Compound Interest Challenge You have been given \( \$ 1,000 \) to invest for 5 years. Your goal is to maximize your returns using compound interest. *Investment Options* 1. *High-Yield Savings Account*: 4\% annual interest rate, compounded monthly 2. *Certificate of Deposit (CD)*: 5\% annual interest rate, compounded quarterly 3. *Stock Market Investment*: 7\% annual interest rate, compounded annually *Questions* 1. Calculate the future value of each investment option after

To calculate the future value of each investment option using compound interest, we can use the formula: \[ FV = P \left(1 + \frac{r}{n}\right)^{nt} \] Where: - \( FV \) = future value of the investment - \( P \) = principal amount (initial investment) - \( r \) = annual interest rate (decimal) - \( n \) = number of times interest is compounded per year - \( t \) = number of years the money is invested Let's calculate the future value for each investment option: 1. **High-Yield Savings Account**: - \( P = 1000 \) - \( r = 0.04 \) - \( n = 12 \) (monthly) - \( t = 5 \) 2. **Certificate of Deposit (CD)**: - \( P = 1000 \) - \( r = 0.05 \) - \( n = 4 \) (quarterly) - \( t = 5 \) 3. **Stock Market Investment**: - \( P = 1000 \) - \( r = 0.07 \) - \( n = 1 \) (annually) - \( t = 5 \) Now, I will calculate the future value for each option. Calculate the value by following steps: - step0: Calculate: \(1000\left(1+\frac{0.07}{1}\right)^{1\times 5}\) - step1: Divide the terms: \(1000\left(1+\frac{7}{100}\right)^{1\times 5}\) - step2: Add the numbers: \(1000\left(\frac{107}{100}\right)^{1\times 5}\) - step3: Calculate: \(1000\left(\frac{107}{100}\right)^{5}\) - step4: Simplify: \(1000\times \frac{107^{5}}{100^{5}}\) - step5: Rewrite the expression: \(10^{3}\times \frac{107^{5}}{10^{10}}\) - step6: Reduce the numbers: \(1\times \frac{107^{5}}{10^{7}}\) - step7: Multiply: \(\frac{107^{5}}{10^{7}}\) Calculate or simplify the expression \( 1000 * (1 + 0.04 / 12)^(12 * 5) \). Calculate the value by following steps: - step0: Calculate: \(1000\left(1+\frac{0.04}{12}\right)^{12\times 5}\) - step1: Divide the terms: \(1000\left(1+\frac{1}{300}\right)^{12\times 5}\) - step2: Add the numbers: \(1000\left(\frac{301}{300}\right)^{12\times 5}\) - step3: Multiply the numbers: \(1000\left(\frac{301}{300}\right)^{60}\) - step4: Simplify: \(1000\times \frac{301^{60}}{300^{60}}\) - step5: Rewrite the expression: \(1000\times \frac{301^{60}}{100^{60}\times 3^{60}}\) - step6: Rewrite the expression: \(10^{3}\times \frac{301^{60}}{10^{120}\times 3^{60}}\) - step7: Reduce the numbers: \(1\times \frac{301^{60}}{10^{117}\times 3^{60}}\) - step8: Multiply the fractions: \(\frac{301^{60}}{10^{117}\times 3^{60}}\) Calculate or simplify the expression \( 1000 * (1 + 0.05 / 4)^(4 * 5) \). Calculate the value by following steps: - step0: Calculate: \(1000\left(1+\frac{0.05}{4}\right)^{4\times 5}\) - step1: Divide the terms: \(1000\left(1+\frac{1}{80}\right)^{4\times 5}\) - step2: Add the numbers: \(1000\left(\frac{81}{80}\right)^{4\times 5}\) - step3: Multiply the numbers: \(1000\left(\frac{81}{80}\right)^{20}\) - step4: Simplify: \(1000\times \frac{81^{20}}{80^{20}}\) - step5: Rewrite the expression: \(40\times 25\times \frac{81^{20}}{80^{20}}\) - step6: Rewrite the expression: \(40\times 25\times \frac{81^{20}}{40^{20}\times 2^{20}}\) - step7: Reduce the numbers: \(25\times \frac{81^{20}}{40^{19}\times 2^{20}}\) - step8: Rewrite the expression: \(25\times \frac{81^{20}}{5^{19}\times 8^{19}\times 2^{20}}\) - step9: Rewrite the expression: \(5^{2}\times \frac{81^{20}}{5^{19}\times 8^{19}\times 2^{20}}\) - step10: Reduce the numbers: \(1\times \frac{81^{20}}{5^{17}\times 8^{19}\times 2^{20}}\) - step11: Multiply the fractions: \(\frac{81^{20}}{5^{17}\times 2^{77}}\) Now, let's summarize the future values for each investment option after 5 years: 1. **High-Yield Savings Account**: \[ FV \approx 1220.996594 \] 2. **Certificate of Deposit (CD)**: \[ FV \approx 1282.037232 \] 3. **Stock Market Investment**: \[ FV \approx 1402.5517307 \] ### Conclusion: - The future value of the High-Yield Savings Account is approximately **$1221.00**. - The future value of the Certificate of Deposit (CD) is approximately **$1282.04**. - The future value of the Stock Market Investment is approximately **$1402.55**. Among the three options, the Stock Market Investment yields the highest future value after 5 years.