A portfolio manager with an investment portfolio containing $500 million of fixed-rate US Treasury bonds with an average duration of five years. Suppose the manager wants to reduce this duration to three over the next year but does not want to sell any of the securities.
One way to do this would be with a pay-fixed interest rate swap.
Duration of the swap used by the manager is 1.5. This duration is less than the existing portfolio duration, so adding the swap to the portfolio will reduce the overall average duration. By properly choosing the notional value of the swap, the portfolio manager can achieve a combination of the existing portfolio duration and the swap duration that sets the overall duration to the target duration.
Question: Using duration of 1.5, how would you calculate the notional value of the swap to set the overall duration to the target duration? (Show work)