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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)
  • σ : 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:

Example 1

Asset daily standard deviation is 1.6%
Market Value is USD 10 mn
What is VaR (%) at 99% confidence?
 

Solution

Daily VaR = 0.016 x 10 x 2.33 = 0.3728 mn


 

Example 2

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

Solution

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


 

Example 3

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% ?
 

Solution

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
 

 

Example 4

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?
 

Solution

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
 


 

Example 5

For an uncorrelated portfolio what is the VaR if:
VaR asset A is Rs 10 mn
VaR asset B is Rs. 20 mn
 

Solution

The VaR comes out to be 22.36 mn

 





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