<blockquote>
  <p>Does the logic go as follows? </p>
  
  <p>short term rate + expected inflation= long term rates?</p>
</blockquote>

<p>Not exactly. There is some confusion. Inflation distinguishes between <em>real</em> and <em>nominal</em> interest rates, whereas the discrepancy between short- and long-term interest rates is derived from the preclusion of <em>arbitrage opportunities</em>. It is neither actual nor expected inflation, but the <em>forward rate</em>, what explains the discrepancy between short- and long-term interest rates. </p>

<p>By way of example, suppose the short- and long-term rates currently are 3% and 5%, with short- and long-term referring to 1 and 2 years, respectively. The interest rate $r_{f}$ for the 2nd year should be such that $(1+0.03)(1+r_{f})=(1+0.05)^2$. Any other rate $r_{f}$ would allow either the borrower or the lender to make profit with no risk (or somewhat equivalently, to make profit without having to invest his own funds).</p>

<p>It is true that a lender requires higher return for long-term loans than for short-term ones because a longer term entails greater exposure to risk. But the inflation is just <em>one of multiple</em> risks that the lender factors in when determining at what rate (of return) he is willing to lend money.</p>

<p><strong>Edited on 7/22/2018 (at 3:29PM EDT) to supplement as per 2nd-to-last comment</strong></p>

<p>There is no robust or widely-adopted method for calculation of long-term rates based on estimated future inflation (and/or short-term rates). Nor is there a mathematical expression describing how --or even whether-- <em>market forces</em> use the estimated future inflation for determining the rate at which lenders are willing to commit funds for a long-term period. I personally doubt that such method or expression could ever exist.</p>

<p>One can only resort to <em>qualitative reasoning</em> to sketch the <em>generic</em> effect that some information tends to have on long-term rates. In the case of expected/estimated future inflation, it might be <em>implicitly</em> (at most, but perhaps not at all) reflected in other <em>factors and preferences</em> that influence those market forces. Uncertainty --as you rightly point out in one of your comments--, risk aversion, and <em>cost of opportunity</em> are three of many other factors and preferences that influence how much a long-term rate would differ from the short-term rate.</p>

<p>Although it is reasonable to conjecture that these estimates, factors, and preferences are positively correlated with the duration of a long-term period, it would be very hard (or I would say futile as well as impossible) to reliably model <em>by how much</em> they cause a long-term rate to differ from the short-term rate.</p>

CC BY-SA 4.0
2018-07-21T12:55:20.223Z
But this what "Inverstopedia" writes right? And you are saying it is oversimplifying?
user123124
CC BY-SA 4.0
2018-07-21T14:31:44.887Z
@user1 It is right in that a lender's perceived risk of future inflation prompts him to require a higher return. But it would be inaccurate --perhaps an oversimplification-- to interpret the article as if short-term rate + expected inflation = long-term rate.Note how Investopedia says that (1) "Inflation is a bond's *worst* enemy" (rather than its *only* enemy), and (2) market forces (rather than the central bank) set the long-term rates. I think the article errs in that current high inflation necessarily increases the expected/future inflation, but that goes beyond the scope of your question.
Iñaki Viggers
CC BY-SA 4.0
2018-07-21T16:53:37.463Z
"Market forces (supply and demand) determine equilibrium pricing for long-term bonds, which set long-term interest rates. If the bond market believes that the FOMC has set the fed funds rate too low, expectations of future inflation increase, which means long-term interest rates increase relative to short-term interest rates" . This is the passage that made me think the above. How about if I change "expected interest" to "what the market expects in terms of interest" ...
user123124
CC BY-SA 4.0
2018-07-21T17:06:59.430Z
...to summerize, given that the market thinks that the interest rate will be some number(depending on the expected inflation) and then do the math of "forward" interest given these expectations we get the long rate. And inflation is the worst enemy due to the uncertainty it adds to the calculation of the forward rate. Moreover it is uncertainty we get paid for, not just the loss of value due to inflation. Maybe I also should write "and" instead of "+" to more precise.
user123124
CC BY-SA 4.0
2018-07-21T18:27:29.263Z
@user1 See the explanation in the answer: Long-term rate is given, and from there the forward rate is calculated, not vice versa. "*Moreover it is uncertainty we get paid for, not just the loss of value due to inflation*": Yes, that's correct, plus other considerations such as *cost of opportunity*, which could be positively correlated with the length or duration of the long-term.
Iñaki Viggers
CC BY-SA 4.0
2018-07-21T20:29:37.173Z
I think I got it now, the market forces(due to for example expected inflation among other rings) decides what they think the value $r_{f}$ should be, and those who get along create some contracts. If there would be no uncertainty it could equal 1.07xx due to lack of arbitrage and what the market agree that $r_{f}$ should be becomes the long rate?
user123124
CC BY-SA 4.0
2018-07-21T20:39:27.177Z
@user1 We're getting there: Market forces determine the *long-term* rate (see item (2) of my 1st comment), and from there we obtain 1+ $r_{f}$ = 1.070388. In other words, $r_{f}$=7.0388% is the [short-term rate] that should apply in the 2nd year, as otherwise there would be an arbitrage opportunity given the current short- and long-term rates of 3% and 5%, respectively.
Iñaki Viggers
CC BY-SA 4.0
2018-07-22T05:48:44.643Z
We might have misunderstood each other, Id like to know how the long term rate is determined. My impression from the text is that it is a function of short rate and expected inflation (i.e future short term rates) by the market. Hence this is what determine the bond prince. So the long term cant be given :/ . It looks to me that we have instead answered, how does arbitrage enter the relation between short and long term rates and more explicitly we computed the second year short term rate in a deterministic setting.
user123124
CC BY-SA 4.0
2018-07-22T06:38:12.617Z
If we use your equation $(1+0.03)(1+r_{f})=(1+0.05)^{2}$, I am basically asking if by estimating inflation can we determine the long rate via what we think what the market expects the central banks to set $r_{f}$ to in the future(given the estimate of the inflation and how the cetnral banks will respond to it) and hence obtain the long rate, now being $(1+0.x)^{2}$, depending on what we estimate or how we think $r_{f}$ will develop.
user123124
CC BY-SA 4.0
2018-07-22T12:19:08.680Z
There is no concrete or general expression for how short-term rates and/or estimated inflation determine long-term rates. Basically, one can only resort to *qualitative* reasoning. Market forces will indicate a consensus (or equilibrium) rate at which lender are willing to commit funds for that long period. But there is no general model of how market forces reach that consensus either. As mentioned previously, estimated inflation is *not the only factor* for setting a long-term rate. Other factors and preferences intervene, and they would be very hard (if not impossible) to model reliably.
Iñaki Viggers
CC BY-SA 4.0
2018-07-22T12:35:13.257Z
Sorry about being hard to understand. But is using qualitative reasoning in the way I have suggested above way to crude, even to the point that it becomes misleading or even nonsense? i.e by just including the expected inflation.
user123124
CC BY-SA 4.0
2018-07-22T12:37:05.990Z
By the way if we add your last comment as answer, id be happy to accept it. Unless you cant possibly interpret my question in a way such that it becomes a suitable answer in your opinion.
user123124
CC BY-SA 4.0
2018-07-22T21:04:13.327Z
@user1 Just FYI, done.
Iñaki Viggers
CC BY-SA 4.0
2018-07-21T11:10:57.350Z
2018-07-22T19:29:59.953Z
2018-07-22T19:29:59.953Z
Iñaki Viggers
<p>I just read about the "dynamics" of long-, short-term intreset rates and inflation. Does the logic go as follows?</p>

<p>Centrals banks determine interest rates which in turn determine the short term rates. Inflation and expected inflation on the other hand determine how the central bank will most likely change the rates in the future(i.e what the market expects the central banks to do), hence</p>

<p>short term rate + expected inflation= long term rates?</p>

<p>Found the info here,</p>

<p><a href="https://www.investopedia.com/articles/bonds/09/bond-market-interest-rates.asp" rel="nofollow noreferrer">https://www.investopedia.com/articles/bonds/09/bond-market-interest-rates.asp</a></p>

<macroeconomics><interest-rate>
Long-,short-term interest rates and inflation
user123124
# Long-,short-term interest rates and inflation

(Asked by: username_0 on 2018-07-21)

I just read about the "dynamics" of long-, short-term intreset rates and inflation. Does the logic go as follows?

Centrals banks determine interest rates which in turn determine the short term rates. Inflation and expected inflation on the other hand determine how the central bank will most likely change the rates in the future(i.e what the market expects the central banks to do), hence

short term rate + expected inflation= long term rates?

Found the info here,

<https://www.investopedia.com/articles/bonds/09/bond-market-interest-rates.asp>

**Tags:** macroeconomics, interest-rate

---

## 1 answers

### Answer by username_1 on 2018-07-21. Score: 2 | Accepted answer

> Does the logic go as follows?
>
> short term rate + expected inflation= long term rates?

Not exactly. There is some confusion. Inflation distinguishes between *real* and *nominal* interest rates, whereas the discrepancy between short- and long-term interest rates is derived from the preclusion of *arbitrage opportunities*. It is neither actual nor expected inflation, but the *forward rate*, what explains the discrepancy between short- and long-term interest rates.

By way of example, suppose the short- and long-term rates currently are 3% and 5%, with short- and long-term referring to 1 and 2 years, respectively. The interest rate $r\_{f}$ for the 2nd year should be such that $(1+0.03)(1+r\_{f})=(1+0.05)^2$. Any other rate $r\_{f}$ would allow either the borrower or the lender to make profit with no risk (or somewhat equivalently, to make profit without having to invest his own funds).

It is true that a lender requires higher return for long-term loans than for short-term ones because a longer term entails greater exposure to risk. But the inflation is just *one of multiple* risks that the lender factors in when determining at what rate (of return) he is willing to lend money.

**Edited on 7/22/2018 (at 3:29PM EDT) to supplement as per 2nd-to-last comment**

There is no robust or widely-adopted method for calculation of long-term rates based on estimated future inflation (and/or short-term rates). Nor is there a mathematical expression describing how --or even whether-- *market forces* use the estimated future inflation for determining the rate at which lenders are willing to commit funds for a long-term period. I personally doubt that such method or expression could ever exist.

One can only resort to *qualitative reasoning* to sketch the *generic* effect that some information tends to have on long-term rates. In the case of expected/estimated future inflation, it might be *implicitly* (at most, but perhaps not at all) reflected in other *factors and preferences* that influence those market forces. Uncertainty --as you rightly point out in one of your comments--, risk aversion, and *cost of opportunity* are three of many other factors and preferences that influence how much a long-term rate would differ from the short-term rate.

Although it is reasonable to conjecture that these estimates, factors, and preferences are positively correlated with the duration of a long-term period, it would be very hard (or I would say futile as well as impossible) to reliably model *by how much* they cause a long-term rate to differ from the short-term rate.

> **Comments:**
> **username_0**: But this what "Inverstopedia" writes right? And you are saying it is oversimplifying?  
> **username_1**: @user1 It is right in that a lender's perceived risk of future inflation prompts him to require a higher return. But it would be inaccurate --perhaps an oversimplification-- to interpret the article as if short-term rate + expected inflation = long-term rate.Note how Investopedia says that (1) "Inflation is a bond's *worst* enemy" (rather than its *only* enemy), and (2) market forces (rather than the central bank) set the long-term rates. I think the article errs in that current high inflation necessarily increases the expected/future inflation, but that goes beyond the scope of your question.  
> **username_0**: "Market forces (supply and demand) determine equilibrium pricing for long-term bonds, which set long-term interest rates. If the bond market believes that the FOMC has set the fed funds rate too low, expectations of future inflation increase, which means long-term interest rates increase relative to short-term interest rates" . This is the passage that made me think the above. How about if I change "expected interest" to "what the market expects in terms of interest" ...  
> **username_0**: ...to summerize, given that the market thinks that the interest rate will be some number(depending on the expected inflation) and then do the math of "forward" interest given these expectations we get the long rate. And inflation is the worst enemy due to the uncertainty it adds to the calculation of the forward rate. Moreover it is uncertainty we get paid for, not just the loss of value due to inflation. Maybe I also should write "and" instead of "+" to more precise.  
> **username_1**: @user1 See the explanation in the answer: Long-term rate is given, and from there the forward rate is calculated, not vice versa. "*Moreover it is uncertainty we get paid for, not just the loss of value due to inflation*": Yes, that's correct, plus other considerations such as *cost of opportunity*, which could be positively correlated with the length or duration of the long-term.  
> **username_0**: I think I got it now, the market forces(due to for example expected inflation among other rings) decides what they think the value $r_{f}$ should be, and those who get along create some contracts. If there would be no uncertainty it could equal 1.07xx due to lack of arbitrage and what the market agree that $r_{f}$ should be becomes the long rate?  
> **username_1**: @user1 We're getting there: Market forces determine the *long-term* rate (see item (2) of my 1st comment), and from there we obtain 1+ $r_{f}$ = 1.070388. In other words, $r_{f}$=7.0388% is the [short-term rate] that should apply in the 2nd year, as otherwise there would be an arbitrage opportunity given the current short- and long-term rates of 3% and 5%, respectively.  
> **username_0**: We might have misunderstood each other, Id like to know how the long term rate is determined. My impression from the text is that it is a function of short rate and expected inflation (i.e future short term rates) by the market. Hence this is what determine the bond prince. So the long term cant be given :/ . It looks to me that we have instead answered, how does arbitrage enter the relation between short and long term rates and more explicitly we computed the second year short term rate in a deterministic setting.  
> **username_0**: If we use your equation $(1+0.03)(1+r_{f})=(1+0.05)^{2}$, I am basically asking if by estimating inflation can we determine the long rate via what we think what the market expects the central banks to set $r_{f}$ to in the future(given the estimate of the inflation and how the cetnral banks will respond to it) and hence obtain the long rate, now being $(1+0.x)^{2}$, depending on what we estimate or how we think $r_{f}$ will develop.  
> **username_1**: There is no concrete or general expression for how short-term rates and/or estimated inflation determine long-term rates. Basically, one can only resort to *qualitative* reasoning. Market forces will indicate a consensus (or equilibrium) rate at which lender are willing to commit funds for that long period. But there is no general model of how market forces reach that consensus either. As mentioned previously, estimated inflation is *not the only factor* for setting a long-term rate. Other factors and preferences intervene, and they would be very hard (if not impossible) to model reliably.  
> **username_0**: Sorry about being hard to understand. But is using qualitative reasoning in the way I have suggested above way to crude, even to the point that it becomes misleading or even nonsense? i.e by just including the expected inflation.  
> **username_0**: By the way if we add your last comment as answer, id be happy to accept it. Unless you cant possibly interpret my question in a way such that it becomes a suitable answer in your opinion.  
> **username_1**: @user1 Just FYI, done.  

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