Using more stringent test assets and more formal model diagnostic tools, the first essay demonstrates the importance of higher-order comoment risks in asset pricing by assessing the performance of the most commonly used asset pricing models with and without these risks incorporated. Specifically, we find that higher-order comoment risks help the Fama and French serial pricing kernels to be closer to the admissible pricing kernel and that the newly developed Fama and French five-factor model (Fama and French, 2015), when augmented by the quadratic and cubic terms of the market return and with momentum incorporated, requires the least adjustment to be admissible.