Dynamic shortfall constraints for optimal portfolios

We consider a portfolio problem when a Tail Conditional Expectation constraint is imposed. The financial market is composed of n risky assets driven by geometric Brownian motion and one risk-free asset. The Tail Conditional Expectation is calculated for short intervals of time and imposed as risk constraint dynamically. The method of Lagrange multipliers is combined with the Hamilton-Jacobi-Bellman equation to insert the constraint into the resolution framework. A numerical method is applied to obtain an approximate solution to the problem. We find that the imposition of the Tail Conditional Expectation constraint when risky assets evolve following a log-normal distribution, curbs investment in the risky assets and diverts the wealth to consumption.