# A question regarding eurodollar futures rate

I am reading a paper, and cannot understand this part. In the second line, it is said that the pay off on this futures contract equals one minus the current annualized 3 month LIBOR rate in the contract expiration month. Right after the sentence, however, the equation is using a policy rate of the central bank. Shouldn't r_t+n be a LIBOR rate instead of policy rate? Please explain why if anybody knows about this.

• You may want to cite and link to the paper.
– user18
Nov 29, 2020 at 3:34
• Also you should not post pictures of a text, but just type it out.
– 1muflon1
Nov 29, 2020 at 9:23
• @1muflon1 I see. Thank you. I will be careful next time. Nov 29, 2020 at 13:33
• Just for clarification - are you asking why there is CB rate in this specific paper or are you asking if outside economics in some model used to manage some business portfolio it would be better to have there LIBOR rate instead for using simplification that CB rate is LIBOR rate?
– 1muflon1
Nov 30, 2020 at 18:58

## 2 Answers

In theoretical literature LIBOR rate and central bank's rate (for example Fed's fund rate) are often used interchangeably (see many examples in Freixas and Rochet: Microeconomics of Banking).

The reason why they are used interchangeably is that there is quite a strong proportional relationship between them and LIBOR rate depends on the Fed's fund rate. Hence:

$$r_{\text{LIBOR}} = f(r_{\text{Fed}})$$

In fact they are approximately equal so that:

$$r_{\text{LIBOR}} \approx r_{\text{Fed}}$$.

You can see that this holds in real life pretty well from the graph below based on Fred data. Also as explained here the correlation between them is almost 1. Hence, for purposes of theoretical modeling we will often just simplify and say that $$r_{\text{LIBOR}} = r_{\text{Fed}}$$ even though in reality this holds only approximately (there is really no point in explicitly modeling $$r_{\text{LIBOR}} = f(r_{\text{Fed}})$$ - models are meant to be simplified version of reality).

• This may be common practice in the economics literature, but in mathematical finance, LIBOR instruments are calibrated against each other - they are two separate yield curves. The spread between the risk-free rate and LIBOR is not constant, and it is possible to lose lot of money in ED futures despite being correct about the policy rate (e.g,, 2008). Nov 29, 2020 at 13:24
• @BrianRomanchuk but the question seems to be about model in a theoretical paper
– 1muflon1
Nov 29, 2020 at 13:26
• It also refers to Eurodollar futures, where it is possible to lose money as a result of not understanding the product, and thinking the payoff is based on the risk-free rate instead of LIBOR. Nov 29, 2020 at 16:26
• @BrianRomanchuk right but the question asks why they did in that theory paper. Well this is likely the answer. In a theoretical model which is simplification of reality you can abstract from this for the sake of parsimony. If you want to argue that this makes the model conclusions invalid fine but it is still answer why it is routinely done in literature. Unless mathematical finance also makes this equality as in the paper above but for some alternative reason how does the fact that this issue exist help answer the question? If you are just providing an interesting tidbit thats of course fine
– 1muflon1
Nov 29, 2020 at 16:31
• @BrianRomanchuk because the spread is approximately zero and in terms of economic theory it is fine to replace that approximation by equality in many models. Also with all due respect, I am surprised you are taking such a cheap shot at econ literature. Many physics papers assume particles are infinitesimal points or assume that there is zero friction in model or zero resistance etc. All are trivially present in reality. Are you now going to rhetorically ask that you don’t understand why would physicists choose to publish papers based on incorrect information? I mean common
– 1muflon1
Nov 29, 2020 at 17:30

There are two main fitted curves currently used in US dollar fixed income - the LIBOR curve, and a risk-free curve. The instruments comprising the risk-free curve include:

• Fed Funds, settlement balances at the Fed
• General collateral Treasury repo.
• Fed Funds futures.
• Treasury bills/bonds/notes, futures.
• Overnight Index Swaps (OIS).

The Federal policy rates (Fed Funds target, interest on reserves) are tied to the risk-free curve.

Meanwhile, the Eurodollar futures are LIBOR. There is a spread between the LIBOR and risk-free curve, which varies over time, and needs to be accounted for by financial market participants.

Finally, in response to:

Shouldn't r_t+n be a LIBOR rate instead of policy rate? Please explain why if anybody knows about this.

Yes, as explained above, the payoff is based on LIBOR, not the policy rate. See the CME contract specifications: contract description. (Note: at the time of writing, LIBOR is being phased out. I have no idea what the implications are the reference change.)

Based on comments made here, conflating the two curves (as was done in the above example) is often done as a simplifying assumption in parts of the economic literature. That assumption assumes away the spread, which is admittedly stable outside of crises. That said, this would likely not happen in any literature related to financial crises, where the LIBOR/risk-free rate spread is discussed.

• This seems to be a comment rather than answer. In addition, economic literature does not conflate them but all science is based on approximations and all models approximate reality instead of capturing it faithfully. Many trade models will often assume no transportation costs, people will be assumed to be rational etc. This is common practice in any field of science.
– 1muflon1
Nov 29, 2020 at 17:37
• I spelled out exactly what I answered. Note that I directly answer the question, which is about ED futures. I modified my comments about the literature. Nov 29, 2020 at 17:57
• But now there is a mistake you say "Yes, as explained above, the payoff is based on LIBOR, not the policy rate." But the paper says that the payoff is function of policy rate not LIBOR rate, since paper states $r_{t+n}$ is the CB policy rate, if it should be function of LIBOR it should be some composite function as likely the model will need CB rate for some results and wont work without it. Since, this is not mistake by authors but intentionally defined that way and since question asks about model in the paper itself it would be mistake to put there just LIBOR without additional parameters
– 1muflon1
Nov 29, 2020 at 18:04
• The text literally says it “pays off on the LIBOR rate”, and the real-world ED futures pays off based on LIBOR (I added the link.) If economics textbooks want to argue that there are hypothetical “eurodollar futures” with a different payoff, that’s great for them. I don’t see why readers visiting this page - who are likely interested in real-would ED futures, should be given incorrect information that can lead to potentially large losses. I answered the question, which is about eurodollar futures, not a model that would be rejected by financiers. Nov 29, 2020 at 21:50
• 1. but that central bank rate might be necessary in that model for it to solve so if you want to argue that there should be $r_L\neq r$ you still cannot just say there has to be $r_L$ you would have to include $r_L=\beta r$ or some complex function. Unless you want to argue they included $r$ there instead of $r_L$ by mistake and not by intent. 2. Models are allowed to make simplifying assumptions whether you approve of that or not. If in this model simplifying assumption is that the spread is zero then the correct answer is that no there is no mistake and there should be central bank rate.
– 1muflon1
Nov 29, 2020 at 22:09