Multiple Choice
To what amount will $P invested for n years at an annual interest rate of r percent, compounded annually, accumulate?
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10. Time Value of Money
Time Value of Money Equations
Multiple Choice
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}
Verified step by step guidance1
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 = \(\dfrac{P \times r/12}{1 - (1 + r/12)^{-36}\)}.
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.
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.
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.
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.
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