Calculating Real Interest Rates with the Fisher Equation

Introduction

In our previous lesson, we established how to measure inflation using the Consumer Price Index (CPI), giving us a robust tool to quantify changes in the cost of living. However, the interest rates quoted by banks or on financial assets—the nominal interest rates—do not account for inflation's erosive effect on money's value. To understand the true return on an investment or the real cost of a loan, we must adjust for inflation.

This lesson focuses on precisely that adjustment. Our goal is to calculate the real interest rate using the Fisher equation. We will explore the crucial distinction between nominal and real returns, derive the mathematical relationship connecting them, and apply it to solve quantitative problems. Given your background in statistics, you will find the derivation and the distinction between the exact formula and its linear approximation particularly relevant. This skill is foundational for analyzing investment decisions, monetary policy, and the long-term consequences of inflation.

The Real vs. Nominal Interest Rate

At its core, the distinction is simple: the nominal interest rate tells you how fast the amount of money in your account grows, while the real interest rate tells you how fast the purchasing power of that money grows. It is the growth in purchasing power that ultimately matters to individuals and firms.

To build intuition, let's start with a simple conceptual example.

Nominal & Real Interest Rate (Fisher Equation)

This video from Iris Franz provides a clear definition of nominal and real interest rates and illustrates the concept with a straightforward example.

Please watch the first two segments. First, watch the introduction from the beginning, which defines the two rates. Then, watch the example of purchasing chalk. Notice how the increase in the price of chalk (inflation) reduces the number of packs you can actually buy, even though your money has grown by the nominal interest rate.

The video demonstrates that what you can do with your money depends on both the interest you earn and how prices have changed. The real interest rate captures this net effect.

Let's define our terms formally:

Nominal Interest Rate ( $i$ ): The rate of growth in the monetary value of an asset. It's the rate quoted without any adjustment for inflation.
Inflation Rate ( $π$ ): The rate at which the general level of prices is increasing, as measured by an index like the CPI.
Real Interest Rate ( $r$ ): The rate of growth in the purchasing power of an asset. This is the nominal interest rate adjusted for inflation.

The Fisher Equation: Exact and Approximate Forms

The relationship between these three variables was formally described by the economist Irving Fisher. The precise relationship, known as the exact Fisher equation, is multiplicative.

Imagine you lend €1 today.

In one year, you receive back $1 + i$ euros.
However, during that year, the price level has increased by a factor of $1 + π$ . So, what €1 could buy today costs $1 + π$ euros in a year.
The real value (purchasing power) of your $1 + i$ euros in a year's time is therefore $\frac{1 + i}{1 + π}$ .
The real rate of return, $r$ , is the growth in this purchasing power. So, $1 + r = \frac{1 + i}{1 + π}$ .

Rearranging this gives the standard form of the exact Fisher equation:

(1 + i) = (1 + r) (1 + π)

This equation states that the growth factor of your money is the product of the growth factor of your purchasing power and the growth factor of the price level.

While this form is exact, for quick calculations and theoretical models, a simpler linear approximation is often used. Your mathematical background will appreciate seeing exactly where this approximation comes from.

How can I derive Fisher's equation? - Economics Stack Exchange

This Stack Exchange answer provides a clear mathematical derivation of the approximate Fisher equation from the exact form.

Please read the first answer on the page, starting from the derivation. Focus on the expansion of the term (1+r)(1+\pi) and the justification for why the product term r\pi can be neglected.

As the reading shows, by expanding the right side of the exact equation, we get:

1 + i = 1 + r + π + r π

Subtracting 1 from both sides gives:

i = r + π + r π

When the real interest rate ( $r$ ) and the inflation rate ( $π$ ) are small, their product ( $r π$ ) is a very small number. For example, if $r = 0.02$ and $π = 0.03$ , then $r π = 0.0006$ . By treating this term as negligible ( $r π \approx 0$ ), we arrive at the approximate Fisher equation:

i \approx r + π

This is most commonly rearranged to solve for the real interest rate:

r \approx i - π

This simple subtraction is a useful rule of thumb, but for your CB2 exam, you should be prepared to use both the exact and approximate formulas.

Calculating the Real Interest Rate

Now, let's focus on the practical calculation. The exact formula for the real interest rate, derived by rearranging the multiplicative equation, is:

r = \frac{1 + i}{1 + π} - 1

Let's see this in action.

Nominal Interest Rate and Real Interest Rate Vs. Inflation. Calculation Fisher Effect. CFA Exam

This video from Farhat Lectures demonstrates how to apply the accurate formula to calculate the real interest rate and contrasts it with the approximation.

Please watch the section from this timestamp. Pay close attention to how the formula r = (i - \pi) / (1+\pi), which is an algebraic rearrangement of our exact formula, is used to find the precise real rate.

Worked Example

Let's work through a problem step-by-step.
Assume the nominal interest rate on a one-year government bond is $i = 7%$ and the expected inflation rate for the year is $π = 4%$ .

Calculate the approximate real interest rate:
Using the linear approximation, $r \approx i - π$ :
$r \approx 7% - 4% = 3%$
Calculate the exact real interest rate:
Using the precise formula, $r = \frac{1 + i}{1 + π} - 1$ :
$r = \frac{1 + 0.07}{1 + 0.04} - 1 = \frac{1.07}{1.04} - 1 \approx 1.0288 - 1 = 0.0288$
So, the exact real interest rate is 2.88%. As you can see, the approximation (3%) is very close to the exact value, which is why it is widely used.

Practice Problem

Now it's your turn. Suppose you deposit money in a savings account that pays a nominal interest rate of 5% per year. The CPI at the beginning of the year is 250, and it is projected to be 260 at the end of the year.

(a) Calculate the expected inflation rate for the year.
(b) Calculate the approximate real interest rate.
(c) Calculate the exact real interest rate.

Click here to check your solution.

(a) Calculate the expected inflation rate:
The inflation rate ( $π$ ) is the percentage change in the CPI:

π = \frac{CPI _{end} - CPI _{start}}{CPI _{start}} = \frac{260 - 250}{250} = \frac{10}{250} = 0.04

The expected inflation rate is 4%.

(b) Calculate the approximate real interest rate:
Using the formula $r \approx i - π$ :

r \approx 5% - 4% = 1%

The approximate real interest rate is 1%.

(c) Calculate the exact real interest rate:
Using the formula $r = \frac{1 + i}{1 + π} - 1$ :

r = \frac{1 + 0.05}{1 + 0.04} - 1 = \frac{1.05}{1.04} - 1 \approx 1.0096 - 1 = 0.0096

The exact real interest rate is 0.96%.

The Fisher Effect and Negative Interest Rates

Irving Fisher also proposed a long-run hypothesis known as the Fisher effect. It posits that the real interest rate is relatively stable over time, being determined by fundamental factors like the productivity of capital and people's time preferences. If this is true, then changes in the inflation rate should be fully reflected in the nominal interest rate.

[PDF] 6. Real interest rate, Fisher equation, arbitrage and speculation

This document summarizes the Fisher equation and its implications.

Please read section 3, "The Fisher equation", which reinforces the approximation. Then, read section 4 on the "Components of the nominal interest rate", which explains the Fisher effect. Finally, review section 5 on "Negative interest rates".

The Fisher effect essentially argues that in the long run, nominal interest rates adjust on a one-for-one basis with expected inflation. If expected inflation rises by 1 percentage point, the nominal interest rate will also tend to rise by 1 percentage point to keep the real rate of return for lenders unchanged.

i = r_{stable} + π_{e}

where $π_{e}$ is expected inflation.

A crucial insight from the Fisher equation is that real interest rates can be negative. This occurs whenever the inflation rate is higher than the nominal interest rate ( $π > i$ ). In such a scenario, the purchasing power of your savings or investment actually decreases over time, even if the nominal amount of money is growing. This is a common situation during periods of high, unexpected inflation.

Conclusion

In this lesson, we have moved beyond nominal figures to understand the real economic return on an investment. By adjusting the nominal interest rate for inflation, we can calculate the real interest rate, which is a far more meaningful measure of changes in purchasing power.

Key Takeaways:

The nominal interest rate ( $i$ ) is the quoted rate of return, while the real interest rate ( $r$ ) is the return after accounting for inflation ( $π$ ).
The exact Fisher equation is multiplicative: $(1 + i) = (1 + r) (1 + π)$ . From this, the exact real rate is $r = \frac{1 + i}{1 + π} - 1$ .
The approximate Fisher equation is a linear simplification, $r \approx i - π$ , which works well when inflation and interest rates are low.
The Fisher effect is the long-run tendency for nominal interest rates to adjust one-for-one with expected inflation.
A negative real interest rate ( $r < 0$ ) occurs when the inflation rate exceeds the nominal interest rate, causing a loss in purchasing power.

This understanding is critical for the next step in our analysis. In the upcoming lesson, we will apply the concept of the real interest rate to analyse the wealth redistribution effects of unexpected inflation, exploring who gains and who loses when price levels change unpredictably.

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