Proof that value at risk is monotonic

Question

can u help me? anybody can exlpanie me about the proof of the monotonic

() is a monotonic

If ≤ , so () ≤ ().

Take = where ≥ 1, so,

() = inf{: ( > ) ≤ }

() = inf{: ( > ) ≤ }

= inf{: ( > y/c ) ≤ }

= inf{: ( > ) ≤ } (assume y/c = )

= inf{: ( > ) ≤ }

= ().

If I take != , how come the () is a monotonic?

Answer 1

The conclusion follows quite directly from the definitions. Let $X\leq Y$ be two stochastic variables. Then, for each $c \in \mathbb{R}$ we have that $P(X>c) \leq P(Y>c)$.

Fixing $c^{*} = \inf\{c \in \mathbb{R} | P(X>c) \leq{\alpha}\} := VaR_{\alpha}(X)$ and using the fact that $P(Y> c^{*}) \geq P(X>c^{*})$, the conclusion follows.