This has been posted a few times now, so I will invest the time on a full response.
FRA / Futures convexity has nothing to do with profits/losses being immediately recognised on the future through margin settlement, and potential reinvestment, whilst deferred on the FRA.
Although the opposite seems to be a very common belief amongst many practitioners (possibly due to bad wording in Hull:Options, Futures, Derivatives) it is not correct.
Let me characterise this with some different intuitive arguments:
FRAs and Futures have similar reinvestment patterns. If you own a future and it makes \$10000 on day 1 you will receive \$10000 margin, which you invest overnight at OIS and receive some interest, say \$1. On day 2 the future is worth \$10000 and you have \$1 in your pocket.
If you own an FRA and it makes \$10000 you will receive \$10000 collateral. You invest this overnight and receive \$1 OIS interest, but you owe \$1 in interest to the collateral poster. On day 2 the discount factors for the FRA cashflows are 1 day closer to expiry and discounted by 1-days OIS interest less. The FRA is valued at \$10001 and you have \$0 in your pocket.
Imagine that your future was instead a fixed CFD and settled on the same date as the FRA. But it was a collateralised CFD. This would result in exactly the same convexity value, but it is independent of having an immediate margin exchange of profits/gains and any reinvestment.
Assume that everyday the future closed at the same price so that no profit exchange was ever made (but this was coincidental). Then the reinvestment argument would not be able to make any gains since there is never any gain to reinvest. However there is an important difference if the intraday volatility is either zero, low, or high, and a portfolio continuously delta hedges, even if at the end of the day it remains at where it started.
FRA / Futures convexity is risk based and its value can be intuitively understood in a way similar to the replicating portfolio of an option, to determine its price.
What does this mean? First consider the formula for the settlement of a paid (bought) FRA:
$$ P = v N d \frac{r - R}{1+d r} $$
Where $v$ is the (ois) discount factor for the settlement date of the FRA, $r$ is the floating rate, $R$ the fixed rate and $d$ the day count fraction. The key here is the term $(1+dr)^{-1}$; when rates go up you make money on the position but this term discounts your contract settlement more heavily. In fact the settlement is paid upfront but is implicitly assumed to be discounted from the end of the FRA at the settled FRA rate, and this is the term that gives the product asymmetry.
So how is this risk related? Consider a portfolio where you have paid (bought) a FRA and hedged it by buying a future. The value of your portfolio is as follows:
$$ P = v N d\frac{r-R}{1+rd} - \tilde{N}d(F-r) $$
where $F$ is the equivalent futures rate at which you have traded a nominal amount, $\tilde{N}$ of futures to delta hedge. Your portfolio will satisfy two properties initially:
$$ P_{t=0} = 0, \quad \frac{\partial P}{\partial r}_{t=0} = 0 $$
So this means that you trade a specific amount of futures initially to be delta hedged:
$$ \tilde{N}|_0 = f(N, d, r, v)|_0 $$
But the issue now is what happens to the risk of your portfolio as rates change?
The risk of your futures always remains constant:
$$\frac{\partial P_{futures}}{\partial r} = \tilde{N} d $$
but the risk on your FRA changes to be dependent upon the prevailing rate, $r$:
$$\frac{\partial P_{fra}}{\partial r} = \frac{vNd}{1+dr} \left (1-\frac{d(r-R)}{(1+dr)} \right ) $$
In fact, more than that it actually depends on previous rates (and the OIS rate) that impacts the discount factor $v$, whilst the risk on the future always remains constant.
You end up with the scenario that when rates increase for this portfolio you need to buy more FRA to remain delta hedged, but you are buying at higher prices. If the price falls again then you sell it back to remain delta hedged costing yourself money. So this is a process that is entirely dependent upon the volatility.
Conversely if you sell a FRA then when rates fall you have too large a position and can pay the FRA back to lower your delta, since you are doing this at more favourable rates your continuous delta hedging is generating an arbitrage profit against the future.
Hence futures are always oversold relative to FRAs that are oversold. The rates on FRAs are naturally lower than those for futures and the difference is called the convexity bias.