Definition problem. Someone esp. magician talked about this in previous thread.
You cannot use a simple equation like these two to find out ED and EC for bond with options, it involves volatility assumptions, OAS, and building “tree”…
First, why are you asking about bongs when you should be studying for your exam? Leave the recreational activities till June 2.
The formula with ½ occurs in the reading on credit analysis: different author, different formula.
I’d suggest using it only in the context of credit analysis.
(The overarching problem is that finance people cannot agree whether the ½ should be part of the convexity number or not. I think it should be, but whether it is or not isn’t important: the finance community should agree on one or else the other and be done with it.)