Multiple Choice
Which of the following are common methods for calculating the time value of money?
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10. Time Value of Money
Time Value of Money Equations
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Approximately what annual interest rate, compounded annually, is needed to double an investment in 5 years?
A
About 10.0%
B
About 14.9%
C
About 20.0%
D
About 7.2%
Verified step by step guidance1
Step 1: Understand the problem. The goal is to determine the annual interest rate (r) required to double an investment in 5 years, using the formula for compound interest. The formula is: \( FV = PV \times (1 + r)^t \), where FV is the future value, PV is the present value, r is the annual interest rate, and t is the time in years.
Step 2: Set up the equation based on the problem. Since the investment needs to double, \( FV = 2 \times PV \). Substituting this into the formula gives: \( 2 \times PV = PV \times (1 + r)^5 \).
Step 3: Simplify the equation. Divide both sides of the equation by PV (assuming PV is not zero): \( 2 = (1 + r)^5 \).
Step 4: Solve for r. Take the fifth root (or raise both sides to the power of \( \frac{1}{5} \)) to isolate \( 1 + r \): \( 1 + r = 2^{\frac{1}{5}} \).
Step 5: Subtract 1 from both sides to find r: \( r = 2^{\frac{1}{5}} - 1 \). This will give the annual interest rate required to double the investment in 5 years.
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