Non-Concave Portfolio Optimization with Average Value-at-Risk

Average Value-at-Risk (AVaR) is a potential alternative to Value-at-Risk in the financial regulation of banking and insurance institutions. To understand how AVaR influences a company's investment behavior, we study portfolio optimization under the AVaR constraint. Our main contribution is to derive analytical solutions for non-concave portfolio optimization problems under the AVaR constraint in a complete financial market by quantile formulation and the decomposing method, where the non-concavity arises from assuming that the company is surplus-driven. We find that the company takes three different investment strategies corresponding to a stringent, a medium, and a loose AVaR constraint. Under each investment strategy, we derive the fair return for the company's debt holders fulfilling the risk-neutral pricing constraint in closed form. Further, we illustrate the above analytical results in a Black-Scholes market. We find that the fair return varies drastically, e.g., from 2.59\% to 35.3\% in different situations, implying that the company's strategy intimately determines the default risk faced by its debt holders. Our analysis and numerical experiment show that the medium AVaR constraint provides the best protection for the debt holders in reducing the company's default risk