A Note on P- vs. Q-Expected Loss Portfolio Constraints

We consider portfolio optimization problems with expected loss constraints under the physical measure P and the risk neutral measure Q, respectively. Using Merton's portfolio as a benchmark portfolio, the optimal terminal wealth of the Q-risk constraint problem can be easily replicated with the standard delta hedging strategy. Motivated by this, we consider the Q-strategy fulfilling the P-risk constraint and compare its solution with the true optimal solution of the P-risk constraint problem. We show the existence and uniqueness of the optimal solution to the Q-strategy fulfilling the P-risk constraint, and provide a tractable evaluation method. The Q-strategy fulfilling the P-risk constraint is not only easier to implement with standard forwards and puts on a benchmark portfolio than the P-risk constraint problem, but also easier to solve than either of the Q- or P-risk constraint problem. The numerical test shows that the difference of the values of the two strategies (the Q-strategy fulling the P-risk constraint and the optimal strategy solving the P-risk constraint problem) is reasonably small