An alternative to portfolio selection problem beyond Markowitz's: Log Optimal Growth Portfolio

This paper constructs an alternative investment strategy to portfolio optimization model in the framework of the Mean‐Variance portfolio selection model. To dierentiate it from the ubiquitously applied Mean‐Variance model, which is constructed on an assumption that returns are normally distributed, our model makes two assumptions: Firstly, that asset prices follow a Geometric Brownian Motion and that secondly asset prices are Log-normally distributed meaning that continuously compounded returns are normally distributed. The traditional Mean‐Variance optimization approach has only one objective, which fails to capture the stochastic nature of asset returns and their correlations. This paper presents an alternative approach to the portfolio selection problem. The proposed optimization model which is an optimal portfolio strategy is produced for investors of various risk tolerance, taking into account the stochastic nature of the returns. Detailed analysis based on log‐ optimal growth optimization and the application of the model are provided and compared to the standard Mean‐Variance approach. 1 Introduction to Portfolio Optimization In this research paper, we construct the growth optimal portfolio (GOP) which is a strategic asset allocation process more suited for those investors with a long term investment view and wish to maximize their expected utility of terminal wealth. Growth optimal portfolio arise from the notion of computing the investment internal rate of return which in essence is bent on constructing those portfolio that have maximal growth. In principle we build a portfolio of risky asset that maximizes the geometric mean. The paper has primarily been inspired and written in the framework of Modern Portfolio Theory (MPT). Portfolio optimization in the context of portfolio theory is a classical problem in mathematical finance which has spawned a great amount of important academic work. In particular it is one of the well studied classical problems. Central to (MPT) is the Mean‐Variance optimization theory(MVO), an important model which was a major breakthrough developed in the 50s and 60s by Markowitz (1952, 1959), his paper opened a new era in the theory of portfolio selection. It plays a important and critical role in determining passive portfolio investment strategies for rational investors and quantifies precisely the relationship between risk and return. His theory set out a way of diversifying investment portfolios so that for any degree of risk, the investor got the best return possible, or alternatively, for any risk, the investor bore the lowest risk. Tobin (1958) built on Markowitz work with the formulation of the “Capital market line” portfolio. Following Markowitz and Tobin, the theory generated a lot of interest and the general equilibrium model capital asset pricing model “CAPM” was independently developed by Sharpe (1964), Lintner (1965), Mossin (1966) . Markowitz theories pervade the finance industry and are well-known to almost everyone vested in portfolio management. Nonetheless, the Mean‐Variance theory suers from certain well-known drawbacks. The most notable one is that, it is a static optimization problem and is only concerned with