Convexity for putable vs Option-free

[wecfanow]

Hi guys,
Struggling with one of the CFAI mock’s questions - Under what scenario is the putable’s convexity lower than equivalent option-free?
I was thinking as rates go down, putable and option-free behave similarly, so let’s ignore that. As rates go up, putable doesn’t go down in price as much, since investor can put the bond back. Graphing this in my head, I see the putable bond price curve more than straight, being bounded by its put price. Feels like the convexity would be greater
But the mock says it can be less. Under what scenario?
Thanks!

 

[S2000magician]

As the bond starts to move into the money, so near the coupon rate.

 

[Gurifissu]

Less convexity = flatter curve. As rates go up, the profile of the putable bond becomes less convex (rates do not affect the price as much as in an option-free equivalent, and the one-sided upside duration is less than the one-sided downside duration).

 

[Portfolio Manager]

I don’t get it.

 

[lammy]

A putable bond protects an investor.
What does an investor want protection from? A decline in the price of the bond.
When does a bond’s price decline? When interest rates rise.
The put serves as a floor on the bond’s price. So the curve for a putable bond flattens at higher interest rates relative to straight bonds than can continue falling in price. Flatter curve = less convexity.

 

[Rasec]

BAM! LAMMY!