Problem 73 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

Georgia purchased a house in January 2000 for $$\$ 200,000$$. In January 2006 she sold the house and made a net profit of $$\$ 56,000$$. Find the effective annual rate of return on her investment over the 6 -yr period.

Step-by-Step Solution

Answer

The effective annual rate of return on Georgia's investment over the 6-year period is approximately \(4.23\%\).

Step 1: Identify the given values, and calculate the final value of the investment

The problem gives us the following information: - Initial Value: $200,000 - Profit: $56,000 - Years: 6 We need to find the final value of the investment. To do this, we add the initial value and the profit: \(Final Value = Initial Value + Profit\) \(Final Value = \$200,000 + \$56,000\) \(Final Value = \$256,000\) Now we have all the information needed to substitute into the compound interest formula and solve for r.

Step 2: Use the compound interest formula and solve for r

We will now substitute the values we have into the compound interest formula: \(Final Value = Initial Value * (1 + r)^{years}\) Substitute the values we found into the formula: \(\$256,000 = \$200,000 * (1 + r)^6\) Now we will solve for r by following these steps: 1) Divide both sides by $200,000 2) Take the 6th root of both sides to remove the exponent 3) Subtract 1 from both sides to solve for r Divide both sides by $200,000: \(\frac{\$256,000}{\$200,000} = (1 + r)^6\) Now, take the 6th root of both sides to remove the exponent: \(\sqrt[6]{\frac{\$256,000}{\$200,000}} = 1 + r\) Subtract 1 from both sides to solve for r: \(r = \sqrt[6]{\frac{\$256,000}{\$200,000}} - 1\) Calculate the value: \(r \approx 0.0423\) Now, to express the result as a percentage, multiply the decimal value by 100: Effective annual rate of return: \(0.0423 * 100\% = 4.23\%\) The effective annual rate of return on Georgia's investment over the 6-year period is approximately 4.23%.

Key Concepts

Compound Interest Formula

The compound interest formula is a crucial tool in finance, allowing us to understand how investments grow over time. This formula helps us calculate the total amount of money accumulated after a certain number of years, considering the initial principal amount, the interest rate, and the number of compounding periods. The general form of the compound interest formula is:\[A = P \times (1 + r)^n\]where:

• A is the future value of the investment or loan, including interest.
• P is the principal investment amount (initial deposit or loan amount).
• r is the annual interest rate (decimal).
• n is the number of years the money is invested or borrowed for.

For our specific problem, Georgia's final home value can be calculated using this formula. By adjusting the formula to solve for the rate, we obtain insights into the investment's effective annual growth rate, providing a clear picture of its performance over time.

Investment Calculations

Investment calculations help us evaluate the performance and value of investments. These calculations determine gains or losses and project future earnings. In the case of Georgia's house, you start by identifying all essential values:

• Initial investment cost: \\(200,000
• Total profit made from the sale: \\)56,000
• Total time held: 6 years

With these figures, you calculate the final value of the investment by adding the initial value and the profit:\[\text{Final Value} = \text{Initial Value} + \text{Profit} = \\(200,000 + \\)56,000 = \$256,000\]This final value represents what the asset is worth at the end of the investment period. Such calculations are fundamental in guiding investment decisions and determining financial outcomes.

Profit Calculation

Profit calculation involves determining how much money has been made or lost in an investment. It is an essential measure of success for any financial endeavor. In Georgia's case, profit was straightforwardly computed by the sale of her house. She purchased it for \\(200,000 and later sold it, realizing a net profit of \\)56,000. Here's how the profit calculation looks:

• Net Profit = Sale Price - Purchase Price\[\text{Net Profit} = \text{Final Value} - \text{Initial Investment}\]
• \[\text{Net Profit} = \\(256,000 - \\)200,000 = \$56,000\]

Understanding this basic calculation is key for investors to measure whether their investments are yielding positive outcomes, helping to determine strategy shifts or affirm successful ventures.

Annual Rate of Return

The annual rate of return is a measure of the profitability of an investment expressed as a percentage per year. It indicates how much an investment has grown or shrunk over a particular period. To calculate it effectively, you can use the compound interest formula rearranged for the interest rate, as demonstrated in Georgia's example.In her scenario, the formula was set to solve for r:\[A = P \times (1 + r)^{n}\]Rearrange to find r:\[1 + r = \left(\frac{A}{P}\right)^{\frac{1}{n}}\]Solve for r:\[r = \left(\frac{A}{P}\right)^{\frac{1}{n}} - 1\]Here, after calculations for Georgia:

• \[(1 + r)^6 = \frac{\\(256,000}{\\)200,000}\]
• \[r = \sqrt[6]{1.28} - 1 \approx 0.0423\]
• \[r \times 100\% = 4.23\%\]

This rate shows how well Georgia's investment performed on an annual basis. It's a key metric for investors to evaluate investment efficiency and benchmark against other opportunities.