Liu Chang, Chang Chuo, Chang Zhe
Institute of Chinese Finance Studies, Southwestern University of Finance and Economics, Chengdu 611130, China.
PBC School of Finance, Tsinghua University, Beijing 100083, China.
Entropy (Basel). 2020 Jun 16;22(6):663. doi: 10.3390/e22060663.
It is well known that Markowitz's mean-variance model is the pioneer portfolio selection model. The mean-variance model assumes that the probability density distribution of returns is normal. However, empirical observations on financial markets show that the tails of the distribution decay slower than the log-normal distribution. The distribution shows a power law at tail. The variance of a portfolio may also be a random variable. In recent years, the maximum entropy method has been widely used to investigate the distribution of return of portfolios. However, the mean and variance constraints were still used to obtain Lagrangian multipliers. In this paper, we use Conditional Value at Risk constraints instead of the variance constraint to maximize the entropy of portfolios. Value at Risk is a financial metric that estimates the risk of an investment. Value at Risk measures the level of financial risk within a 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 is a single number that indicates the extent of risk in a given portfolio. This makes the risk management relatively simple. The Value at Risk is widely used in investment bank and commercial bank. It has already become an accepted standard in buying and selling assets. We show that the maximum entropy distribution with Conditional Value at Risk constraints is a power law. Algebraic relations between the Lagrangian multipliers and Value at Risk constraints are presented explicitly. The Lagrangian multipliers can be fixed exactly by the Conditional Value at Risk constraints.
众所周知，马科维茨的均值 - 方差模型是开创性的投资组合选择模型。均值 - 方差模型假定收益的概率密度分布是正态的。然而，对金融市场的实证观察表明，该分布的尾部比对数正态分布衰减得更慢。该分布在尾部呈现幂律。投资组合的方差也可能是一个随机变量。近年来，最大熵方法已被广泛用于研究投资组合收益的分布。然而，仍使用均值和方差约束来获得拉格朗日乘数。在本文中，我们使用条件风险价值约束而非方差约束来最大化投资组合的熵。风险价值是一种估计投资风险的金融指标。风险价值衡量投资组合内的金融风险水平。该指标最常用于投资银行以确定投资组合中潜在损失的程度和发生率。风险价值是一个单一数字，表明给定投资组合中的风险程度。这使得风险管理相对简单。风险价值在投资银行和商业银行中被广泛使用。它已成为资产买卖中公认的标准。我们表明，具有条件风险价值约束的最大熵分布是幂律。明确给出了拉格朗日乘数与风险价值约束之间的代数关系。拉格朗日乘数可以通过条件风险价值约束精确确定。
