Calculate Investment Maturity Value After 3 Years At 12% Compounded Annually

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Introduction: Understanding Compound Interest and Investment Growth

In the realm of business and finance, understanding the power of compound interest is crucial for making informed investment decisions. Compound interest allows your investment to grow exponentially over time, as the interest earned is added to the principal, and subsequent interest is calculated on the new, higher balance. This article delves into a specific scenario involving Jose's investment of $50,000 at a 12% annual interest rate, compounded annually, for a period of 4 years. Our primary objective is to determine the maturity value of this investment at the end of Year 3. To achieve this, we will meticulously break down the calculations involved in compound interest, providing a clear and comprehensive understanding of how Jose's investment grows year by year. The concepts discussed here are fundamental to financial planning and investment management, making this a valuable exercise for anyone looking to grasp the dynamics of wealth accumulation. Understanding how investments grow through compound interest is essential for making sound financial decisions, whether you're planning for retirement, saving for a major purchase, or simply looking to grow your wealth over time. This article aims to provide a clear and detailed explanation of the calculations involved, ensuring you can apply these principles to your own financial scenarios.

Problem Breakdown: Jose's Investment Scenario

To accurately calculate the maturity value of Jose's investment, we must first dissect the problem and identify the key components. Jose's initial investment, known as the principal, is $50,000. The investment earns interest at an annual rate of 12%, which is compounded annually. This means that the interest earned each year is added to the principal, and the next year's interest is calculated on this new, larger sum. The investment period is 4 years, but our focus is on determining the maturity value at the end of Year 3. This requires us to calculate the accumulated value of the investment after each of the first three years. To solve this problem effectively, we will utilize the compound interest formula, which provides a structured approach to calculating the future value of an investment. The compound interest formula takes into account the principal amount, the interest rate, the compounding frequency, and the investment period. By applying this formula, we can accurately determine the value of Jose's investment at the end of Year 3. Understanding the problem setup is crucial for avoiding errors and ensuring the accuracy of our calculations. This step-by-step breakdown will guide us through the process and lead us to the correct solution.

The Compound Interest Formula: A Mathematical Tool for Investment Growth

The cornerstone of our calculation is the compound interest formula, a powerful tool for understanding investment growth. The formula is expressed as follows:

A = P (1 + r/n)^(nt)

Where:

  • A represents the future value of the investment/loan, including interest.
  • P is the principal investment amount (the initial deposit or loan amount).
  • r is the annual interest rate (as a decimal).
  • n is the number of times that interest is compounded per year.
  • t is the number of years the money is invested or borrowed for.

In Jose's case, we have the following values:

  • P = $50,000
  • r = 12% or 0.12 (as a decimal)
  • n = 1 (compounded annually)
  • t = 3 years (since we want the value at the end of Year 3)

This formula encapsulates the essence of compound interest, demonstrating how interest earned in previous periods contributes to future interest calculations. The compounding frequency, denoted by 'n,' plays a significant role in the overall growth of the investment. The more frequently interest is compounded, the higher the future value will be. In our scenario, interest is compounded annually, meaning 'n' is equal to 1. The variable 't' represents the investment period, which in this case is 3 years. By carefully applying this formula and plugging in the relevant values, we can accurately determine the maturity value of Jose's investment at the end of Year 3. The compound interest formula is a fundamental concept in finance, and mastering its application is essential for making informed investment decisions.

Year-by-Year Calculation: Tracking Jose's Investment Growth

To determine the maturity value at the end of Year 3, we will calculate the investment's value year by year, demonstrating the effect of compound interest. Let's break down the calculations:

Year 1:

  • Interest earned = $50,000 * 0.12 = $6,000
  • Value at the end of Year 1 = $50,000 + $6,000 = $56,000

At the end of the first year, Jose's investment has grown by $6,000, bringing the total value to $56,000. This increase reflects the 12% annual interest earned on the initial principal of $50,000. This new value becomes the principal for the next year's interest calculation, illustrating the compounding effect. The interest earned in Year 1 contributes to the principal for Year 2, leading to a higher interest earning in the subsequent year. This process continues throughout the investment period, accelerating the growth of the investment over time. Understanding this year-by-year progression provides a clear picture of how compound interest works and its impact on investment growth.

Year 2:

  • Interest earned = $56,000 * 0.12 = $6,720
  • Value at the end of Year 2 = $56,000 + $6,720 = $62,720

In the second year, the interest earned is calculated on the increased principal of $56,000, resulting in $6,720 in interest. This demonstrates the power of compounding, as the interest earned in Year 1 contributes to the growth in Year 2. The total value at the end of Year 2 reaches $62,720, reflecting the accumulated interest over the two years. This growth trajectory highlights the accelerating nature of compound interest, where the investment grows at an increasing rate over time. By tracking the value year by year, we can clearly see the impact of compounding on the investment's overall performance. The higher the principal, the greater the interest earned, leading to exponential growth over the investment period.

Year 3:

  • Interest earned = $62,720 * 0.12 = $7,526.40
  • Value at the end of Year 3 = $62,720 + $7,526.40 = $70,246.40

By the end of the third year, Jose's investment has grown significantly, thanks to the compounding effect. The interest earned in Year 3 is calculated on the principal of $62,720, resulting in $7,526.40 in interest. This brings the total value of the investment to $70,246.40 at the end of Year 3. This final calculation demonstrates the cumulative impact of compound interest over the three-year period. The investment has grown steadily each year, with the interest earned increasing due to the growing principal. This highlights the importance of long-term investing and the benefits of allowing compound interest to work its magic over time. The maturity value at the end of Year 3 represents a substantial return on Jose's initial investment, showcasing the power of compounding in wealth accumulation.

Applying the Formula Directly: A Simplified Approach

Alternatively, we can directly apply the compound interest formula for 3 years:

A = $50,000 (1 + 0.12/1)^(1*3) A = $50,000 (1 + 0.12)^3 A = $50,000 (1.12)^3 A = $50,000 * 1.404928 A = $70,246.40

This direct application of the formula confirms our year-by-year calculations, providing a more concise method for determining the maturity value at the end of Year 3. The result is the same, $70,246.40, demonstrating the consistency and accuracy of both methods. Using the formula directly can save time and effort, especially when calculating the future value over longer periods. However, understanding the year-by-year calculations provides valuable insight into the compounding process and how interest accumulates over time. Both approaches are valid and can be used depending on the specific context and the level of detail required. The key takeaway is that the compound interest formula provides a powerful tool for financial planning and investment analysis.

Answer and Explanation: Selecting the Correct Option

Based on our calculations, the maturity value of Jose's investment at the end of Year 3 is $70,246.40. Therefore, the correct answer is:

D. $70,246.40

This option accurately reflects the value we calculated using both the year-by-year method and the direct application of the compound interest formula. The other options provided are incorrect and do not align with the results of our calculations. Understanding the steps involved in calculating compound interest is crucial for selecting the correct answer and avoiding errors. This problem demonstrates the practical application of compound interest concepts in real-world investment scenarios. By carefully analyzing the problem, applying the appropriate formula, and performing the necessary calculations, we can arrive at the correct solution. This exercise reinforces the importance of financial literacy and the ability to make informed investment decisions.

Conclusion: The Power of Compound Interest and Long-Term Investing

In conclusion, Jose's investment of $50,000 at 12% compounded annually for 4 years will have a maturity value of $70,246.40 at the end of Year 3. This demonstrates the significant impact of compound interest on investment growth. Understanding the principles of compound interest is essential for effective financial planning and wealth accumulation. By reinvesting earnings and allowing interest to compound over time, individuals can achieve substantial financial gains. This example highlights the importance of starting early and investing consistently to maximize the benefits of compounding. The power of compound interest lies in its ability to generate exponential growth over the long term, making it a cornerstone of successful investment strategies. Whether you're saving for retirement, a down payment on a house, or any other financial goal, understanding and utilizing compound interest is key to achieving your objectives. This article has provided a comprehensive guide to calculating compound interest and its application in a real-world investment scenario. By mastering these concepts, you can make informed decisions and take control of your financial future.