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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 fulfilling the P -risk constraint and the optimal strategy solving the P -risk constraint problem) is reasonably small. [ABSTRACT FROM AUTHOR]