Gross interest rate ($R$) v. Net interest rate ($R-1$)

Question

I am reading a macro-finance article that deals with asset financing.

While it defines the gross interest rate to be $R$, it defines the net interest rate to be $r\equiv R-1$. In equilibrium, the interest rate on one-period loan is said to be $R=\frac{1}{\beta}$, where $\beta$ is the discount factor applied to the dividends firms obtain over an infinite horizon.

$\textbf{My Question}:$ How do you interpret the difference between $R$ and $r$? Is it conventional to define net interest in macro-finance literature in this way?

Answer 1

It is often conventional. R is the amount of the loan plus the interest and r is just the interest rate. So R=1+r.

Suppose you borrow 100€ at a 10% interest rate. The interest you pay is 10€, since 10% of 100 is 10. That is where r comes in, with an interest rate of r=10%=0.1.

On the other hand, R is the rate implied by that interest rate r plus the original amount, so it is relevant for the total you pay back. Before adding interest payments, you must pay back 100% of the original loan or 1x the original loan, so R=1+r=110%=1.1.

So, in our example, with R you pay back 100×1.1= 110€, while with r we have a payment of 100×0.1=10€.