Question

Mr. Riley loaned his friend, Mr. Gillis, \(\$ 400\) to purchase a second-hand car. One year later Mr. Gillis returned \(\$ 420\) to Mr. Riley. During that period, the consumer price index rose by \(5 \%\). a) What is the nominal rate of interest in this example? b) What is the real rate of interest?

Short answer

a) The nominal rate of interest in this example is \(5 \%\). b) The real rate of interest is \(0 \%\).

Step-by-step solution

Identify the given values

Initial loan amount: \(\$ 400\) Total amount returned: \(\$ 420\) Consumer price index rise (inflation rate): \(5 \% = 0.05\)

Calculate the nominal interest rate

Apply the formula: Nominal interest rate = (Total amount returned - Initial loan amount) / Initial loan amount Nominal interest rate = (\(420 - \)400) / \(400 = \)20 / $400 = 0.05 So, the nominal interest rate is \(5 \%\).

Calculate the real interest rate

Apply the formula: Real interest rate = (1 + Nominal interest rate) / (1 + Inflation rate) - 1 Real interest rate = (1 + 0.05) / (1 + 0.05) - 1 To find the real interest rate, first, add 1 to the nominal interest rate: 1 + 0.05 = 1.05 Next, add 1 to the inflation rate (consumer price index rise) 1 + 0.05 = 1.05 Now, divide the first result by the second result: 1.05 / 1.05 = 1 Finally, subtract 1 from the result: 1 - 1 = 0 The real interest rate is \(0 \%\). #Conclusion:# a) The nominal rate of interest in this example is \(5 \%\). b) The real rate of interest is \(0 \%\).

Key concepts

Nominal Interest Rate

The nominal interest rate is the basic percentage increase in money that a lender earns from a borrower over a period of time. Think of it as the sticker price on a loan agreement. It’s straightforward, as it does not take into account any other factors like inflation.
In our example, Mr. Riley loaned Mr. Gillis \(400. After a year, Mr. Gillis paid back \)420. This means Mr. Riley earned \(20 as interest. The nominal interest rate can be calculated with the formula:\[\text{Nominal interest rate} = \frac{\text{Total amount returned} - \text{Initial loan amount}}{\text{Initial loan amount}}\]Inserting the numbers:\[\frac{\\)420 - \\(400}{\\)400} = \frac{\\(20}{\\)400} = 0.05\]This gives us a nominal interest rate of 5%. This means Mr. Riley earned 5% more than he lent out, purely in monetary terms.

• It is expressed as a percentage.
• Does not account for inflation.

Real Interest Rate

The real interest rate reflects the true earning power of money when adjusted for inflation, or the rise in the cost of goods and services. It shows how much more in real terms a lender makes after inflation is considered. This is what economists are really interested in, since it provides a more accurate picture of purchasing power.
Using our example, the consumer price index increased by 5% over the period. We calculate the real interest rate using the formula:\[\text{Real interest rate} = \frac{1 + \text{Nominal interest rate}}{1 + \text{Inflation rate}} - 1\]So plugging in the numbers:\[\frac{1 + 0.05}{1 + 0.05} - 1 = \frac{1.05}{1.05} - 1 = 0\]Thus, the real interest rate is 0%. This means that after considering inflation, Mr. Riley did not gain any additional purchasing power from the interest.

• Takes inflation into account.
• Provides a clearer view of actual interest gains.

Consumer Price Index

The Consumer Price Index (CPI) is a critical economic metric that measures the average change over time in the prices paid by urban consumers for a market basket of consumer goods and services. Inflation, one of the key components in our real interest rate calculation, is often gauged by observing changes in the CPI.
In the exercise, the CPI rose by 5%. This indicates that on average, prices increased by 5% over the year. But what does this mean for individuals?

• CPI is used to adjust real income.
• Influences monetary policy and inflation adjustment.

Understanding CPI fluctuations helps individuals and businesses make informed decisions about spending, saving, and investing.
Changes in the CPI can significantly impact the real value of monetary gains, like in our example, where a nominal interest rate of 5% was completely offset to a real interest rate of 0% due to a 5% rise in CPI, implying no real earnings after adjusting for inflation.