The VaR of a single security is the value of the position multiplied by the volatility. The VaR of a 2 security portfolio can be described as the amount of position multiplied by the volatility plus a diversification factor. The standard deviation of a two security portfolio is expressed as

                     [11]
 
Where α value of the security, a and b represent two different securities and ρ refers to correlation. The first two terms within the equation are value of the portfolio multiplied by the volatility. The third term refers to the diversification benefit.  Adjusting the parameters we define the value at risk of a two security portfolio as:

                     [12]

Example 1.
Calculate the 95% one tail Value at Risk for one day and 1 week for the following position:-

IBM stock trading at $113 with a position in a 100 stocks with a one year price return variance of 0.0441.

Solution:- 
The 1 year standard deviation of the of the IBM stock is the square root of the variance.
 
 

The 1 year standard deviation of price returns for IBM is 21%.

A 95% confidence interval implies that there is a 5% chance that the VaR of IBM will be greater than the VaR that we will arrive at. In a normal distribution the 95%

 confidence interval is 1.646 standard deviations away from the mean. Using

1 year standard deviation* # of standard deviations from mean = Range

0.21 * 1.646 = 0.346

1 year VaR = 0.346 * 113*100 = $3906

The 1 day VaR is simply the 1 year VaR divided by the square root of 250

Therefore 1 day VaR = $247.06

The two week VaR is simply the 1 day VaR multiplied by the square root of the number of trading days. There are 10 trading days in two weeks.

Two week VaR = 3.16 * 247.06 = $781.27  

Example 2.
Calculate the portfolio VaR using the variance / covariance method adopted by the Riskmetrics model for the following portfolio:

Security 1: Long 100 shares of SPN, price at 35.61, 1 year volatility 18%
Security 2: 50   shares of CAM , price at 71.15,  year volatility 16%
Correlation between two stocks : 40%

Also calculate the diversified risk.

Solution
In the above example volatility is defined as the % of value that may be lost within a certain probability. In our case we used the 95% probability that more would not be lost. This is 1.646 standard deviations. However as the probability is incorporated we do not need to calculate a range.

Using equation [11]

VaR1 =100 * 35.61* 0.18 = 640.98
VaR2 = 50  * 71.15 *0.16 = 569.2

=$1’013.30 is the total portfolio VaR

The sum of the individual VaR’s are 640.98+569.2 =1210.18

This implies that the diversification benefit is $1210.18-$1013.30= $196.91

August 23,2007