Hedging With Swaps

You are in risk management at a bank and are trying to hedge the interest rate risk of a $75 billion bond portfolio with a duration of 8. To do so, you would like to purchase 10-year swaps with a swap rate of 4% (semiannual payments and compounding).

1. What is the duration of a bond with coupon rate and yield equal to the swap rate? Use this to compute the dollar duration of a swap with a notional value of $X.
2. Solve for the $X that would make the dollar duration of your portfolio combined with the swap equal to 0.