Estimation of Portfolio Return and Value at Risk Using a Class of Gaussian Mixture Distributions

This paper deals with the estimation of portfolio returns and Value at Risk (VaR), by using a class of Gaussian mixture distributions. Asset return distributions are frequently assumed to follow a normal or log normal distribution. It also can follow Brownian motion or Geometric Brownian motion based upon the Gaussian process. However, many empirical studies have shown that return distributions are usually not normal. They often find evidence of non-normality, such as heavy tails, excess kurtosis, finite moments, etc. We propose a class of Gaussian mixture distributions to approximate the return distributions of assets. This class of Gaussian mixture distributions, having good statistical properties, can accurately capture the above-mentioned statistical characteristics of return distributions. The model is applied easily to estimate the return distribution of a portfolio, and to evaluate the VaR. We demonstrate the model theoretically and provide some applications