# Fisher Equation Calculator

Created by Tibor Pal, PhD candidate
Reviewed by Jack Bowater and Dominik Czernia, PhD candidate
Last updated: Jun 21, 2022

Use the Fisher equation calculator to study the relationship between the nominal interest rate, the real interest rate, and expected inflation, also referred to as the Fisher effect.

• What is the Fisher equation in economics?
• What is the Fisher equation formula?
• How to carry out the Fisher equation derivation?

In addition, we also provide you with some practical Fisher equation examples to help you better understand the topic.

## What is the Fisher equation in economics?

The Fisher equation explains the relationship between nominal and real interest rates in relation to inflation. The relatedness of the three variables was first observed by Fisher in 1933, who noticed that it is the real rather than the nominal interest rate that affects real expenditure decisions in the economy.

To see why, let's consider an investment project. Higher interest rates do not imply a greater real burden on the firm if it's offset by a proportionately higher inflation rate. This is because the expected profits from the investment project will be higher in nominal terms. In other words, the greater return in money terms will offset the higher interest cost, leaving the real return unaffected.

So, what does the Fisher equation tell us? It states that inflation expectations drive the divergence between the real and nominal interest rates. In other words, the equation tells us that the nominal interest rate is the sum of the real interest rate and expected inflation.

## What is the Fisher equation formula? Fisher equation derivation

According to the relations described above, the Fisher equation formula takes the following form:

$r \approx i - \pi^E$

where:

• $r$ - Real interest rate;
• $i$ - Nominal interest rate; and
• $\pi^E$ - Expected inflation rate.

However, it is useful to see how to solve the Fisher equation to understand the fundamentals behind the Fisher equation. A good starting point for the Fisher equation derivation is the following equation:

$(1 + r) = (1 + i) \times \frac{P}{P_{t+1}^E}$

where $P$ and ${P_{t+1}^E}$ is the current and the future expected price level at time $t$, respectively, since we do not know what the price level will be at $t+1$.

The equation states that the real interest rate is defined as the expected price of a good plus the nominal interest rate. For clarity, let's think about it in terms of a specific essential consumer good - flour. In this context, the real interest rate, $r$, is how much extra flour, $1+r$, would have to be given up (or paid) in the future to get (or borrow) one unit of the same flour today.

If the price of goods doesn't change, the real and nominal interest rates are equal: if you lent one dollar today, you would be able to buy $(1+r)$ goods in the future. It is the future expected price level (${P_{t+1}^E}$) that is crucial for an accurate calculation since at time $t$,
we do not know what the price level will be at $t+1$. To learn how to solve the Fisher equation, start by using the following definition of expected inflation:

$\pi^E = \frac{P_{t+1}^E - P}{P}$

then rearrange:

$\frac{P}{P_{t+1}^E} = \frac{1}{1 + \pi^E}$

By substituting the rearranged formula above into the initial Fisher equation derivation, we get:

$(1 + r) = \frac{1 + i}{1 + \pi^E}$

which we can rearrange to find:

$r = \frac{i - \pi^E}{1 + \pi^E}$

When expected inflation is low, the denominator of the above equation is close to one, which gives the standard approximation for the relationship between the real and the nominal rate of interest, formulating the basic Fisher equation.

In our Fisher equation calculator, you can use both the basic and the more recently introduced, more explicit form of the Fisher equation for the computation.

## How to use the Fisher equation calculator

To apply the Fisher equation calculator, you need to set the nominal interest rate and the expected inflation rate to get the real interest rate. You will see both the approximated rate computed by the basic formula and the more accurate one given by the explicit form of the Fisher equation.

Alternatively, you can set one of the real interest rates and set either the nominal interest rate or the expected inflation rate. Our Fisher equation calculator will find the missing variable for you.

## FAQ

### What does deflation imply in the Fisher equation?

When an economy experiences deflationary pressures, such as that faced by many advanced countries occurred after the 2008 financial crisis, the Fisher equation suggests that the negative expected inflation increases the real interest rate.

The higher real interest rate, in turn, constitutes an additional drag on the economy due to its negative affect on investment and consumption.

### What is the real interest rate when the expected inflation rate is 5 percent?

According to the Fisher equation, a 5 percent expected inflation rate with, for example, a 2 percent nominal interest rate, implies a 3 percent real interest rate.

The real interest rate is a simple difference between the expected inflation and nominal interest rates.

### What is Fisher debt deflation?

The Fisher debt deflation is an economic theory formulated by Irving Fisher. In essence, it implies that a decreasing price level caused by the massive selling of financial assets increases the real debt burden in the economy, leading to a further rise in loan defaults and bank insolvencies.

This process may initiate a deflationary spiral which can push the economy into a vicious downward circle.

### How do I calculate the real interest rate using the Fisher equation?

To find the real interest rate using the Fisher equation:

1. Determine nominal interest rate.
2. Estimate the expected inflation rate.
3. You can approximate the real interest rate by subtracting the expected inflation rate from the nominal interest rate.
Tibor Pal, PhD candidate
Nominal interest rate
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Expected inflation rate
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Real interest rate
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Approximated real interest rate
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