What is the formula to calculate the monthly payment (PMT) on a 20-year loan with principal amount \(P\), annual interest rate \(r\) (compounded monthly), and total number of monthly payments \(n\)?

• A. PMT = \dfrac{P}{n}
• B. PMT = P \cdot r \cdot n
• C. PMT = \dfrac{P \cdot \left(\dfrac{r}{12}\right)}{1 - (1 + \dfrac{r}{12})^{-n}}
• D. PMT = P \cdot (1 + r)^{n}

Answer: (C)

Verified step by step guidance

1. Step 1: Understand the components of the formula. The monthly payment (PMT) is calculated using the loan principal amount (P), the annual interest rate (r), the number of monthly payments (n), and the monthly interest rate (r/12). The formula accounts for the compounding effect of interest over time.
2. Step 2: Break down the formula. The correct formula is PMT = \dfrac{P \cdot \left(\dfrac{r}{12}\right)}{1 - (1 + \dfrac{r}{12})^{-n}}. Here, \dfrac{r}{12} represents the monthly interest rate, and (1 + \dfrac{r}{12})^{-n} accounts for the present value of the loan payments over the total number of months.
3. Step 3: Calculate the monthly interest rate. Divide the annual interest rate (r) by 12 to find the monthly interest rate. This step ensures the interest rate is adjusted for monthly compounding.
4. Step 4: Compute the denominator of the formula. The denominator is 1 - (1 + \dfrac{r}{12})^{-n}, which represents the discount factor for the loan payments over the total number of months. Use exponentiation to calculate (1 + \dfrac{r}{12})^{-n}.
5. Step 5: Combine the components. Multiply the principal amount (P) by the monthly interest rate (\dfrac{r}{12}) and divide the result by the denominator calculated in Step 4. This will give the monthly payment (PMT).