# Tag Info

Accepted

### Rockafellar-Uryasev mean-CVaR optimiztion

$VaR_\alpha$ is a scalar choice variable in the minimization problem. In the Rockafeller-Uryasev paper, it is simply called $\alpha\in R$. (C.f., the program described in Theorem 2 of that paper, or ...
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### Question on Rockafellar's Paper for optimisation of CVaR

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, ...
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### CVAR alternatives for optimization

Following the comments and the edits to the question, I'll try to show how conditional Value-at-Risk (aka Expected Tail Loss) can be minimised for a portfolio. We start with the implementation ...
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### Confidence Interval on Monte-Carlo-CVaR

One: Your VaR CI relies on normal approximation and might be (very) bad depending on the number of samples and the target function (P&L). Often it is better to use the exact approach based on the ...

### CVaR is concave risk measure or convex?

CVaR is a convex function in the underlying portfolio (measured as for instance absolute value or profit). I won't get into proving anything so instead I am going to link the first result from Google ...
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### CVaR formulation

If $Y=-\pi(\mu,D)$ then the first formula is $$\mathrm{CVaR}_\eta(-Y)=\max_{\nu\in R}\left\{\nu+\frac1\eta E((-Y-\nu)^-)\right\}$$ where $X^-=\min (X,0)$ and $X^+=\max(X,0)$. Note that $(-X)^-=-(X^+)$....
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### How to minimize $CVaR_{\alpha}(\min(X,d))$, where $X$ is a random variable and d is the decision variable?

The minimum value is always attained at $d=0$. In this proof, I will assume that the distribution of the random variable $X$ is absolutely continuous and monotonically increasing, and thus the CDF ...

### What are the advantages of $EVaR$ over $CVaR$?

The entropic value at risk (EVaR) is a coherent risk measure, developed to tackle some computational inefficiencies of the CVaR. It is the tightest possible upper bound for traditional VaR and CVaR, ...
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### Questions about VaR and CVaR. Is there any relation between $VaR_{\alpha}(X)$ and $VaR_{\alpha}(-X)$, or $CVaR_{\alpha}(X)$ and $CVaR_{\alpha}(-X)$?

We consider the case where the distribution function $F$ of $X$ is strictly increasing. Then \begin{align*} VaR_{\alpha}(X) &= \inf\{x: P(X >x) \le \alpha \}\\ &=\inf\{x: F(x)\ge 1-\alpha \...
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### Calculate CVaR for a portfolio

This sounds correct, however step 2 is a little vague, so I will try to restate the steps here for you. The assets in your portfolio must be priced with respect to a set of risk factors (e.g. ...
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### How to calculate the distortion function for CVaR?

I have solved it myself. The key was to realize that for $X \geq 0$ and $S_X(t) = \mathbb{P}(X>t)$ $$\int_0^\infty S(t) dt = \int_0^1 F_X^{-1}(u) du = \mathbb{E}\left[X \right].$$ This is ...
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### How to prove the following relation of Conditional Value-at-Risk and Value-at-Risk?

A slightly different take here:

### How to prove the following relation of Conditional Value-at-Risk and Value-at-Risk?

Let $F$ be the cumulative distribution function of $X$. We assume that $F$ is continuous. Then, for $x\ge 0$, \begin{align*} F^{-1}(x) = \inf\{s: F(s) \ge x \}. \end{align*} Moreover, \begin{align*} \...

### CVaR is concave risk measure or convex?

It does not even matter if it’s concave or convex wrt global optimisation, both concave and convex functions have global optimal points albeit the only difference is maximum vs minimum which is easily ...
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### When is the VAR equal to the CVAR

That is incredibly unlikely for a continuous distribution -- though possible for a distribution with a part that is not absolutely continuous, i.e. is atomic. The way to see this is to remember that ...