Risk-based power dispatch has been proposed as a viable alternative to Security-Constrained Dispatch to reduce power grid costs and help to better understand of prominent hazards. In contrast to classical approaches, risk-based frameworks assign different weights to different contingencies, quantifying both their likelihood occurrence and severity. This leads to an economically profitable operational schedule by exploiting the trade-off between grid risks and costs. However, relevant sources of uncertainty are often neglected due to issues related to the computational cost of the analysis. In this work, we present an efficient risk assessment frameworks for power grids. The approach is based on the Line-Outage Distribution Factors for the severity assessment of post-contingency scenarios. The proposed emulator is embedded within a generalized uncertainty quantification framework to quantify: (1) The effect of imprecision on the estimation of the risk index; (2) The effect of inherent variability, aleatory uncertainty, in environmental-operational variables. The computational cost and accuracy of the proposed risk model are discussed in comparison to traditional approaches. The applicability of the proposed framework to real size grids is exemplified by several case studies.