Before revision:   Finally, we illustrate how investors with different  risk preferences invest and improve upon the benchmark using the Tail Value-at-Risk, inverse S-shaped distortion risk measures , and lower- and upper-tail  risk measures as examples.
After revision:  We further propose a simulation approach to calculate the optimal strategy's terminal wealth, making our approach applicable to a wide range of market models. Finally, we illustrate how investors with different risk preferences invest and improve upon the benchmark using the Tail Value-at-Risk, inverse S-shaped distortion risk measures , and lower- and upper-tail risk measures as examples.