Kellys Criterion in Portfolio Optimization: A Decoupled Problem

📝 Abstract

Kelly’s Criterion is well known among gamblers and investors as a method for maximizing the returns one would expect to observe over long periods of betting or investing. These ideas are conspicuously absent from portfolio optimization problems in the financial and automation literature. This paper will show how Kelly’s Criterion can be incorporated into standard portfolio optimization models. The model developed here combines risk and return into a single objective function by incorporating a risk parameter. This model is then solved for a portfolio of 10 stocks from a major stock exchange using a differential evolution algorithm. Monte Carlo calculations are used to verify the accuracy of the results obtained from differential evolution. The results show that evolutionary algorithms can be successfully applied to solve a portfolio optimization problem where returns are calculated by applying Kelly’s Criterion to each of the assets in the portfolio.

💡 Analysis

Kelly’s Criterion is well known among gamblers and investors as a method for maximizing the returns one would expect to observe over long periods of betting or investing. These ideas are conspicuously absent from portfolio optimization problems in the financial and automation literature. This paper will show how Kelly’s Criterion can be incorporated into standard portfolio optimization models. The model developed here combines risk and return into a single objective function by incorporating a risk parameter. This model is then solved for a portfolio of 10 stocks from a major stock exchange using a differential evolution algorithm. Monte Carlo calculations are used to verify the accuracy of the results obtained from differential evolution. The results show that evolutionary algorithms can be successfully applied to solve a portfolio optimization problem where returns are calculated by applying Kelly’s Criterion to each of the assets in the portfolio.

📄 Content

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Kelly’s Criterion in Portfolio Optimization: A Decoupled Problem Zachariah Peterson1,* 1Adams State University School of Business 208 Edgemont Blvd. Alamosa, CO 81101 petersonz1@grizzlies.adams.edu

• Correspondence: petersonz1@grizzlies.adams.edu Abstract: Kelly’s Criterion is well known among gamblers and investors as a method for maximizing the returns one would expect to observe over long periods of betting or investing. This paper will show how Kelly’s Criterion can be incorporated into standard portfolio optimization models that include a risk function. The model developed here combines the risk and return functions into a single objective function using a risk parameter. This model is then solved for a portfolio of 10 stocks from a major stock exchange using a differential evolution algorithm. Monte Carlo calculations are used to directly simulate and compare the average returns from the Mean Variance and Kelly portfolios. The results show that Kelly’s Criterion can be used to calculate optimal returns and can generate portfolios that are similar to results from the Mean Variance model. The results also show that evolutionary algorithms can be successfully applied to solve this unique portfolio optimization problem. Keywords: portfolio optimization; Kelly criterion; differential evolution; mean variance; logarithmic utility Key Messages:  A decoupled form of logarithmic utility can be used to optimize portfolios  The decoupled Kelly model generates portfolios with similar returns as the MV model  The results were reached using test data with an intuitive solution JEL Classification: C6 Mathematical Methods; Programming Models; Mathematical and Simulation Modeling

1. Introduction In general, portfolio optimization problems aim to determine an optimal allocation of wealth among a pool of candidate securities. Portfolio optimization was first discussed in 1952 by Harry Markowitz in his work on modern portfolio theory (MPT) (Markowitz 1952). According to MPT, an optimum portfolio can be arranged such that return is maximized for a specified level of risk, or vice-versa, where risk is minimized for a specified level of return. Many formulations of portfolio optimization problems are linear or quadratic, depending on the definition of portfolio risk that is used in the particular problem. The original formulation of Markowitz is known as the Mean Variance (MV) model and treats return on a portfolio of investments using historical averages of changes in market prices for each asset. The total portfolio return was defined to be a weighted sum of returns from individual investments. Risk was defined as variance of returns and is found by taking the inner product of the covariance matrix for the assets in the portfolio. Later models for asset pricing, such as the Capital Asset Pricing Model (CAPM) (Fama and French 2004), would continue to use the

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covariance among changes in asset values to quantify risk. Other models, for example in (Bichpuriya and Soman 2016; El Ghaoui, et. al. 2003) use value-at-risk (VaR) or conditional value-at-risk (CVaR) to model the variation in portfolio returns. The MV approach to portfolio optimization, where returns are defined using average changes in market prices of assets over time, over-simplifies the problem. A better reflection of reality is to determine the probability distribution of price changes for each of the assets in the portfolio and reformulate the return function in terms of these probabilities. This is done in (Yang and Liu 2016), where returns are treated as fuzzy numbers. Portfolio optimization based on MPT has also been used in electricity generation and distribution, where electricity demand is treated as a random variable (Bichpuriya and Soman 2016). Both of these examples formulate the portfolio return function in terms of expectation values and covariant risk, generating a linear (Bichpuriya and Soman 2016; El Ghaoui, et. al. 2003) or quadratic (Yang and Liu 2016) objective function problem that has the same form as the MV formulation. The linear return function in the MV model has been used by many authors in the automation literature (Bichpuriya and Soman 2016; El Ghaoui, et. al. 2003; Yang and Liu 2016; Kamili and Riffi 2016; Chen, et. al. 2012; Zaheer and Pant 2016; Korczak and Roger 2000; Ma, et. al. 2012; Chang, et. al. 2009) and is treated as something of a standard model for portfolio optimization. However, one can show from probability theory that the optimum return on an investment (or portfolio of investments) is not a linear function of the fraction of wealth placed in each investment. Kelly’s Criterion is well known among gamblers as a betting strategy (Rotando and Thorpe 1992; Browne and Whitt 1996; Thorpe 1997). Kelly’s result is, in its simplest sense, a solution to an optimization problem which maximizes logarithmic utility and was originally applied

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