Although multi-asset portfolios are central in modern finance, the multivariate statistical estimation involved in portfolio selection and management is not an easy task. This article focuses on the problem of estimating the probability of multi-asset portfolio large losses. We present a semi-parametric Extreme Value Theory (EVT) estimator of the probability of multivariate large losses. This estimator has two significant advantages over other estimators. First, it does not impose a parametric family on the dependence between large losses. Second, it does not suffer from the curse of dimensionality. An empirical implementation illustrates its relevance for portfolio selection and management practice.