# Eurodollar future vs Eurodollar forward contracts

You are considering two contracts: a Eurodollar futures contract with six months to maturity, selling at 5%, settled on three-month LIBOR, marked to market every day; and a Eurodollar forward contract with six months to maturity, selling at 5%, settled on three-month LIBOR

Which contract do you prefer?

I don't understand the answer given in Crack's Book Heard on the Street.

Could anyone else explain this to me please.

• I guess I would select the future because it would be easier to trade (standardized contract). – DOMiguel Jun 26 '20 at 18:33

I guess the author's argument is that, because of the frequent settlements, one needs to invest the mark-to-market gains and fund the losses. As the exchange traded futures contract is negatively correlated to interest rates, the mark-to-market gain happens when interest rates are low, so not a great time to invest, while the loss happens when interest rates are high, so not a great time to look for funding.

Over-the-counter (OTC) forward contract does not have to go through this pressure (even though these days some forms of initial margins, variation margins and collateralization - all of them needing to account for some funding/investing - are accompanying most OTC contracts).

Edit: In general, theoretically, there is an expectation that futures price is less than forward price, if the uknown amount (at expiry) is positively correlated to the (stochastic) discount factor.

Given $$T$$ expiry date and $$S$$ payment date and unknown $${\cal F}_T$$-measurable amount $$X$$, $$\beta_t = \exp (-\int_0^t r_udu)$$ stochastic discount factor (its inverse, bank account value, being the standard numeraire here) and (zero-coupon) bond price $$B(t,T)=\beta_t^{-1}\mathbf{E}[\beta_T | \cal{F}_t]$$, we have:

$${\rm Fwd}_t^X = B(t,S)^{-1}\beta_t^{-1}\mathbf{E}_t[\beta_S X]$$

and, due to (continuous) resettlement (and other technical assumptions),

$${\rm Fut}_t^X = \mathbf{E}_t[X]$$

(making futures prices a martingale).

It can then be proved that the futures convexity correction is:

$${\rm Fut}_t^X = {\rm Fwd}_t^X - \beta_t^{-1} B(t,S)^{-1} \mathrm{Cov}_t(X, \beta_S).$$ Time $$0$$ relation is: $${\rm Fut}_0^X = {\rm Fwd}_0^X - B(0,S)^{-1} \mathrm{Cov}(X, \beta_S).$$

In the Libor futures/forward price context above (long futures contract), the covariance is positive.

(Proofs are available in Hunt and Kennedy's book, Financial derivatives in Theory and Practice.)

• ".. as exchange-traded futures contract is negatively correlated to interest rates.. " what does this mean? which contract is that exactly? IR swap? how does that compare to forwards? how does low IR environment translate into "bad time to invest"? – John Jun 28 '20 at 1:18
• @Jonat You long \$10000*(100 - 90/360*r), where r tends toward 3m Libor at maturity. Contract value is negatively correlated to r which in turn is positively correlated to the environment rates at which you can invest or raise cash. – ir7 Jun 28 '20 at 1:32
• @ir7: pls see my answer below. I believe your argument is only valid for being long the futures contract. When you are short the futures, you benefit when rates go up: so you should receive positive MtM in a higher rates environment and you need to post collateral in a lower rates environment. – Jan Stuller Jun 28 '20 at 7:20

I have written a response here:

Why are FRA/futures convexity adjustments necessary?

The credit risk is eliminated in either case since, these days the future will trade on the exchange and the FRA contract will be settled with the clearing house.

There might actually be some value created dependent upon which futures exchange you trade: LCH versus EUREX can (at time) have significant value differentials solely due to the positioning of dealers looking to reduce concentrated exposure. This effect can outweigh convexity and can be in either direction. However, if you traded a future and a FRA across different exchanges you would want to ensure the carry charge on the required margin at each institution was smaller than the gain - if the two were traded at the same settlement venue then the margin would be much more reduced.

didn't read the book, but I would guess the future contract all else being equal.

the point being credit risk.

as mentioned the future contract is settled daily by the exchange. forwards are traded over the counter and settled at expiration, by that time, the party owing to the other could default on the payment.

The below is an open-ended answer: more of an additional question on top of the original question. I am looking for answers and am thinking of merging the below into the original question asked by Permian.

Interesting, been asked this very question in a recent interview, and having given the "clearing removes credit risk" answer, the interviewer gave the following additional point:

"Futures are entirely linear, whilst Forwards have an additional convexity affect: i.e. when you receive a Forward, rates go down, your terminal pay-off increases, but also your discount factor decreases, so you get a second order effect from the discount factor".

Not sure I entirely follow that - did he assume single-curve discounting? Or did he assume that the OIS curve will move together with the forward curve?

Also: The futures are settled daily, but the MtM calculation should be the same, specifically: the MtM of your futures contract is your terminal pay-off discounted to today. If you use OIS discounting for the Futures as well as the Forward, then the MTM should be affected the same for both, right?

Regarding the answer made by @ir7: if you are short the futures, you benefit when rates go up. So you receive positive MtM when rates go up and this MtM should be remunerated by the higher prevailing rate via interest at the margin account (by the exchange I presume). When rates go down and you are short, you need to post collateral, but you should be able to fund it at the lower prevailing rates: so the argument given is only true for being long the futures, no?

Last but not least: obviously, when the Forward is collateralized, one needs to make (even daily) MtM payments with the counterparty. So when you're long the forward, rates go up, you receive money, but if you assume the collateral (usually OIS) rate is correlated to the Forward rate, you will need to remunerate the collateral from the counterparty at the higher rate. And vice versa: rates go down, you will need to fund collateral (at the lower prevailing rate), but your lower funding rate will likely be offset by the lower OIS interest rate on your collateral.

So when the Forward is collateralized, there is still some difference to the Futures contract: namely, you need to worry about the rate at which you remunerate the collateral posted by the counterparty, which should offset any effects from "receiving MtM when rates are higher" (or lower).

• The answer by @ir7 is correct. Yes, the assumption is that the ois curve is highly positively correlated with the libor curve (True in practice). Yes, the argument presented is that there is a disadvantage to being LONG the Eurodollar futures. No, the futures price is not just the PV of its terminal payoff. The Futures price is linear in its underlying, by definition. That’s the main difference. – dm63 Jun 28 '20 at 13:22
• Thank you @dm63. I get everything you wrote except for "the argument presented is that there is a disadvantage to being LONG the Eurodollar futures". So do you agree that there is an advantage to being short the Eurodollar futures? If the disadvantage and advantage is symmetrical around being long / short, then is there really an advantage? (obviously, as we just discussed: being short the Forward might be better due to convexity vs. linearity of the Futures). – Jan Stuller Jun 28 '20 at 13:54
• Yes, an advantage to be short the Eurodollar futures vs FRA – dm63 Jun 28 '20 at 14:30
• Yes, the answer the author (interviewer) would probably be looking for to the original question would be: go long the forward AND short the futures. (Assuming the same discounting applies to both contracts.) – ir7 Jun 28 '20 at 15:00
• @ir7: thanks for coming back to respond. Sorry to be a pest, but I see the following problem with going long the forward: when rates go up, your terminal pay-off goes up, but also discounting rates go up, so the PV of your terminal pay-off does not benefit from the second order effect: it actually suffers from the second order effect, I'd say being short the Forward is better than being short the future because of second order effects, and being long the forward is worse than being long the future because of second order effects. – Jan Stuller Jun 28 '20 at 15:19