Compounding Interest: Calculating Full Loan Repayment with Annual Interest

Understanding how compound interest works is crucial for personal and business financial planning. In this article, we will explore the mathematical process of calculating the total amount owed for a loan when interest is compounded annually. We will use a concrete example to illustrate the steps involved.

Understanding Compound Interest

Compound interest is the interest calculated on the initial principal and also on the accumulated interest of previous periods of a deposit or loan. In our scenario, we have a loan of $16,000 borrowed for 6 years at an annual interest rate of 9%, compounded annually. This means that the interest earned in each year is added to the principal, and the next year's interest is calculated on the new total.

Step-by-Step Calculation

The formula for compound interest is:

A P(1 r)^n

Where:

A is the amount of money accumulated after n years, including interest. P is the principal amount (the initial amount of money). r is the annual interest rate (decimal). n is the number of years the money is borrowed for.

Given Values:

P 16000 r 0.09 (9% as a decimal) n 6 years

Calculation Process:

1. Convert the interest rate to a decimal and add 1:

1 0.09 1.09

2. Raise the result to the power of the number of years:

1.09^6 ≈ 1.6771

3. Multiply the principal by this result:

16000 * 1.6771 ≈ 26833.60

4. Round to the nearest dollar:

A ≈ 26834

Explanation of Each Step

First, we convert the interest rate of 9% to a decimal by dropping the % and dividing by 100. So, 9% becomes 0.09. We then add 1 to this decimal to incorporate the principal amount:

1 0.09 1.09

Next, we need to determine the number of compounding periods, which is 6 years in this case. We raise our result to the power of the number of years:

1.09^6 ≈ 1.6771

We then multiply the principal amount by this result:

16000 * 1.6771 ≈ 26833.60

Rounding to the nearest dollar, the total amount owed is:

A ≈ 26834

Verification and Importance of Understanding

To verify, we can check the interest accumulation over each year:

Year 1: 16000 * 0.09 1440.00

Year 2: (16000 1440) * 0.09 1569.60

Year 3: (16000 1440 1569.60) * 0.09 1710.86

Year 4: (16000 1440 1569.60 1710.86) * 0.09 1864.84

Year 5: (16000 1440 1569.60 1710.86 1864.84) * 0.09 2032.68

Year 6: (16000 1440 1569.60 1710.86 1864.84 2032.68) * 0.09 2215.62 Total interest: 1440 1569.60 1710.86 1864.84 2032.68 2215.62 10833.60

Total amount paid: 16000 10833.60 26833.60, rounding to 26834.

Understanding compound interest is vital for managing loans and investments. Many financial services and loans use this method to calculate interest, so knowing the formula and the steps involved can help individuals make informed financial decisions.