The majority of financial data exhibit asymmetry and heavy tails, which makes forecasting the entire density critically important. Recently, a forecast combination methodology has been developed to combine predictive densities. We show that combining individual predictive densities that are skewed and/or heavy-tailed results in significantly reduced skewness and kurtosis. We propose a solution to overcome this problem by deriving optimal log score weights under Higher-order Moment Constraints (HMC). The statistical properties of these weights, such as consistency and asymptotic distribution, are investigated theoretically and through a simulation study. An empirical application that uses the S&P 500 daily index returns illustrates that the proposed HMC weight density combinations perform very well relative to other combination methods.