Maximum Varma Entropy Distribution with Conditional Value at Risk Constraints.

Area of Science:

• Quantitative Finance
• Financial Risk Management
• Information Theory

Background:

• Markowitz's mean-variance model, a foundational portfolio selection tool, assumes normal return distributions.
• Empirical financial data reveals power-law tail behavior, deviating from the log-normal distribution assumption.
• Traditional maximum entropy methods often rely on mean and variance constraints, which may not fully capture market realities.

Purpose of the Study:

• To investigate portfolio return distributions using maximum entropy with Conditional Value at Risk (CVaR) constraints.
• To replace the conventional variance constraint with CVaR in portfolio entropy maximization.
• To analyze the resulting distribution and the relationship between Lagrangian multipliers and CVaR constraints.

Main Methods:

• Application of the maximum entropy method to portfolio optimization.
• Incorporation of Conditional Value at Risk (CVaR) as a constraint, replacing variance.
• Derivation of algebraic relations between Lagrangian multipliers and CVaR constraints.

Main Results:

• The maximum entropy distribution under CVaR constraints exhibits a power-law characteristic.
• Explicit algebraic relationships are established between Lagrangian multipliers and the imposed CVaR constraints.
• Lagrangian multipliers are shown to be precisely determined by the CVaR constraints.

Conclusions:

• Conditional Value at Risk constraints provide a more suitable framework for maximizing portfolio entropy compared to variance.
• The derived power-law distribution offers a more realistic model for financial market returns.
• This approach enhances the understanding and management of portfolio risk through a robust theoretical framework.