The disappearing convexity bias

The convexity bias is a technical trade-off between either a strip of STIR futures versus an interest rate swap, or in single period terms, a trade off between a single STIR future and an equivalent term forward rate agreement (FRA). A FRA is an over-the-counter bilateral product that allows the buyer /seller to notionally borrow/lend a specified amount at a LIBOR linked rate over a forward period.

What is the convexity bias?

The rationale for the convexity bias exists due to the FRA’s convex payoff compared to STIR futures linear payoff. As underlying rates move up and down, a position in STIR futures will have the same payoff but that of the FRA will change. The convexity of the FRA comes about due to its discounted payoff.

  • The payoff of a short STIR future is given by (L%-F%)* 10000*$25
  • The payoff of a short FRA is given by ((F%-L%)* Notional*0.25)/(1+L%*0.25)


  • F% is the FRA forward rate or rate implied by STIR
  • L% is the FRA maturity date LIBOR rate or futures EDSP rate.
  • 0.25 is the 3-month accrual (in reality measured by ACT/360 convention for USD and EUR)
  • $25 assumes Eurodollar tick value per basis point.


The table shows the convexity bias between a position of short 1000 Eurodollar (ED) futures and an offsetting short $1005m 3-month FRA (slightly more than $1000m to compensate for discounting methodology), both instigated at a rate of 2%.


  • An increase in underlying rates from 2% to 2.10% would result in a credit to the variation margin account of  short 1000 ED STIR position of $250,000 and a debit of slightly less than that in the discounted equivalent of $1005m 3M FRA collateral account (assuming zero threshold – zero threshold means every dollar of value change has to be made good.).
  • A decrease in underlying rates of 10 basis points to 1.9% would result in a debit to the variation margin account of a short 1000 ED STIR position of $250,000 and a credit of slightly more than that in the $1005m 3-month FRA collateral account (assuming zero threshold).

If the rates on both products were the same, there would be an advantage or bias in borrowing (selling) via the STIR future and lending via a short FRA to take advantage of the FRA’s convexity. This means that STIR futures should in theory trade at a higher rate compared to the equivalent term FRA. The amount of this difference is the convexity bias which is dependent on and driven by term and short rate volatility.

Monetarising the bias

The convexity bias is monetarised via margining. The short STIR future position would result in beneficial variation margin flows. If rates moved higher, the variation margin account would be credited and this could be lent out at higher overnight rates. If rates moved lower, the variation margin deficit could be financed at lower rates. Time and rate volatility would give rise to what is effectively a long gamma payoff.

Disappearing act

However, as a consequence of the financial crisis, many over-the-counter derivatives like FRA’s and swaps are migrating to an exchange traded and central counterparty cleared model, the convexity bias has largely disappeared as both products will be margined in a similar futures like way.

Furthermore, today’s low interest rate environment means that margins are subject to higher funding rates that often exceed the amounts that can be earned on margin. Also, funding margin accounts can be very capital intensive for products that don’t benefit from cross margining benefits. For example, a Euribor STIR future trade against a EUR swap/FRA might attract two sets of independent margins whereas a Eurodollar STIR future trade against a USD swap/FRA would be more capital efficient since both products are cleared by CME Group.

The chart shows the convexity bias that used to exist on Eurodollar futures prior to 2011, when central counterparty clearing for swaps and FRA’s gained traction. Now it has largely disappeared due to the cross margining benefit of both products being cleared and margined by CME Group.


ED Convexity bias

In Europe, the bias went slightly negative this summer mainly since margins for LIFFE Euribor futures could not be netted against margins for Euro FRA’s cleared at SwapClear.  Any trader doing the European convexity bias trade would have to post two sets of margin and as the rate paid on margin is lower than  the trader’s own funding costs,  a substantial funding adjustment results which would be greater than value of the convexity bias. Consequently, there was nothing to prevent the bias going negative!

Short term rates and curve steepening or flattening

Many short term rate traders trade the ever changing term structure of rates, called the (yield) curve. Generally, most curve movements can be classified as steepeners (where the differential between long term rates minus short term rates widens) or flatteners (the opposite). They can be further differentiated by bull/ bear steepening/flattening:


One big question remains though – will the curve steepen or flatten? There are many curve drivers but an interesting one is the relationship between short rates like 3-month LIBOR or EURIBOR and steepening or flattening phases.

The chart below shows this relationship for 3-month USD LIBOR (RH scale) and the RedGreen one year Eurodollar calendar spread (LH scale)


The RedGreen one year Eurodollar calendar spread is the spread between the first red Eurodollar futures and the first green Eurodollar future, rolled every quarterly expiry, expressed in rate terms


An increase in this spread means the curve is steepening and a decrease means it is flattening.

There is a causal link relationship between the short rate and the state of the curve.


  •  when short rates rise, the curve flattens
    • As policy rates increase, pushing money market rates higher, the short end of the curve increases more quickly than the longer end, hence flattening. This is technically a bear flattening.
    • And when short rates fall, the curve steepens for the opposite reasons. This is a bull steepening.

Here’s the same for the Euribor.


Central banks and short rate trending

In the G3 80% of base rate changes are in the same direction as the previous change.

Tapering talk and improved G3 economics leads to one inexorable conclusion:  We are probably in the endgame of historically low rates and the consequence of improved economic output and  cutbacks in US asset purchase programs should result in steepening yield curves and a general increase in rates.

Although it is unlikely that there will be base rate changes for some time, STIR futures are largely about expectations and the prospect of some eventual central bank action prompted me to revisit an old study from UBS written by Antony Morris called Using trend strategies to benefit from gradualism in monetary policy, October 2006.

This study put forward an interesting hypothesis that central banks create trends in short-term rates and often STIR markets do not price in these trends. Indeed, the table below shows that over a significant period of time, one central bank rates change is followed by one in the same direction.

central bank action

You can’t get much better than that for a long term trending market, something in scarce supply for systematic traders in recent years. Indeed, the trends are actually enforced by the very nature of central banks when compared to other markets like equity.

  • Central bankers face few restrictions on timing or tone of their comments on policy  compared to that  of equity corporate insiders who face considerable restrictions.
  • Central Bankers can decide exactly when to make changes at the short end of the curve whereas corporate equity insiders cannot completely control earnings reports
  • Central Bankers exercise significant control over the market at the short end,whereas equity corporate insiders cannot control reaction to new new information.


The charts below show these effects. First is the ECB MRO and second Fed Funds both with 9 day and 124 moving averages highlighting the stable long term trends evident in rate cycles.


ECB MRO (black), Euribor front month STIR future 9 day MA (green) Euribor front month STIR future 124 day MA (purple)


Fed Funds (black), Eurodollar front month STIR future 9 day MA (green) Eurodollar front month STIR future 124 day MA (purple) Both charts from Reuters Eikon.

Curve rolling

Roll yield is more commonly associated with the $250 billion commodity markets than STIR futures. Here, traders and fund managers replicate commodity investment and commodity indices by buying futures and rolling each (monthly ) expiry. When the commodity forward curve is backwardated (longer dated forwards < near dated forwards), there is a positive roll pick-up which is called the roll yield. This is a consequence of the long futures position being allowed to expire and the long exposure being reinstated by buying the next futures contract which is trading lower than the expiring series. Various studies have shown that roll yield has added a significant amount to commodity returns over the last couple of decades.

A couple of years ago, I wondered whether this might apply to STIR futures. Admittedly, there is a very different set of curve dynamics to that of commodity markets but there are some observable knowns:

  • STIR forward curve are generally positive (longer dated futures > shorter dated futures – in yield terms).
  • The nearer parts of the STIR curve are often steeper, in yield terms, than the further parts.

The chart below shows this generalisation – curves are usually positive but will experience prolonged periods where they are negative, usually at the perceived top of a rate cycle and/or in expectation of an economic downtown. Most of the negative curves in the chart occurred post Lehman in late 2008.



 Monthly implied forward curves from LIFFE Euribor futures June 2002 to July 2012

(Dotted line – latest curve in 2012)


The following chart puts this in perspective.


 Euribor Stir futures curve steepness 2002 – 2012

(15th quarterly expiry – 1st quarterly expiry)
(Higher value = steeper curve, Negative value = negative sloping curve)

Rather than look at the effects of instigating a rolling long or short position, like the commodity replication trades, which will be largely driven by market returns as well as  roll yields, I thought it might be interesting to look at the returns on  buying a 3-month spread and letting it roll down the curve. The general idea being to buy the spread where the curve is flat, in which case the purchase price should be close to zero and sell it when it has aged and rolled down the curve to a steeper point, usually closer to expiry, where it is sold, hopefully at a higher price. A spread is sold each month and so the process is on-going over time.

Based on my data sample (Dec 1999- Dec 2011), the optimised strategy was the following:

  • Buy one 3M spread
    • E.g. the 18th 3M spread
    • Starting in  December 1999, this was  the  M4U4 at 0.06
  • Sell spread 36 months later
    • M4U4 spread sold in December 2002 for 0.195 
  • Roll every month for 10 years

The table shows the result of the backtest.


The horizontal axis shows the Nth spread  to use (1 is the front spread e.g if today is Oct 13, then Z13H14 and 2 would be H14M14) and the vertical axis is selling N months later.

Therefore, the table shows that based upon this data sample (Dec 1999 – Dec 2011) optimal strategy to use is to buy the 18th spread along the curve and to sell it 36 months later. Such a strategy would have returned 7.4 ticks over the period representing an approximated 4.3% return on expected margin financing (fees and transaction expenses not included).


Not exactly a killing but perhaps something for a systematic trader to run with, especially one who is willing to include a bit of trading intuition to avoid  periods where the curve is negative or trending towards that state.