Duration of portfolio with pay fixed position

Five-Year Pay-Fixed Swap Overlay
In this example, the manager enters into a pay-fixed swap overlay with a notional principal equal to one-half of the size of the total bond portfolio. We will focus solely on first-order effects of yield changes on price (ignoring coupon income and swap carry) to determine the active and index portfolio impact. As the pay-fixed swap is a “short” duration position, it is a negative contribution to portfolio duration and therefore subtracted from rather than added to the portfolio. Recall the $100 million “index” portfolio has a modified duration of 5.299, or (1.994 + 4.88 + 9.023)/3. If the manager enters a $50 million notional 5-year pay-fixed swap with an assumed modified duration of 4.32, the portfolio’s modified duration falls to 3.139, or [(5.299 × 100) − (4.32 × 50)]/100.

Why was the duration calculated like this? You are long 100mn, and short 50mn due to the pay fixed. So, wouldn´t you calculate it as: 5,299 x 100/(100-50) + 4,32 x (-50)/(100-50)? And then the result would be 6,27, which would actually mean a higher duration. Which I know is wrong, duration drops if you enter a pay fixed swap. I just don´t get how they calculated the duration of 3.139.

Yes you are right. I am also facing the same issue. Your method is correct w.r.t. weighted average duration but cannot understand why it’s increasing the new duration :persevere:

The value of the portfolio doesn’t change by adding the swap, any more than it would change by adding a futures position.