It has been shown that the price distribution of financial assets is of the form of power law. Mandelbrot first noticed the scaling properties of financial markets. In particular, the power-law tails of empirical observations on financial markets have been studied extensively (). GARCH models are commonly employed in modeling financial time series that exhibit time-varying volatility and volatility clustering. If an autoregressive moving average model is assumed for the error variance, the model is a generalized autoregressive conditional heteroscedasticity (GARCH) model (see ). The autoregressive conditional heteroscedasticity (ARCH) model (see ) describes the variance of the current error term or innovation as a function of the actual sizes of the previous time periods’ error terms. It has been suggested that the volatility of an empirical financial market should be a stochastic quantity (see ). There is empirical evidence indicating that volatility is driven by a mean-reverting stochastic process (see ). A variety of models have been proposed to improve Markowitz’s mean-variance model. However, empirical observations on financial markets show that the tails of the distribution decay slower than the log-normal distribution (see ). The mean-variance model is based on the assumption that return of assets follows a normal distribution. Markowitz’s mean-variance model is the pioneer portfolio selection model. The Lagrangian multipliers can be fixed exactly by the Conditional Value at Risk constraints. Algebraic relations between the Lagrangian multipliers and Value at Risk constraints are presented explicitly. We show that the maximum entropy distribution with Conditional Value at Risk constraints is a power law. It has already become an accepted standard in buying and selling assets. The Value at Risk is widely used in investment bank and commercial bank. This makes the risk management relatively simple. Value at Risk is a single number that indicates the extent of risk in a given portfolio. The metric is most commonly used by investment bank to determine the extent and occurrence ratio of potential losses in portfolios. Value at Risk measures the level of financial risk within a portfolio. Value at Risk is a financial metric that estimates the risk of an investment. In this paper, we use Conditional Value at Risk constraints instead of the variance constraint to maximize the entropy of portfolios. However, the mean and variance constraints were still used to obtain Lagrangian multipliers. In recent years, the maximum entropy method has been widely used to investigate the distribution of return of portfolios. The variance of a portfolio may also be a random variable. The distribution shows a power law at tail. However, empirical observations on financial markets show that the tails of the distribution decay slower than the log-normal distribution. The mean-variance model assumes that the probability density distribution of returns is normal. It is well known that Markowitz’s mean-variance model is the pioneer portfolio selection model.
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