An investor has a portfolio of three positions. The 1-day $95\%$ VaRs for positions 1, 2 and 3 are $\$250$, $\$180$ and $\$480$ respectively. The correlation matrix is given as follows.
Position 1 2 3
1 1.0 0.6 -0.5
2 1.0 0.8
3 1.0
How do i compute 1 day $95\%$ VaR portfolio without the volatility?
I tried finding the price volatility of each position and put them in a vector. After that I multiply with the correlation matrix with both vectors on each side and get 146923.02. With that,I find the VaR by multiplying 1.645 by square root of 146923.02. I'm not sure if it's correct.
Answer:
$$1.645\sigma_1 = 250=>\sigma_1 =\frac{250}{1.645}$$ $$1.645\sigma_2 = 180=>\sigma_2 =\frac{180}{1.645}$$ $$1.645\sigma_3 = 480=>\sigma_3 =\frac{480}{1.645}$$
$\rho_{12} = .6, \rho_{13} =- .5,\rho_{23} = 0.8$
$$\sigma_{combined}^2 = \sigma_1^2 + \sigma_2^2 + \sigma_3^2 +2\rho_{12}\sigma_1\sigma_2 + 2\rho_{13}\sigma_1\sigma_3 + 2\rho_{23}\sigma_2\sigma_3$$
If you compute, you get $\sigma_{combined} = 383.2874$
$$Var_{portfolio} = 383.2874*1.645 = 630.5$$
It is normal to assume that the mean of the distributions are $0$
Goodluck