# Measuring Value-at-Risk (VAR)

- ZX% : the normal distribution value for the given probability (x%) (normal distribution has mean as 0 and standard deviation as 1)
- Ïƒ : standard deviation (volatility) of the asset (or portfolio)
- VAR in absolute terms is given as the product of VAR in % and Asset Value:

- This can also be written as:

- VAR for n days can be calculated from daily VAR as:

- This comes from the known fact that the n-period volatility equals 1-period volatility multiplied by the square root of number of periods(n).

- As the volatility of the portfolio can be calculated from the following expression:

The above written expression can also be extended to the calculation of VAR:

Asset daily standard deviation is 1.6%

Market Value is USD 10 mn

What is VaR (%) at 99% confidence?

Daily VaR = 0.016 x 10 x 2.33 = 0.3728 mn

What is the VaR value for 10 day VaR in the earlier case?

10 day VaR = 0.3728 x (10)^0.5 = 1.1789

What is the daily portfolio VaR at 97.5% confidence level?

Investment in asset A is Rs. 40 mn

Investment in asset B is Rs. 60 mn

Volatility of asset A is 5.5% and asset B is 4.25%

Portfolio VaR if correlation between A and B is 20% ?

VaR(A)(in %) = 5.5 x 1.96 = 10.78%; VaR(B)(in %) = 4.25 x 1.96 = 8.33%;

Portfolio VaR = [(40 x 0.1078)2 + (60 x 0.0833)2 + 2x0.1078x0.833x40x60x0.20]0.5 = 7.22 mn

Market Value of asset Rs. 10 mn

Daily variance is 0.0005

What is the annual VaR at 95% confidence with 250 trading days in a year?

Daily VaR = 10 x (0.0005)^{0.5} x 1.65 = 0.36895 mn

Annual VaR = 0.36895 x (250)0.5 = 5.834 mn

For an uncorrelated portfolio what is the VaR if:

VaR asset A is Rs 10 mn

VaR asset B is Rs. 20 mn

The VaR comes out to be 22.36 mn