Optimal Portfolio Estimation for Dependent Financial Returns with Generalized Empirical Likelihood

This paper proposes to use the method of generalized empirical likelihood to find the optimal portfolio weights. The log-returns of assets are modeled by multivariate stationary processes rather than i.i.d. sequences. The variance of the portfolio is written by the spectral density matrix, and we seek the portfolio weights which minimize it. 1. Introduction The modern portfolio theory has been developed since circa the 1950s. It is common knowledge that Markowitz [1, 2] is a pioneer in this field. He introduced the so-called mean-variance theory, in which we try to maximize the expected return (minimize the variance) under the constant variance (the constant expected return). After that, many researchers followed, and portfolio theory has been greatly improved. For a comprehensive survey of this field, refer to Elton et al. [3], for example. Despite its sophisticated paradigm, we admit there exists several criticisms against the early portfolio theory. One of them is that it blindly assumes that the asset returns are normally distributed. As Mandelbrot [4] pointed out, the price changes in the financial market do not seem to be normally distributed. Therefore, it is appropriate to use the nonparametric estimation method to find the optimal portfolio. Furthermore, it is empirically observed that financial returns are dependent. Therefore, it is unreasonable to fit the independent model to it. One of the nonparametric techniques which has been capturing the spotlight recently is the empirical likelihood method. It was originally proposed by Owen [5, 6] as a method of inference based on a data-driven likelihood ratio function. Smith [7] and Newey and Smith [8] extended it to the generalized empirical likelihood (GEL). GEL can be also considered as an alternative of generalized methods of moments (GMM), and it is known that its asymptotic bias does not grow with the number of moment restrictions, while the bias of GMM often does. From the above point of view, we consider to find the optimal portfolio weights by using the GEL method under the multivariate stationary processes. The optimal portfolio weights are defined as the weights which minimize the variance of the return process with constant mean. The analysis is done in the frequency domain. This paper is organized as follows. Section 2 explains about a frequency domain estimating function. In Section 3, we review the GEL method and mention the related asymptotic theory. Monte Carlo simulations and a real-data example are given in Section 4. Throughout this paper, and indicate the transposition and