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6 votes
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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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5 votes
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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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5 votes
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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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3 votes
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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 ...
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  • 1,933
3 votes

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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  • 1,585
3 votes
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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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2 votes
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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 ...
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  • 136
2 votes

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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2 votes
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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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  • 20.4k
2 votes
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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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  • 1,335
2 votes
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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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  • 81
2 votes
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How to prove the following relation of Conditional Value-at-Risk and Value-at-Risk?

A slightly different take here:
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2 votes

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*} \...
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  • 20.4k
2 votes

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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2 votes
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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 ...
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  • 2,740
2 votes
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Showing that the shortfall-to-quantile ratio of a normal distribution goes to one

$$ \lim_{x \to \infty} \frac{\sigma - \sigma \Phi(x)}{(\mu + \sigma x)\phi(x)} $$ $$\stackrel{0/0}{=} \lim_{x \to \infty} \frac{-\sigma \phi(x)}{\sigma\phi(x) + (\mu + \sigma x)\phi'(x) } $$ $$ =\...
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  • 4,963
2 votes

Mean-EVaR efficient frontier

It makes very little sense compare efficient frontiers across different risk measures, as per your attached picture. That is because efficient Mean-CVaR portfolios are always sub-optimal compared to ...
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  • 349
1 vote

Mean-EVaR efficient frontier

Yes, you can check the paper Entropic Portfolio Optimization: a Disciplined Convex Programming Framework in SSRN.
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  • 11
1 vote
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Estimating CVaR for non-Gaussian distributions

Using a bunch of Dirac delta functions would not be a good idea; you essentially would be assuming a distribution of point masses instead of a continuous distribution. If you work with the integral of ...
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  • 2,740
1 vote

Subadditivity of cvar(R)، R is random vector

Yes, conditional VaR (aka Expected Shortfall) is a coherent risk measure and thus, satisfies Monotonicity, Translation invariance, Positive homogeneity and Subadditivity. The latter means that $CVaR(...
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1 vote
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Why no median-CVaR optimization for portfolios?

Mean-CVaR portfolio optimization is an alternative to the more widely known and simpler mean-variance model. Since there doesn't seem to be any median-variance model out there, the familiarity ...
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  • 2,815
1 vote

Why Can I not estimate a CVAR from Heston Model

Given the main uses of the VaR relate to risk management such as limit management, and measurement of P&L volatility, it is usually calculated under the physical/real world measure. Reason being ...
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1 vote

How to calculate $\frac{\partial\ \text{CVaR}_{\alpha}(\min(X,d))}{\partial d}$ and $\frac{\partial\ \text{VaR}_{\alpha}(\min(X,d))}{\partial d}$?

It really depends on the way you calculate your Var and CvaR. If your are able to get a closed form solutions for the derivative then you must use those, for faster results. Otherwise, you can use ...
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  • 221
1 vote

How to calculate the distortion function for CVaR?

Maybe prove that $$CVaR_\alpha (X) = \frac{1}{\alpha} \int_0^\alpha F^{-1}_X(u) du$$ has the distortion function $$ g(u)= \begin{cases} \frac{u}{\alpha}, \quad \; u \leq \alpha \\ 1, \qquad ...
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1 vote
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Elicitability of risk measures

From Ziegel (2013) : The risk of a financial position is usually summarized by a risk measure. As this risk measure has to be estimated from historical data, it is important to be able to verify and ...
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