# C Real vs. nominal interest rate

Back to chapter 4: “The effect of the interest rate on investment demand”

Why is the real interest rate (\(r\)) relevant for consumption and investment and not just the nominal interest rate (\(i\))?

The nominal interest rate indicates the monetary value that must be paid for a loan (per period of time). The real interest rate indicates instead the value of this payment in terms of actual goods or, simply said, in real terms. If the future expected price level corresponds to the current price level, there is no difference between the nominal and the real interest rate. However, since a loan is usually spread over several periods, it is unlikely that the price level will remain constant over time. It is therefore more important to look at the real value of interest payments, i.e. the amount of goods that can be purchased with the value of the interest payments. We need to adjust the nominal interest rate for the expected change in the price level, \(\pi^e\):

\[\begin{equation} 1 + r = \frac{1 + i}{1 + \pi^e} \tag{C.1} \end{equation}\]

This means that the change in the price level, i.e. inflation, plays an important role in determining the purchasing power of future interest payments.

We can simplify equation (C.1) and obtain:

\[r = \frac{1 + i}{1 + \pi^e} - 1\] \[r = \frac{1 + i}{1 + \pi^e} - \frac{1 + \pi^e}{1 + \pi^e}\]

\[\begin{equation} r = \frac{i - \pi^e}{1 + \pi^e} \tag{C.2} \end{equation}\]

In the case that the expected inflation rate is relatively low (\(1+\pi^e \approx 1\)), the relationship between the nominal interest rate, the real interest rate and expected inflation can be approximated by the so-called Fisher equation:

\[\begin{equation} r \approx i - \pi^e \tag{C.3} \end{equation}\]