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I've just learned the following way of calculating the real return rate $R$ of an investment:

$$R=\frac{P(1+N)-P(1+I)}{P(1+I)}$$

Where $P$ is the initial value invested, $N$ is the nominal interest rate and $I$ is the inflation rate.

However, I've seen an alternative formula for calculating the same thing, which is

$$R=N-I$$

So my question is, which one gives a better result for the effective rate of return of an invesment and why? Thanks very much in advance.

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The second expression is an approximation which works reasonably well for small inflation $I$ but badly for high $I$

Suppose the inflation rate is $I=300\%$ (i.e. prices this year are four times what they were a year ago) and nominal interest rates were $N=100\%$ (i.e. nominally your savings doubled in a year). Then, in real terms after interest, your savings could buy half the stuff it previously could and the real interest rate was $\frac{N-I}{1+I}=-50\%$ while $N-I=-200\%$

Instead suppose the inflation rate is $I=3\%$ and nominal interest rates were $N=1\%$. Then the real interest rate was $\frac{N-I}{1+I}\approx-1.94\%$ while $N-I=-2\%$, fairly close

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First, it's better to cancel $P$ in your first definition. Then $$ R=\frac{N-I}{1+I} $$ And the second expression is $$ R=N-I $$ Actually, it is the famous $\textit{Fisher Equation}$.

The difference that there is a $1+I$ on denominator. Of course they are different. But economists sometimes use both since they are approximately equal when $R$ and $I$ are small. If you look at the data, both real interest rate and inflation rate are just a little above 2%.

Instead of using your first definition, let's start from an alternative form $$ 1+N = (1+R)(1+I) = 1+R+I+R\cdot I $$ When $R,I$ are both small, their product can be ignored. Say $2\%\times 2\%=0.0004$. Hence, you can say $$ N= R+I \Leftrightarrow R= N-I $$

Here is the Wikipedia Link. https://en.wikipedia.org/wiki/Fisher_equation

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  • $\begingroup$ So, I understand now that the first equation is the more accurate way of calculating the real return, and that the second one is just a good approximation when $R$ and $I$ are small. However, $N-I$ makes perfect intuitive sense to me, and I really can't see why we have to divide it by $1+I$ to get the actual rate of return. Could you provide some intuition on this? Thanks. $\endgroup$ – Sasaki Nov 23 '17 at 4:34
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    $\begingroup$ The most accurate way is to write the equation as $1+N=(1+R)(1+I)$, which, mathematically, gives you the first defnition. It implies that the norminal income $P(1+N)$ is inflated by $(1+I)$, so your real income is $P(1+N)/(1+I)$ and the real return rate is $(1+N)/(1+I)-1$. It's why there is always an $I$ on the denominator. I'm not sure if it is the intuition you need. But for me, if I use the first definition, I prefer using the expression: $(1+N)/(1+I)-1$. $\endgroup$ – Andy Xu Nov 23 '17 at 4:40

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