Open Question
You invest \$4,545 in Clutch Bank today earning a juicy 10% annual interest. What is the value of your investment in one year? What is the value of the investment after two years?
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10. Time Value of Money
Time Value of Money Equations
Multiple Choice
Which of the following formulas represents the present value (PV) of a single future sum (FV) discounted at an interest rate \(r\) for \(n\) periods?
A
PV = \(\dfrac{FV}{(1 + r)^n}\)
B
PV = FV \(\times\) r \(\times\) n
C
PV = FV - (FV \(\times\) r \(\times\) n)
D
PV = FV \(\times\) (1 + r)^n
Verified step by step guidance1
Understand the concept of present value (PV): The present value is the current worth of a future sum of money (FV) given a specific interest rate (r) and time period (n). It accounts for the time value of money, which states that money today is worth more than the same amount in the future due to its earning potential.
Recall the formula for the present value of a single future sum: The correct formula is PV = \(\dfrac{FV}{(1 + r)^n}\). This formula discounts the future value (FV) back to its present value by dividing it by the compound growth factor (1 + r)^n.
Analyze the options provided: Compare each formula to the correct formula. For example, PV = FV \(\times\) r \(\times\) n is incorrect because it does not account for compounding. Similarly, PV = FV - (FV \(\times\) r \(\times\) n) is also incorrect because it subtracts a simple interest calculation rather than applying a discount factor. Lastly, PV = FV \(\times\) (1 + r)^n is incorrect because it represents the future value, not the present value.
Identify the correct formula: The correct formula is PV = \(\dfrac{FV}{(1 + r)^n}\), as it properly discounts the future value using the compound interest factor.
Apply the formula: To calculate the present value, substitute the given values for FV, r, and n into the formula PV = \(\dfrac{FV}{(1 + r)^n}\). Simplify the denominator by calculating (1 + r)^n, then divide FV by this value to find the present value.
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