We extend the semi-nonparametric (SNP) density of León, Mencía and Sentana (2009) to time-varying higher-order moments for daily asset return innovations of stock indexes and foreign-exchange rates. We estimate robust tail-indexes for testing the existence of the unconditional higher-order moments. We obtain closed-form expressions of partial moments and expected shortfall under the time-varying SNP density with the GJR-GARCH for modeling returns. A comparative study between SNP and Hansen's skewed-t, based on skewness-kurtosis frontiers, in-sample and backtesting analyses, is also implemented. Finally, we conduct an out-of-sample portfolio selection exercise for the stocks of the S&P 100 index through an equity screening method based on our parametric one-sided reward/risk performance measures and compare with the Sharpe ratio portfolio.