A note on - vs. -expected loss portfolio constraints

Abstract

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