<p>You can find the derivation in <a href="https://books.google.co.in/books/about/Macroeconomics.html?id=hrgQuwEACAAJ&amp;redir_esc=y" rel="nofollow noreferrer">Macroeconomics by Olivier Blanchard</a></p>
<p>We want to prove the relation, <span class="math-container">$$(1+i)=(1+r)(1+\pi)$$</span>
where <span class="math-container">$i$</span> means the nominal interest rate,</p>
<p><span class="math-container">$r$</span> stands for the real interest rate and</p>
<p><span class="math-container">$\pi$</span> for the expected inflation</p>
<p>We also know that from the definition of real interest rate,
<span class="math-container">$$r=i-\pi$$</span>
<span class="math-container">$$\Rightarrow i=r+\pi$$</span></p>
<p>If we expand the RHS of the above equation, we get
<span class="math-container">$$(1+r)(1+\pi)=1+r+\pi+(r\pi)$$</span></p>
<p>Observe that as both <span class="math-container">$r \in (0,1), \pi \in (0,1)$</span>
and are usually small numbers, we can approximate their product to zero (<span class="math-container">$r\pi \approx 0$</span>)</p>
<p>Therefore, we get <span class="math-container">$(1+r)(1+\pi)=1+r+\pi+(r\pi)=1+r+\pi=1+i$</span></p>
<p>Hence this proves the claim.</p>

CC BY-SA 4.0
2024-11-24T07:09:28.670Z
The equation in the question and first equation in your answer is an identity (it defines the real interest rate). There is nothing to prove. This answer has things backwards: the equation $r=i-\pi$ is an approximation derived from the equation in the question
smcc
CC BY-SA 4.0
2024-04-15T05:26:04.967Z
2024-04-15T05:26:04.967Z
SGP
<p>The inflation represents the loss of purchasing power of the monetary unit, terefore, the expected inflation rate will have to be subtracted as a first approximation from the yield on government bonds which is normally expressed in nominal terms.</p>
<p>Let</p>
<p><span class="math-container">$C $</span> : capital</p>
<p><span class="math-container">$\ i_{r} $</span>:  real interest rate</p>
<p><span class="math-container">$\ i_{n} $</span>: nominal interest rate</p>
<p><span class="math-container">$\ \pi_{e} $</span>:   expected inflation</p>
<p>We can write</p>
<p><span class="math-container">$ C \cdot\ \left (1 + i_{r} \right )= C \cdot\   \left(1 + i_{n} \right ) \cdot\ \left(1 - \pi_{e} \right ) $</span></p>
<p>Eliminating the factor <span class="math-container">$C$</span> from both members and carrying out the products we obtain <span class="math-container">$ 1 + i_{r} = 1 - \pi_{e} + i_{n} - i_{n}\cdot\ \pi_{e} $</span> that is <span class="math-container">$\ i_{r} =  i_{n} - \pi_{e} - i_{n}\cdot\ \pi_{e} $</span></p>
<p>Generally,  we have <span class="math-container">$\  i_{n} \gg \  i_{n}\cdot\ \pi_{e} $</span>, so the term <span class="math-container">$\  i_{n}\cdot\ \pi_{e} $</span> is negligible in first approximation. As a results we get</p>
<p>(*) <span class="math-container">$\ i_{r} \simeq \  i_{n} - \pi_{e} $</span></p>
<p>In conclusion, by subtracting only the contribution of the expected inflation rate from the nominal rate as indicated by the formula (*), we will commit a <span class="math-container">$\varepsilon$</span> error.</p>
<p>Some details on the estimation and calculation errors of the real discount rate are provided below.
As is known, interest rates are numerical values truncated/rounded to a certain decimal place or are known with a given uncertainty (think of the variance of expected inflation); the error in estimating <span class="math-container">$i_{r}$</span> in approximating formula will be due to two contributions: the truncation/rounding error <span class="math-container">$\varepsilon_{1}$</span> and <span class="math-container">$\varepsilon_{2}$</span> approximation error introduced in the formula obtained above.
Let denote by <span class="math-container">$ \ \varepsilon_{n} $</span> the estimate of the rounding error in the nominal interest rate and by <span class="math-container">$\ \varepsilon_{\pi} $</span> the estimate of the rounding error in the expected inflation rate. We will better represent the rate values as follows: <span class="math-container">$ i_{n} \pm \varepsilon_{n}$</span> and <span class="math-container">$ \pi_{e} \pm \ \varepsilon_{\pi} $</span>. If the starting rates are truncated to the k+1-th decimal place and rounded to the k-th decimal place it means that the starting interest rates are known to exactly k-2 decimal places and we will write: <span class="math-container">$\ i_{n } \pm 0.5\cdot\ 10^{-k} $</span> and <span class="math-container">$\ \pi_{e} \pm 0.5\cdot\ 10 ^{-k} $</span>. Using the general formula of <strong>error propagation</strong>, it will be possible to evaluate the number of exact digits with which <span class="math-container">$\ i_{r} $</span> is knowable and therefore ultimately the actualized capital. It is cleare that if the expected inflation rate is estimated with an error <span class="math-container">$\ \varepsilon_{\pi} $</span> greater than <span class="math-container">$\varepsilon_{n} $</span> the estimate of the rate in a more accurate form is useless   since the precision of the result is determined by the quantity with the lowest number of exact digits. For completeness, the evaluation of errors is reported. The error resulting from the exact formula is due only to the propagation of rounding errors and is quantifiable in <span class="math-container">$\varepsilon_{r} $</span>=<span class="math-container">$\sqrt{\varepsilon_n^2 + \varepsilon_{\pi} ^2 + \left( - \pi_{e} \cdot \ \varepsilon_{n}\right )^2 + \left( -i_n \cdot \ \varepsilon_{\pi}\right )^2 }$</span> , while the error resulting from the approximate formula can be quantified in <span class="math-container">$\varepsilon_{r}^\prime $</span>=<span class="math-container">$\sqrt{\varepsilon_n^2 + \varepsilon_{\pi}^2 + \left( i_{n} \cdot \ \pi_{e}\right )^2 }$</span>. Now assume that <span class="math-container">$\ \varepsilon_{\pi} \gg \ \varepsilon_{n} $</span> i.e. <span class="math-container">$\ \varepsilon_{\pi } $</span> is larger by two orders of magnitude ( i.e. larger by a factor of 100) than <span class="math-container">$\ \varepsilon_{n} $</span>. We say that <span class="math-container">$\ \varepsilon_{\pi} $</span> is to be considered an infinitesimal of lower order than <span class="math-container">$\ \varepsilon_{n} $</span>.</p>
<p>Reasoning in infinitesimal terms and taking into account that in a sum of infinitesimals the lower order infinitesimal prevails, we can neglect the terms that converge to zero faster and obtain: <span class="math-container">$\varepsilon_{r} \sim \ \varepsilon_{\pi} $</span> and <span class="math-container">$\varepsilon_{r}^\prime \sim \ \varepsilon_{\pi}$</span> where the symbol <span class="math-container">$\sim \ $</span> denotes the <strong>asymptotic estimate</strong>.</p>
<p>Finally, it turns out that <span class="math-container">$\varepsilon_{r}^\prime \simeq \ \varepsilon_{r} $</span> or the values of the real interest rate can be known with equal uncertainty: what has just been described justifies the extensive use of the formula (*) .</p>

CC BY-SA 4.0
2024-04-15T07:17:43.060Z
2024-04-15T07:17:43.060Z
marco tognoli
<p>Guys help me derive Fisher's equation or suggest references that have the derivation.</p>
<p>I am looking for this relation: <strong>(1+i)=(1+r)(1+π)</strong></p>

CC BY-SA 4.0
2024-04-15T10:58:00.443Z
Does this answer your question? [In fisher equation, Should I use "expected inflation" not "actual inflation"?](https://economics.stackexchange.com/questions/14057/in-fisher-equation-should-i-use-expected-inflation-not-actual-inflation)
Giskard
<macroeconomics><econometrics><mathematical-economics><monetary-policy><finance>
How to derive Fisher's equation?
user205152
# How to derive Fisher's equation?

(Asked by: username_0 on 2024-04-15)

Guys help me derive Fisher's equation or suggest references that have the derivation.

I am looking for this relation: **(1+i)=(1+r)(1+π)**

**Tags:** macroeconomics, econometrics, mathematical-economics, monetary-policy, finance

> **Comments:**
> **username_1**: Does this answer your question? [In fisher equation, Should I use "expected inflation" not "actual inflation"?](https://economics.stackexchange.com/questions/14057/in-fisher-equation-should-i-use-expected-inflation-not-actual-inflation)  

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## 2 answers

### Answer by username_2 on 2024-04-15. Score: 2

You can find the derivation in [Macroeconomics by Olivier Blanchard](https://books.google.co.in/books/about/Macroeconomics.html?id=hrgQuwEACAAJ&redir_esc=y)

We want to prove the relation, $$(1+i)=(1+r)(1+\pi)$$
where $i$ means the nominal interest rate,

$r$ stands for the real interest rate and

$\pi$ for the expected inflation

We also know that from the definition of real interest rate,
$$r=i-\pi$$
$$\Rightarrow i=r+\pi$$

If we expand the RHS of the above equation, we get
$$(1+r)(1+\pi)=1+r+\pi+(r\pi)$$

Observe that as both $r \in (0,1), \pi \in (0,1)$
and are usually small numbers, we can approximate their product to zero ($r\pi \approx 0$)

Therefore, we get $(1+r)(1+\pi)=1+r+\pi+(r\pi)=1+r+\pi=1+i$

Hence this proves the claim.

> **Comments:**
> **username_3**: The equation in the question and first equation in your answer is an identity (it defines the real interest rate). There is nothing to prove. This answer has things backwards: the equation $r=i-\pi$ is an approximation derived from the equation in the question  

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### Answer by username_4 on 2024-04-15. Score: 1

The inflation represents the loss of purchasing power of the monetary unit, terefore, the expected inflation rate will have to be subtracted as a first approximation from the yield on government bonds which is normally expressed in nominal terms.

Let

$C $ : capital

$\ i\_{r} $: real interest rate

$\ i\_{n} $: nominal interest rate

$\ \pi\_{e} $: expected inflation

We can write

$ C \cdot\ \left (1 + i\_{r} \right )= C \cdot\ \left(1 + i\_{n} \right ) \cdot\ \left(1 - \pi\_{e} \right ) $

Eliminating the factor $C$ from both members and carrying out the products we obtain $ 1 + i\_{r} = 1 - \pi\_{e} + i\_{n} - i\_{n}\cdot\ \pi\_{e} $ that is $\ i\_{r} = i\_{n} - \pi\_{e} - i\_{n}\cdot\ \pi\_{e} $

Generally, we have $\ i\_{n} \gg \ i\_{n}\cdot\ \pi\_{e} $, so the term $\ i\_{n}\cdot\ \pi\_{e} $ is negligible in first approximation. As a results we get

(\*) $\ i\_{r} \simeq \ i\_{n} - \pi\_{e} $

In conclusion, by subtracting only the contribution of the expected inflation rate from the nominal rate as indicated by the formula (\*), we will commit a $\varepsilon$ error.

Some details on the estimation and calculation errors of the real discount rate are provided below.
As is known, interest rates are numerical values truncated/rounded to a certain decimal place or are known with a given uncertainty (think of the variance of expected inflation); the error in estimating $i\_{r}$ in approximating formula will be due to two contributions: the truncation/rounding error $\varepsilon\_{1}$ and $\varepsilon\_{2}$ approximation error introduced in the formula obtained above.
Let denote by $ \ \varepsilon\_{n} $ the estimate of the rounding error in the nominal interest rate and by $\ \varepsilon\_{\pi} $ the estimate of the rounding error in the expected inflation rate. We will better represent the rate values as follows: $ i\_{n} \pm \varepsilon\_{n}$ and $ \pi\_{e} \pm \ \varepsilon\_{\pi} $. If the starting rates are truncated to the k+1-th decimal place and rounded to the k-th decimal place it means that the starting interest rates are known to exactly k-2 decimal places and we will write: $\ i\_{n } \pm 0.5\cdot\ 10^{-k} $ and $\ \pi\_{e} \pm 0.5\cdot\ 10 ^{-k} $. Using the general formula of **error propagation**, it will be possible to evaluate the number of exact digits with which $\ i\_{r} $ is knowable and therefore ultimately the actualized capital. It is cleare that if the expected inflation rate is estimated with an error $\ \varepsilon\_{\pi} $ greater than $\varepsilon\_{n} $ the estimate of the rate in a more accurate form is useless since the precision of the result is determined by the quantity with the lowest number of exact digits. For completeness, the evaluation of errors is reported. The error resulting from the exact formula is due only to the propagation of rounding errors and is quantifiable in $\varepsilon\_{r} $=$\sqrt{\varepsilon\_n^2 + \varepsilon\_{\pi} ^2 + \left( - \pi\_{e} \cdot \ \varepsilon\_{n}\right )^2 + \left( -i\_n \cdot \ \varepsilon\_{\pi}\right )^2 }$ , while the error resulting from the approximate formula can be quantified in $\varepsilon\_{r}^\prime $=$\sqrt{\varepsilon\_n^2 + \varepsilon\_{\pi}^2 + \left( i\_{n} \cdot \ \pi\_{e}\right )^2 }$. Now assume that $\ \varepsilon\_{\pi} \gg \ \varepsilon\_{n} $ i.e. $\ \varepsilon\_{\pi } $ is larger by two orders of magnitude ( i.e. larger by a factor of 100) than $\ \varepsilon\_{n} $. We say that $\ \varepsilon\_{\pi} $ is to be considered an infinitesimal of lower order than $\ \varepsilon\_{n} $.

Reasoning in infinitesimal terms and taking into account that in a sum of infinitesimals the lower order infinitesimal prevails, we can neglect the terms that converge to zero faster and obtain: $\varepsilon\_{r} \sim \ \varepsilon\_{\pi} $ and $\varepsilon\_{r}^\prime \sim \ \varepsilon\_{\pi}$ where the symbol $\sim \ $ denotes the **asymptotic estimate**.

Finally, it turns out that $\varepsilon\_{r}^\prime \simeq \ \varepsilon\_{r} $ or the values of the real interest rate can be known with equal uncertainty: what has just been described justifies the extensive use of the formula (\*) .


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