Why CVaR Is Subadditive but Copula-Based CCVaR Can Be Superadditive
Summary
The discussion distinguishes standard conditional value at risk (CVaR), also called expected shortfall, from a separate copula-based measure called CCVaR. Standard CVaR is described as a coherent risk measure, whose properties include subadditivity: the risk of a sum is no greater than the sum of the individual risks. This captures how diversification can reduce measured risk, and the inequality allows equality.
The apparent opposite result comes from confusing CVaR with the paper’s CCVaR. The cited paper defines a different measure and reports that it is superadditive under its formulation. The replies state that this property follows from the relationship between CVaR and CCVaR. The distinction depends on the precise measure being used; the brief discussion does not derive the coherence properties or specify conventions for losses versus returns, so those should be checked when applying the result.
Key ideas
- Standard CVaR, also known as expected shortfall, is subadditive.
- Subadditivity means the risk of a portfolio sum is at most the sum of its component risks.
- The inequality may hold with equality, so strict diversification benefit is not guaranteed.
- Copula-based CCVaR is a distinct measure and may be superadditive under the cited formulation.
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# Subadditivity of cvar(R)، R is random vector
# Subadditivity of cvar(R)، R is random vector
$R=(R_1,\ldots,R_n)$ is random vector in $L^1(\mathcal{R}^n)$. Then is it true that $$ \operatorname{Cvar}(R_1+ \cdots + R_n) \le \operatorname{Cvar}(R_1) + \cdots +\operatorname{Cvar}(R_n)? $$ Can we say $\operatorname{Cvar}(R)$ is subadditive? I see in the paper portfolio optimization with copula based extention conditional value at risk that $$ \operatorname{Cvar}(R_1+ \cdots + R_n) \ge \operatorname{Cvar}(R_1)+ \cdots +\operatorname{Cvar}(R_n). $$
## Answer by Kevin (score 1)
https://quant.stackexchange.com/a/46656
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 $CVaR(R_1+R_2) \leq CVaR(R_1) + CVaR(R_2)$ which directly extends to sums of $n$ random variables. Sub-additivity captures the notion that diversification is beneficial. Note that volatility is also subadditive, but Value-at-Risk is not.
Finally, subadditivity involves $\leq$ and not $<$.
### Edit
The paper you mention, Krzemienowski and Szymczyk (2016), does not deal with Conditional Value-at-Risk (CVaR). Their paper introduces a new risk measure which is not coherent. Their risk measure is named Copula-based conditional value-at-risk (CCVaR). Thus, the properties of CVar do not apply. In Section 4 of the paper (Proposition 2), the authors list several properties of CCVar, one of them is super-additivity. Thus, $CCVaR(R_1+R_2) \geq CCVaR(R_1) + CCVar(R_2)$. This property is proven on page 224 which follows from the relationship between $CVaR$ and $CCVaR$.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.