Difference between two ways of calculating real return 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.
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