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Volume 9, Issue 4


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Skew Densities and Ensemble Inference for Financial Economics
H. D. Vinod

1. Introduction

Excess return from an asset (e.g., a mutual fund) is defined as the return over and above the risk-free rate (e.g., interest rate earned on the three-month U.S. Treasury Bills). Recent literature in financial economics recognizes that the density of excess returns is often nonnormal due to skewness, kurtosis, and fat tails. While some extensions of the normal called log-normal and inverse Gaussian have been used, Azzalini's [1] SN density appears to have been ignored. The aim of this article is to introduce mathStatica implementation from [3]. This software also enables easy estimation of several other nonnormal for myriad potential uses in finance. We discuss ML estimators for the parameters of the SN density.

Wall Street investors, bankers, and government regulators often want a dollar figure on the potential loss in a worst-case scenario. VaR is one such measure developed by statisticians at RiskMetrics Group, a private company in New York. VaR is often described as based on historical volatility and tells the investor what might happen if some unusual event made the asset more volatile than normal. Although actual measurements differ among analysts, VaR is often obtained from a low (e.g., 1%) quantile of a parametric or nonparametric . Since mutual fund investors have seen three consecutive down years, both investors and regulators are concerned about fund VaR values. Morgenstern [4] notes that risky funds in the aggressive growth stock category have recently experienced $5.41 billion in redemptions compared to May 2002, whereas fund managers have increased assets in the aggressive category from $430 billion to $1.3 trillion. Our point is that VaR is important in finance, and we propose simple tools for calculating it with the help of mathStatica.

Section 2 considers the Alliance All-Asia Investment Advisors Fund mutual fund, with the ticker symbol AAAYX, for the period of months from January 1987 to December 1997 from [5]. Descriptive statistics reveal that the density is negatively skewed. Section 3 provides a generalised SN model adjusted for location and scale. Section 4 discusses ML estimation of the parameters of the generalised SN density. Section 5 briefly discusses VaR.

In future work, we expect to develop mathStatica software for a new maximum entropy algorithm (ME-alg) for the time series inference suggested in [6] for inference regarding the VaR. Classical time series inference methods treat the observed time series as one random realization from an unobservable ensemble CapitalOmega. These were developed in the 1930s before computers and before Shannon information and entropy ideas were developed. The ME-alg suggests a computer intensive construction of an approximate CapitalOmega built up from elements for with a large number. Section 6 contains our conclusions.

We begin by loading the mathStatica package.

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