The shape of the yield curve (at least in the GBP Rates market) is upward sloping from the front end up to the long end (i.e. 30y), but then begins to become downward sloping as we go beyond 30y and 40y. (Although, at the time of writing, and I think for the first time ever, the 30s50s curve has become upward sloping.)
I know that the reason for this is: convexity.
However, I'm not entirely sure why the ultra-long end has this extra convexity, and why it is necessary that the extra convexity implies an inversion of the slope.
The extra convexity: In a normal scenario, as we go further out of the curve, our certainty about what kind of interest rate regime we might be in reduces, meaning that the long end has to be flatter than the front end. In order to maintain this flatness, there has to be a point of reasonably high convexity as the front end morphs into the long end. (I.e. steepness manifesting as reasonable expectations morphs into 'who-knows-what-rates-will-be-in-20-plus-years'.) It's reasonable to have a view on where rates will be in 1 to 5 years, maybe even 10, but beyond that you can pretty much forget about it, so there is no need for significant steepness.
But why do we see a change in convexity from 30y+? Why does the curve not just level out beyond 30y? Why does it dip lower?