|
...portfolio strategy, among others. CAPM relates the expected rate of return of an individual security with a measure of its systematic risk. To test the validity of the CAPM, researchers test the security market line given as: E([R.sub.i])=[R.sub.f]+[[beta].sub.im]{E([R.sub.m]-[R.sub.f]} where [R.sub.i], [R.sub.f] and [R.sub.m] are return on risky asset i, risk-free asset, and market portfolio, respectively and [[beta].sub.im] is a measure of systematic risk defined as the ratio of Cov([R.sub.1], [R.sub.m]) and Var([R.sub.m]). The security market line suggests an overall positive relationship between beta and expected returns. Studies, however, provide weak empirical evidence (1) on this relationship. See, for example, Fama and French (1992), He and Ng (1994), Davis (1994), and Miles and Timmermann (1996).
Pettengill, Sundaram, and Mathur (1995) observe that the studies of beta and cross-sectional returns that use realized return as a proxy for the expected may have produced bias results due to aggregation of positive and negative market excess return periods. Pettengill, Sundaram, and Mathur argue that when the market return in excess of the risk-free rate is negative, an inverse relationship between the beta and portfolio returns should exist and test for a systematic conditional relationship between the realized portfolio returns and the beta. Their empirical investigation of U.S. data reveals a positive slope on beta in the up market and a negative relationship in the down market. An analysis of the unconditional and the systematic conditional relationship between returns and beta on the Brussels Stock Exchange reveals that unconditional betas are unable to explain the cross-sectional observed returns, where as the conditional model does (Crombez and Vander Vennet, 2000). Friend and Westerfield (1980) examine beta and co-skewness in the up- and down-markets and report that while beta is significant in both markets and its signs are consistent with the CAPM theory, the co-skewness is significant only in the up-market.
To date, we believe, no study has adopted the Pettengill, Sundaram, and Mathur approach to investigate the extended CAPM with higher-order co-moments (systematic variance, systematic skewness and systematic kurtosis) and for high frequency data. We investigate the CAPM with higher-order co-moments using a sample of securities listed in the Australian Stock Exchange and with daily data. As it is clear from stylized facts that the skewness and kurtosis of the returns distribution become prominent in the high frequency data, our study is more likely to uncover any unconditional relationship between return and higher moments.
Higher-Order Pricing Models
In this section, we include two versions of the four-moment CAPM and specify the cross-sectional model reflecting the relationship between the asset returns and higher-order co-moments conditioned on market movements.
Four-Moment CAPM
The following are two versions of the four-moment CAPM, in which it is assumed that only the risks measured by systematic variance, systematic skewness, and systematic kurtosis are priced.
Kraus and Litzenberger (1976) version:
(1) E([R.sub.i])-[R.sub.f] = [[alpha].sub.1][[beta].sub.im] + [[alpha].sub.2][[gamma].sub.im] + [[alpha].sub.3][[theta].sub.im]
where [R.sub.f] and [R.sub.i] are returns on the risk-free asset and risky asset i, respectively,
(2) [[beta].sub.im] = E[([R.sub.it] - E([R.sub.i]))([R.sub.mt] - E([R.sub.m]))]/[E([R.sub.mt] - E([R.sub.m])).sup.2] = beta,
(3) [[gamma].sub.im] = E[([R.sub.it] - E([R.sub.i])) [([R.sub.mt] - E([R.sub.m])).sup.2]]/E[([R.sub.mt] - E([R.sub.m])).sup.3] = co-skewness,
(4) [[theta].sub.im] = E[([R.sub.it] - E([R.sub.i]))[([R.sub.mt] - E([R.sub.m])).sup.3]]/E[([R.sub.mt] - E([R.sub.m])).sup.4]] = co-kurtosis.
Due to the desirable properties of the utility function, we expect the market price of beta reduction by one unit to be [[alpha].sub.1], which is expected to be positive as in the conventional CAPM. The market price of co-skewness is [[alpha].sub.2], which is expected to have the opposite sign to the skewness of the market return distribution. The market price of co-kurtosis is [[alpha].sub.3], which is an additional measure of degree of dispersion in returns and is expected to be positive.
Sears and Wei (1985) version:
(5) E([R.sub.i]) - [R.sub.f] = ([c.sub.1][[beta].sub.im] + [c.sub.3][[gamma].sub.im] + [c.sub.3][[theta].sub.im] (E([R.sub.m]) - [R.sub.f].
A derivation of equations (1) and (5) is available in Hwang and Satchell (1999).
Four-Moment conditional Model
When testing the significance of parameters in empirically testable forms of equation (1), (2) previous studies use realized returns in lieu of expected returns. Pettengill, Sundaram, and Mathur argue that when the market return is lower than the risk-free rate and the realized returns for high beta portfolios are lower than the realized returns for low beta portfolios an inverse relationship between the beta and returns could reasonably be inferred during the time periods when the realized market return is less than the risk-free return. In what follows, these ideas are extended to the CAPM with higher-order co-moments. We argue that using the empirical value of market return could lead to bias results because the relationship between co-moments and realized returns may differ from that of the relationship between co-moments and expected return.
The four-moment CAPM version given by equation (1) does not provide a direct relationship between security co-moments and security returns when the realized market return is less than the risk-free rate. Equating (3) the coefficients of the co-moments in equations (1) and (5), we obtain [[alpha].sub.1] = [c.sub.1](E([R.sub.m]) - [R.sub.f]), [[alpha].sub.2] = [c.sub.2](E([R.sub.m]) - [R.sub.f]) and [[alpha].sub.3] = [c.sub.3](E([R.sub.m]) - [R.sub.f]). These relationships suggest that the sign of the parameters of equation (1) could be affected, and, consequently, the tests of the unconditional asset pricing model when realized returns instead of expected returns are used (Sears and Wei, 1985). When the market return is lower than the risk-free rate, we hypothesize an inverse relationship between security returns and higher-order co-moments and suggest a systematic conditional relationship between the security return and the higher-order co-moments given as:
(6) [R.sub.it] = [[delta].sub.0t] + [[delta].sub.1t][kappa][[beta].sub.im] + [[delta].sub.2t](1-[kappa])[[beta].sub.im] + [[delta].sub.3t][kappa][[gamma].sub.im] + [[delta].sub.4t](1-[kappa]) [[gamma].sub.im] + [[delta].sub.5t][kappa][[theta].sub.im] + [[delta].sub.6t] (1-[kappa])[[theta].sub.im] + [[epsilon].sub.it]
where [kappa] = 1 when ([R.sub.mt] - [R.sub.ft]) > and [kappa] = when ([R.sub.mt] - [R.sub.ft]) < 0. We refer to equation (6) as the four-moment conditional model. (4)
The sign expected for the coefficients of equation (6) can be inferred from the hypothesized inverse relationship and the signs expected for [[alpha].sub.1], [[alpha].sub.2], and [[alpha].sub.3] in the unconditional model given in equation (1). That is, we expect [[bar.[delta]].sub.1], > 0, [[bar.[delta]].sub.2] and [[bar.[delta]].sub.4] when up market return distribution is positively skewed, [[bar.[delta]].sub.5] > and [[bar.[delta]].sub.6] < 0. [[bar.[delta]].sub.1], [[bar.[delta]].sub.3], and [[bar.[delta]].sub.5] are the average risk premium corresponding to systematic variance, systematic skewness and systematic kurtosis respectively in the up market and [[bar.[delta]].sub.2], [[bar.[delta]].sub.4], and [[bar.[delta]].sub.6] are the average risk premium corresponding to systematic variance, systematic skewness and systematic kurtosis respectively in the down market. We test the existence of a conditional relationship by examining the coefficients of the cross-sectional regression model in equation (6).
Hypotheses of Interest
In order to see if there is supportive empirical evidence of a conditional relationship between expected return and higher-order co-moments, the following pairs of hypotheses are tested.
Test for a systematic conditional relationship between beta and realized returns: {[H.sub.0] : [[bar.[delta]].sub.1] =...
NOTE: All illustrations and photos
have been removed from this article.
Looking for additional articles?
Search our database of over 3 million articles.
Looking for more in-depth information on this industry?
Search our complete database of Industry & Market reports by text, subject, publication
name or publication date.
About Goliath
Whether you're looking for sales prospects, competitive information, company
analysis or best practices in managing your organization,
Goliath can help you meet your business needs.
Our extensive business information databases empower business
professionals with both the breadth and depth of credible,
authoritative information they need to support their business
goals. Whether it be strategic planning, sales prospecting,
company research or defining management best practices -
Goliath is your leading source for accurate information.
|
