ریسک و بازگشت در یک مدل تعادل عمومی دینامیکی

In this paper we examine the relationship between risk and return on productive assets using the intertemporal general equilibrium model of Brock (1982, Asset Prices in a Production Economy, the University of Chicago Press, Chicago, pp. 1–42) as a basis for a simulation study. Current computational techniques are used to solve the growth model of Brock (1979, An Integration of Stochastic Growth and the Theory of Finance — Part I: The Growth Model, Academic Press, New York, pp. 165–192) in order to analyze the underlying financial model. Contrary to recent empirical findings, we find that there is a theoretical basis for the linear relationship between risk and return. This apparent contradiction is due in part to the fact that the dynamic relationship between risk and return depends on the level of output.

Over the last two decades researchers have spent a great deal of time to evaluate the performance of the Capital Asset Pricing Model (CAPM) by testing how well the model fits the data. The empirical evidence on the validity of the CAPM is mixed. While some studies have concluded that the model is misspecified others have found support for the predictions of the model. All of these studies, however encountered serious and difficult econometric problems in their efforts to provide the best empirical tests of the model. To what extent their results are derived by these methods has been a source of controversy. However, many researchers have taken the mixed empirical evidence to imply that the CAPM is not the correct model of risk and have attempted to test other determinants of expected stock returns. In this study, we examine the prediction of the CAPM, the linear relationship between risk and return, using the intertemporal general equilibrium model of Brock (1982) as a basis for a simulation study. The dynamic structure of the model provides some insights about the dynamic relationship between risk and return which shed light on to the problems in empirical testing of the model. More specifically, we find that the dynamic relationship between risk and return depends on the level of output in the economy. In other words, the position of the economy on the business cycle matters in testing the relationship between risk and return. Contradictory to the predictions of the CAPM, factors other than beta have been found to explain the cross-section of expected stock returns. These factors include market equity or in other words size (Banz, 1981; Reinganum, 1981), earnings price ratios (Basu, 1983), firm's book value of common equity to its market equity (Rosenberg et al., 1985), and leverage (Bhandari, 1988). Recently, Fama and French (1992) reconsider these different effects and find that size and book-to-market equity ratio provide the best characterization of the cross-section of stock returns and conclude that beta does not explain the cross-section of expected stock returns. This empirical evidence has led researchers to deduce that the pure theoretical form of the CAPM does not agree well with reality. Although Fama and French (1992) make a persuasive case against the CAPM, their study itself has been challenged. Kothari et al. (1995) show that Fama and French (1992) findings are crucially contingent on the methodology and data used. Black (1993) finds that the size effect, that is significant in some periods, disappears in others; therefore, Fama and French's results may simply result from their select sample. Jagannathan and Wang (1996) show that the CAPM is able to explain the cross-sectional variation in average stock returns when betas and expected returns are allowed to vary over the business cycle and when human capital is included in measuring wealth. Stimulated by these empirical findings, a number of researchers have sought to find alternative explanations for equity premia. One line of attack has been that of Fama and French, 1993 and Fama and French, 1995 who conclude that fundamental variables found to explain the variation in returns must be proxies for some unidentified risk. Another line has been that of Lakonishok et al. (1994), who argue that due to mispricing of assets, there are excess returns which are not accounted for by the standard measures of risk. As Fama and French (1993, p. 3) point out, this line of research relies on `….variables that have no special standing in asset-pricing theory. . . ..'. The reason researchers have taken this direction stems from problems that have been encountered in attempts to empirically verify the theoretical predictions of the CAPM. Is it really the CAPM that is misspecified or is it the empirical tests of the CAPM that are performed erroneously? The CAPM is a two-period, linear model expressed in terms of expected return and expected risk. Since these expectations cannot be measured, empirical studies use observed data to test for this linear relationship. However, in this study we are able to calculate both expected return and the true beta at a given period in time; thus, we are able to theoretically test the CAPM in its ex ante form. In such tests, we find that there is a linear relationship between beta and the expected return at any given period in time, as predicted by the CAPM. In addition, we have also found that the intercept as well as the slope of the Security Market Line shift up and down with fluctuations in output in the economy, thereby suggesting that in the empirical tests of the CAPM, one needs to control for the fluctuations in the output level in order to properly test the model. In other words, in estimating the relationship between risk and return, only those observations that correspond to similar output levels should be used. Such findings gave us the impetus to pursue an empirical testing of the model using simulated data. We have performed two empirical tests of the CAPM. First, we have used the full data set (240 months) in Fama–MacBeth regressions (1973) of the cross-section of stock returns on beta and size (stock's price times shares outstanding). Second, we have controlled for the output level, therefore, used only those periods (20 months) that have output levels that are close to the mean output level. Without controlling for the output level, we have found that the size variable is significant in explaining the cross-section of expected stock returns while beta is not. However, once we control for the output level the size effect vanishes and only beta remains significant. The conclusions of this paper are determined from a simulation study of Brock's asset pricing model. Except for the specific case of logarithmic utility and Cobb–Douglass production functions and carefully paired constant relative risk aversion utility functions and constant elasticity of substitution production functions, there are no closed form solutions to Brock's model.1 As Judd (1995) points out, the simulation methods provide a strong complement to economic theory for those models that are not analytically tractable. Brock's model has been frequently used and cited in the literature over the past 15 years. However, some researchers have only used the specification mentioned above which is characterized by a linear investment function. Others starting with Kydland and Prescott (1982) have used a quadratic approximation to the value function which also results in a linear policy function.2 Thus, these studies failed to produce the cyclical variation in equity premia. The paper proceeds as follows: Section 2 presents the stochastic growth model of Brock (1979); Section 3 introduces the financial model; Section 4 presents the parameters and describes the simulation; Section 5 discusses the results; and Section 6 concludes the paper.

In this study, we showed that a properly parameterized version of Brock's model provide a theoretical explanation of the observed fluctuations in asset prices. In particular, Brock's model provides a theoretical basis for the observed variations of the equity premia over the business cycles. Also, by using Brock's model, we have shown that the CAPM predictions hold in a fully dynamic general equilibrium framework. Particularly, we found a linear relationship between expected risk and expected return at any given period of time, which implies that beta is the sole determinant of the expected returns. Furthermore, from Fig. 1, the model (even with iid shocks) replicates the typical pattern of widely fluctuating output levels and relatively constant levels of consumption over time. Several implications for the empirical testing of the relationship between risk and return come from the dynamic nature of this study. In Section 4, we have shown that the intercept as well as the slope of the security market line fluctuate over the business cycle. This means that the empirical estimation of the SML, averaging over long enough periods to hold main portion of the business cycle, will result in serious errors. Thus, the empirical estimation of the SML must be done at similar levels of output. This phenomenon is further supported by Fig. 5, which indicates that, if the time series of returns are regressed on betas, one will get a much steeper relationship with the possibility of a wrong sign on the coefficient. In ongoing research, we examine various extensions and applications of Brock's model. Mainly, we model the shocks as Markov process in order to more closely approximate observed business cycle fluctuations of output. We also incorporate a growth factor to capture the trends in output. As further extensions to Brock's model, Black (1995, p. 159) suggests incorporating labour, adjustment cost of capital and human capital. Prescott (1982, p. 45) makes similar suggestions with regard to labour in order to more closely approximate the business cycle. There are many potential applications of this model. It can be used to price assets other than those that are in Brock's two papers (1979, 1982). For example, by calculating the value of pure discount bonds, the term structure of interest rates can be determined over the business cycle. The dynamic implications of corporate tax policy can also be analyzed by using this model. It is our belief that with the advent in the use of simulation methods, the full richness of the applications of Brock's model can be developed.