Multiple Choice
What is the present value of a \$300 annuity payment received annually for 5 years if the interest rate is 8\% per year? (Assume payments are made at the end of each year.)
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10. Time Value of Money
Time Value of Money Equations
Multiple Choice
Which of the following equations correctly calculates the future value (\(FV\)) of \$1,000 invested today for 3 years at 5\% annual interest compounded annually?
A
\(FV = 1,000 \times (1 - 0.05)^3\)
B
\(FV = 1,000 \times (1 + 0.05)^3\)
C
\(FV = 1,000 \div (1 + 0.05)^3\)
D
\(FV = 1,000 \times (1 + 0.05 \times 3)\)
Verified step by step guidance1
Understand the concept of future value (FV): Future value is the value of a current amount of money at a future date, based on a specific interest rate and compounding frequency. The formula for FV with annual compounding is \(FV = PV \times (1 + r)^n\), where \(PV\) is the present value, \(r\) is the annual interest rate, and \(n\) is the number of years.
Analyze the given options: The first option, \(FV = 1,000 \times (1 - 0.05)^3\), incorrectly uses subtraction instead of addition in the formula, which is not valid for compounding interest. The second option, \(FV = 1,000 \times (1 + 0.05)^3\), correctly follows the formula for future value with annual compounding. The third option, \(FV = 1,000 \div (1 + 0.05)^3\), represents a formula for present value, not future value. The fourth option, \(FV = 1,000 \times (1 + 0.05 \times 3)\), assumes simple interest rather than compounding, which is incorrect for this scenario.
Identify the correct formula: The correct formula for future value with annual compounding is \(FV = PV \times (1 + r)^n\). In this case, \(PV = 1,000\), \(r = 0.05\), and \(n = 3\). This matches the second option: \(FV = 1,000 \times (1 + 0.05)^3\).
Explain why compounding is used: Compounding means that interest is calculated not only on the initial principal but also on the accumulated interest from previous periods. This is why the formula includes \((1 + r)^n\), where the exponent \(n\) accounts for the number of compounding periods.
Summarize the solution: The correct formula for calculating the future value of \$1,000 invested today for 3 years at 5% annual interest compounded annually is \(FV = 1,000 \(\times\) (1 + 0.05)^3\). This formula accounts for the compounding effect over the specified time period.
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