Equally Weighted Market Return and Value Weighted Market Return as a Proxy of the Market Index: An Empirical Study
Il-woon Kim., Akron Business and Economic Review
Equally Weighted Market Return And Value Weighted Market Return As A Proxy Of The Market Index: An Empirical Study(*)
Beta has been extensively used by both practitioners and academicians as a measure of the the systematic risk of a firm. Investors use beta to choose a portfolio with a specific level of risk. Performance of the portfolio can be more accurately measured using the risk adjusted returns. Researchers often use beta to control market-wide factors in estimating abnormal returns associated with a specific event. Beta is also important in understanding risk-return relations in capital market theory.
The beta can be measured from the following market model:
[R bar.sub.it] = [a.sub.i] + [b.sub.i].sup.*][R bar.sub.mt] + [U bar.sub.it],
where [R.sub.it] = return on security i in period t,
[R.sub.mt] = return on the market portfolio in period t,
[U.sub.it] = a residual, where E([U.sub.it]) = 0 and E([U.sub.it], [U.sub.it] - 1) = 0 for
t [is not equal to] 0,
[a.sub.i] = the intercept, and
[b.sub.i] = the measure of systematic risk of a security. The tilde "-" denotes a random variable.
A key assumption of the model is that [U.sub.i] is independent of [U.sub.j], where i and j are different companies. This implies that stock prices move together systematically only because of common movement with the stock market. Adding the assumption that the expected value of the residual is zero, the model asserts that a linear relationship exists between the expected return on each security and the expected return of a market portfolio.
Theoretically, the market portfolio should consist of all assets in the world, marketable and nonmarketable. Since the true market portfolio is not observable, a proxy must be used. There is both theoretical and empirical evidence that the choice of market index has impact on estimated betas. The reasons for the difference in betas estimated using different indices and the magnitude of the difference have yet to be investigated. In addition, there is more than one technique to estimate beta. The use of different methods may result in different betas.
The purpose of this study is to identify the characteristics of different market indices, to examine the differences in estimated beta using different techniques. The implications of using different market indices in empirical research will also be discussed.
THEORETICAL AND EMPIRICAL EVIDENCE
Roll provides a mathematical proof that beta is a function of the market proxy against which the stock is measured. Roll also shows that a market index can be found to produce a beta of any desired magnitude and that, for every asset, judicious choice of the index can produce any desired performance in the market.
There is empirical evidence that estimated betas are affected by the market index used. Saniga, McInish, and Gouldey used three different market indices to investigate the effect of differing interval length on beta. They found that the relationship between estimated beta and the time intervals were dependent upon the choice of the market index. Roden also demonstrated that the homogeneity of beta rankings varies according to the market index used.
Ross[15, p.892], however, supports the use of any market index in empirical tests:
The bulk of the tests...rely on explicit proxies of the market portfolio. The
commonest such proxies are the value weighted, and the equal weighted portfolio,
about which, unfortunately, the CAPM has absolutely nothing to say. The equal
weighted portfolio may or may not be efficient, but even if it is, such a finding
would have no implications whatsoever for acceptance or rejection of the CAPM
theory.... This is an unfortunate state of affairs; the market portfolio itself is
much harder to observe in practice than in principle. …