Academic journal article Quarterly Journal of Business and Economics

Empirical Evidence on the Conditional Relation between Higher-Order Systematic Co-Moments and Security Returns

Academic journal article Quarterly Journal of Business and Economics

Empirical Evidence on the Conditional Relation between Higher-Order Systematic Co-Moments and Security Returns

Article excerpt

Introduction

Since the seminal paper by Markowitz (1959), the capital asset pricing model (CAPM) of Sharpe (1964) and Lintner (1965) has become an important tool in finance for assessment of cost of capital, portfolio performance and diversification, valuing investments, and choosing portfolio strategy, among others. The 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. …

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