<ol>
<li>The paper to which you link actually uses actual inflation not expected inflation. They use 5 year moving average inflation in their measure. Quoting from the paper:</li>
</ol>
<blockquote>
<p>As per Furman and Summers (2020) the
inflation rate is taken as the five-year moving average to give a measure closer to the
concept of expected inflation</p>
</blockquote>
<p>Both of the models thus use real past inflation averaged over 5 years. This is concept that is closer to expected inflation, as the quote says, but it is not an expected inflation itself.</p>
<ol start="2">
<li>Expected inflation is used in models that model people's/institution's behavior since current rate of inflation is simply unobservable.  The real interest rate thus depends on what inflation people expect (since nominal interest rate is effectively set by central bank).</li>
</ol>
<p>They do not simply calculate real interest rate as nominal rate minus inflation as far as I can see they take estimates of real interest rate using the 10y inflation protected securities. In that case relevant variable to use is inflation rate that is close to expected inflation.</p>

CC BY-SA 4.0
2023-01-21T15:56:07.773Z
Thanks for engaging. But I'm still unclear. Yes, we do not know what the real interest rate will be this year as inflation is unknown. But we can calculate the real interest rate retrospectively. So if I want to calculate what the real rate of interest on a loan was, say, five years ago I would take the nominal rate that year and subtract the actual inflation rate NOT a proxy for what inflation was expected to be. Similarly, if we want to calculate the real burden of servicing debt retrospectively, I don't see why we don't use actual inflation.
R.S.
CC BY-SA 4.0
2023-01-21T20:53:12.467Z
@R.S.  1. but we are not calculating $r$ here. We have actual data for $r$ so calculating it retrospectively is unnecessary.  2. Well but the inflation rate you find in some dataset is not the actual inflation rate its an estimate from CPI, so the inflation rate is a proxy for what inflation was.
1muflon1
CC BY-SA 4.0
2023-01-21T22:09:13.840Z
But the real rate of return is exactly the variable I want to calculate. As I said, I was using the expressions 'to calculate real rates of return, or equivalently the real debt burden'. The linked papers similarly were trying to calculate the real burden of debt service. As an aside, statistical agencies typically do define inflation to be a change in the CPI. So, I still don't follow you on either count.
R.S.
CC BY-SA 4.0
2023-01-21T23:39:34.223Z
@R.S. 1. those papers are not trying to calculate real rate of return. They already have data for that. 2. $r \approx i - \pi$ holds for actual inflation, not on how statistical agencies calculate it. If statistical agency would lets say only calculate CPI from food prices it would mean $r \not\approx i - \pi$. For example, we can show mathematically that all incomes $w+i+\Pi + r = Y = C+I+G +NX$ but when statistical offices calculate them they are no longer equal due to various errors and the fact that stat offices just use estimates not real values
1muflon1
CC BY-SA 4.0
2023-01-21T23:45:30.263Z
hence although you can theoretically calculate wages by subtracting profit, interest rate and return on capital from GDP IRL that would be less precise than directly using data for wages
1muflon1
CC BY-SA 4.0
2023-01-22T00:03:29.540Z
Per paper (p40) "In order to estimate Equation (1)...we use GDP forecasts from the latest
QEC, gross debt and interest payment forecasts from Budget 2021 and inflation forecasts
from the European Commission (averaged over five years). The real interest rates as a
share of GDP as well as the debt-to-GDP ratio are presented in Figure B". In other words GDP, debt, nominal interest payments, and inflation are the input variables, real interest payment is the calculated variable. Yes, wages may be better calculated directly, but I don't see the relevance of your analogy.
R.S.
CC BY-SA 4.0
2023-01-22T00:30:47.473Z
@R.S. I am quoting from the paper you linked: "Real
interest rates are based on 10-year Treasury Inflation Protected Securities (TIPS) from January 2000 and February 2020." so they did not estimated the real interest rate in that paper.
1muflon1
CC BY-SA 4.0
2023-01-21T13:59:36.720Z
2023-01-21T13:59:36.720Z
1muflon1
<p>The real rate of return is given by:</p>
<p>1+r=(1+i)/(1+π)</p>
<p>and it can easily be shown that:</p>
<p>1+r ≈ i-π+1 so that r ≈ i-π</p>
<p>This can be used to calculate real rates of return, or equivalently the real debt burden multiplying across by previous year debt D_(t-1):</p>
<p>rD_(t-1)=(i-π) D_(t-1)=iD_(t-1)-πD_(t-1)</p>
<p>However, <a href="https://www.brookings.edu/wp-content/uploads/2020/11/furman-summers-fiscal-reconsideration-discussion-draft.pdf" rel="nofollow noreferrer">some</a> <a href="https://www.econstor.eu/bitstream/10419/237300/1/qec20211.pdf" rel="nofollow noreferrer">economists</a> use expected inflation, rather than actual inflation to do this, which I don't understand.
If an investor has a required rate of return, expected inflation is the appropriate indicator used to meet that RRR. So if I want a return of 10%, and I believe inflation will be 2% then I charge a nominal of 10% on a loan:
r= i - expected inflation. 5-year average of inflation would be useful for that.
But if we want to see what rate of real return was actually realised, looking back or ex post, then it is r=i-inflation. So I don't understand why in the above equation for the real burden of servicing debt, one would use expected inflation. That seems to be measuring what the real debt burden of servicing debt would be if inflation turned out as expected, not what the real burden actually was. I discussed the real debt burden on this <a href="https://economics.stackexchange.com/questions/54126/real-debt-burden">forum</a> recently. Can anyone clarify?</p>

<inflation><interest-rate><government-debt><debt>
Expected vs actual inflation and the real debt burden
R.S.
# Expected vs actual inflation and the real debt burden

(Asked by: username_0 on 2023-01-21)

The real rate of return is given by:

1+r=(1+i)/(1+π)

and it can easily be shown that:

1+r ≈ i-π+1 so that r ≈ i-π

This can be used to calculate real rates of return, or equivalently the real debt burden multiplying across by previous year debt D\_(t-1):

rD\_(t-1)=(i-π) D\_(t-1)=iD\_(t-1)-πD\_(t-1)

However, [some](https://www.brookings.edu/wp-content/uploads/2020/11/furman-summers-fiscal-reconsideration-discussion-draft.pdf) [economists](https://www.econstor.eu/bitstream/10419/237300/1/qec20211.pdf) use expected inflation, rather than actual inflation to do this, which I don't understand.
If an investor has a required rate of return, expected inflation is the appropriate indicator used to meet that RRR. So if I want a return of 10%, and I believe inflation will be 2% then I charge a nominal of 10% on a loan:
r= i - expected inflation. 5-year average of inflation would be useful for that.
But if we want to see what rate of real return was actually realised, looking back or ex post, then it is r=i-inflation. So I don't understand why in the above equation for the real burden of servicing debt, one would use expected inflation. That seems to be measuring what the real debt burden of servicing debt would be if inflation turned out as expected, not what the real burden actually was. I discussed the real debt burden on this [forum](https://economics.stackexchange.com/questions/54126/real-debt-burden) recently. Can anyone clarify?

**Tags:** inflation, interest-rate, government-debt, debt


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## 1 answers

### Answer by username_1 on 2023-01-21. Score: 0

1. The paper to which you link actually uses actual inflation not expected inflation. They use 5 year moving average inflation in their measure. Quoting from the paper:

> As per Furman and Summers (2020) the
> inflation rate is taken as the five-year moving average to give a measure closer to the
> concept of expected inflation

Both of the models thus use real past inflation averaged over 5 years. This is concept that is closer to expected inflation, as the quote says, but it is not an expected inflation itself.

2. Expected inflation is used in models that model people's/institution's behavior since current rate of inflation is simply unobservable. The real interest rate thus depends on what inflation people expect (since nominal interest rate is effectively set by central bank).

They do not simply calculate real interest rate as nominal rate minus inflation as far as I can see they take estimates of real interest rate using the 10y inflation protected securities. In that case relevant variable to use is inflation rate that is close to expected inflation.

> **Comments:**
> **username_0**: Thanks for engaging. But I'm still unclear. Yes, we do not know what the real interest rate will be this year as inflation is unknown. But we can calculate the real interest rate retrospectively. So if I want to calculate what the real rate of interest on a loan was, say, five years ago I would take the nominal rate that year and subtract the actual inflation rate NOT a proxy for what inflation was expected to be. Similarly, if we want to calculate the real burden of servicing debt retrospectively, I don't see why we don't use actual inflation.  
> **username_1**: @username_0  1. but we are not calculating $r$ here. We have actual data for $r$ so calculating it retrospectively is unnecessary.  2. Well but the inflation rate you find in some dataset is not the actual inflation rate its an estimate from CPI, so the inflation rate is a proxy for what inflation was.  
> **username_0**: But the real rate of return is exactly the variable I want to calculate. As I said, I was using the expressions 'to calculate real rates of return, or equivalently the real debt burden'. The linked papers similarly were trying to calculate the real burden of debt service. As an aside, statistical agencies typically do define inflation to be a change in the CPI. So, I still don't follow you on either count.  
> **username_1**: @username_0 1. those papers are not trying to calculate real rate of return. They already have data for that. 2. $r \approx i - \pi$ holds for actual inflation, not on how statistical agencies calculate it. If statistical agency would lets say only calculate CPI from food prices it would mean $r \not\approx i - \pi$. For example, we can show mathematically that all incomes $w+i+\Pi + r = Y = C+I+G +NX$ but when statistical offices calculate them they are no longer equal due to various errors and the fact that stat offices just use estimates not real values  
> **username_1**: hence although you can theoretically calculate wages by subtracting profit, interest rate and return on capital from GDP IRL that would be less precise than directly using data for wages  
> **username_0**: Per paper (p40) "In order to estimate Equation (1)...we use GDP forecasts from the latest QEC, gross debt and interest payment forecasts from Budget 2021 and inflation forecasts from the European Commission (averaged over five years). The real interest rates as a share of GDP as well as the debt-to-GDP ratio are presented in Figure B". In other words GDP, debt, nominal interest payments, and inflation are the input variables, real interest payment is the calculated variable. Yes, wages may be better calculated directly, but I don't see the relevance of your analogy.  
> **username_1**: @username_0 I am quoting from the paper you linked: "Real interest rates are based on 10-year Treasury Inflation Protected Securities (TIPS) from January 2000 and February 2020." so they did not estimated the real interest rate in that paper.  

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