  Let $\rho$ be a general law--invariant convex risk measure, for instance the
average value at risk, and let $X$ be a financial loss, that is, a real random
variable. In practice, either the true distribution $\mu$ of $X$ is unknown, or
the numerical computation of $\rho(\mu)$ is not possible. In both cases, either
relying on historical data or using a Monte-Carlo approach, one can resort to
an i.i.d.\ sample of $\mu$ to approximate $\rho(\mu)$ by the finite sample
estimator $\rho(\mu_N)$ (where $\mu_N$ denotes the empirical measure of $\mu$).
In this article we investigate convergence rates of $\rho(\mu_N)$ to
$\rho(\mu)$. We provide non-asymptotic convergence rates for both the deviation
probability and the expectation of the estimation error. The sharpness of these
convergence rates is analyzed. Our framework further allows for hedging, and
the convergence rates we obtain depend neither on the dimension of the
underlying assets, nor on the number of options available for trading.
