# 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 $$\operatorname{Cvar}(R_1+ \cdots + R_n) \ge \operatorname{Cvar}(R_1)+ \cdots +\operatorname{Cvar}(R_n).$$

• Hi Farzin and welcome. I added latex to your question, I hope I got it right. When you say "a paper" wouldn't it be good to tell the title? – Ric Jul 16 '19 at 13:15
• Hi, thank you. Yes it's good. I edit now – Farzin Jul 16 '19 at 13:26

Yes, conditional VaR (aka Expected Shortfall) is a coherent risk measure and thus, satisfies

• Monotonicity,
• Translation invariance,
• Positive homogeneity and

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 $$CCVaR$$.

• Thank you for your answers. I edit my question... Have you commente about last sentence? In portfolio optimization with copula based extention conditional value at risk 2016 paper. link.springer.com/article/10.1007/s10479-014-1625-3 – Farzin Jul 16 '19 at 12:52
• I'll add my answer. Give me 5 min ;) – Kevin Jul 16 '19 at 12:58
• Thank you so much. In page 224. Author write cvar is superaddative – Farzin Jul 16 '19 at 13:03
• So, a risk-measure can be sub-additive, additive, super-additive or neither. CVaR and Volatility are examples of risk measures which are sub-additive. CCVaR is an example of a super-additive risk measure. – Kevin Jul 16 '19 at 13:07
• , and the fact that CVaR is supperadditive when larger outcomes are preferred (see, e.g., Pflug 2000). Therefore we have \begin{aligned} \hbox {CCVaR}_\beta (\mathbf {R}) \ge \hbox {CVaR}_\beta (\mathbf {1}^T \mathbf {R}) \ge \sum _{i=1}^n \hbox {CVaR}_\beta (R_i) = \sum _{i=1}^n \hbox {CCVaR}_\beta (R_i). \end{aligned} – Farzin Jul 16 '19 at 13:11