Which of the following equations correctly calculates the present value (PV) of a single future sum (FV) to be received in \(n\) periods at an interest rate \(r\) per period?

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

Answer: (D)

Verified step by step guidance

1. Step 1: Understand the concept of present value (PV). Present value is the current worth of a future sum of money or cash flow, discounted at a specific interest rate over a certain number of periods. It accounts for the time value of money, which states that money available today is worth more than the same amount in the future due to its earning potential.
2. Step 2: Identify the formula for calculating the present value of a single future sum (FV). The correct formula is: PV = FV / (1 + r)^n, where FV is the future value, r is the interest rate per period, and n is the number of periods.
3. Step 3: Break down the formula. The denominator (1 + r)^n represents the compounding factor, which adjusts the future value to account for the interest rate and the number of periods. Dividing the future value by this factor gives the present value.
4. Step 4: Compare the given equations. The first equation, PV = FV - (FV × r × n), incorrectly subtracts a simple interest calculation from FV, which does not account for compounding. The second equation, PV = FV × r × n, incorrectly multiplies FV by the interest rate and number of periods, which also does not account for compounding. The third equation, PV = FV × (1 + r)^n, incorrectly multiplies FV by the compounding factor instead of dividing by it.
5. Step 5: Conclude that the correct formula is PV = FV / (1 + r)^n, as it properly accounts for the time value of money and the compounding effect over n periods.