So this monotonicity (IMO) is saying: if the future value of portfolio (Y) is greater than the future value of portfolio (X), then ceteris paribus, a monotonic risk measure will be lower Y than X; i.e., it will say that X is riskier. Notice that absolute VaR does this: if absolute VaR = -μ + ασ, then for unchanged volatility (σ), higher return lowers the risk (absolute VaR).
@Detective, Thank you very much.
I have tried to assume that rvs X>Y and therefore Z=X-Y>0
It follows that Var(Z)=\[ {sigma^2}_x-2sigma_{x,y}+{sigma^2}_y>0 ].
It looks OK to me, but I am not 100% sure
Could you share your thoughts please?
Thank you @Detective, I thought the same about Z=Y-X, but unfortunately this was the best I could come up with mathematically.
Analytically one could say that in case the Variance is a measure of risk and X is 'bigger' (riskier) than Y, than Var(X)>Var(Y), But I am not too sure if this is defined enough to prove monotonicity.
Please share your thoughts.
Thank you
Thank you @Detective. This is a Market Risk problems I am trying to tackle and the actual problem was to show that Variance is not a coherent Risk measure however I have tried to check every assumption of coherency for Variance so I can better understand it myseld.