Why CVaR Is Coherent Even for Discrete Loss Distributions
Summary
The document addresses whether Conditional Value-at-Risk (CVaR), also called Expected Shortfall, remains coherent when losses follow a discrete distribution. It presents the optimization definition that takes a threshold value and adds a scaled expectation of losses above that threshold. The cited discussion concludes that CVaR is coherent for general loss distributions, including discrete ones.
Coherence is described through monotonicity, sub-additivity, positive homogeneity, and translation invariance. The answers contrast CVaR with Value-at-Risk, which can fail sub-additivity, and distinguish CVaR from Tail Conditional Expectation (TCE): these measures need not agree for general distributions, although they coincide in the continuous case described. The exchange summarizes cited papers rather than proving the properties, and terminology can vary across sources, so the precise definition matters when comparing risk measures.
Key ideas
- CVaR, also known as Expected Shortfall, is coherent for discrete as well as general loss distributions under the definition given.
- Coherence requires monotonicity, sub-additivity, positive homogeneity, and translation invariance.
- Value-at-Risk can fail sub-additivity and therefore is not generally coherent.
- Tail Conditional Expectation and CVaR are distinct for general distributions but coincide for continuous distributions in the account given.
- The chosen definition matters because risk-measure terminology is not used consistently across sources.
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Full text
# Is Conditional Value-at-Risk (CVaR) coherent?
# Is Conditional Value-at-Risk (CVaR) coherent?
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.
## Answer by Chang (score 12, accepted)
https://quant.stackexchange.com/a/1358
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.
## Answer by Richard Herron (score 9)
https://quant.stackexchange.com/a/1354
$VaR^\alpha$ 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.
## Answer by vonjd (score 7)
https://quant.stackexchange.com/a/1353
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.Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.