Minimum capital requirement portfolios according to the new Basel framework for market risk

In the new Basel framework for market risk finalized in January 2019, the minimum capital requirement for banks involves a liquidity-adjusted Stressed Expected Shortfall and a penalization factor that depends on the outcome of a Value at Risk-based backtesting procedure. This paper examines the optimization problem of a bank aiming to minimize its capital requirement, expressed analytically by a non-convex and non-differentiable function, being the penalization a discrete function of the number of Value at Risk violations. To address the portfolio selection problem, we implement an algorithm based on the Particle Swarm Optimization metaheuristic. In the empirical analysis, we compare portfolios obtained by minimizing the capital requirement with those selected by standard risk measures, i.e. Expected Shortfall, Value at Risk, and variance. Two data samples are considered: one from the recent Covid-19 pandemic period and the other from a quieter phase. Our findings suggest that minimizing the capital requirement during highly volatile periods, such as the global pandemic, would help contain the number of backtesting violations compared to other strategies. By contrast, during tranquil periods the inclusion of the penalization multiplier in the objective function would have no impact. However, portfolios selected to minimize Expected Shortfall, Value at Risk, and variance in both cases are far from the regulatory capital efficient frontier, resulting in varying degrees of efficiency losses.