Empirical estimation of the parameters of the consumption-based capital asset pricing model with 20 different market indices.

Author:Dondeti, V. Reddy


The fundamental concepts of the Consumption-based Capital Asset Pricing Model (CCAPM) linking personal consumption and asset prices are described in Lucas (1978). Power utility functions (Xie, 2000) play a significant role in the definition of an individual consumer's risk aversion and form the foundation for the CCAPM. Among the several utility functions used in deriving the CCAPM, the one that is most commonly used is the power-utility function (Cochrane, 2005) given below:

U (C) = [C.sup.1-[gamma]] -1/1 - [gamma], [gamma] [not equal to] 1. (1)

The CCAPM, based on the utility function described by (1), can be stated as (Cochrane, 2005):

E[[[beta]([C.sub.t+1]/[C.sub.t]).sup.-[gamma]] R ] = 1 (2)

Equation (2) is often called the Euler Equation (Hansen and Singleton, 1982) and [gamma] is called the relative risk aversion coefficient. The Euler Equation approach is used in Brandt (1999) is analyzing the relation between the portfolio and consumption choices of individual consumers. Estimation of the value of y has been the focus of many studies and a concise review can be found in Savov (2011). It will also be the primary focus of this study. The parameter [beta] is the subjective discount factor, 0


The parameter [gamma] is the reciprocal of the intertemporal elasticity of substitution of future consumption for the current consumption (Romer, 2006). The basic question deals with the effect of the changes in the interest rates (or, in general, the changes in the returns on assets held by consumers) on the growth of consumption (and concomitantly, on personal savings and investment as well). If G denotes the gross rate of change in personal consumption, we get the intertemporal elasticity of substitution (Romer, 2006) as follows:

[xi] = dG/G/dR/R = 1/[gamma] (3)

First, we have to note that y is always positive, but cannot be equal to 1. If [gamma]=1, the utility function becomes U (C) = ln(C) and equations (1), (2), and (3) are no longer valid. If the value of the parameter y is between 0 and 1 (i.e., 0


In one of the earliest studies dealing with the estimation of [gamma], Mehra and Prescott (1985) found that the value of [gamma] has to be very large to justify the returns on the equities. On the other hand, if we assume that the value of [gamma] is between 1 and 10, the risk-free rate of return (on the T-bills) has to be much higher or the average equity returns have to be much lower than the observed values, according to the conclusions of Mehra and Prescott (1985), based on the data covering the 90-year period of 1889 through 1978. Mehra and Prescott (1985) called this divergence between the expected and observed values of the equity returns as the "equity premium puzzle." There have been several studies to estimate [gamma] and resolve the equity premium puzzle, and the results of some of these studies are briefly reviewed in Mehra (2003). As indicated in Mehra (2003), some authors have called it the risk-free rate puzzle (Weil, 1989) rather than the equity premium puzzle. Hall (1988) has given estimates of the intertemporal elasticity of substitution [xi], and also summarized the results from some of the previous studies, in which, the values of [xi] ranged from 0.03 to 0.34. In other words, the value of [gamma] (which is the reciprocal of [xi]) was estimated to be between 3 and 33. Other estimates of [gamma] have varied between 3 and 85. Some empirical results based on different utility functions are provided in Epstein and Zin (1991). In a related study, Kocherlakota (1996) concludes that the equity premium puzzle will remain a puzzle. Campbell and Cochrane (1999) explain the reasons for the puzzle and the low level of variation in consumption as habit formation among the consumers. Constantinides (2002) offers "incomplete markets" and "heterogeneous consumers" as possible reasons in explaining the equity premium puzzle.

Regardless of the results, the basic question remains: what is a reasonable value for [gamma]? The review of literature indicates that a reasonable value for [gamma] may be between 3 and 10, but it depends on the type of the asset returns used in the analysis. According to Savov (2011), who uses the amount of garbage generated by the whole population as a proxy for consumption, the estimates of y were between 7 and 26, and these values were considered to be more reasonable than the ones obtained by the others in earlier studies. The value of [gamma] also depends on the value of [beta]. Savov uses a value of [beta] =0.95 in estimating [gamma]. This implies that the amount of garbage generated is, perhaps, a better proxy for consumption, than the consumption itself, because the level of consumption is very smooth and may not accurately reflect the movements in asset markets. A more extensive review of literature on the CCAPM and Equity Premium Puzzle can be found in Mehra (2003) and Savov (2011)


In almost all the previous studies dealing with the estimation of the CCAPM parameters, S&P 500 index has been used as a proxy for the market return and also data spanning longer time periods (50 years or longer) has been used. However, the consumers today have a vast array of investment choices and they do explore several other market indices before making their consumption and investment decisions. Further, use of data covering longer time periods may mask the impact of the medium-term (and short-term) volatility in the asset returns and personal consumption as well. This study has two objectives. The first objective is to use, in addition to the S&P 500 index, 19 other indices which are popular in the financial press, in the estimation of [beta] and [gamma]. To the best of our knowledge, no other study has considered these 19 indices before. In other words, this study looks at consumers not as one monolithic group, but as a collection of several heterogeneous groups. The second objective of the study is to use the data covering two different time spans (an intermediate horizon of 20 years and a longer horizon of 50 years) and compare the results to...

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