forward - Why are FRA/futures convexity adjustments necessary?

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.

Answer

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, whilst deferred on the FRA.

Although this seems to be a very common belief amongst many practitioners it is not correct.

Let me characterise this with 5 different intuitive arguments:

1) The profits on a FRA/Future are dependent upon LIBOR, and the interest accrued on immediate cash is OIS. Assume there is no correlation between these rates. The argument above does not hold and then asserts that convexity would be zero, but this is not the case.

2) Assume that OIS was fixed at 0% so that there was no interest costs or gains on a daily basis. Then being short/long futures is symmetrical. If LIBOR increases or declines any profits/losses have no associated overnight gain/costs. This assumption asserts there is zero convexity bias, however this is false.

3) Assume that everyday the future closed at the same price so that no profit exchange was ever made (but this was coincidental). This asserts by the above argument that the convexity adjustment is zero, however there is an inherent difference and importance 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.

4) An FRA has a particular form of interest accrual, firstly an asset is collateralised so you receive cash to insure your position. You can invest this cash at OIS. You owe this interest back on the collateral agreement to the liability holder but your FRA asset increases in value overnight (since the discount factor now has one less day to discount your asset). So this is a comparable mechanic to a future in any case.

5) 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.

FRA / Futures convexity is risk (2nd derivative) based and intuitively can be represented by the reinvestment of profit over the term of the FRA

What does this mean? First consider the formula for 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:

$$ P = v N d\frac{r-R}{1+rd} - \tilde{N}d(F-r_F) $$

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.