Multiple Choice
Given a cash flow of \$1,000 to be received at the end of year 3 and a discount rate of 8\% per year, what is the present value of this year 3 cash flow?
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10. Time Value of Money
Time Value of Money Equations
Multiple Choice
Which of the following equations correctly computes the present value (PV) of a single future cash flow (FV) to be received in \(n\) periods, discounted at an annual interest rate \(r\)?
A
PV = FV \(\times\) r \(\times\) n
B
PV = \(\dfrac{FV}{(1 + r)^n}\)
C
PV = FV \(\times\) (1 + r)^n
D
PV = \(\dfrac{FV}{r^n}\)
Verified step by step guidance1
Step 1: Understand the concept of present value (PV). Present value is the current worth of a future cash flow (FV) discounted at a specific interest rate (r) over a certain number of periods (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.
Step 2: Identify the correct formula for computing the present value of a single future cash flow. The formula is: PV = FV / (1 + r)^n. This formula discounts the future value (FV) by dividing it by the compound factor (1 + r)^n, which represents the growth of money over n periods at an annual interest rate r.
Step 3: Compare the given equations to the correct formula. The first equation, PV = FV × r × n, is incorrect because it does not account for compounding and simply multiplies the future value by the interest rate and number of periods. The second equation, PV = FV × (1 + r)^n, is incorrect because it multiplies the future value by the compound factor instead of dividing. The third equation, PV = FV / r^n, is incorrect because it uses r^n instead of (1 + r)^n, which does not properly account for compounding.
Step 4: Recognize that the correct formula is PV = FV / (1 + r)^n. This formula properly discounts the future value by dividing it by the compound factor, which reflects the time value of money and the effect of compounding over n periods.
Step 5: Apply the correct formula in practice. To compute the present value, substitute the values of FV (future cash flow), r (annual interest rate), and n (number of periods) into the formula PV = 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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