What is the formula for a monthly installment loan?

Decoding the Monthly Installment Loan Formula: More Than Just Simple Division

The statement “Monthly loan repayments are determined by dividing the total loan amount, including accrued interest, evenly across the loan’s duration” is a simplification. While it captures the essence of consistent monthly payments, it omits the crucial role of compound interest, which significantly impacts the actual calculation. True, the repayment is even across the loan’s life, but the calculation to determine that even payment is far from simple division.

The seemingly straightforward monthly installment calculation actually involves a more complex formula rooted in the principles of compound interest. This formula considers several key factors:

• Loan Principal (P): The original amount of money borrowed.
• Annual Interest Rate (r): The yearly interest rate charged on the loan, expressed as a decimal (e.g., 5% becomes 0.05).
• Number of Years (n): The loan’s total repayment period in years.
• Number of Payments per Year (m): Typically 12 for monthly payments.

The formula for calculating the monthly payment (M) is:

M = P [ i(1 + i)^nt ] / [ (1 + i)^nt – 1 ]

Where:

• i = r/m (the monthly interest rate)

Let’s break it down:

• i(1 + i)^nt: This part calculates the future value of a single monthly payment compounded over the loan term.

• (1 + i)^nt – 1: This represents the total growth of the principal and interest over the loan’s lifetime.

• P [ i(1 + i)^nt ] / [ (1 + i)^nt – 1 ]: This combines the elements above to provide the monthly payment that will amortize the loan (pay off both principal and interest) over the specified period.

Example:

Let’s say you borrow $10,000 (P) at an annual interest rate of 6% (r), for 3 years (n), with monthly payments (m = 12).

1. Calculate the monthly interest rate: i = 0.06 / 12 = 0.005

2. Plug the values into the formula:

   M = 10000 [ 0.005(1 + 0.005)^(123) ] / [ (1 + 0.005)^(123) – 1 ]

3. Solve the equation: This calculation will yield the monthly payment (M), which will be approximately $304.22. Note that a slight variation may occur depending on the rounding used in the calculation.

Why simple division is insufficient:

Simple division of the total loan amount (principal + total interest) by the number of months ignores the compounding effect of interest. Compound interest means interest is calculated not only on the principal but also on accumulated interest from previous periods. Therefore, a simple division would underestimate the monthly payment needed to repay the loan fully.

Understanding the correct formula ensures accurate budgeting and avoids potential financial surprises during the loan repayment process. Using online loan calculators that employ this formula simplifies the calculation, but understanding the underlying principles provides valuable financial literacy.