive at a widely used valuation model and obtain new relationships on expected sample variances of time series of stock prices and time series of ex-post rational stock prices. Simulations are run to support these relationships. Contrary to Shiller, we find that in an efficient market the expected sample variance of a time series of stock prices should exceed the expected sample variance of a time series of ex-post rational stock prices. Observed data agrees with this. The paper is organized as follows. First we replicate Shiller’s (1981a) argument in the next section and we highlight his assumptions by rederiving his variance relationship in a time series framework. In the following section we obtain new relationships under the framework of a different dividend process and present the results of a simulation. In the final section we present our conclusions. zyxwvutsrqp zy zyxwvuts 0143-6570/86/020119-04$05.00 0 1986 by John Wiley & Sons, Ltd. SHILLER’S (1981a) DERIVATION AND ITS IMPLICIT ASSUMPTION To highlight that implicitly Shiller assumes that stock prices follow a stationary process we replicate his 120 zyxwvuts zyxwvutsrqp zyxwvu zyxwvut zyxwvutsrqp zyxwv zyxwvutsr zyxwvuts R. KUMAR A N D A. K. MAKHIJA derivation of the condition that the variance of the stock price should be smaller than the variance of the present value of subsequent dividends. That is, Var (P,) d Var (P:) series means that (1) 1 +-E where P, is the price at the beginning of the period t and P: is the present value of subsequent actual dividends. If we define d, to be the dividend paid in t and r to be the discount rate, T ( t1I ~ ) (h-$2 where and 1 7 I T (3) Now it follows that r'f = Er(P:) (4) Equation (2) implies that the dividend valuation model of stock prices is correct and is implicitly used by investors in an efficient market to value stocks. The equation also assumes that the appropriate discount rate is known and is expected to remain constant. Since P,is the best forecast of P: at time t and the difference between them is the forecast error, i.e. u, = P: - PI (5) c o v (P,, u,) = 0 (6) it must be that In an efficient market the forecast and the forecast error should be independent otherwise the forecast can be improved. Writing variances across Eqn. (5) we get + Var (P): = Var ( P , ) Var (u,) + 2Cov ( P , ,u , ) (7) Since the variance of the error term will always be positive and Cov (P,,u,)is zero from Eqn. (6),Shiller concludes that Eqn. (1) must hold. Shiller tests Eqn. (1) against observed data (reproduced in part in Table 1 here). He finds that stock prices are too volatile when compared to the ex-post rational prices. He concludes that movements in stock prices are excessive for an efficient market or that valuation models in Eqns (2) and (3) are inadequate. However, some assumptions have crept in when we applied Eqn. (7) to the sample of observed time series. Returning to Eqn. (7) and applying it to observed ~ U(P') u, Tr=1 If the variance of the error term is positive and T E 1 (P,- P)(&- ii) (r=1 ) =0 (9) then Eqn. (8) reduces to Shiller's relationship (1) above. Note that Eqn. (9) is necessary to conclude that the variance of stock prices should be less than the variance of the ex-post rational stock prices. For Eqn. (9) to be valid requires that not only the error be independent of the forecast but also that prices at each time are being drawn from the same distribution with a fixed mean and that B is an unbiased estimate of that mean. Shiller must implicitly assume this to obtain Eqn. (9) and consequently conclude to Eqn. (1). However, in an efficient market setting with a zero growth rate the best forecast for future price is the current price, but with time, expected prices will fluctuate depending, at each time, on the outcome in the previous period. We can illustrate the inconsistency of Shiller's stationarity assumption with the EMH using the random walk model, which states that the current stock price is an unbiased estimate of its future price, when detrended for growth, i.e., zy E , ( P , + i )= P, for i = 1,2,. . ., cc and (10) EI+l(P,+i)=P,+lfor i = 2 , 3 , ..., 03 At time t, P, + is a random variable with an expected value of P, and at time t 1, P, + is a random variable with an expected value of P,,,.The expectation of future prices keeps changing and hence (in a zero growth, constant discount-rate model of stock prices) the random walk model implies that7 + ~~ Table 1. Sample Statistics for Price and Dividend Series Sample period 5(P) u=- Data Set 1 Standard and Poor s Data Set 2 Modified Dow Industrial 1871-1 979 50 12 8 968 1928-79 355.9 26 80 Data definitions are in the text. This table is reproduced from information in Table 2 of Shiller ( 1 981 a). P , = P , - l +AP, (1 1) and E,-,(AP,) = 0 (12) When stock prices are drawn from distributions with zyxw zyxwvuts zy zyxwv zyxwv 121 VOLATILITY OF STOCK PRICES A N D MARKET EFFICIENCY different expected values at different points in time, Eqn. (9) need not be true. ALTERNATIVE DERIVATION ASSUMING CHANGES IN DIVIDENDS FROM A STATIONARY DISTRIBUTION and 1T E [ (P, - P)(u,- q] r=i r=1 In this section we adopt an alternative assumption compatible with the EMH. We assume that dividend changes are drawn from a stationary distribution. This leads us to show that the left-hand side of Eqn. (9) has a negative expected value and that the expected sample variance of the ex-post rational stock price is smaller than the expected sample variance of stock prices. This reverses Shiller’s inequality in Eqn. (1). Like Shiller (1981a),we assume a zero growth and a constant discount rate.* All future dividends are expected to equal current dividend, so that zyxwvu (17) (1 + r ) , Note that the expected sample variance and covariance terms depend on time T. The right-hand side of Eqn. (17) for all feasible values of r and T is n e g a t i ~ eThis . ~ violates Eqn. (9) which was essential to arrive at Shiller’s results. Evaluating the right-hand sides of Eqns (15) and (16) for all feasible values of r and T, we get zyxwvu zyxwvutsrq zyxwvu E ( d , + J 4 , )= d, i = 1,. . ., 00 (13) In a zero growth model Eqn. (13) is a standard assumption which leads to the valuation price (P:-P*)2 1 (18) Expression (18) states that the expected sample variance of time series of stock prices will be greater than the expected sample variance of ex-post rational stock prices. This is consistent with the observed data. For estimates of the left-hand sides of Eqns ( I 5)-( 17) we take r to be 5% and we simulate for dividends for 300 periods, assuming that - It can be easily shown that Eqns(13) and (14) are compatible with the random walk model of EMH, with zero growth. Our use of the dividend valuation model in Eqns (1 3) and (14) does not suggest that it is necessarily the correct specification of the valuation process. It remains, nevertheless, the most commonly used dividend valuation model under the assumption of zero growth. The only difference between this model and Shiller’s is that in the former, the expected value of dividend changes is zero, a reasonable assumption in a zerogrowth world. All other assumptions of this model, including a world of zero growth, are identical to those of Shiller. Next we assume that changes in dividends, Ad,( = d, - d,- 1), are drawn from a distribution with an expected value of zero and a standard deviation, a. It can then be shown that terms analogous to those in Eqn. (8) are k [,Il C -E (P,-P)’ ] T + l)(T- I =1 -E T “1 and stock pricelo r In each simulation run, Ad,’s and d,’s were generated for t = 1,. . .,300. These were used to calculate the P I , P: and u, for t = 1,. .., 100. For each simulation 2 1 1) Table 2. Simulations for Sample Variances Expression - (1 + r)(T - 1) + r2(2 + r) + -1 r3(2+ r)(l + (1 Average of sirnulattonsb 2564 1 6 Expected valuea J (P: - P*)2 r=i zyx p, = d2 zyxwvutsrq T(T a2 =-r 2 T 2 [ Ad N(O,1) (19) The choice of a standard normal distribution for Ad is motivated by the requirement that E(Ad) = 0, and that it results in simpler computations. Any other stationary distribution or any other value for the variance would lead to similar results as long as the mean of the distribution is zero. Now we can calculate the ex-posf rational price 3906 3665 6666 6290 ’ Expected values are as per R.H.S. of Eqns (1 7), (1 6 ) and (1 5), respectively. bAveragesof 1000 simulations. * r is assumed to be 5%. T is taken to be 100. See text for a detailed description of the simulation. a 122 zyxwvutsrq zyxwvutsrq zyxw zyxwvu R. KUMAR A N D A. K. MAKHIJA the left-hand sides of Eqns (1 5)417) were calculated. Averages of lo00 simulations are reported in column (2) ofTable 2. Note that the left-hand side of Eqn. (1 7) is negative. Hence Eqn. (9) is violated. Both the simulation results in Table 2 and the observed data in Table 1 support the reverse of Shiller’s inequality, i.e. they support Eqn. (1 8). CONCLUSIONS We have argued in this paper that Shiller’sresults when applied to a time series of price data require the assumption that stock prices follow a stationary process. We show that this assumption is not compatible with the random walk of EMH. Hence violation of Shiller’s results by observed data does not lead to a rejection of EMH. Using a non-stationary dividend process compatible with the random walk model of EMH leads to a reversal of Shiller’s results. It can be easily demonstrated that the other variance relationships derived by Shiller and others are specifically a result of their assumption about the nature of dividend process and do not hold up to the alternate assumptions of this paper. It is not our assertion that our set of assumptions is superior but that the assumptions regarding stationarity of the price and dividend process are crucial to the results derived by Shiller and that these assumptions may be incompatible with the random walk model of EMH. Acknowledgements We are grateful to the attendees of seminars at the University of Pittsburgh. Virginia Polytechnic Institute and State University. University of North Carolina and Rutgers University for comments. NOTES zyxwvutsr zyxwv Samuelson (1965). Mandelbrot (1 966). Alexander (1961 ) , Fama (1965) and Fama and Blume (1 966) are examples in support of a weak form of EMH. Fama etal. (1969). Ball and Brown (1968). Beaver (1968) and Scholes (1972) are examples in support of a semi-strong form of EMH. It can be argued that a non-zero growth rate (trend) would not alter the basic results of this paper or of Shiller (1981 a) While working on this paper, we came across Copeland (1983). which raised the same issue but does not demonstrate the necessity of the stationarity assumption and its possible incompatibility with EMH. While working on the final draft of this paper we came across Marsh and Merton (1983), which raises this issue. Our demonstration of effect of the stationarity assumption of Shiller is pointed and concise. Unlike Merton and Marsh, we have concentrated on a specific widely used dividend process which has enabled us to provide a simple and direct refutation of the Shiller inequalities. 5. The dividend is announced at the beginning of the period and paid at the end of the period. This assumption simplifies subsequent derivations. It is, however, not necessary to derive the results. 6. Assuming a constant r makes derivations simple. The effects of this assumption are discussed in notes 7 and 8. 7. Adding growth to the model and taking into account a stochastic discount rate will not alter the implication of the ’random walk’ property of stock prices. 8. It can be argued that a non-zero growth rate will not detract from the results of this paper and consideration of stochastic discount rates will strengthen the results. 9. Wetested theright-hand sideof Eqn. (17) forrrangingfrom 0.5% to 30% and for T ranging from t w o periods to 1000 periods. The result is consistently negative. 10. We checked that the present value of dividends after 200 periods did not affect our results. REFERENCES S. S . Alexander (1961 ). 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