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.
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
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
