Editing Subadditivity (section)

===Finance===
Subadditivity is one of the desirable properties of [[coherent risk measure]]s in [[risk management]].<ref name="Rau-Bredow">{{Cite journal | doi = 10.3390/risks7030091| title = Bigger Is Not Always Safer: A Critical Analysis of the Subadditivity Assumption for Coherent Risk Measures| year = 2019| last1 = Rau-Bredow | first1 = H. | journal = Risks| volume = 7| issue = 3|pages = 91| doi-access = free| hdl = 10419/257929| hdl-access = free}}</ref> The economic intuition behind risk measure subadditivity is that a portfolio risk exposure should, at worst, simply equal the sum of the risk exposures of the individual positions that compose the portfolio. The lack of subadditivity is one of the main critiques of [[Value at risk|VaR]] models which do not rely on the assumption of [[Normal distribution|normality]] of risk factors. The Gaussian VaR ensures subadditivity: for example, the Gaussian VaR of a two unitary long positions portfolio <math> V </math> at the confidence level <math> 1-p </math> is, assuming that the mean portfolio value variation is zero and the VaR is defined as a negative loss,
<math display="block"> \text{VaR}_p \equiv z_{p}\sigma_{\Delta V} =  z_{p}\sqrt{\sigma_x^2+\sigma_y^2+2\rho_{xy}\sigma_x \sigma_y} </math>
where <math> z_p </math> is the inverse of the normal [[cumulative distribution function]] at probability level <math> p </math>, <math> \sigma_x^2,\sigma_y^2 </math> are the individual positions returns variances and <math> \rho_{xy} </math> is the [[Pearson correlation coefficient|linear correlation measure]] between the two individual positions returns. Since [[variance]] is always positive,
<math display="block"> \sqrt{\sigma_x^2+\sigma_y^2+2\rho_{xy}\sigma_x \sigma_y} \leq \sigma_x + \sigma_y  </math>
Thus the Gaussian VaR is subadditive for any value of <math> \rho_{xy} \in [-1,1] </math> and, in particular, it equals the sum of the individual risk exposures when <math> \rho_{xy}=1 </math> which is the case of no diversification effects on portfolio risk.