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 decomposition method, where the non-concavity arises from assuming that the company is surplus-driven. Given the AVaR constraint, the company takes three investment strategies depending on its initial budget 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 4.99% to 37.2% 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 AVaR constraint cannot eliminate the company's default risk but can reduce it compared with the benchmark portfolio. However, the protection for the debt holders is poor if the company has a low initial budget.