This would be my explanation for the reason that convexity adjustments must exist:

Futures are margined daily, such that if a trader is paid a future and rates goes up then money is paid into their margin account, and if rates goes down then money is taken from their margin account, daily, so that we have two outcomes from a position: 

1. *Paid position*: Rates go up, so money is paid into the margin account, and said money can be reinvested at the newer, higher rates. Rates go down, so money is taken from the margin account, but can be borrowed back at the newer, lower rates.
2. *Received position*: Rates go up, so money is taken from the margin account, and said money can be borrowed back at lower rates. Rates go down, so money is paid into the margin account, and can be reinvested at higher rates.

Is this where the idea of convexity arises from? The fact that the daily margining creates a clear advantage to utilising futures instead of the corresponding FRA?

Thus, in order to offset this advantage from investing in futures over FRAs, a convexity adjustment is implemented such that (in a naive sense):

$$FRA=Futures-Convexity.$$

If this is not correct or I haven't fully understood, then please correct me.