Question

1).Consider a zero coupon bond with 10 years maturity with face value $100. The internal rate of return (IRR, y) is 8% and the daily volatility of the IRR is 0.0007 (7 basis points). Required: • Compute the percentage VaR over 1-day and under 99% confidence level. • Compute the dollar VaR over 1-day and under 99% confidence level.

2).Consider the risk of a short position on a call option. The VaR of its underlying asset is 2000. The delta of this option is 0.5. Required: • Compute the linear VaR of the option • Discuss how the gamma effect would affect the linear VaR of the option.

3).A local linear valuation method is appropriate to compute VaR if the portfolio contains complex options. Required: • Discuss whether the above statement is correct or not. • Discuss which methodology is more suitable to compute VaR

4).Discuss how the backtesting is used to test the accuracy of VaR over 1-day and under 99% confidence level.

Answer 1

1) VAR is used to quantify Value at Risk of a volatile asset assuming normal distribution of asset prices or asset returns. In the given question, IRR has a daily volatility of 7 basis points. Since, the present value of the bond is only dependent on the par value of the bond, time to maturity and IRR and since first two factors are constant, we can say that the value of the zero coupon bond also has daily volatility of 7 basis points.

We first calculate the present value of the zero coupon bond which is simply:

We then calculate the VAR using the below formula: Please note that Z-score of the given confidence level can be obtained from the standard normal probabilities table or from excel using the function NORM.S.INV(0.99) which is equal to 2.32635

% 1 day VAR (99% CI) = Z-score of 0.99 * 1 day volatility = 2.32635 * 0.0007 = 0.00163 = 0.163%

$ 1 day VAR (99% CI) = Present Value of security * % VAR = 46.31935 * 0.163% = 0.07543

Above result can be interpreted as we can be 99% certain that the value of the bond will not reduce by more than $0.075 on any 1 day period.

2) VAR of the underlying asset = 2000

Delta of the option = 0.5  

VAR of 2000 indicates that the value of underlying asset will not reduce by more than 2000 for the given confidence level. Delta of 0.5 implies that if the underlying price changes by $1 the option price changes by $0.5.

If the value of underlying reduce by 2000 then using the delta of the option, we can say that the value of option will reduce by 2000 * 0.5 i.e 1000. Therefore the linear VAR of the option can be calculated as the corresponding

Option's Linear VAR = VAR of underlying * delta of option = 1000