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I just read that standard deviation is a coherent risk measure, and therefore it should satisfy the monotonicity property:

$X_1 \geq X_2 \implies \rho(X_1) \leq \rho(X_2)$ where $X_1,X_2$ are asset positions.

We could define $X_1$ being greater than $X_2$ for every $\omega \in \Omega$, but $X_1$ could still have a greater standard deviation than $X_2$. Am I missing something with the definition of monotonicity?

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    $\begingroup$ Precisely because of that it is not a coherent risk measure. $\endgroup$
    – fes
    Jul 1 at 17:24
  • $\begingroup$ @fes what is the problem with monotonicity proof at sfu.ca/~poitras/standard-deviation-coherent.pdf for example? $\endgroup$
    – Andrei
    Jul 1 at 23:56
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    $\begingroup$ The fact that the standard deviation the difference of the two variables is positive does not make the measure monotonic. $\endgroup$
    – fes
    Jul 2 at 5:57

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