A note on - vs. -expected loss portfolio constraints

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