18
$\begingroup$

When the risk is defined by a discrete random variable, is CVaR a coherent risk measure? I stick to the following definition of CVaR:

$$ CVaR_\alpha(R) = \min_v \quad \left\{ v + \frac{1}{1-\alpha} \mathbb{E}[R-v]^+ \right \}$$

where $R$ is the DISCRETE random variable for the loss and $\alpha$ is the confidence level.

Improve this question
$\endgroup$
1
  • 2
    $\begingroup$ I'm almost sure CVaR has been created to satisfy the sub-additivity property which isn't valid for normal VaR. The whole point of this new measure was to "enhance" VaR and have a coherent measure. I must say that I am unsure of what the fact that it is DISCRETE changes? $\endgroup$
    – SRKX
    Jun 24, 2011 at 10:51

3 Answers 3

Reset to default
12
$\begingroup$

I found this paper: Conditional value-at-risk for general loss distributions by Rockafellar and Uraysev http://dx.doi.org/10.1016/S0378-4266(02)00271-6

which says CVaR is coherent for general loss distributions, including discrete distributions.

I think that I was confused by other authors who were also confused with the definitions of CVaR. In particular, in the following paper, the author mistakenly stated that Tail Conditional Expectation (TCE) is same as CVaR, and they are not coherent.

http://dx.doi.org/10.1016/S0378-4266(02)00281-9

However, TCE is not same as CVaR in general. If the underlying distribution is continuous, they are same.

Improve this answer
$\endgroup$
1
  • $\begingroup$ For calculating CVaR, should the distribution a Gaussian? $\endgroup$
    – GoingMyWay
    Aug 16, 2020 at 16:05
9
$\begingroup$

$VaR^\alpha$ is not a coherent risk measure because it fails sub-additivity (a coherent risk measure is monotonic, sub-additive, positive homogenous, and translation invariant). The expectation operator $E[\cdot]$ is linear, so it meets sub-additivity, as well as the other three properties, so $CVaR$ is a coherent risk measure.

Improve this answer
$\endgroup$
2
  • $\begingroup$ Why can't use "minimum"? I can't think of a good example without a minimum. Could you suggest one? $\endgroup$
    – Chang
    Jun 24, 2011 at 20:01
  • $\begingroup$ @chang -- Good catch! You're right. I read too quickly and misinterpreted discrete $R$ as $R$ in a finite set, which would require infimum. $\endgroup$
    – Richard Herron
    Jun 25, 2011 at 15:58
7
$\begingroup$

Conditional VaR (CVaR), which is also called Expected Shortfall, is a coherent risk measure (although being derived from a non-coherent one, namely VaR).

See this paper:

Expected Shortfall: a natural coherent alternative to Value at Risk
from Carlo Acerbi and Dirk Tasche

http://www.bis.org/bcbs/ca/acertasc.pdf

EDIT: I just saw that you emphasized discrete but that shouldn't change the general situation.

Improve this answer
$\endgroup$
2
  • 1
    $\begingroup$ [2002 - Acerbi] Spectral measures of risk: a coherent representation of subjective risk aversion This paper actually says that CVaR is not a coherent measure in general. But, now I think the author was confused with different names for similar concepts. $\endgroup$
    – Chang
    Jun 24, 2011 at 19:46
  • 1
    $\begingroup$ I agree to @vonjd's answer. In his link in equation (12) that's the thing to do if you have atoms in the distribution (could be continuous with points of mass - for example at 0 in an insurance/operational loss example). Formula (12) applies for discrete distributions and the "correction term" vanishes for continuous ones (or when there is no atom). $\endgroup$
    – Richi Wa
    Jul 24, 2012 at 12:04

Your Answer

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

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