Numerical on Duration, Modified Duration, Volatility and Yield

Numerical
 
Bank has a €5m position in 3 a year zero coupon bond with face value €5,955,080. The bond is trading at an YTM of 6%. The historical mean change in daily yield is 0.00% with a standard deviation of 8 basis points (0.0008).

 a) What is the duration, and the modified duration of the bond?

 Solution: Duration of Zero Coupon Bond is maturity of period of Bond. Hence duration is 3 Year. 

Modified Duration= Duration/(1+YTM)
                              =3/(1+.06)
                              =2.83

 b) What is the maximum adverse bad daily yield change given that we do not want no more than 5% change that yield changes will be greater than this maximum?

Solution: In respect of 5% change in yield, means it can lie from +5% to -5%, taking 90% confidence interval , maximum adverse yield  change = 1.65*.0008 = .00132

 c) What is the daily earning at risk for this bond? Use modified duration.

Solution:
 
      %change in price= -(modified  duration ) X yield change
                                 = -2.83*.00132 => .0037356
                                  =50,00,000*.0037356==>>18678
 

 d) What is the 10 day VaR for this bond?

Solution:

Daily Volatility = € 18678 

for 10 Day of Volatility =  Daily Volatility  X √10 =>18678* √10 ==>59065.02