Summary: The Fast Fourier Transform provides an alternative approximate method to evaluate the distribution of aggregate losses in insurance and finance. The efficiency of this method has already been proved for univariate and bivariate insurance models; therefore, in this paper, we extend it to a multivariate setting by considering its application to a particular model that includes losses of different types and dependency between them. Since the Fourier transform method works with truncated claims distributions, it can generate aliasing errors by wrapping around the probability mass that lies at the truncation point below this point. To eliminate this problem, we also discuss a suitable change of measure called exponential tilting that forces the tail of the distribution to decrease at exponential rate. Other possible errors are also discussed. We also illustrate the method on several numerical examples.