The probabilities considered in value-at-risk (VaR) are typically of moderate deviations. However, the variance reduction techniques developed in the literature for VaR computation are based on large deviations methods. Modeling heavy-tailed risk factors using multivariate $t$ distributions, we develop a new moderate-deviations method for VaR computation. We show that the proposed method solves the corresponding optimization problem exactly, while previous methods produce approximations to the exact solution. Thus, the proposed method consistently outperforms existing methods derived from large deviations theory under various settings. The results are confirmed by a simulation study.