Multiple Choice
If Ida invests \$10 at an annual interest rate of 7% compounded annually, approximately how many years will it take for her investment to double?
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10. Time Value of Money
Time Value of Money Equations
Multiple Choice
A 1-year loan with a 5\% APR will have the most interest charged with which type of compounding?
A
Semiannual compounding
B
Continuous compounding
C
Quarterly compounding
D
Annual compounding
Verified step by step guidance1
Understand the concept of compounding: Compounding refers to the frequency with which interest is calculated and added to the principal balance. The more frequent the compounding, the higher the total interest accrued over time.
Review the types of compounding mentioned: Semiannual compounding means interest is compounded twice a year, quarterly compounding means four times a year, annual compounding means once a year, and continuous compounding means interest is compounded infinitely often.
Recall the formula for compound interest: \( A = P \cdot e^{r \cdot t} \) for continuous compounding, and \( A = P \cdot (1 + \frac{r}{n})^{n \cdot t} \) for other types of compounding, where \( A \) is the future value, \( P \) is the principal, \( r \) is the annual interest rate, \( n \) is the number of compounding periods per year, and \( t \) is the time in years.
Compare the impact of compounding frequency: Continuous compounding results in the highest amount of interest because it assumes interest is being compounded at every possible moment, whereas semiannual, quarterly, and annual compounding occur less frequently and result in lower total interest.
Conclude that continuous compounding will result in the most interest charged for the 1-year loan with a 5% APR, as it compounds infinitely often compared to the other options.
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