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  • #587
    Anonymous
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    Hello,

    If we have 2 bonds and 1 stock in a portfolio..

    Can someone explain me how to calculate VAR using an example.

    Thanks,
    Arun

    #110
    Arun
    Member

    EXAM-3 Market VAR

    #588
    Anonymous
    Guest

    If V denotes variance and SD denotes standard deviation and rho(m,n) denotes the correlation between Xm and Xn, then the following statistical relationship is true for the Variance of X1+X2+X3:
    V(X1+X2+X3) = V(X1)+V(X2)+V(X3)+ 2*rho(1,2)*SD(X1)*SD(X2)+ 2*rho(1,3)*SD(X1)*SD(X3)+2*rho(2,3)*SD(X2)*SD(X3).
    Now think of a portfolio with 2 bonds and 1 stock. For simplicity let’s consider the example of zero coupon bonds, and look at p.101 in the Handbook. This example is actually a single bond which is mapped to 3 and 12 month cashflow. But for our case let’s imagine we actually have 2 distinct zero coupon bonds which mature in 3 and 12 months time. Further suppose that we have $719,219 face value maturing in 3 months and $654,189 maturing in 12 months (just so as we can use the figures in the Handbook). The daily dollar volatilities for those bond positions are $17.79×5.625 (=$100.07)and $62.2253×5.000 (=$311.13) respectively. These numbers are explained in the Handbook. Now suppose that we also have in our portfolio $100,000 of a stock whose Beta is 1.5 in a stock market whose volatility is 20%. So the specific stock volatility is 30% pa (1.5 x 20%) which is equivalent to about 1.9% daily [30/sqrt(250)]. In dollar terms this volatility of 1.9% is $1,900 on a stock position of $100,000. We have been given that the correlation between the 3 and 12 month zero rates is 0.85. Let’s further suppose that the correlation between the 3 month and the stock market returns is minus 0.5 and between the 12 month and stock market returns is minus 0.6. Using the formula above we can express the portfolio volatility as:

    V(X1+X2+X3) = 100.07-squared + 311.13-squared + 1900-squared + 2*0.85*100.07*311.13 + 2*(-0.5)*100.07*1900 + 2*(-0.6)*311.12*1900 = 2,870,258. This is the portfolio variance, and the standard deviation (volatility) is the square root of this which is $1,694.18. So the portfolio volatility is actually less than the volatility of the stock on its own ($1,900). This is portfolio diversification. We have zero rates that are negatively correlated with the stock, which further helps to reduce risk.
    So the 95% daily VaR would be 1.645 * $1,694.18 = $2,786.93

    The Formula above with 3 variables can be extended to N variables using matrix methods. Let say we represent the N standard deviations (volatilities) in a column vector V with elements Sigma1, Sigma2, ….SigmaN. Let V-transpose be the equivalent row vector of volatilities. And suppose that C is the Correlation matrix for the N variables where the i-j element is the correlation between the i-th and j-th variable. Then the portfolio variance is the matrix multiplication of: [V-transpose][C][V]

    Jim

    #589
    Anonymous
    Guest

    Thanks a lot Jim.

    Really nice and detailed explanation.

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