Question

VAR Calculation

A firm has a portfolio composed of stock A and B with normally distributed returns. Stock A has an annual expected return of 15% and annual volatility of 20%. The firm has a position of $100 million in stock A. Stock B has an annual expected return of 25% and an annual volatility of 30% as well. The firm has a position of $50 million in stock B. The correlation coefficient between the returns of these two stocks is 0.3.

a.       Compute the 5% annual VAR for the portfolio. Interpret the resulting VAR.

b.       What is the 5% daily VAR for the portfolio? Assume 365 days per year.

c.       If the firm sells $10 million of stock A and buys $10 million of stock B, by how much does the 5% annual VAR change?

Answer 1

a. Compute the 5% annual VAR for the portfolio. Interpret the resulting VAR.
Expected Return of the Portfolio=(100*15%+50*25%)/(100+50)=18.333%
       Standard Deviation of the Portfolio=sqrt((100/(100+50)*20%)^2+(50/(100+50)*30%)^2+2*(100/(100+50))*(50/(100+50))*20%*30%*0.3)=18.915%
       Worst 5% return=18.333%-18.915%*1.65=-12.877%
       VaR=12.877%*(100+50)=19.3155 million
       On 95% best cases, the highest loss is 19.3155 million
       On 5% worst cases, the least loss is 19.3155 million

b. What is the 5% daily VAR for the portfolio? Assume 365 days per year.
       Expected Return of the Portfolio=18.333%/365
       Standard Deviation of the Portfolio=18.915%/sqrt(365)
       Worst 5% return=18.333%/365-1.65*18.915%/sqrt(365)=-1.583%
       VaR=1.583%%*(100+50)=0.023745 million
      
c. If the firm sells $10 million of stock A and buys $10 million of stock B, by how much does the 5% annual VAR change?
       Expected Return of the Portfolio=(90*15%+60*25%)/(100+50)=19.000%
       Standard Deviation of the Portfolio=sqrt((90/(100+50)*20%)^2+(60/(100+50)*30%)^2+2*(90/(100+50))*(60/(100+50))*20%*30%*0.3)=19.349%
       Worst 5% return=19.000%-19.349%*1.65=-12.926%
       VaR=12.926%*(100+50)=19.389 million
       VaR would increase by 19.389-19.3155=0.0735 million