I don’t know why, but this one took me a while to figure out too. The key to this is to remember that we are talking about a classical immunization for a single liability. This liability has a static termination date.
Let’s assume the liability is due in 10 years and I’ve matched the duration of my assets to my liabilites and the PV of my assets equals the PV of my liabilties. Cool. To keep it simple let’s say I used a 10 year semi-annual pay vanilla bond on the asset side.
In six months I am going to receive a coupon payment. I need to re-invest this coupon.
Upward Sloping Yield Curve: In six months, I no longer have a 10 year bond. My 10 year bond has become a 9.5 year bond. If you look at the yield curve, assuming no other movement, I have moved to the left a little bit. The rates available to me now are for 9.5 year bonds and the rates are lower than the 10 YTM that I started with. I have rolled down the yield curve. My reinvestment rate will continue to fall as time passes and I keep having to reinvest into shorter and shorter maturity securities.
Downward Sloping - The exact opposite is true. I am now moving left back up the curve and all of my reinvestment is occuring at a rate higher than the YTM that I started with.