$R=(R_1,\ldots,R_n)$ 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 $$ \operatorname{Cvar}(R_1+ \cdots + R_n) \ge \operatorname{Cvar}(R_1)+ \cdots +\operatorname{Cvar}(R_n). $$

  • $\begingroup$ 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? $\endgroup$ – Ric Jul 16 '19 at 13:15
  • $\begingroup$ Hi, thank you. Yes it's good. I edit now $\endgroup$ – 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
  • 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 $<$.


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 $CCVaR$.

  • 1
    $\begingroup$ 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 $\endgroup$ – Farzin Jul 16 '19 at 12:52
  • $\begingroup$ I'll add my answer. Give me 5 min ;) $\endgroup$ – Kevin Jul 16 '19 at 12:58
  • $\begingroup$ Thank you so much. In page 224. Author write cvar is superaddative $\endgroup$ – Farzin Jul 16 '19 at 13:03
  • $\begingroup$ 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. $\endgroup$ – Kevin Jul 16 '19 at 13:07
  • $\begingroup$ , 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} $\endgroup$ – Farzin Jul 16 '19 at 13:11

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