In Rockafellar and Uryasev's Paper about CVaR Optimisation they showed in Equation (17) that using Monte-Carlo-Simulation one can use $$\tilde F_{\beta}(x,\alpha)=\alpha+\frac{1}{q(1-\beta)}\sum_{k=1}^qmax(-x^Ty_k-\alpha,0)$$ to approximate the CVaR approximation, where $\alpha$ denotes the percentile of $\beta-$CVaR, $x\in\Bbb R^n$ the portfolio and $y_k$ the return in k-th scenario. In the following paragraph he introduced some auxiliary real variable $u_k$ for $k=1,...,r$ and claim it is equivalent to minimising the linear expression $$\alpha+\frac{1}{q(1-\beta)}\sum_{k=1}^qu_k$$ subject to $u_k\geq0$ and $x^Ty_k+\alpha+u_k\geq0$.


  1. Why is the indices of $u_k$ running from 1 to r and not to q?
  2. Why are the both problem equivalent? I can rewrite the last condition in the second problem as $u_k\geq-x^Ty_k-\alpha$ and together with $u_k\geq0$ I am allowing $-x^Ty_k-\alpha\leq0$ in the sum, which has value 0 in the first problem with $max(-x^Ty_k-\alpha,0)$ in the summand.

1 Answer 1


On 1, I suspect that is a typo and that the second formula should sum to r.

On 2, that is applying well-known techniques in how to handle piece-wise linear functions in an optimizer. For instance, see page 4 of these lecture notes. It's basically doing the same thing with a few additional complications. In CVaR optimization, there are more things to sum and also $\alpha$ is also part of the optimization (following the optimization, $\alpha$ should equal the Value-at-Risk).

Finally, there's an issue with your math: $-x^Ty_k-\alpha\leq0$ only follows if $u_k$ is always $0$.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.