A note on P- vs. Q-expected loss portfolio constraints

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