Question

Suppose that the only good you purchase is premium bottled water and that at the beginning of the year, the price of a bottle is \(\$ 2.00\). Suppose you lend \(\$ 1,000\) for one year at an interest rate of 5 percent. At the end of the year, the price of premium bottled water has risen to \(\$ 2.08\). What is the real rate of interest you earned on your loan?

Short answer

The real rate of interest you earned on your loan is 1%.

Step-by-step solution

Identify the given variables

The initial price of bottled water, denoted as \( P1 \), is \(\$2.00\) while the final price, denoted \( P2 \), is \(\$2.08\). The nominal interest rate, denoted \( r \), is given as 5 percent or 0.05 in decimal form.

Calculate the Inflation Rate

The inflation rate can be calculated by subtracting the initial price from the final price, dividing the result by the initial price, and then multiplying by 100. This can be represented as \( \frac{P2 - P1}{P1} × 100\% \). Using the provided values, the inflation rate (\( r_I \)) is therefore \( \frac{2.08 - 2.00}{2.00} × 100\% = 4\% \). The inflation rate is 4%.

Calculate the Real Interest Rate

The real interest rate can be calculated by subtracting the inflation rate from the nominal interest rate. This can be represented as \( r - r_I \). Using the calculated and given values, the real interest rate is therefore \( 0.05 - 0.04 = 0.01 \) or 1%.

Key concepts

Inflation Rate

Inflation rate measures how much prices of goods and services increase over a specific period. It affects purchasing power, meaning the higher the inflation, the less you can buy with the same amount of money. Understanding inflation is vital because it impacts financial decisions and economic stability.
To calculate the inflation rate, you compare the initial price of an item to its price at a later time. For example, if a bottle of water prices increase from \( \\(2.00 \) to \( \\)2.08 \), you will find the rate using the formula:

• Subtract the initial price from the final price: \( \\(2.08 - \\)2.00 \)
• Divide by the initial price: \( \frac{\\(0.08}{\\)2.00} \)
• Multiply by 100 to convert into a percentage.\( \frac{\\(0.08}{\\)2.00} \times 100\% = 4\% \)

This indicates a 4% inflation rate, meaning prices overall increased by 4% over the given time.

Nominal Interest Rate

The nominal interest rate is the percentage increase in money that the lender expects to receive, without considering inflation. It's the rate you commonly see advertised by banks or lenders. Nominal interest rates are important because they help you understand the growth of your money in simple monetary terms.
If you lend someone \( \\(1,000 \) with a nominal interest rate of 5%, you expect to receive \( \\)1,050 \) after one year. However, this doesn't guarantee your purchasing power will grow in proportion if inflation is high during that period.

• Starting amount: \( \\(1,000 \)
• Nominal interest rate: 5% or 0.05 in decimal
• Future amount received: \( \\)1,000 \times 1.05 = \$1,050 \)

This rate helps calculate expected earnings, but it's crucial to analyze alongside the inflation rate to understand the real value gained.

Real Interest Rate

Real interest rate provides insight into the true increase in purchasing power from an investment or savings after adjusting for inflation. It's important because it reflects the actual economic gain rather than just nominal figures. You calculate it by subtracting the inflation rate from the nominal interest rate.
If the nominal interest rate is 5% and the inflation rate is 4%, the formula to find the real interest rate would be:

• Real Interest Rate Formula: \( r - r_I \)
• Substitute the values: \( 0.05 - 0.04 = 0.01 \)
• Convert to percentage: 1%

So, while the nominal rate suggests a 5% gain, the real rate of 1% shows your earning's true value when you consider inflation's impact. Understanding this difference is crucial for long-term financial planning and decision-making.