Based on the alpha (α) we wanted to estimate, we also proposed a different method to test the market efficiency. We also wanted to find out whether the monthly or daily data with the same estimator will yield better alpha estimation. In light of recent research of Le, Kim, and Su (2018), we wanted to study the efficiency of alpha (α) estimation with alternative methods, the robust maximum likelihood type mestimator (MM estimator) developed by Huber (1964, 1973) and Yohai (1985, 1987) and the Bayes estimator with conjugate prior, comparing with OLS estimator in this paper. This raised some concerns about the validity of alpha (α) estimate and investment decision making process with the ordinary least square estimator. However, the returns of security are known to be not normally distributed, especially with small sample size data set as Fama (1965), McDonald and Nelson (1989), and Martin and Simin (1999) pointed out in their research. In practices as well as in finance literature, the alpha (α) has often been estimated with ordinary least square (OLS) estimator and monthly data set. Hence, alpha estimation methods are critical to those investors. This strategy becomes one of the investment strategies for active investors. Some investors spend great time to locate a security or a portfolio of securities that has a positive alpha (α), especially in non-efficient markets. Therefore, alpha becomes one of the key risk metrics used in the modern portfolio theory as stated in Association for Investment Management and Research (AIMR) performance presentation standards handbook (1996) even though it has deficiency as discussed in Black, Jensen, and Scholes (1972). The existence of this alpha (α) or abnormal return of a securities or portfolio of securities in worldwide financial markets has been documented by Jensen (1967 & 1969) himself, Kothari and Warner (1997a, 1997b), Liang (2008), Gerber and Hensz (2009), and many other researchers in the literature. The higher the alpha, the better performance of security or a portfolio of securities since it has earned more than expected return in Capital Asset Pricing Model. This metric is used to measure the risk-adjusted return of a security or a portfolio of securities in line with the expected market return from Capital Asset Pricing Model (CAPM) discussed by Treynor (1961), Sharpe (1964), and Lintner (1965). The Jensen’s alpha (α) introduced by Jensen (1967, 1969). More important, those findings above are checked with and validated by Jackknife resampling results. We also proposed an alternative market efficiency test with the hypothesis testing Ho: α = 0 and was able to prove the S&P 500 index is efficient, but not perfect. Interestingly, we also found that daily return data set would give more accurate alpha estimation than monthly return data set in all three MM, OLS, and Bayes estimators. The Bayes estimator did not perform better than the OLS estimator as expected. This research showed that the robust MM estimator performed well better than the OLS and Bayes estimators in terms of efficiency. Their daily and monthly returns were collected over a period of the last five years. A sample of 50 securities is randomly selected from the list of the S&P 500 index. According to finance literature and practices, alpha has often been estimated using ordinary least square (OLS) regression method and monthly return data set. This research examined the alternatives of Jensen's alpha (α) estimation models in the Capital Asset Pricing Model, discussed by Treynor (1961), Sharpe (1964), and Lintner (1965), using the robust maximum likelihood type m-estimator (MM estimator) and Bayes estimator with conjugate prior.
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