<h3>Argument 1:</h3>

<p>$$ 1+r_{t+1} = \frac{1+i_{t}}{1+\pi_{t+1}}$$</p>

<p>Take the <a href="https://en.wikipedia.org/wiki/Natural_logarithm" rel="nofollow noreferrer">natural log</a> of both sides.</p>

<p>$$ \log(1+r_{t+1}) = \log(1+i_{t}) - \log(1+\pi_{t+1})$$</p>

<p>Observe that $\log(1 + x) \approx x$ for $x$ near zero because that's the <a href="https://stats.stackexchange.com/questions/244199/why-is-it-that-natural-log-changes-are-percentage-changes-what-is-about-logs-th/244237#244237">first order taylor expansion</a>. Hence:</p>

<p>$$ r_{t+1} \approx i_{t} - \pi_{t+1}$$</p>

<h2>Argument 2:</h2>

<p>$$ 1+r_{t+1} = \frac{1+i_{t}}{1+\pi_{t+1}}$$
$$  1+r_{t+1} + \pi_{t+1} + r_{t+1} \pi_{t+1} = 1+i_{t}$$</p>

<p>And if $r_{t+1} \pi_{t+1}$ is closer to zero</p>

<p>$$ r_{t+1} + \pi_{t+1} \approx i_{t}$$</p>

CC BY-SA 3.0
2017-07-28T02:33:08.960Z
2017-07-28T15:06:17.653Z
2017-07-28T15:06:17.653Z
Matthew Gunn
<p>I don't understand how one can get from:</p>

<p>$1+r_{t+1}$=$\frac{1+i_{t}}{1+\pi_{t+1}}$</p>

<p>to</p>

<p>$1+r_{t+1}$$\approx1+i_{t}-\pi_{t+1}$</p>

<p>by applying the following rule of thumb:</p>

<p>$Z$=$\frac{X\cdot{Y}}{Q}$</p>

<p>implies</p>

<p>$\frac{\Delta{Z}}{Z}$=$\frac{\Delta{X}}{X}$+$\frac{\Delta{Y}}{Y}$-$\frac{\Delta{Q}}{Q}$</p>

<p>Note:</p>

<p>$r_{t+1}$: real interest rate</p>

<p>$i_{t}$: nominal interest rate</p>

<p>$\pi_{t+1}$: inflation rate</p>

<p>I hope someone will help :)</p>

<macroeconomics><interest-rate>
An approximate measure of the real interest rate applying rule of thumb for growth rates
frankfranfrank
# An approximate measure of the real interest rate applying rule of thumb for growth rates

(Asked by: username_0 on 2017-07-27)

I don't understand how one can get from:

$1+r\_{t+1}$=$\frac{1+i\_{t}}{1+\pi\_{t+1}}$

to

$1+r\_{t+1}$$\approx1+i\_{t}-\pi\_{t+1}$

by applying the following rule of thumb:

$Z$=$\frac{X\cdot{Y}}{Q}$

implies

$\frac{\Delta{Z}}{Z}$=$\frac{\Delta{X}}{X}$+$\frac{\Delta{Y}}{Y}$-$\frac{\Delta{Q}}{Q}$

Note:

$r\_{t+1}$: real interest rate

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

$\pi\_{t+1}$: inflation rate

I hope someone will help :)

**Tags:** macroeconomics, interest-rate

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## 1 answers

### Answer by username_1 on 2017-07-28. Score: 3

### Argument 1:

$$ 1+r\_{t+1} = \frac{1+i\_{t}}{1+\pi\_{t+1}}$$

Take the [natural log](https://en.wikipedia.org/wiki/Natural_logarithm) of both sides.

$$ \log(1+r\_{t+1}) = \log(1+i\_{t}) - \log(1+\pi\_{t+1})$$

Observe that $\log(1 + x) \approx x$ for $x$ near zero because that's the [first order taylor expansion](https://stats.stackexchange.com/questions/244199/why-is-it-that-natural-log-changes-are-percentage-changes-what-is-about-logs-th/244237#244237). Hence:

$$ r\_{t+1} \approx i\_{t} - \pi\_{t+1}$$

## Argument 2:

$$ 1+r\_{t+1} = \frac{1+i\_{t}}{1+\pi\_{t+1}}$$
$$ 1+r\_{t+1} + \pi\_{t+1} + r\_{t+1} \pi\_{t+1} = 1+i\_{t}$$

And if $r\_{t+1} \pi\_{t+1}$ is closer to zero

$$ r\_{t+1} + \pi\_{t+1} \approx i\_{t}$$


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