Two state Markov model of the S&P 500 stock return distribution
ABSTRACT
By
Anthony Tessitore and Nilufer Usmen
November 9, 2013
Objective: The goal is to extend the research papers by Harry Markowitz and Nilufer Usmen (1996) and a later paper by Artun Alparslan, Anthony Tessitore and Nilufer Usmen (2013), whose results using Type IV distributions were derived under the assumption that S&P 500 returns were independent. The present paper postulates a dependent structure of returns through the Markov regime variable.
Data/Methods: The sample consists of daily returns of the S&P 500 stock index from July 2, 1962 to August 30, 2013. The source of the data is Bloomberg and the sample represents all the available data on the series. This paper presents a Markov model of the probability distribution of S&P 500 daily log returns. We assume an unobservable Markov process that selects a distribution based on the state of the process at a point in time. These state contingent distributions are modeled as Pearson Type IV densities, a four parameter family of distributions which contain the Student’s t distribution as a special case. Thus, the present paper postulates a dependent structure of returns through the Markov regime variable. The estimation procedure used in this paper is based on the principals developed in Lindgren (1978) who derived the recursion formula for Markov mixed distribution models.
Results: Our results so far show that the distribution that best fits the empirical data is a mixture of two conditional Type IV densities. The first represents a regime of moderate risk. The density has positive mean, small standard deviation, some negative skewness and kurtosis around 5, somewhat larger than the Gaussian density. The second conditional indicates more risky regime. It has negative mean, twice the level of standard deviation as the first, has smaller but still negative skewness but high kurtosis of around 11.4. We find that both densities have finite moments which stand in contrast to the infinite third and fourth moments derived under the assumption of independence. Moreover, the likelihood ratio test shows that the mixed model is hundreds of order of magnitude more likely as a model of stock return distributions than the independent case.
JEL CATEGORY: C11/G11