A predictive approach to quantiles: Application to Value at Risk and Tail Value at Risk

We prove that quantiles are best predictors in a special metric. The best predictor turns out to coincide with the notions of generalized arithmetic mean, exponential barycenter and certainty equivalent. We also show that the computation of tail value at risk (TVaR) reduces to the computation of a quantile with a higher level of confidence. This point of view makes the analysis of the statistical properties of TVaR easier.