The stochastic shortest path problem (SSPP) asks to resolve the non-deterministic choices in a Markov decision process (MDP) such that the expected accumulated weight before reaching a target state is maximized. This paper addresses the optimization of the variance-penalized expectation (VPE) of the accumulated weight, which is a variant of the SSPP in which a multiple of the variance of accumulated weights is incurred as a penalty. It is shown that the optimal VPE in MDPs with non-negative weights as well as an optimal deterministic finite-memory scheduler can be computed in exponential space. The threshold problem whether the maximal VPE exceeds a given rational is shown to be EXPTIME-hard and to lie in NEXPTIME. Furthermore, a result of interest in its own right obtained on the way is that a variance-minimal scheduler among all expectation-optimal schedulers can be computed in polynomial time.