Which of the following formulas correctly calculates the future value of an ordinary annuity when additional equal investments are made at the end of each period for \(n\) periods at an interest rate \(r\) per period?

• A. \(FV = PMT \times \dfrac{1 - (1 + r)^{-n}}{r}\)
• B. \(FV = PV \times (1 - r)^n\)
• C. \(FV = PV \times (1 + r)^n\)
• D. \(FV = PMT \times \dfrac{(1 + r)^n - 1}{r}\)

Answer: (D)

Verified step by step guidance

1. Step 1: Understand the concept of an ordinary annuity. An ordinary annuity involves equal payments (PMT) made at the end of each period for a specified number of periods (\(n\)) at a given interest rate (\(r\)). The future value (FV) represents the total value of these payments compounded at the interest rate.
2. Step 2: Recall the formula for the future value of an ordinary annuity. The correct formula is \(FV = PMT \times \dfrac{(1 + r)^n - 1}{r}\). This formula accounts for the compounding effect of the interest rate on the periodic payments.
3. Step 3: Analyze why the other formulas provided are incorrect. For example: - \(FV = PMT \times \dfrac{1 - (1 + r)^{-n}}{r}\) is incorrect because it represents the present value of an ordinary annuity, not the future value. - \(FV = PV \times (1 - r)^n\) is incorrect because it does not account for compounding correctly. - \(FV = PV \times (1 + r)^n\) is incorrect because it calculates the future value of a single lump sum, not an annuity.
4. Step 4: Break down the correct formula. The term \((1 + r)^n\) represents the compounding factor over \(n\) periods, and subtracting 1 accounts for the accumulation of payments. Dividing by \(r\) adjusts for the periodic interest rate applied to each payment.
5. Step 5: Apply the formula to calculate the future value of an ordinary annuity when given specific values for PMT, \(r\), and \(n\). Substitute the values into \(FV = PMT \times \dfrac{(1 + r)^n - 1}{r}\) and simplify step by step to find the future value.