Using the DVO$1($or modified duration$),$ answer the following questions:

a$)$ A bond portfolio manager has $$25$ million invested in a $5$ year zero coupon

bond and $$100$ million invested in a $10$ year zero coupon bond. What is the dollar impact of a one basis point parallel shift upwards in the yield curve on the value of the portfolio?

b$)$ Assume now that the bond portfolio contains coupon paying bonds $($total of

$$125$ million face value$)$ and the portfolio's modified duration is equal to $8.$ Using a $10-$year zero coupon bond, show how the portfolio manager can hedge the portfolio against interest rate risk $($explain how many $10$ year zero coupon bonds the portfolio should buy or go short$)$ such that the profit or loss remains the same.