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Financial Market Volatility

Author(s):
International Monetary Fund. Research Dept.
Published Date:
January 1991
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Volatility in financial markets, particularly stock markets, is an issue that concerns government policymakers, market analysts, corporate managers, and economists. During the 1980s new financial markets developed around the world, and exchanges introduced futures and option contracts on interest rates, stock indices, and foreign exchange rates. These markets experienced impressive growth until the crash of stock markets around the world in October of 1987. Serious questions were raised following the crash concerning volatility in financial markets and the role of the new financial futures and options. As a result, a number of reforms for financial markets were proposed, some of which have been instituted.

During the period when financial markets were experiencing dramatic growth, and asset price volatility was becoming more noticeable, economists were re-examining the efficient markets hypothesis–the theory that states that financial markets always price securities correctly. The evidence that has accumulated over the last ten years suggests that there may be excess volatility in stock markets and that stock prices regularly deviate from their fundamental values. The empirical results have stimulated several alternative views to explain the observed volatility; these include the speculative bubbles model, fads, and noise trading. The stock market crash of 1987 forced economists to reassess the validity of the efficient markets hypothesis, and it is fair to say that it remains an open issue. The purpose of this paper is to survey the economic theory and the empirical evidence on pricing and volatility in financial markets, to synthesize the literature, and to address the necessity for some of the market reforms that have been recently proposed. Section I is a review of the relevant economic theory of how securities are priced in financial markets; it includes the efficient markets hypothesis as well as recent models that incorporate either speculative bubbles or noise trading. Section II is a survey of the empirical evidence on excess volatility and noise trading. The principal issue is whether asset prices deviate significantly from fundamental value. Section III offers some concluding remarks.

The review of the empirical evidence on financial market volatility reveals evidence that stock prices deviate from fundamental value, but there is no evidence that prices deviate from fundamental value in another important market, the bond market. At a superficial level, the evidence of excess volatility and deviations of prices from fundamental value in the stock market would imply the need to impose additional restrictions and regulations on stock markets and related markets, such as stock index futures markets. The goal for any new market regulation should be the elimination of discrepancies between market prices and fundamental values. Many of the proposed market reforms have not addressed this issue.

I. A Review of the Economic Theory

A careful analysis of financial market volatility requires a theory of how capital assets are valued in the marketplace. The theory must explain how prices are determined, and in a dynamic world, it must explain how and why prices change. Without a theory or an explanation for asset price changes, one can do little more than catalogue the sequence of events. A theory of price changes is essential for determining appropriate forms of market regulations. There are several models available for pricing capital assets and several competing explanations or theories for the price changes observed in financial markets. Asset-pricing models are used to determine the market fundamental or fundamental value–that is, the asset value based on economic fundamentals only. A more complete, rigorous definition is presented below.

The alternative theories for price changes deal with changes in the fundamental value and deviations of market prices from it. These theories are organized under the following broad categories: the efficient markets hypothesis, speculative bubbles, noise, and overreaction. The efficient markets hypothesis is the rational pricing theory, and the last three can be classified as irrational pricing theories. Although irrational pricing theories can incorporate selected aspects of rational expectations and the rational determination of fundamental value, they take the position that market prices deviate from rationally determined fundamental values.

Valuation Models

The model that investment analysts typically use for estimating fundamental value is the familiar present value model that was developed by John B. Williams in 1938:

where V is the value of the asset, Dt is the future cash flow or dividend, and k is the required rate of return or the discount rate for the asset. In words, the fundamental value of an asset is equal to the discounted present value of the future cash flows. If the future cash flows are risky Dt, is replaced with its expected value, E(Dt). In the case of bonds, there is a fixed maturity and a schedule of promised cash flows (interest plus principal). For relatively safe bonds, the cash flows are easy to predict, and investment analysts typically set the present value equal to the current price in order to compute yields. In the case of stocks, the analyst must forecast future dividends, including a long-term growth rate, and determine an appropriate discount rate in order to value the asset. Some observers have noted that earnings are more important than dividends in the valuation of common stocks, and there is a temptation to value earnings directly. If one uses a valuation of earnings approach, there must be an adjustment for the retained earnings. Williams (1938) and others have shown that if the analyses are done correctly, the two approaches should produce the same answer. Indeed, a dividend forecast should start from an earnings forecast.

In recent years, more sophisticated asset-pricing models have been developed in the financial economics literature. The familiar present value model can be extended to incorporate random variation in both cash flows and discount rates. Let k, be the discount rate from period t + 1 back to period t. Then the present value model becomes

Another approach is to start with a dynamic equilibrium asset-pricing model, tike the model of Lucas (1978). The fundamental asset-pricing equation in Lucas’s model is

where U’(Ct) is the marginal utility of consumption at time r, and β, which is a discount factor from the representative agent’s intertemporal utility function, is a number between zero and unity. At this point, it is necessary to make a distinction between real and nominal variables. This distinction is not necessary in models (1) and (2) if one remembers to discount nominal cash flows with nominal rates and real cash flows with real discount rates. The Lucas model is derived from an intertemporal consumption-investment problem, and the cash flows and prices are all denominated in consumption units. For this reason, the model formally applies to real interest rates, real cash flows, and real prices, with consumption and wealth in real terms. The model is a first-order difference equation, and the solution that ignores the bubble solution is

This expression simplifies under the assumption of risk neutrality–a linear utility function:

Equation (4) states that fundamental value is equal to the discounted present value of expected future dividends, where β is the discount factor, which is related to the discount rate used above via the relation β= 1/(1 + k). In this case, the discount rate k, is the same for all assets, regardless of their risk. The more complicated expression in equation (3)

incorporates risk aversion by weighting the cash flow with the ratio of marginal utility of future consumption to marginal utility of consumption today.1

A more general version of this model is one in which marginal utility of consumption is replaced with marginal utility of wealth:

where U’(W) now represents marginal utility of (real) wealth. In the Lucas model, the intertemporal utility function is separable over time, and marginal utility of wealth equals marginal utility of consumption. In models that relax the assumption of time-separable utility, marginal utility of wealth is no longer equal to marginal utility of current consumption only, and models like equation (5) must be used to determine fundamental value for assets. This model can be extended to price or value assets in nominal terms. Let pt be the consumption price deflator at time t, and replace Vt and Dt+j with their deflated nominal counterparts: Vt = ptVt and Dt+j=pt+j vt and Dt+j=pt+j Dt+j. After some rearrangement

Now define a new variable, λt=Uʹ(Wt) pt, Drop the tilde (̃) on Vt and Dt and simply interpret these variables as nominal quantities:

This asset-pricing model is applied to nominal cash flows and nominal prices; λt+jt is the marginal rate of substitution (MRS) between $1 at time t + j and $1 at time t, and the model is sometimes called an MRS model.

There are other capital asset-pricing models (CAPMs) in the finance literature: the Sharpe-Lintner CAPM, Ross’s arbitrage pricing theory, Mcrton’s intertemporal CAPM, and Breeden’s version of the consumption-based CAPM. The principal results for these models are expressions for equilibrium expected returns, and none of them directly addresses the problem of determining equilibrium asset prices. In fact, these models provide very little insight into the determination of equilibrium asset prices. Take, for example, the Sharpe-Lintner CAPM: its basic result is the following statement about equilibrium expected returns: E(Ri)=RFi(E(RM)-RF),

where the expected return on the market and the risk-free rate are determined outside the model. By using the identity

one can derive the following expression for Vit:

where the risk-free rate, RF,t+1 is known at time t, To solve this difference equation for the asset price, one must place some additional structure on changes in the risk-free interest rate and the market risk premium, Et(RM,t+1)- RF,t+1. Some investment analysts use the Sharpe-Lintner CAPM and Ross’s arbitrage pricing theory to calculate expected returns, which are then used to determine discount rates for valuing assets with equation (1). These equilibrium expected-return models are not directly useful in the analysis of fundamental value for capital assets, but they are useful for comparing returns across different assets and different portfolios over time.

The continuous-time asset pricing of Cox, Ingersolt, and Ross (1985a) is another more recent model that does address the issue of pricing assets. This model goes beyond the analysis in the continuous-time models of Merton (1973) and Breeden (1979); it derives the equilibrium interest rate endogenously, and a method for determining equilibrium asset prices is developed. Cox, Ingersoll, and Ross show that prices of capital assets and related contingent claims must satisfy a fundamental partial differential equation, plus a set of boundary conditions, and the solution is a risk-adjusted expectation of the cash flow discounted by an integral of the instantaneous interest rate. This model is extremely useful in the valuation of contingent claims and bonds that have finite lives, but it has not been used to value assets with extremely long (or infinite) lives, such as common stocks.

Prices, Fundamental Value, and the Efficient Markets Hypothesis

The financial models discussed above serve as precise statements or models of market fundamentals. An important issue in financial markets is the relationship between market prices and fundamental value. Do market prices reflect fundamental value only? If there are deviations from fundamental value, are the deviations large? Does the market incorporate all relevant information in forming the expectations that determine fundamental value? A related issue, which has been studied extensively for the last 30 years, concerns the informational efficiency of financial markets. If financial markets do not incorporate all relevant information in the formation of expectations, traders can act to earn either arbitrage profits or excess profits on the basis of other available information. However, the availability of this information does not account for or define the relationship between market prices and fundamental value. It is possible to have markets in which all information is reflected in current prices–that is, expectations are formed rationally– and the market price can still deviate substantially from fundamental value. Examples are contained in the theories discussed below.

The perfect markets theory is commonly known as the efficient markets hypothesis (EMH). There are several versions of EMH, the most frequently cited version of which was presented by Fama (1976, Chap. 5), which is a revision, with some corrections, of his original treatment of the EMH (Fama (1970)). According to Fama, the EMH states that asset prices reflect all available information. In forming expectations about next period’s price or rate of return, the market uses the correct probability distributions and all available information.

Formally, let FM(Pt+1) be the market’s subjective probability distribution function for next period’s price; E(Pt-1|IMt) is the market’s subjective expectation; and there are corresponding definitions for rates of return, FʹM(Rt+1) E(Pt-1|IMt) and. Now, consider the actual or objective distributions, F(Pt+1) and FʹM(Rt+1), as well as the objective expectations, E(Pt+1|It) and E(Rt+1|It). In the EMH, the market uses all the relevant information, and the market’s subjective distributions equal the objective distributions: FM=F,FʹM=Fʹand IMt=It. The market uses the correct distributions in forming expectations and arriving at equilibrium or market-clearing prices. The empirical implications are that price changes and rates of returns should possess the fair game property. Let ϵt+1= Pt+1- E(Pt+1|IMt and ϵʹt+1=Rt+1-E(Rt+1|IMt). The fair game property implies that the innovations, ϵt+1 and ϵʹt+1, cannot be predicted using any available information at time t. If an empirical researcher could find information in It, that was useful in predicting ϵt+1 or αʹt+1, the EMH would have to be rejected. The difficult aspect of this empirical research is specifying the behavior of expected returns or expected prices, A variety of models have been used to test this version of the EMH, but when the empirical tests result in rejection, it is not possible to determine whether the model for expected returns or the EMH has been rejected by the data.

Fama (1976) provides a review of the early studies through the early 1970s. Most studies, or at least those cited by Fama, generally support his definition of an efficient market. Specifically, stock returns have very little serial correlation, and it is difficult to find variables that are useful in predicting future returns. Or, from an investor’s perspective, there are no simple trading rules that can produce above-normal profits. There are, however, some exceptions. For a brief review of these exceptions, see LeRoy’s (1989. Section VII) survey. First, there is a small amount of serial correlation in stock returns, but Fama notes that this result can be induced by a plausible variation in expected returns. More recent evidence on the predictable variation in stock returns can be found in the work of Fama and French (1988a, 1988b), Poterba and Summers (1988), and Lo and MacKinlay (1988). The literature on the January effect, the size effect, and the price-earnings ratio effect is considerable. The principal findings are that stock returns tend to be high in January for small stocks, and a trading strategy based on price-earnings ratios (buy stocks with the lowest price-earnings ratio) can outperform the market. More recently, DeBondt and Thaler (1985, 1987) and Lehmann (1990) have presented profitable trading strategies that suggest overreaction in the stock market.

These studies of stock return behavior and the numerous studies on market efficiency in the finance literature can be interpreted as empirical studies of the informational efficiency of financial markets, and they represent tests of the implications of Fama’s definition for market efficiency. The other version of the EMH states that market prices are always equal to fundamental value. This view of the EMH is implicit in much of the finance literature, and an unequivocal statement can be found in Sharpe and Alexander (1990, p. 79). As new information comes to the market, it is quickly incorporated and reflected in a new set of prices. Price changes can be explained by the arrival of new information, which causes changes in the expectations of future dividends or cash flows. Thus, the EMH implies a theory for determining prices and the dynamics for price changes.

Speculative Bubbles and Noise

The other theories focus on deviations between market prices and fundamental value. The theory of rational bubbles is an example of a model in which expectations are formed rationally and the market is informationally efficient, but there are large deviations between market prices and fundamental value. This point can be most easily demonstrated in a model with risk neutrality and constant discount rates. The fundamental dynamic asset-pricing relation in such a model is the difference equation:

where β can be restated as 1/(1 + k).The market fundamental presented in equation (4) is only one possible solution for this difference equation. Since this is a first-order difference equation, one can add an arbitrary solution, as follows:

where At is a martingale: Et(At+k) = At for any k >0. It is easy to verify that equation (8) for prices also satisfies the difference equation in (7). The term At/β (i is also known as a rational bubble. Because 1 /β is greater than unity, the bubble term is expected to grow, and one can construct bubble processes that simultaneously satisfy the martingale property, on the one hand, and, on the other, have a small probability of experiencing a large drop, or a crash each period.

This theory is unsettling because it suggests that there is no unique equilibrium price and there can be large deviations from the market fundamental. The price solution in equation (8) satisfies Fama’s definition for market efficiency because all information that is relevant for forming expectations on future dividends and future paths for the bubble is incorporated in the current price. The bubble satisfies the fair game property.

At this point, it is worth noting that tests on price changes or rates of returns may be able to identify information inefficiencies in the market, but they cannot detect deviations of prices from market fundamentals. The rates of return from the price process in equation (8) are serially uncorrelated, and any empirical test that relies on the predictability of rates of return would have absolutely no power to detect the bubble or the deviations from the market fundamental. Some financial economists have suggested the use of runs tests on rates of returns to look for evidence of speculative bubbles, but the model here implies that such exercises are useless.2 Even though the bubble, At /β, is expected to grow, it can be random and experience the same kind of variation that we attribute to rates of return in the EMH.

All of the analysis on bubbles presented here in a model with constant discount rates can be extended to the intertemporal models that do not have this restriction. The solution in equation (5) can be modified as follows:

where At, again satisfies the martingale property. With risk aversion, U’(Wt) decreases as wealth grows, and this revised bubble term also has the property that it is expected to grow. The consumption-based version of this model follows by replacing Uʹ(Wt) with Uʹ(Ct). A similar result holds for the MRS model in equation (6).

At the theoretical level, a variety of arguments can be used to rule out bubbles in some models. In terms of mathematical modeling, the price must satisfy a first-order difference equation, but there are not enough boundary conditions to pin down a unique price. If the difference equation (7) is taken from Lucas’s model, a transversality condition from the representative agent’s dynamic optimization problem can be used to rule out the possibility of a bubble. This follows from the infinite horizon in the agent’s intertemporal utility function; intuitively, the agent considers the long-run consequences of the bubble and knows that it cannot be sustained. Tirole (1982, 1985) has examined this issue, noting that bubbles arise if agents have myopia; otherwise, rational bubbles cannot develop in markets with agents who pay attention to the long run. Ad hoc models, in which agents maximize a utility of wealth function over a finite time horizon, can produce difference equations for asset prices like equation (7), but these models do not have the necessary boundary conditions to rule out bubbles. Most of the trading in financial markets is done by institutional investors, firms that manage funds for individuals and other organizations, and a myopic utility function may very well represent an accurate description of their behavior. Even though conditions exist under which rational bubbles can be theoretically eliminated, there are models in which rational bubbles can develop.

More recent models or explanations for departures of prices from market fundamentals have been offered by Shiller (1984,1989), Summers (1986), and Black (1986). These theories are less formal in the sense that they are not derived from careful models of optimizing behavior by individuals. In these models, prices include both the fundamental and a noise term:

Shiller suggested that investors in the marketplace ignored fundamental value and followed fads or popular trends. In Summers’s simple model, the noise term, Nt, is driven by an autoregressive process with extremely slow mean reversion. He noted that in this model there can be large deviations of prices from fundamental value for long periods of time and that the conventional tests for market efficiency have extremely low power for detecting these departures. More recently, in work with other economists (Cutler, PoterbAt and Summers (1989, 1990) and De Long and others (1989, 1990)), he has developed models that demonstrate that it is possible for noise traders to influence market prices without being driven out by individuals who trade on fundamentals only. In De Long and others (1990), a simple model is constructed in which rational trading by fundamental traders can push the price even further away from fundamental value. There are noise traders and rational speculators in the noise trading models, and the equilibrium price contains a noise component– that is, a deviation from fundamental value. The noise traders are not driven out of the market in these models, and the noise persists. Black (1986) also presented a model in which prices include some noise around the fundamental value. An important issue concerns the size of a potential noise term. If noise accounted for only 5 percent of the variation in asset prices, it could be considered innocuous, but if it accounted for over half of the variation, one could make a strong case for imposing regulations and restrictions on financial markets. In the next section the empirical work that addresses these issues is reviewed.

II. Market Volatility and Noise: A Review of the Empirical Research

This section presents a review of the empirical literature on financial market volatility, particularly the work on excess volatility and noise. This literature includes studies of volatility in both the stock market and the bond market, but most have focused on the stock market. The empirical tests for both markets are important, and at the end of this section some interesting insights are obtained by contrasting the results for these two markets. There have also been studies of volatility in foreign exchange markets, and at least one paper, Meese (1986), has attempted a test for bubbles in foreign exchange markets. The major difficulty in applying these tests to foreign exchange markets is the specification of market fundamentals, and there seems to be little agreement on the appropriate form. Therefore, this paper concentrates on bond and stock markets, on which there is general agreement on the form of the market fundamental; the disagreements arise over the specification of discount rates.

In the review of the literature, the original variance bounds tests on the stock market and the subsequent criticism are covered first. This is followed by a review of what some have called the second generation tests, which were developed in response to the criticism. The discussion then turns to the tests for mean reversion, the tests on long horizon returns, and some recent tests that incorporate discount rate variability. Recent work on the connection between stock market volatility and margin requirements is briefly reviewed, as well as the tests that have been applied to the bond market. There are already several good surveys of the literature on bubbles and excess volatility. These include West (1988b), Shilter (1989, Chap. 4). LeRoy (1989), and Camerer (1990). Additional tests presented in recent working papers by Manktvv, Romer, and Shapiro (1989), Durlauf and Hall (1988, 1989), and Scott (1990) are reviewed, as well as recent research on stock market volatility and margin requirements.

The Variance Bounds Tests and Criticism

The variance bounds literature began with the original work of Shiller (1981b) and LeRoy and Porter (1981). These two papers examined the variance restrictions that are implied by the present value model of stock prices. Define the ex post market fundamental, as follows:

where Pt* is based on actual dividends or cash flows. If asset prices are determined by market fundamentals alone, as in equation (4) of Section I, and expectations are rational, then Pt = Et(Pt*) and P*t= Pt +et, where e, is a forecast error that should be uncorrected with anything in the time (information set, It. This observation implies the following variance relations:

Shiller and LeRoy and Porter presented tests of this variance restriction. Shiller constructed a time series for P* and computed sample variances for detrended versions of P* and P LeRoy and Porter estimated bivariate time-series models and used the parameter estimates to calculate the relevant variances. Shiller’s sample variance for P* was so much greater than the sample variance for P* that he did not bother with formal statistical tests. LeRoy and Porter also found dramatic rejection of the variance restrictions in their point estimates, but many of their tests were not statistically significant at conventional levels. Shiller concluded that the stock market was too volatile. LeRoy and Porter suggested several possible explanations: (1) the market could be too volatile; (2) the present value model with constant discount rates had been rejected; or (3) the tests were invalid. Subsequent papers by LeRoy and LaCivita (1981) and Michener (1982) presented intertemporal models with risk aversion as possible explanations for the volatility of stock prices. In these models the discount rates vary over time.

The original tests sparked a lively debate as critics focused on some of their weaknesses. For example, if P and P* are not stationary time series, then the variances do not exist and the corresponding sample variances are meaningless. Shiller removed a deterministic time trend from his data series before computing the sample variances, while LeRoy and Porter concluded that their data–earnings and stock prices with an adjustment for retained earnings–appeared to be stationary. Much of the criticism, particularly that of Kleidon (1986at1986b) and Marsh and Merton (1986), has focused on this part of the analysis. If dividends and stock prices need to be differenced in order to have stationary time series, they have argued, then tests based on Shiller’s method of detrending are invalid.

The issue is ultimately related to the recent debate in the econometrics literature concerning unit roots in time series, and several tests have been conducted to determine whether there are unit roots in dividends and stock prices. For the reader who is unfamiliar with the literature on unit roots, here is a brief introduction. Consider a time series, yt with the following representation: yt = σyt-1 where ut is a stationary time series that can have some serial correlation. One of the roots for y is 1 /σ, and if σ= 1, y has a root on the unit circle and is a nonstationary time series. Kleidon and Marsh and Merton showed that the variance bounds tests were extremely sensitive with respect to assumptions about station-arity. These authors used plausible models for dividends and stock prices, in which the growth rates are stationary time series, to show that the original tests are extremely biased. Kleidon conducted Monte Carlo simulations in which the present value model holds and the variance bounds tests lead to rejection. Marsh and Merton. employing a logarithmic random walk for dividends and Shiller’s method for computing the terminal value in the P* series, showed that the variance bound inequalities were reversed if the present value model holds. In his original tests. Shiller calculated the P* series recursively by starting at the end of the sample and working backwards:

For the terminal value, P*t+1 he used the sample mean from the price series; this is the procedure criticized by Marsh and Merton. Formally, the important feature of the model, Pt = Et(P*t ), no longer holds if P* is calculated in this manner. A better method for calculating P* is to use the terminal price, as was done in the paper by Grossman and Shiller (1981). If the terminal stock price is used for P* at the end of the sample, then the relation Pt= Et(Pt*) is preserved.

The Second Generation of Tests

The criticism motivated a second round of tests, the second generation, which are contained in papers by Mankiw, Romer, and Shapiro (1985), Scott (1985), Campbell and Shiller (1987), and West (1988a). All of these papers addressed the statistical problems that arise when the levels for P and P* are not stationary time series. Mankiw, Romer, and Shapiro developed a straightforward extension of the variance bounds test by considering the variability of P and P* relative to a naive forecast–P0 in their notation:

Their test was based on the following two inequalities:

For their naive forecast, they assumed that dividends follow a random walk, and they used Pot = (β/(l – β))Dt. Sample variances can be applied directly to these time series; the use of the naive forecast eliminates the need to detrend or to remove sample means. There was a possibility that the variances of these time series were growing (a heteroscedasticity problem), and the authors ran a second set of tests on the data series, deflated by the price, Pt .Their test results implied rejection of the present value model.

West (1988a) also developed a revised variance bounds test by considering the variability of innovations in the dividend process. He used a subset, Htof the information set It, used by the market to derive the following inequality:

The variance of the forecast error for the left-hand side of the inequality is greater because less information is used to form the forecast. By specifying a time-series model for dividends, West calculated the variance for the upper bound, the left-hand side, by using the methods of Hansen and Sargent (1980). Since Pt-1= βE(Pt + Dt|It-1), the variance for the right-hand side can be calculated by using the price series and an estimate for p. This relationship holds even if one needs to difference the dividend series to make it stationary. This variance inequality is also rejected by the data but both Mankiw, Romer, and Shapiro and West noted that rejection of their inequalities was not nearly as dramatic as the rejection in the original tests. West also presented some Monte Carlo simulations for his test and he found that although there was a small bias, it was not large enough to explain the rejections observed in the actual data.

Scott (1985) and Campbell and Shiller (1987) presented alternative tests of the present value model that were not based on variance inequalities. The test I developed in Scott (1985) was based on a simple regression interpretation of the present value model. As pointed out above, if stock prices reflect market fundamentals only, then the following relationship between prices and ex post market fundamentals must hold: P*t= Pt + et .The stock price should be an unbiased predictor of P*t and the series, (P*t - Pt ), should not be correlated with any variables in the time t information set. A simple test is to run the least-squares regression:

and test whether a = 0 and b = 1. The test can be easily modified if growth rates in dividends and prices are the relevant stationary time series: deflate the time series P*t and Pt , by a measure of dividends at time t. 3 1 used D –dividends summed over the previous year. The resulting regression is

If the growth in dividends, (Dt/Dt-1), is a stationary time series and the mean value for this rate is less than the discount rate, then P* /D¯ and P/D¯ are also stationary time series. If (Dt/Dt-1) and Pt/D¯t, are not stationary time series, then the infinite sums of discounted dividends do not converge and there is no solution for the market fundamental. In this regression it is necessary to account for the serial correlation in the error term when constructing the test statistics, and I used a spectral method for this calculation. This particular test exploits two implications of the present value model: (1) stock prices should be unbiased predictors of P* and (2) there should be some positive covariation between P* and P. I applied this test to stock price data and found that the model restrictions were strongly rejected by the data. My results suggest that there is little or no covariability between the price series and the ex post market fundamental. I ran a Monte Carlo simulation for this test to verify that there were no biases against the present value model; there was a small bias in the point estimates of the slope coefficient, but it was not large enough to bias the tests based on individual t-statistics.

Campbell and Shiller (1987) used the theory of cointegration to derive testable implications of present value models. Their results on the bond market are discussed below in the subsection on tests for the bond market. The present value model for stock prices implies that P* and P are cointegrated. Campbell and Shiller defined a spread variable, St = Pt – βDt (l - β); if ΔDt is stationary, then 5, and Δ Pt, are also stationary time series. They specified and estimated a vector autorcgres-sion (VAR) for Δ Dt and St. 4 The present value model applied to the spread variable implies that St= Et(S*t), where

This implication produces a nonlinear restriction on the coefficients in the VAR for Δ Dt and St. Campbell and Shiller also used the unrestricted VAR to calculate some interesting test statistics. They defined the theoretical spread:

where Ht indicates that the expectations of future dividend changes are calculated from the unrestricted VAR. If the present value model holds, then S and S1 should differ only by some sampling error, which implies that Var(S)/Var(S1) = 1 and Corr(S, S1) = 1. Campbell and Shiller used two different discount rates, or values for beta, in their tests on stock prices: 5 in one case the discount rate was set equal to the average rate of return, 8.2 percent, and in the other case, it was estimated from the cointegrating regression for an estimate of 3.2 percent. They found that the nonlinear restriction on the VAR is rejected at conventional significance levels and that the variance of S is much greater than the variance of S1, the theoretical spread from the unrestricted VAR. The correlation between St and S1t is negative with the higher discount rate, but it is 0.911 with the lower discount rate. In the latter case, St seems to track S1t, but in the first case, it does not. Although their results were somewhat mixed and dependent on the discount rate used, they concluded that the restrictions implied by the present value model were rejected by the stock price data.

Long-Horizon Returns and Mean Reversion in the Stock Market

A review of the excess volatility literature would not be complete without some mention of recent tests carried out by Fama and French (1988a) and Poterba and Summers (1988) for long-horizon returns. Fama and French found that serial correlation in stock returns was much greater if the returns are calculated over longer time horizons. They calculated autocorrelations by running the following regressions on rates of return:

for different return horizons, T, ranging from one year to ten years. They calculated these autocorrelations for a wide variety of stock portfolios and found a very interesting pattern. The autocorrelations were close to zero for short horizons (one year or less), but at two years were consistently negative, with the magnitude reaching a peak at about five years with values between -0.3 and -0.6. As the horizon extended to ten years, the autocorrelations returned to zero. According to Fama and French, these results could be interpreted in two ways: (1) they were consistent with the model of Summers (1986), which included a noise component with slow mean reversion, or (2) they were consistent with variation in expected returns.

Poterba and Summers (1988) examined both autocorrelation tests such as those of Fama and French and a variance ratio test. The variance ratio test compares variances for rates of return over different time horizons:

where the returns are monthly returns and the benchmark is the variance for one-period returns. If there is no serial correlation in the return series, then all of the variance ratios should be close to unity. If there is some mean reversion in the return series, then the variance ratios will drop below unity as the horizon is extended. Poterba and Summers applied the variance ratio test, which they argued was more powerful than the autocorrelation test to aggregate stock return series for the United States and 17 other countries. Their tests generally showed evidence of positive serial correlation over short horizons (less than one year), and they found evidence of mean reversion, since the variance ratios for long-horizon (eight years) returns dropped well below unity. Poterba and Summers emphasized the noise interpretation of their results. Their tests provided evidence that there is definitely some serial correlation in stock returns, but, as has already been noted, serial correlation in stock returns is not necessarily evidence of noise or bubbles in asset prices. Campbell and Shiller (1987, 1988a, 1988b), in several of their recent papers, have commented on the connection between these tests on long-horizon returns and the previous tests in the excess volatility literature. Several of the latter can be reinterpreted as tests on the predictability of the series, P*t– Pt. If the present value model is correct, this series should be uncorrelated with any variable in the time t information set. The regression test can be rearranged as follows:

where a and b should be zero, and the coefficient on any time t auxiliary variable should also be zero. My results indicate a negative b coefficient that is very significant. Look closely at P*t Pt; it is a very long horizon return, and it should be no surprise that the results of Fama and French and Poterba and Summers are very similar to the results found earlier in the excess volatility literature.

Tests that Incorporate Discount Rate Variability

Many economists have tried to rationalize the results from such tests by observing that all of them have a maintained hypothesis that discount rates are constant, or equivalently, that expected returns are constant.

How much variation in interest rates and discount rates is necessary in order to explain the observed variability of stock prices? Shiller (1981b) indicated that his original results could not be explained by real interest rate variability, but since his analysis was based on a linear approximation, some important second-order effects may have been omitted. Several papers have presented tests that attempt to account for interest rate variability; for example, Grossman and Shiller (1981), Campbell and Shiller (1988a, 1988b), and recent working papers by Flood, Hodrick, and Kaplan (1986), Mankiw, Romer, and Shapiro (1989), and Scott (1990). Grossman and Shiller used a consumption-based CAPM to incorporate discount rate variability; their model was essentially equation (3) in Section 1. The corresponding ex post market fundamenta is:

Grossman and Shiller used a constant relative risk-aversion utility function, U(C) = C1-γ/(1-γ) with values for γ, the risk-aversion parameter, that range from 1 to 4. They presented the calculations for P and P* graphically, and did not calculate any sample variances or other statistics. They found that a large risk-aversion parameter increases the variability of P*t but large persistent deviations of P from P* remain. These results are suggestive rather than conclusive.

Flood, Hodrick, and Kaplan (1986) suggested a procedure that falls somewhere between the Fama-French serial correlation tests and the Grossman-Shiller application of the consumption-based CAPM. Their idea was to estimate iterated Euler equations by using the econometric techniques of Hansen and Singleton (1982). Using the constant relative risk-aversion utility function, they estimated the p and y parameters by applying the following restrictions to real consumption and real stock return data:

The overall fit of these models can be tested by applying a x2 goodness-of-fit test; that is, increasing; effectively increases the return horizon over which the returns are calculated and tested. This approach, which Flood, Hodrick, and Kaplan adopted, is similar to the analysis of Fama and French, adjusted for changes in the marginal rate of substitution as measured by ratios of the marginal utility of consumption. The X2 est incorporates a form of discount rate variability, or variability in expected returns. Flood, Hodrick, and Kaplan found that rejection by the x2-test became progressively worse as the time horizon was extended, and they interpreted the results as overwhelming rejection of the consumption based CAPM, which has a long history of failing miserably in empirical tests.

Another method, more in the spirit of standard finance models, is to model discount rate variability as a function of the short-term interest rate. One simple model is to assume that the required rate of return on stock, the relevant one-period discount rate, is equal to the short-term interest rate, plus a constant risk premium. This approach has been applied by Campbell and Shiller (1988a, 1988b), Mankiw, Romer, and Shapiro (1989), and Scott (1990). In my paper I also presented some tests in which the risk premium was allowed to vary with volatility of one-period stock returns. The review of these models begins with the test in Scott (1990).

When discount rates vary, one must use a model of the following form for ex post market fundamentals:

and if prices reflect market fundamentals, only then does Pt = Et(P*t). The required or expected rate of return, Et(Rt,l ), is kt and it is modeled as follows: kt =RF.t-1 + RPt where RF,j+1 is the risk-free rate known at time f, and RPt is a risk premium. If the risk premium is constant, it can be easily estimated from the sample mean for (Rt - RFs). If nominal interest rates and nominal discount rates are used, then the tests can be applied directly to nominal prices and cash flows, and there is no need to deflate by a consumption price index.

I considered both a constant-risk premium model for the discount rate and one in which the risk premium varies with stock return volatility. This latter model was motivated by recent research in which expected returns on stock market aggregates were linked to the underlying return volatility.6 In both cases a regression test is applied with the new P* / D regressed on P/D, where again the series are deflated by dividends. The estimated coefficients for the price-dividend ratio should be close to unity, but the estimates are negative, and the t statistics for the test that b = 1 indicate rejection at conventional significance levels. The striking feature of the regression is that the R2s are very close to zero, which implies that the price series is an extremely poor forecaster of the ex post market fundamental. This revised ex post market fundamental now accounts for dividend variability and some discount rate variability. Shiller has noted that the regression can be interpreted as a test of the predictability of very long horizon returns. Consider the regression of P*t – Pt, deflated, on the price-dividend ratio. These results imply that the price-dividend ratio is a good predictor of this long-horizon return, even after an adjustment is made for discount rate variability.

Mankiw, Romer, and Shapiro (1989) used the constant-risk premium model to recalculate the P* series, and they applied their variance and regression tests to the data. They also found that the model continues to be rejected when discount rate variability is incorporated. Campbell and Shiller (1988a, 1988b) developed some additional tests based on their dividend-price ratio model, which can be easily adapted to handle discount rate variability. Their dividend-price ratio model follows from a linear approximation for the logarithm of the holding period return, and the model produces the following relationship when the expected return is modeled as the short-term interest rate plus a constant risk premium;

where rt, is the short-term interest rate, and c is a constant. In Campbell and Shiller (1988a), they applied the econometric tests in Campbell and Shiller (1987) to a VAR that included ln(Dt/Pt)(A In D,- r,), and a third variable, which is the log of the ratio of a 30-year moving average of earnings over the price. This last variable is useful in forecasting future dividends and future dividend-price ratios. They found that the restrictions on the VAR were rejected by the data. They also considered different time horizons for the dividend-price ratio model and found that the rejection became more significant as the time horizon was extended.

Interpretation of the Empirical Results

All of the tests that have been discussed so far are tests of the present value model for stock prices, interpreted as tests of the null hypothesis that stock prices reflect market fundamentals only. The alternative hypotheses that are supported by the empirical results are varied. One view is that there are serious specification errors in the models used for market fundamentals. Another view is that stock prices contain a large noise component or a bubble. Flood and Hodrick (1986) have shown that none of these tests can be interpreted as evidence of “rational” bubbles in stock prices. In most of the tests, the empirical researchers used the terminal price as the starting point in the recursive calculation for P*, the ex post market fundamental series. This procedure is consistent with the underlying null hypothesis that stock prices reflect fundamental value, but if there is a rational bubble in the stock prices, this rational bubble is inserted into the P* series as well. As Flood and Hodrick show, the net effect is that the tests presented above should have no power in detecting a rational bubble, and they interpret these test results as evidence against a rational bubble. This, of course, does not rule out the alternative hypothesis that stock prices contain either a large noise component or a near-rational bubble.

One exception to this criticism is the specification test for bubbles developed by West (1987). West set up equations for returns, dividends, and prices and performed a specification test for the nonlinear across-equation restrictions implied by the null hypothesis of no bubbles, which is essentially the present value model for stock prices.7 He noted that under the null hypothesis of no bubbles, all of the parameters are estimated consistently if the equations for returns and dividends have been specified correctly. If there is a rational bubble in stock prices, the parameters of the return and dividend equations can be estimated consistently, but the parameter estimates for the stock price equation are inconsistent because of the missing variable, the bubble, which may be correlated with dividends. The restrictions are tested with a x2 -statistic, and the behavior of this test statistic under the alternative hypothesis with bubbles in stock prices cannot be determined, so that the power of this test is unknown. West ran a variety of diagnostic tests on the return and dividend equations to check for specification errors, and the equations passed the full battery of tests. Although the specification test for the restrictions on the parameters in the stock price equation was strongly rejected by the data, it can be viewed as another test of the present value model, constructed so that a rational bubble can be incorporated as part of the alternative hypothesis. These results may also be interpreted in the same manner as have the other results on tests of the present value model.

Numerous other tests and variations have been developed to determine whether stock prices reflect market fundamentals since the initial variance bounds tests were conducted by Shiller, LeRoy, and Porter. The issue of stationary time series is obviously important, because some form of stationarity in the time series is necessary in order to have reliable large-sample properties for the test statistics, and the different transformations needed to obtain stationary time series have very different implications for the potential variability of the series. Much of the early debate focused on unit-root tests for the dividend series. Subsequent tests, which account for unit roots in the dividend process, have shown that the present value model continues to be rejected, but the rejections are not nearly as dramatic as those of the original tests.

Several studies have presented tests for unit roots in the dividend scries; in most cases, the null hypothesis of a unit root is not rejected for the level of dividends, but it is rejected for the change in dividends. Another possibility for nonstationarity is that the variance of the change in dividends is growing over time. If this were true, one would need to work with percentage changes or growth rates, instead of first differences. The tests presented by Mankiw, Romer, and Shapiro (1985, 1989), Scott (1985, 1990), and Campbell and Shiller (1988a, 1988b) follow from the assumption that growth rates in dividends and earnings are stationary. Time series plots of U.S. data are presented inFigures 1-4; the data are Shiller’s annual time series from 1890 to 1985, reproduced in his book Market Volatility (Shiller (1989)). The earnings series is included because most firms set dividends as a proportion of their earnings, and earnings are useful in forecasting future dividends. Figures 1 and 2 contain plots of changes in, respectively, real dividends and real earnings, deflated by the consumption deflator. Figures 3 and 4 contain plots of the percentage changes in real dividends and real earnings. In all four figures, the time series have the appearance of being stationary.

A skeptic may raise the issue, however, that the growth rates are not stationary time series;8 but this claim can be countered. If the growth rates are not restricted, the fundamental value runs the risk of being either infinite or undefined; there would be no market fundamental in such an economy, and asset prices would bounce around without any meaningful variation. A growth rate for dividends and earnings that exceeds the discount rate, or the interest rate, for the economy makes no sense at all. If the return on capital were that great, then competition in capital markets would push interest rates and discount rates up. Finally, take the identity for returns, 1 + Rt = (Pt + Dt)/Pt-1 P/Pt+1+Dt/Pt The return is a combination of the growth rate in the price and the dividend yield. If one wants to argue that growth rates in dividends, earnings, and prices are not stationary, then all of the research of the last 30 years on returns must also be discarded.

Figure 1.Changes in Real Dividends

Stock Market Volatility and Margin Requirements

Since the stock market crash of 1987, there has been renewed interest in stock market volatility and margin requirements. A series of papers on the connection between volatility and margins was stimulated by results published in a study by Hardouvelis (1988). Hardouvelis examined this relationship by running regressions of stock market volatility on a set of explanatory variables that included initial margin requirements. The other variables–variability of industrial production and bond returns, plus a measure of recent stock price movements–were included as controls. Hardouvelis found a statistically significant negative relation between stock market volatility and initial margin requirements. The policy implication is that one can reduce volatility by increasing margin requirements. Several observations are necessary. First, the initial margin requirements, set by the U.S. Federal Reserve Board, are not changed frequently, and the time series resembles a step function. Second, the initial margin requirement is not really a measure of the amount of margin that investors and speculators are actually using in the stock market, but is a limit only on the amount of borrowing when an investor initially buys the stock. The exchanges set the maintenance margin requirements, which determine margin calls.

Figure 2.Changes in Real Earnings

Hardouvelis’s paper stimulated further studies by Salinger (1989), Schwert (1989b), Kupiec (1989), and Hsieh and Miller (1990). (Roll (1989b, Section 3.1) provides a good review of this recent work.) This later research re-examined the regression analysis and introduced a more relevant variable–actual margin credit, specifically the ratio of margin credit to total value on the New York Stock Exchange (NYSE). If the use of margin credit affects volatility, then there should be a positive relation between margin credit and volatility. These studies showed that whether initial margin requirements or a measure of margin credit was used, the margin had no significant effect on volatility either if the regressions were run on first differences (changes) or the 1930s were eliminated from the data set.

Figure 3.Changes in Real Dividends

(In percent)

Numerous regressions have been run on this issue, but the real insight can be seen in a graph of the ratio of margin credit to value on the NYSE. Such a graph is reproduced in Figure 5, with data taken from Table 1 in Salinger (1989) for the period 1926 to 1987, Schwert (1989b) constructed a similar data series, beginning from 1917; he found that this ratio varied between 15 percent and 28 percent for the period 1917 to 1929. Prior to the stock market crash of 1929, investors in the United States used much wider margins in financing their stock portfolios than they have since the depression of the 1930s. During the 1930s when margin regulations and other reforms were first introduced at the federal level, there was a sharp reduction in the use of margins. Since World War II, the margin credit ratio has been very stable, with values fluctuating between 1 percent and 2 percent. Stock market volatility was high during the 1930s, and it increased for short periods during the 1974-75 downturn and after the crash of 1987. A regression of volatility on margin credit will produce a positive relation if the 1930s are included in the data set. Otherwise, there is no significant relationship in the data. The most interesting feature of the data is the sharp decrease in the relative use of margins during the 1930s and the low level since then.

Figure 4.Changes in Real Dividends

(In percent)

Hardouvelis (1990) presented some additional evidence on the relationship between margin requirements and excess volatility. In the first half of this study he claimed that his additional work continued to produce a significant negative relation between initial margin requirements and volatility, but a careful analysis of his results does not support this claim.

Figure 5.Ratio of Margin Credit to New York Stock Exchange Value

First, consider a simple description of the transmission of the effect of margin requirement changes on volatility. An increase in the initial margin requirement reduces the use of margin credit. As margin credit is restricted, speculators leave the market and volatility decreases. If the change in initial margin requirements is to have an effect on volatility, then it should produce a change in the actual use of margin. Under this hypothesis, a positive relation between the use of margin credit and volatility should be observed. If there is no relationship between initial margin requirements and the actual use of margin credit, then it would be difficult to argue that initial margin requirements have an effect on market volatility. By using ordinary regressions of volatility on initial margin requirements, the margin credit ratio, and several control variables, Hardouvelis was able to continue to produce significant negative coefficients on the initial margin requirement variable. He corrected for the serial correlation in the residuals, but did not include lagged values of volatility, the dependent variable. There is evidence of persistence in volatility–that is, some mean reversion in volatility–and this period’s volatility is a good predictor for next period’s volatility. A time-series approach that accounts for this serial dependence in the data is a more appropriate technique for analyzing the relationship among these variables, and Hardouvelis (1990, Table 4D) presented the results of VARs on a system for volatility, initial margin requirements, real stock returns, and the margin credit ratio. He claimed that this analysis also supported the negative relationship between margin requirements and volatility, but a careful analysis of Table 4D reveals that this is an overstatement. In the VAR for volatility, the sum of the coefficients on the initial margin requirement variable is negative and significant at the 5 percent level, but a joint test that all of the coefficients are zero cannot be rejected at the 5 percent level. The x2-statistic for this test is, however, significant at the 5.9 percent level. The sum of the coefficients on the margin credit ratio is also negative and significant at the 5 percent level, but the x2 -test that all of the coefficients are zero is not significant at the 10 percent level.

There is, nevertheless, strong evidence that lagged values of volatility and lagged values of stock returns are useful in predicting volatility. In the time-series regression, the effects of the margin variables on volatility is weak. In the VAR for the margin credit ratio, the coefficients on the initial margin requirements are significantly different from zero, but the sum of the coefficients is effectively zero. Changes in initial margin requirements arc useful in predicting changes in the margin credit ratio, but the long-run effect is zero. The sample for the VARs is monthly data from 1935 to 1987. Initial margin requirements were set at 45 percent in 1934, increased to 55 percent in 1936, and then lowered to 40 percent in 1937, where they remained until 1945. Since 1945, the initial margin requirement has fluctuated between 50 percent and 100 percent. Recall from Figure 5 that the margin credit ratio was higher during the 1930s, when the initial margin requirements were at historically low levels. Since 1945, the margin credit ratio has been low, while the initial margin requirements have generally been higher. If the 1930s were removed from the sample, the significant coefficients on initial margin requirements in the margin credit ratio equation might disappear. The hypothesis that initial margin requirements affect the use of margin credit and margin credit affects volatility is not strongly supported in the VAR analysis.

In the second half of the paper, Hardouvelis did present some evidence of a connection between initial margin requirements and excess volatility. The most interesting results are his regressions of long-horizon returns on price-dividend ratios, in which he re-examined the results of Fama and French (1988b). A large proportion–roughly 30 percent to 50 percent–of the variation of long-horizon returns (two to five years) can be predicted by the price-dividend ratio. Hardouvelis introduced dummy variables for high and low margin requirements and found that the predictable component in stock returns (as predicted by price-dividend ratios) is smaller when margin requirements are higher. These results are evidence of a relationship between stock return predictability and margin requirements, and suggest a possible connection between excess volatility and margin requirements. Hardouvelis concluded that the U.S. Federal Reserve had been effective in using margin requirements to dampen the effects of destabilizing speculation. It would be worthwhile to extend this analysis with actual margin credit.

Additional research has been carried out on the behavior of volatility over time. Schwert (1989a) constructed monthly estimates of stock return volatility in the United States that go back to 1859, and from this data he presented some interesting stylized facts regarding stock return volatility. Stock return volatility was much higher during the Depression of the 1930s, and it tends to increase during recessions. Most of the macroeeonomie time series were more volatile during the Depression, but none of the economic variables experienced increases in volatility that were similar to the increase in stock return volatility, which was two to three times greater during the Depression. There seems to be some association between stock return volatility and the volatility in macroeeonomie variables and between stock return volatility and financial leverage, but all of these effects are weak. There is, however, a strong positive correlation between volume and volatility, and Schwert found that the number of trading days had a small positive effect on stock return volatility. However, he concluded by confessing that he was unable to explain changes in aggregate stock market volatility with simple valuation models.

French and Roll (1986) also discovered a trading-day effect by studying the volatility of daily returns during the second half of 1968 when the stock market was closed on Wednesdays; variances of stock price changes were smaller on Wednesdays when the market was closed, even though the information flow of the economy was the same.

Prices, Interest Rates, and Market Fundamentals in the Bond Market

Tests for excess volatility have been applied to bond markets by Shiller (1979), Singleton (1980), and Campbell and Shiller (1987). All of these studies examined the following present value relation for interest rates:

where rt is the short-term, one-period interest rate; Rt(N) is the long rate (the yield to maturity on an N period bond); b, v is a constant liquidity premium; and -y = 1/(1 + R ), where R is the coupon rate.9 This model is derived from a linear approximation of the holding period return on a bond. The relationship represents a version of the expectations theory of the term structure of interest rates that places more weight on expected short rates in the near future and less weight on the distant future. The typical statement of the expectations theory is

where the long rate is a geometric average of the corresponding expected short rates. This relationship applies when Rt(N) is the yield to maturity on an N-period discount bond. These relationships are not exact and do not follow directly from asset-pricing models, even if risk neutrality is assumed. Equation (9), which was tested by Shiller and others, is an approximation at best, and if it is rejected by the data it is possible that the approximation error may be responsible.

A brief review of the empirical work on volatility in the bond market is presented here; a more complete survey can be found in Shiller (1989, Chap. 15). The first variance bounds tests on the relationship in equation (9) were presented by Shiller (1979) and Singleton (1980). Define the ex post series:

Then, Rt(N)= Et[R*t(N)] φN, where φN is assumed to be a constant. Shiller showed that the model implies the following set of variance restrictions: Var(R)⩽ Var(R*) ⩽ Var(r). In his initial work (Shiller (1979)), he found that the long rates, Rt, were too volatile when compared with the corresponding R*t series, and for some of his series the sample variance for Rt exceeded the sample variance for the short rate, rt. His general conclusion was that long-term bond rates were too volatile. Singleton (1980) extended this work by constructing formal statistical tests of the variance bounds; he found that the variance of the long rate exceeded the upper bound measured by the variance of R*.

Flavin (1983) subsequently demonstrated that the small-sample properties of this interest rate model are questionable. Using a simple auto-regressive process for the short rate with an autoregressive coefficient of 0.95, she found a serious bias in this variance test if one uses sample sizes comparable to those used by Shiller and Singleton. In chapter 13 of Market Volatility, Shiller presented more recent tests with two very long data sets: one for the United States from 1857 to 1988, and one for the United Kingdom from 1824 to 1987. Shiller’s results for these longer data sets differed from those of the initial tests. The observed variability of R was consistent with the variability of R*. He also ran the following simple regression:

where b should equal 1 and a is an estimate of the negative of the constant liquidity premium. The estimate of the slope coefficient in the U.S. data was 1.156, and the test for b = 1 was not rejected. The estimated slope coefficient for the U.K. data was 0.347, and the test for b = 1 was rejected at conventional significance levels. Despite the mixed results, Shiller concluded that the evidence generally supported this expectations model for the bond market. He attributed the earlier rejections of the model to the smaller sample sizes used in the initial studies and the small sample biases discussed by Flavin.

Additional results on the term structure were presented in Campbell and Shiller (1987), where the theory of cointegration was applied to the present value model for stocks and the expectations theory in equation (9) above. For bonds, they defined the spread variable to be Rt(N) – rt and they estimated a VAR.10 The model for the term structure implied a set of nonlinear restrictions on the coefficients in the VAR. These restrictions were rejected by the data, but Campbell and Shiller did not find a high correlation between the theoretical spread computed from the unrestricted VAR and the actual spread. Despite the rejection of the model restrictions, there is evidence that long-term rates move with rational forecasts of future short-term rates.

In a recent working paper (Scott (1990)), I ran some tests of the present value model applied directly to bond prices. I used the following model, with discount rates that vary with short-term interest rates, to calculate the P* series for bond prices:

In one case, kt+i is set equal to rt+i, and in a second case, kt+i is set equal to rt+i plus a risk premium that declines as time to maturity decreases. The sample consisted of monthly prices on short-, medium-, and long-term U.S. Treasury bonds for the period 1932 to 1985, and the corresponding P* series was calculated for each bond. The result of the test–regressing P* on Pt and a constant–was that the present value model could not be rejected for any of the bonds in the sample. The sample variance of P* was also found to be greater than the sample variance of P for all of the bond series. The results of these recent tests on the term structure models suggest that there is no evidence of excess volatility or noise in the bond market.

Synthesis and Summary

The empirical results reviewed here suggest that prices reflect market fundamentals in the bond market but not in the stock market. For the stock market, the present value model is usually rejected by the data, and there is some evidence of serial correlation and mean reversion in returns. This evidence on stock prices has several interpretations. As previously noted, the evidence on serial correlation is consistent with the EMH, coupled with intertemporal variation in expected returns. The evidence on present value models with variation in discount rates cannot be so easily dismissed. One possibility is that there is a very large noise component in stock prices. Another is that there is a serious specification error in the models for fundamental value. Diba and Grossman (1988) and Durlauf and Hall (1988, 1989) have provided useful interpretations of these models. Diba and Grossman introduced a specification error, so that Pt = Et(P*t) + ut, where ut is the specification error; and suggested omitted tax effects as a possible specification error. Durlauf and Hall presented a view that is observationally equivalent: Pt = Et(P*t) + Nt, where JV, is a noise component. They showed how to estimate lower bounds for the variability of the noise term and suggested that the usefulness of models could be measured by the relative size of the noise term. Their analysis involved regressions of the following form:

where Pt*Pt, is equal to the forecast error, P*t - Et(P*t), plus the noise term. The forecast error should be uncorrelated with any information variable dated at time r. but the noise term (or the specification error) can be correlated with time t variables, particularly the stock price, which would contain the noise term. If there is no noise, then all of the coefficients in the regression should be zero. If there is noise, then some of the coefficients will be nonzero, and the variance of the fitted values will represent an estimate of the lower bound for the variance of the noise term.

Table 1.Regression Results: Stock Prices, 1927-87
(P*t -Pt) = a + b(Pt/Dt) + et
Regression Equation TermsConstant-Risk Premium ModelTime-Varying Risk Premium Model
b-1.1514-1.06322
(0.3802)a(0.4314)a
t(b = 0)-3.032.46
R20.680.68
row(Nt/Dt)7.426.85
row [(P*t- Pt)/Dt]9.008.31
row(Pt/Dt)6.456.45

Standard error.

Standard error.

These regressions are very similar to the regression tests used by Scott and more recently by Shiller. The Durlauf and Hall noise measure can be easily calculated for the regressions in Scott (1990). Two sets of calculations are presented. The first set is for two stock price models in which the discount rates are functions of the short-term interest rate plus a risk premium. The data for the regressions are monthly prices and dividends for the value-weighted NYSE portfolio, 1927-87. The regression equation is

where the series P*t and Pt, are deflated by annual dividends, Dt. The results are summarized in Table 1.11 In both cases, 68 percent of the variance of (P*t – Pt)/ Dt is explained or predicted by the price-dividend ratio; P*t – Pt is a long-horizon return, adjusted for discount rate changes, and the series should be unpredictable given information at time t. The estimates for the noise variability suggest that the noise term is relatively large and that it accounts tor most (68 percent) of the variability in P*t – Pt; the noise also accounts for most of the price variability. Of course, one can interpret these numbers as estimates for the variability of the specification error, and conclude that the specification error for the model is quite large.

Table 2.Regression Results: Long-Term Band Prices, 1932-45
(P*t - Pt) = a + b Pt + et
Regression Equation TermsZero-Risk PremiumRisk Premium as a Function of Maturity
b-0.1024-0.6730
(2.56O0)a(2.0303)a
t(b = 0)-0.04-0.33
R20.0020.101
row(Nt)0.98286.4590
row(P*t - Pt)23.855720.2744
row(Pt)9.59749.5974

Standard error.

Standard error.

Now, contrast these numbers with the same calculations for long-term bond prices (15- to 30-year Treasury bonds) in the following regression:

The sample consists of monthly bond prices for the period 1932-85. The results are summarized in Table 2. In the bond regressions, none of the coefficients are statistically significant, and the resulting estimates for the noise variance are also not statistically significant. Noise or specification error is relatively small for the present value model applied to bond prices. Therefore, the present value models work reasonably well for bonds, but not for stocks.

Do stock prices deviate substantially from fundamental value? The evidence from the numerous studies reviewed suggests that there are significant deviations of stock prices from market fundamentals.12 The inability of economists to explain either stock price movements or changes in stock market volatility provides additional support for this view (see Cutler, Poterba, and Summers (1990) and Schwert (1989a, 1990)).

The challenge to defenders of the EMH is to explain the failure of valuation models when applied to the stock market. The different empirical tests have incorporated dividend variability, interest rate variability, and variability in different measures of the risk premium, but these modifications have been unable to rationalize the observed variability of stock prices. Why do the models work reasonably well for bond prices, but not for stock prices? One answer, from the discussion of bubbles in Section I, is the presence of irrational or near-rational bubbles in stock prices, but not in bond prices. The evidence also supports the notion of noise or fads in stock prices, but noise trading or fads models should not be applied to all markets because the empirical evidence does not warrant their application to the bond market, LeRoy (1989, p. 1616) recently summarized his position on the EMH:

The most radical revision in efficient markets reasoning will involve those implications of market efficiency that depend on asset prices equaling or closely approximating fundamental values. The evidence suggests that, contrary to the assertion of this version of efficient markets theory, such large discrepancies between price and fundamental value regularly occur.

III. Conclusions

Do asset prices reflect fundamental value in financial markets? Is there too much volatility in financial markets? The empirical studies reviewed in Section II provide evidence that stock prices do regularly deviate from fundamental value. The early studies of excess volatility in the stock market undertaken by Shiller, LeRoy, and Porter were criticized and challenged, but more recent studies, which relax some of the restrictive assumptions in the initial studies and address some of the econometric issues, continue to produce evidence against the efficient markets hypothesis and the notion that stock prices reflect fundamental value. The results of the more recent studies are not as dramatic, but they do imply that stock prices deviate significantly from fundamental value. By contrast, the recent work on interest rates and prices in bond markets has been unable to uncover any significant evidence of deviations from fundamental value in the bond market, Shiller, who generated sortie of the initial evidence of excess volatility in long-term interest rates, has recently modified his position with respect to the bond market (see Shiller (1989, Chap. 13)).

In Section I, several alternative models for the formation of prices in financial markets were discussed: the efficient markets hypothesis, the speculative hubbies model, and noise trading models. The evidence of excess volatility in the stock market provides support for the alternatives to the efficient markets hypothesis, and there has been a recent increase in research on noise-trading models and other alternative views in the financial economics literature. The stock market crash of 1987 also stimulated interest in this line of research, since most financial economists have been unable to explain the crash in terms of changes in fundamental value or fundamental economic factors (see Fama (1989) and Roll (1989a. 1989b)).

In his recent survey on efficient capital markets, LeRoy (1989) concluded that it will be necessary to revise the way that efficient markets reasoning is applied. Despite the evidence that stock prices do not always reflect fundamental value and although there may be noise and occasional market crashes, it is not clear that a wide range of restrictions on financial markets are necessary. Financial markets serve an important role in the allocation of capital, and the goal of any new regulations should be to promote reforms or changes that serve to move stock prices closer to their fundamental values.

(In addition to the sources cited in the article, this list includes items that may be of interest to readers wishing to pursue the subject further.)

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Louis O. Scott is Associate Professor of Finance at the University of Georgia, and he holds degrees from Duke University and the University of Virginia. This paper was written while he was a Visiting Scholar in the Research Department.

The author has benefited from discussions with Donald J. Mathieson and William Perraudin.

Note that

where Covt is the covariance conditional on information at time t; is equal to the price of a default-free asset that pays one unit of consumption at time t + j and corresponds to a long-term real interest rate. The first term on the right-hand side represents the expected dividend or cash flow discounted at this real interest rate. The second term represents an adjustment for the risk.

At least two well-known financial economists have discussed the use of runs tests for detecting bubbles. The references have been omitted to protect the innocent.

If one must difference the logarithms of dividends and prices to get stationary time series, then the corresponding growth rates are stationary time series.

The form of the VAR is

where a(L), b(L), c(L), and d(L) are polynomials in the lag operator.

Recall that β is related to the discount rate k, as follows: (β = 1/(1 + k).

Scott (1990) used the following GARCH (generalized autoregressivc conditional heteroscedasticity) model, one of several models presented in French, Schwert, and Stambaugh (1987):

The risk premium varies with volatility, as measured by the GARCH model, and there is an adjustment for the small amount of serial correlation that remains in the excess return. This model is estimated by the method of maximum likelihood, and the P* series is calculated with the estimated discount rates. This technique has been useful in modeling changes in conditional variances.

The West system of equations in change form is

The present value model implies that the coefficients m, 8,. . . ,8, are functions of the parameters p, |i,<t:i,.. . , d>r, and West uses a x-test for this restriction.

This discussion is motivated by Kleidon’s comment that these tests need to be run on independent cross sections of data. His comments seem to suggest that one cannot apply the theory of stationary time series, in particular ergo die theory, to these series on prices and dividends.

Because the coupon rates on the bonds in his samples change over time, Shiller used the sample mean of R,(N) for R .

The model for the VAR is

The estimate for Var(Nt) is b2 Var(Pt/Dt), which is the variance for the fitted value of the regression.

It should be noted that almost all of these studies have concentrated on data for the U.S. market, although in their study of mean reversion, Poterba and Summers (1988) presented evidence of serial correlation in the stock markets of 18 countries.

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