<p>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).</p>
<p>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:</p>
<p><span class="math-container">$$r_{\text{LIBOR}} = f(r_{\text{Fed}})$$</span></p>
<p>In fact they are approximately equal so that:</p>
<p><span class="math-container">$$r_{\text{LIBOR}} \approx r_{\text{Fed}}$$</span>.</p>
<p>You can see that this holds in real life pretty well from the graph below based on Fred data. Also as explained <a href="http://katchum.blogspot.com/2013/06/correlation-libor-vs-fed-funds-rate.html" rel="nofollow noreferrer">here</a> the correlation between them is almost 1. Hence, for purposes of theoretical modeling we will often just simplify and say that <span class="math-container">$r_{\text{LIBOR}} = r_{\text{Fed}}$</span> even though in reality this holds only approximately (there is really no point in explicitly modeling <span class="math-container">$r_{\text{LIBOR}} = f(r_{\text{Fed}})$</span> - models are meant to be simplified version of reality).</p>
<p><a href="https://i.sstatic.net/fgaUM.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/fgaUM.png" alt="enter image description here" /></a></p>

CC BY-SA 4.0
2020-11-30T18:35:52.220Z
@BrianRomanchuk in my view this question is clearly about specific paper  and if there is a mistake in it or not and clearly not about whether for some pure finance application it would be better. It is completely valid to make approximations in science or assume some spread is zero and so forth - what you are saying is equivalent of telling student that is asking why in a profit function there is a demand equation instead of price that it should not be there and yes there should be price which would during optimization yield incorrect results.
1muflon1
CC BY-SA 4.0
2020-11-30T18:39:22.083Z
@BrianRomanchuk also accusing me of spreading misinformation when I clearly supported in my answer with references to leading texts on microeconomics of banking is low... does finance theory make no simplifying assumptions whatsoever? Next you will accuse Newton of spreading misinformation because in his models he often assumed perfect vacuum when there never was such thing? Clearly in theoretical economic application spread between the rates can be assumed away in many models. If you are not willing to accept that fact fine, but please be so kind and dont accuse me of spreading misinformation
1muflon1
CC BY-SA 4.0
2020-11-30T20:31:59.937Z
I deleted my comment. But I answered the question, that’s the end of the discussion.
Brian Romanchuk
CC BY-SA 4.0
2020-11-29T13:24:37.853Z
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).
Brian Romanchuk
CC BY-SA 4.0
2020-11-29T13:26:47.913Z
@BrianRomanchuk but the question seems to be about model in a theoretical paper
1muflon1
CC BY-SA 4.0
2020-11-29T16:26:58.520Z
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.
Brian Romanchuk
CC BY-SA 4.0
2020-11-29T16:31:42.803Z
@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
CC BY-SA 4.0
2020-11-29T17:21:23.523Z
You write “the theoretical literature” says this. This is not what the correct theoretical literature says. The correct answer is that there is a spread between LIBOR and Fed Funds, and the spread can move for a variety of reasons. I have no idea why economists would publish incorrect information in their literature, as I am outsider.
Brian Romanchuk
CC BY-SA 4.0
2020-11-29T17:30:24.527Z
@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
CC BY-SA 4.0
2020-11-29T09:32:27.843Z
2020-11-29T13:03:14.903Z
2020-11-29T13:03:14.903Z
1muflon1
<p>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:</p>
<ul>
<li>Fed Funds, settlement balances at the Fed</li>
<li>General collateral Treasury repo.</li>
<li>Fed Funds futures.</li>
<li>Treasury bills/bonds/notes, futures.</li>
<li>Overnight Index Swaps (OIS).</li>
</ul>
<p>The Federal policy rates (Fed Funds target, interest on reserves) are tied to the risk-free curve.</p>
<p>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.</p>
<p>Finally, in response to:</p>
<blockquote>
<p>Shouldn't r_t+n be a LIBOR rate instead of policy rate? Please explain why if anybody knows about this.</p>
</blockquote>
<p>Yes, as explained above, the payoff is based on LIBOR, not the policy rate. See the CME contract specifications: <a href="https://www.cmegroup.com/trading/interest-rates/stir/eurodollar_contractSpecs_futures.html" rel="nofollow noreferrer">contract description.</a> (Note: at the time of writing, LIBOR is being phased out. I have no idea what the implications are the reference change.)</p>
<p>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.</p>

CC BY-SA 4.0
2020-11-30T15:22:14.890Z
The question title refers to Eurodollar futures. The question states: “ Shouldn't r_t+n be a LIBOR rate instead of policy rate?” The answer is that yes, it should be LiBOR. That’s what the CME says, that’s what the quoted text says. The objective is to answer the question, not cover up incorrect assumptions made in the literature.
Brian Romanchuk
CC BY-SA 4.0
2020-11-30T15:44:16.143Z
but the quoted text clearly states in equation it uses policy rate so the question is - is this a mistake of an authors or it is an intention - in some comments in past you mentioned you have PhD in mathematics, have you never heard of substitution? If students comes to you with text that says monopoly profit is price times quantity minus cost but then equation says $\Pi=(100-q)q-cq$ and student asks you shouldn’t there be p instead of 100-q you would tell that student yes there should be p - instead of p depends on demand? That would lead to wrong solution to model and would be wrong answer
1muflon1
CC BY-SA 4.0
2020-11-30T15:50:34.980Z
Like I am completely mind boggled that you would seriously suggest something like that instead of providing insight that in economic problems we use substitutions you insist this is typo and has to be interpreted literary - like any normal text will sometimes skip a step and directly substitute for price like in the monopoly example above where you might say in text revenue is price times quantity but then in equation just directly use inverted demand $(100-q)q$ instead of $pq$... I can’t believe that you are serious right now
1muflon1
CC BY-SA 4.0
2020-11-30T16:01:58.830Z
Your whole argument just boils down that you disapprove of the substitution because it is not common in your field outside economics so you say it is wrong. That would be like saying in the monopoly example above no it should be p*q because in real life monopoly cannot observe full  demand function and so in management literature we don’t substitute demand for price so in economics people who do that make mistake - as far as I can see this is exactly an equivalent of your answer.
1muflon1
CC BY-SA 4.0
2020-11-29T17:37:33.037Z
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
CC BY-SA 4.0
2020-11-29T17:57:39.423Z
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.
Brian Romanchuk
CC BY-SA 4.0
2020-11-29T18:04:46.377Z
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
CC BY-SA 4.0
2020-11-29T21:50:45.907Z
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.
Brian Romanchuk
CC BY-SA 4.0
2020-11-29T22:09:44.603Z
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
CC BY-SA 4.0
2020-11-29T17:32:12.790Z
2020-11-29T21:45:26.467Z
2020-11-29T21:45:26.467Z
Brian Romanchuk
<p><a href="https://i.sstatic.net/D1NJP.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/D1NJP.png" alt="enter image description here" /></a></p>
<p>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.</p>

CC BY-SA 4.0
2020-11-30T18:58:59.127Z
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
CC BY-SA 4.0
2020-11-29T09:23:50.997Z
Also you should not post pictures of a text, but just type it out.
1muflon1
CC BY-SA 4.0
2020-11-29T13:33:00.507Z
@1muflon1 I see. Thank you. I will be careful next time.
user14261785
<interest-rate>
A question regarding eurodollar futures rate
user14261785
# A question regarding eurodollar futures rate

(Asked by: username_0 on 2020-11-29)

[![enter image description here](https://i.sstatic.net/D1NJP.png)](https://i.sstatic.net/D1NJP.png)

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.

**Tags:** interest-rate
> **Comments:**
> **username_1**: Also you should not post pictures of a text, but just type it out.  
> **username_0**: @1muflon1 I see. Thank you. I will be careful next time.  
> **username_1**: 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?  

---

## 2 answers

### Answer by username_1 on 2020-11-29. Score: 3

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](http://katchum.blogspot.com/2013/06/correlation-libor-vs-fed-funds-rate.html) 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).

[![enter image description here](https://i.sstatic.net/fgaUM.png)](https://i.sstatic.net/fgaUM.png)

> **Comments:**
> **username_2**: 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).  
> **username_1**: @BrianRomanchuk but the question seems to be about model in a theoretical paper  
> **username_2**: 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.  
> **username_1**: @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  
> **username_2**: You write “the theoretical literature” says this. This is not what the correct theoretical literature says. The correct answer is that there is a spread between LIBOR and Fed Funds, and the spread can move for a variety of reasons. I have no idea why economists would publish incorrect information in their literature, as I am outsider.  
> **username_1**: @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  
> **username_1**: @BrianRomanchuk in my view this question is clearly about specific paper  and if there is a mistake in it or not and clearly not about whether for some pure finance application it would be better. It is completely valid to make approximations in science or assume some spread is zero and so forth - what you are saying is equivalent of telling student that is asking why in a profit function there is a demand equation instead of price that it should not be there and yes there should be price which would during optimization yield incorrect results.  
> **username_1**: @BrianRomanchuk also accusing me of spreading misinformation when I clearly supported in my answer with references to leading texts on microeconomics of banking is low... does finance theory make no simplifying assumptions whatsoever? Next you will accuse Newton of spreading misinformation because in his models he often assumed perfect vacuum when there never was such thing? Clearly in theoretical economic application spread between the rates can be assumed away in many models. If you are not willing to accept that fact fine, but please be so kind and dont accuse me of spreading misinformation  
> **username_2**: I deleted my comment. But I answered the question, that’s the end of the discussion.  

---

### Answer by username_2 on 2020-11-29. Score: 1

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.](https://www.cmegroup.com/trading/interest-rates/stir/eurodollar_contractSpecs_futures.html) (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.

> **Comments:**
> **username_1**: 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.  
> **username_2**: 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.  
> **username_1**: 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  
> **username_2**: 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.  
> **username_1**: 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.  
> **username_2**: The question title refers to Eurodollar futures. The question states: “ Shouldn't r_t+n be a LIBOR rate instead of policy rate?” The answer is that yes, it should be LiBOR. That’s what the CME says, that’s what the quoted text says. The objective is to answer the question, not cover up incorrect assumptions made in the literature.  
> **username_1**: but the quoted text clearly states in equation it uses policy rate so the question is - is this a mistake of an authors or it is an intention - in some comments in past you mentioned you have PhD in mathematics, have you never heard of substitution? If students comes to you with text that says monopoly profit is price times quantity minus cost but then equation says $\Pi=(100-q)q-cq$ and student asks you shouldn’t there be p instead of 100-q you would tell that student yes there should be p - instead of p depends on demand? That would lead to wrong solution to model and would be wrong answer  
> **username_1**: Like I am completely mind boggled that you would seriously suggest something like that instead of providing insight that in economic problems we use substitutions you insist this is typo and has to be interpreted literary - like any normal text will sometimes skip a step and directly substitute for price like in the monopoly example above where you might say in text revenue is price times quantity but then in equation just directly use inverted demand $(100-q)q$ instead of $pq$... I can’t believe that you are serious right now  
> **username_1**: Your whole argument just boils down that you disapprove of the substitution because it is not common in your field outside economics so you say it is wrong. That would be like saying in the monopoly example above no it should be p*q because in real life monopoly cannot observe full  demand function and so in management literature we don’t substitute demand for price so in economics people who do that make mistake - as far as I can see this is exactly an equivalent of your answer.  

---

