Assuming that I calculate a parametric VaR of a portfolio with 3 assets, and I need to assign the amount each asset (equity) contributes to the VaR.
Lets say that:
- $C$: Is the correlation matrix
- $w$: Is a vector of the weight of each asset
- $s$: Is the vector of the standard deviation of each asset (volatility)
- $VaR_t$: Total VaR of the combined portfolio
- $VaR_i$: Is the individual VaR of the Asset $i$
- $VaR_i'$: Is the part of the $VaR_t$ assigned to the Asset $i$
What is the best way to do it?
I thought 2 different ways but I don't feel comfortable with them:
- Using the w vector so each VaR will be $$VaR_i=VaR_t \cdot w_i.$$ The problem is that I don't take into account the volatility
- Using the individual VaR of each asset, so $$VaR_1'=\frac{VaR_t \cdot VaR_1}{VaR_1+VaR_2+VaR_3}.$$ The problem is that I don't take into account the correlation.