# Lesson summary: nominal vs. real interest rates

## Lesson summary

What you see is what you get, right? Not when it comes to interest rates! When you see an ad saying a bank will pay $5\%$ interest on savings accounts, it doesn’t necessarily mean you will be able to buy 5% more stuff with your money after a year.
When you put $\100$ in a savings account, the real value of that $\100$ is what you can buy with it. Therefore the real value of what you earn in interest is what you can buy with that interest. When there is inflation, the purchasing power of the interest you earn decreases. Your real interest is the nominal interest rate (the interest you get paid) minus the rate of inflation (the loss of purchasing power).

## Key Terms

Key termDefinition
nominal interest ratethe interest rate that you earn (or pay) on a loan; this is the amount you see on a sign advertising interest rates.
real interest ratethe nominal interest rate adjusted for inflation; this is the effective interest rate that you earn (or pay).
Fisher effectthe idea that an increase in expected inflation drives up the nominal interest rate, which leaves the expected real interest rate unchanged

## Key Takeaways

### Nominal interest is the sum of the expected real interest rate and the expected inflation rate

How does a bank decide what interest rate to charge? It needs to consider two important things: How much interest is enough to make it worthwhile for the bank to loan the money (the real interest rate they earn)? How much of the interest’s purchasing power might be lost to inflation?
For example, suppose a bank wants to earn $10\%$ interest, but it thinks there will be $3\%$ inflation. If they don’t factor that inflation into what they change in interest, they will effectively earn only $7\%$ (because they will lose $3\%$ of the purchasing power of an interest rate of $10\%$). Instead, banks factor inflation into their interest rates. To account for inflation, this bank would charge $13\%$ interest.
Remember from a previous lesson that inflation results in winners and losers? Suppose the bank thought inflation would be $3\%$, but inflation turned out to be $4\%$. We can figure out the real interest that the bank actually earned in retrospect:
\begin{aligned}\text{real interest rate}&=\text{nominal interest rate}-\text{inflation}\\\\ &=13\%-4\%\\\\ &=9\%\end{aligned}
The bank was hurt by the unexpected inflation because they only got a return of $9\%$, not the $10\%$ they hoped for. On the other hand, the borrower ended up only paying $9\%$ real interest. The borrower got the better deal!
This is an important takeaway: it was the unanticipated aspect of the inflation that hurt the bank and helped the borrower. If the bank had anticipated the higher rate of inflation, they would have simply charged a higher nominal interest rate to ensure they got the real interest rate.
This is the basic idea behind something called the Fisher Effect. When expected inflation changes, the nominal interest rate will increase. However, inflation will not effect the real interest rate.

## Key Equations

### The interest rate borrowers pay and savers earn

$\text{Nominal interest rate} = \text{real interest rate} + \text{expected inflation}$
Sometimes this equation is written using symbols:
$i=r + \text{inf}_e$
Where:
\begin{aligned}i&=\text{nominal interest rate}\\\\ r&=\text{real interest rate}\\\\ \text{inf}_e &= \text{expected inflation}\end{aligned}
Note: sometimes you will see inflation abbreviated using the Greek symbol $\pi$, and expected inflation abbreviated as $\pi_e$.

### The real interest rate in retrospect

$\text{Real interest rate} = \text{nominal interest rate} - \text{inflation rate}$
The actual interest earned (or paid) will depend on the nominal interest rate and how much the inflation rate turned out to be.
For example, the bank expects a real return of $4\%$ to their earnings. They expect the inflation rate to be $1\%$, so they charge a nominal interest rate of $5\%$:
\begin{aligned} i&=r+\text{inf}_e\\\\ &=4\%+1\%\\\\ &=5\%\end{aligned}
However, it turns out that that inflation is 6%. In retrospect they only earned:
\begin{aligned} r&=i-\text{inf}\\\\ &=5\%-6\%\\\\ &=-1\%\end{aligned}
In this case, inflation was higher than they anticipated. They actually lost money, rather than earned it.

## Common misperceptions

• A point of confusion some people have is whether nominal and real interest rates can be negative. Real interest rates can be negative, but nominal interest rates cannot. Real interest rates are negative when the rate of inflation is higher than the nominal interest rate. Nominal interest rates cannot be negative because if banks charged a negative nominal interest rate, they would be paying you to borrow money! This is called the “zero bound” on interest rates: the nominal interest rate can only go down to $0\%$.

## Discussion questions

• Tywin knows he has a debt to repay soon. The bank charges him an interest rate of $6\%$. If the expected rate of inflation is $5\%$, how much interest is he effectively paying? Explain.
• Calculate the nominal rate of inflation that will be charged if the expected rate of inflation is $7\%$ and the real return desired is $5\%$. Show all work.
If the rate of inflation is $3\%$ instead, what happens to the value of the money paid back? Explain.
\begin{aligned}i&=r +\text{inf}_e\\\\ &=7\%+5\%\\\\ &=12\%\end{aligned}
If the actual rate of inflation turns out to be $3\%$, then the real interest earned from a nominal interest rate of $12\%$ is:
\begin{aligned}r&=i-\text{inf}\\\\ &=12\%-3\%\\\\ &=9\%\end{aligned}
In this case, because the inflation was lower than expected, the real amount paid back ($9\%$) is higher than the real amount that was anticipated ($7\%$). Lenders benefits from this unanticipated disinflation and borrowers are hurt.
• The real interest rate paid on an asset was $10\%$, but the nominal rate was $9\%$. What was the rate of inflation?