<p><span class="math-container">$Growth\ in\ Purchasing\ Power=1+r_r = \displaystyle\frac{1+r}{1+i}$</span></p>

<p>Also it is given that <span class="math-container">$real\ interest \ rate=r_r$</span></p>

<p>So :</p>

<p><span class="math-container">$$real\ interest \ rate=(1+r_r)-1=\frac{1+r}{1+i}-1=\frac{1+r}{1+i}-\frac{1+i}{1+i}=\frac{r-i}{1+i}$$</span></p>

CC BY-SA 4.0
2019-06-28T19:25:21.117Z
2019-06-28T19:25:21.117Z
user18214
<p>My textbook gives these two equations including the real interest rate and says that the second equation (b) can be arrived at from the first (a), but doesn't explain how.</p>

<p><em>r<sub>r</sub> = Real Interest Rate, r = Nominal Interest Rate, i = Inflation</em></p>

<p>Equation (a):
<span class="math-container">$$Growth\ in\ Purchasing\ Power=1+r_r = \displaystyle\frac{1+r}{1+i}$$</span></p>

<p>Equation (b):</p>

<p><span class="math-container">$$Real\ Interest\ Rate = r_r = \displaystyle\frac{r-i}{1+i}$$</span></p>

<p><br>I'm feeling slightly obtuse but I've been away from math quite a while, and can't figure out how to rearrange the first formula into the second one.</p>

<p>What are the exact equation steps necessary to get from: <span class="math-container">$ r_r = \displaystyle\frac{1+r}{1+i}-1$</span> ⇒ <span class="math-container">$r_r = \displaystyle\frac{r-i}{1+i}$</span> ?</p>

<interest-rate><purchasing-power-partity>
Relationship between Growth in Purchasing Power and Real Interest Rate equations
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# Relationship between Growth in Purchasing Power and Real Interest Rate equations

(Asked by: username_0 on 2019-06-28)

My textbook gives these two equations including the real interest rate and says that the second equation (b) can be arrived at from the first (a), but doesn't explain how.

*rr = Real Interest Rate, r = Nominal Interest Rate, i = Inflation*

Equation (a):
$$Growth\ in\ Purchasing\ Power=1+r\_r = \displaystyle\frac{1+r}{1+i}$$

Equation (b):

$$Real\ Interest\ Rate = r\_r = \displaystyle\frac{r-i}{1+i}$$

I'm feeling slightly obtuse but I've been away from math quite a while, and can't figure out how to rearrange the first formula into the second one.

What are the exact equation steps necessary to get from: $ r\_r = \displaystyle\frac{1+r}{1+i}-1$ ⇒ $r\_r = \displaystyle\frac{r-i}{1+i}$ ?

**Tags:** interest-rate, purchasing-power-partity


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## 1 answers

### Answer by username_1 on 2019-06-28. Score: 1 | Accepted answer

$Growth\ in\ Purchasing\ Power=1+r\_r = \displaystyle\frac{1+r}{1+i}$

Also it is given that $real\ interest \ rate=r\_r$

So :

$$real\ interest \ rate=(1+r\_r)-1=\frac{1+r}{1+i}-1=\frac{1+r}{1+i}-\frac{1+i}{1+i}=\frac{r-i}{1+i}$$


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