In this assignment, you will practice calculating duration for two government bonds. The first is a $5-$year bond with an annual coupon of $5\%.$ The second is a $10-$year bond with an annual coupon of $5\%.$ Assume that the face value of each bond is $$100$ and that the spot rate curve is flat at $5\%($annual compounding$).$

Part A: For each of the two bonds, calculate the price, Macaulay duration, and modified duration.

Part B: Now suppose that you have a $$100$ million long position in the $10-$year bond. You are trying to hedge the interest rate exposure on this position by selling $5-$year bonds.

Assuming that you rely on modified duration, how much of the $5-$year should you sell?

Part C: After putting on your hedge, the spot curve shifts down by $10$ basis points across all maturities.

What are the new prices of the two bonds? What is the profit$/$loss on your hedged portfolio?

Part D $-($Optional$)*$ Now suppose that instead the spot curve shifts down by $100$ basis points to $4\%$ across all maturities.

What are the prices of the two bonds and the profit$/$loss on your hedged portfolio in this case?