Real vs. Nominal Interest Rates: The Fisher Equation

Introduction

Welcome to the final lesson in our module on measuring the macroeconomy. In our last lesson, you learned how to measure the overall price level and calculate the inflation rate using the Consumer Price Index (CPI). We saw that inflation means your money buys less over time. This has a profound impact on one of the most common financial activities: borrowing and lending.

Imagine you lend money to a friend for a year. The interest they pay you is your return. But if prices rise during that year, the money you get back won't have the same purchasing power it did when you lent it out. How can we account for this?

This lesson addresses that exact question. Our goal is to distinguish between real and nominal interest rates using the Fisher equation. We will explore:

• The fundamental difference between the interest rate quoted by a bank (nominal) and the actual return on your purchasing power (real).
• The mathematical relationship connecting these two rates with inflation, known as the Fisher equation.
• The consequences for borrowers and lenders when actual inflation differs from what they expected.

Understanding this distinction is not just an academic exercise; it's essential for making informed decisions about saving, investing, and borrowing. It is a critical link between the macroeconomic indicators we've studied and the behavior of individuals and firms that drives the economy.

1. Nominal vs. Real Interest Rates: Money vs. Purchasing Power

When you deposit money in a savings account or take out a loan, the bank quotes an interest rate. This is the nominal interest rate. It tells you how many more dollars you will have in the future.

However, what we often care more about is our purchasing power—what we can actually buy with those dollars. The real interest rate measures the change in your purchasing power. It is the nominal interest rate adjusted for inflation.

As the video explains, the relationship is straightforward:

• Nominal Interest Rate ($i$): The stated interest rate; the rate of growth in the monetary value of your investment.
• Real Interest Rate ($r$): The rate of growth in the purchasing power of your investment.
• Inflation Rate ($\pi$): The rate at which the general price level is increasing.

The real interest rate is what ultimately matters for economic decisions. A high nominal interest rate might look attractive, but if inflation is even higher, your purchasing power is actually decreasing. In this case, the real interest rate would be negative.

2. The Fisher Equation

The American economist Irving Fisher formalized the relationship between these three variables. His work gives us the Fisher equation, a cornerstone for understanding interest rates in an inflationary environment.

The Approximate Formula

For most practical purposes, especially when inflation is relatively low, a simple approximation is used. The real interest rate is approximately the nominal interest rate minus the inflation rate.

The approximate Fisher equation can be written in two ways:

1. To find the real rate: $r\approx i-\pi$
2. To set the nominal rate: $i\approx r+\pi$

For example, if a bank wants to earn a real return of 3% ($r=3\%$) and expects inflation to be 2% ($\pi =2\%$), it will set its nominal interest rate at approximately 5% ($i\approx 3\%+2\%$).

The Exact Formula

Your background in technology means you appreciate precision. The formula $i\approx r+\pi$ is an approximation. The exact formula is slightly different and reflects the compounding nature of returns and inflation.

The exact Fisher equation is:
$(1+i)=(1+r)(1+\pi )$
If you expand the right side, you get $1+r+\pi +r\pi$. This gives:
$i=r+\pi +r\pi$
The term $r\pi$ is the product of two small percentages, making it a very small number. For example, if $r=3\%$ (0.03) and $\pi =2\%$ (0.02), then $r\pi =0.0006$, or 0.06%. Because this last term is so small, it is often dropped, leading to the approximation $i\approx r+\pi$.

3. Expected vs. Actual Inflation: Who Wins and Who Loses?

When a loan agreement is signed, the future inflation rate is unknown. Therefore, the nominal interest rate is set based on the expected inflation rate. The actual real interest rate that the lender receives and the borrower pays depends on the actual inflation rate over the life of the loan.

What happens when expectations are wrong?

• If actual inflation > expected inflation: The real interest rate is lower than expected. This benefits the borrower, who repays the loan with money that is worth less than anticipated. It hurts the lender, whose real return is eroded more than they planned.
• If actual inflation < expected inflation: The real interest rate is higher than expected. This benefits the lender, who receives a higher real return. It hurts the borrower, who repays the loan with money that is more valuable than anticipated.

This redistribution of purchasing power is a key reason why high and unpredictable inflation is damaging to an economy. It creates uncertainty and risk in all borrowing and lending contracts.

4. The Fisher Effect

The logic of the Fisher equation leads to an important macroeconomic phenomenon known as the Fisher effect. It states that, in the long run, the nominal interest rate tends to adjust one-for-one with changes in the expected inflation rate.

Think about it from a lender's perspective. If they want to maintain a real return of, say, 3%, they will adjust the nominal rate they charge to fully incorporate whatever they expect inflation to be. If expected inflation rises from 2% to 7%, a lender will raise their nominal interest rate from 5% to 10% to keep their real return constant at 3%.

This relationship can also be visualized using a supply and demand diagram for loanable funds. An increase in expected inflation makes lenders want to charge more (shifting supply left/up) and makes borrowers more willing to pay higher nominal rates since they expect to pay back with cheaper dollars (shifting demand right/up). The result is a higher nominal interest rate, but the same real interest rate.

Conclusion

In this lesson, we have dissected the relationship between nominal interest rates, real interest rates, and inflation. This is a crucial concept that bridges our understanding of macroeconomic data with the real-world decisions of households and firms.

Key Takeaways:

• The nominal interest rate ($i$) is the rate quoted by banks and measures the growth in money.
• The real interest rate ($r$) is adjusted for inflation and measures the growth in purchasing power. It is the rate that truly motivates economic decisions.
• The Fisher equation connects them: $i\approx r+\pi$. Lenders set nominal rates based on their desired real return and the expected rate of inflation.
• Unexpected changes in inflation redistribute wealth: higher-than-expected inflation helps borrowers and hurts lenders, while lower-than-expected inflation does the opposite.
• The Fisher effect describes the tendency for nominal interest rates to move in tandem with expected inflation.

Preview of the Next Lesson:

You have now mastered the three core indicators used to measure the macroeconomy: GDP, unemployment, and inflation. We have also seen how inflation influences the all-important interest rate.

This sets the stage perfectly for our next module, where we will begin building models to explain the short-run fluctuations in the economy. In the next lesson, we will start by describing the phases of a business cycle (expansion, peak, contraction, trough). We will see how variables like GDP, unemployment, and the real interest rate behave during these cycles.