MVO solves the optimization problem $\max_{w}w^{T}r ~~\mathrm{s.\!t.}~ w^{T}\Sigma w= v$for some given $v$. The solution is given by $w \propto \Sigma^{-1}r,$i.e. the risk-adjusted returns. The proportionality is determined by the constraint. In the special case with $\Sigma$ being diagonal, i.e. the returns being uncorrelated, we have $\text{cash volatility}=\Sigma^{1 / 2}w_{i} \propto \Sigma^{-1 / 2}r=\text{Sharpe Ratios},$i.e. the amount of volatility budget each asset gets is proportional to its [[Sharpe Ratio]]. ^0cfff3