What is the formula to calculate the monthly payment (PMT) on a 36-month loan with principal \(P\), annual interest rate \(r\) (compounded monthly), and 36 equal payments?

• A. PMT = \dfrac{P}{36}
• B. PMT = \dfrac{P \times r/12}{1 - (1 + r/12)^{-36}}
• C. PMT = P \times r \times 36
• D. PMT = P \times (1 + r)^{36}

Answer: (B)

Verified step by step guidance

1. Understand the problem: The goal is to calculate the monthly payment (PMT) for a loan with principal \(P\), annual interest rate \(r\) (compounded monthly), and 36 equal payments. The formula provided is PMT = \dfrac{P \times r/12}{1 - (1 + r/12)^{-36}}.
2. Step 1: Break down the formula. The numerator, P \times r/12, represents the monthly interest portion of the loan. Here, r/12 converts the annual interest rate into a monthly interest rate.
3. Step 2: Analyze the denominator. The term 1 - (1 + r/12)^{-36} accounts for the present value of the loan payments over 36 months, factoring in the monthly compounding of interest.
4. Step 3: Substitute the given values into the formula. Replace P with the loan principal, r with the annual interest rate, and ensure r/12 is used for monthly compounding. The exponent -36 represents the 36 monthly payments.
5. Step 4: Simplify the formula step by step. First, calculate r/12, then (1 + r/12), followed by raising it to the power of -36. Finally, subtract this value from 1 and divide the numerator by the resulting denominator.