Calculating Monthly Compound Interest for Becca’s Loan

In this article, we will explore the process of determining the monthly compounded interest rate on a loan through an example provided by Becca. We will go step by step through the calculation, explain the formula used, and provide practical insights to understand and apply the concept of compound interest in financial scenarios.

Introduction to 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. This method of calculating interest is more favorable to lenders and less favorable to borrowers. By understanding how to compute the monthly compounded interest rate, individuals can manage their debts or investments more effectively.

Understanding the Scenario

Let's consider a loan where Becca borrows $35,400 and agrees to pay back $47,500 after 2 years. To calculate the interest rate compounded monthly, we will use the formula for compound interest:

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

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 (in decimal form). n is the number of times that interest is compounded per year. t is the time the money is invested or borrowed for (in years).

For our example:

P $35,400.00 A $47,500.00 n 12 (compounded monthly) t 2 years

Step-by-Step Calculation

To find the interest rate compounded monthly, we need to rearrange the formula to solve for r. The rearranged formula is:

r n[(A/P)^(1/(nt)) - 1]

Now, let's plug in the values:

r 12[(47,500/35,400)^(1/(24)) - 1]

First, calculate the fraction:

47,500 / 35,400 ≈ 1.34

Now, calculate the exponent:

1.34^(1/24) ≈ 1.0132

Substitute this back into the equation:

r 12[1.0132 - 1] ≈ 12 * 0.0132 ≈ 0.1584

To convert r to a percentage:

0.1584 * 100 ≈ 15.84%

Interpreting the Results

The interest rate compounded monthly that Becca is paying on the loan is approximately 15.84% per annum. This means that for every $100 borrowed, $15.84 in interest is added each year, compounded monthly.

Another Method for Precision

Mike Angelo Castillo proposed a method using a financial calculator to find the precise future value and the interest rate. This approach is valid and can be used for more complex calculations or scenarios. The values provided are:

Periods N 24 (2 years x 12 months) Interest rate I 1.23261 per period or 14.79% per year Present value PV -35,400.00 Future value FV 47,500.00 Precise future value 47,498.77

This method uses the compoundinginterest formula to find the precise interest rate:

Divide the future value by the present value to get the total product of 24 monthly compounds. Find the 24th root of this product using the calculator's “yth root of X” button. Convert the decimal amount to a percentage and multiply by 12 to get the nominal annual interest rate.

Conclusion

Both methods provide a clear understanding of how to calculate the monthly compounded interest rate. Understanding these concepts and calculations is crucial for managing personal finances effectively, whether you are borrowing money or investing. Whether you use the compound interest formula or a financial calculator, the principles remain the same, ensuring that you can make informed decisions about your financial future.