**Date Published:** December 29, 2014

**Publisher:** Public Library of Science

**Author(s):** Mihály Ormos, Dávid Zibriczky, Giampiero Favato.

http://doi.org/10.1371/journal.pone.0115742

**Abstract**

**We investigate entropy as a financial risk measure. Entropy explains the equity premium of securities and portfolios in a simpler way and, at the same time, with higher explanatory power than the beta parameter of the capital asset pricing model. For asset pricing we define the continuous entropy as an alternative measure of risk. Our results show that entropy decreases in the function of the number of securities involved in a portfolio in a similar way to the standard deviation, and that efficient portfolios are situated on a hyperbola in the expected return – entropy system. For empirical investigation we use daily returns of 150 randomly selected securities for a period of 27 years. Our regression results show that entropy has a higher explanatory power for the expected return than the capital asset pricing model beta. Furthermore we show the time varying behavior of the beta along with entropy.**

**Partial Text**

We build an equilibrium capital asset pricing model by applying a novel risk measure, the entropy. Entropy characterizes the uncertainty or measures the dispersion of a random variable. In our particular case, it characterizes the uncertainty of stock and portfolio returns. In modern Markowitz [1] portfolio theory and equilibrium asset pricing models [2] we apply linear regressions. This methodology supposes that the returns are stationary and normally distributed; however, this is not actually the case [3]. Entropy, on the other hand, does not have this kind of boundary condition. The main goal of this paper is to apply entropy as a novel risk measure. As a starting point even the density function itself has to be estimated. In the traditional asset pricing model there is equilibrium between expected return the beta parameter, which is the covariance–variance ratio between the market portfolio and the investigated investment opportunity. If the random variable is normally distributed then the entropy follows its standard deviation; thus in the ideal case there is no difference between the two risk measures. However; our results show that there is a significant difference between the standard deviation, or beta, and the entropy of a given security or portfolio. In this paper we show that entropy offers an ideal alternative for capturing the risk of an investment opportunity. If we explain the return of a wide sample of securities and portfolios with different risk measures then on an ordinary least squares (OLS) regression setting the explanatory power is much higher in the case of the entropy measure of risk than in the case of the traditional measures, both in-sample and out-of-sample. We show that entropy reduction in line with diversification behaves similarly to standard deviation; however at the same time it captures a beta-like systematic risk of single securities or non-efficient portfolios as well. For well-diversified portfolios the explanatory power of entropy is 1.5 times higher than that of the capital asset pricing model (CAPM) beta.

In our empirical analysis we apply daily returns from the Center for Research in Security Prices (CRSP) database for the period from 1985 to the end of 2011. We randomly select 150 securities from the S&P500 index components that are available for the full period. The market return is the CRSP value-weighted index return premium above the risk-free rate. The index tracks the return of the New York Stock Exchange (NYSE), the American Stock Exchange (AMEX) and NASDAQ stocks. The risk-free rate is the return of the one-month Treasury bill from the CRSP. We use daily returns because they are not normally distributed (see S1 Table). Erdős and Ormos (2009) [3] and Erdős et al. (2011) [4] describe the main difficulties of modeling asset prices with non-normal returns. The daily return calculation enables us to compare different risk measures.

Entropy is a mathematically-defined quantity that is generally used for characterizing the probability of outcomes in a system that is undergoing a process. It was originally introduced in thermodynamics by Rudolf Clausius [5] to measure the ratio of transferred heat through a reversible process in an isolated system. In statistical mechanics the interpretation of entropy is the measure of uncertainty about the system that remains after observing its macroscopic properties (pressure, temperature or volume). The application of entropy in this perspective was introduced by Ludwig Boltzmann [6]. He defined the configuration entropy as the diversity of specific ways in which the components of the system may be arranged. He found a strong relationship between the thermodynamic and the statistical aspects of entropy: the formulae for thermodynamic entropy and configuration entropy only differ in the so-called Boltzmann constant. There is an important application of entropy in information theory as well, and this is often called Shannon [7] entropy. The information provider system operates as a stochastic cybernetic system, in which the message can be considered as a random variable. The entropy quantifies the expected value of the information in a message or, in other words, the amount of information that is missing before the message is received. The more unpredictable (uncertain) the message that is provided by the system, the greater the expected value of the information contained in the message. Consequently, greater uncertainty in the messages of the system means higher entropy. Because the entropy equals the amount of expected information in a message, it measures the maximum compression ratio that can be applied without losing information.

We present the empirical results in four parts. First, we show how the entropy behaves in the function of securities involved into the portfolio. Second, we present the long-term explanatory power of the investigated models. Third we examine and compare the performance of different risk measures in in upward and downward market trends. Fourth we apply the different risk parameters to predict future returns, thus we test the out of sample explanatory power of the well-known risk parameters and compare their efficiency to the entropy based risk measures.

Entropy as a novel risk measure combines the advantages of the CAPM’s risk parameter (beta) and the standard deviation. It captures risk without using any information about the market, and it is capable of measuring the risk reduction effect of diversification. The explanatory power for the expected return within the sample is better than the beta, especially in the long run covering bullish and bearish periods; the predictive power for the expected return is higher than for standard deviation. Both the Shannon and the Rényi entropies give more reliable risk estimation; their explanatory power exhibits significantly lower variance compared to the beta or the standard deviation. If upward and downward trends are distinguished, the regime dependency of entropy can be recognized: this result is similar to that for the beta. Among the entropy estimation methods reviewed, the histogram-based method proved to be the most efficient in terms of explanatory and predictive power; we propose a simple estimation formula for the Shannon and the Rényi entropy functions, which facilitates the application of an entropy-based risk measure.

Source:

http://doi.org/10.1371/journal.pone.0115742