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We consider the problem of estimating a spectral risk measure (SRM) from i.i.d. samples, and propose a novel method that is based on numerical integration. We show that our SRM estimate concentrates exponentially, when the underlying distribution has bounded support. Further, we also consider the case when the underlying distribution satisfies an exponential moment bound, which includes sub-Gaussian and subexponential distributions. For these distributions, we derive a concentration bound for our estimation scheme. We validate the theoretical findings on a synthetic setup, and in a vehicular traffic routing application.