Use the compound interest formulas [tex]A = P \left(1+\frac{ r }{ n }\right)^{ nt }[/tex] and [tex]A = Pe ^{ tt }[/tex] to solve the problem given. Round answers to the nearest cent.
Find the accumulated value of an investment of [tex]$$ 20,000[/tex] for 7 years at an interest rate of [tex]4.5 %[/tex] if the money is
a. compounded semiannually,
b. compounded quarterly,
c. compounded monthly,
d. compounded continuously
Answer (1)
Calculate the accumulated value when compounded semiannually: A = 20000 ( 1 + 2 0.045 ) 2 × 7 \tapprox 27309.67 .
Calculate the accumulated value when compounded quarterly: A = 20000 ( 1 + 4 0.045 ) 4 × 7 \tapprox 27357.03 .
Calculate the accumulated value when compounded monthly: A = 20000 ( 1 + 12 0.045 ) 12 × 7 \tapprox 27389.05 .
Calculate the accumulated value when compounded continuously: A = 20000 e 0.045 × 7 \tapprox 27405.19 .
Therefore, the final answers are: a. 27309.67 b. 27357.03 c. 27389.05 d. 27405.19
Explanation
Understanding the Problem We are asked to find the accumulated value of an investment of $20,000 for 7 years at an interest rate of 4.5% under different compounding scenarios: semiannually, quarterly, monthly, and continuously. We will use the compound interest formula A = P \tleft(1+\frac{ r }{ n }\right)^{ nt } for the first three scenarios and the continuous compounding formula A = P e r t for the last scenario.
Calculating Semiannual Compounding a. Compounded semiannually: Here, P = 20000 , r = 0.045 , n = 2 , and t = 7 . Plugging these values into the compound interest formula, we get: A = 20000 \tleft(1+\frac{ 0.045 }{ 2 }\right)^{ 2 \times 7 } = 20000 (1 + 0.0225)^{14} = 20000 (1.0225)^{14} Calculating this, we find A ≈ 27309.67 .
Calculating Quarterly Compounding b. Compounded quarterly: Here, P = 20000 , r = 0.045 , n = 4 , and t = 7 . Plugging these values into the compound interest formula, we get: A = 20000 \tleft(1+\frac{ 0.045 }{ 4 }\right)^{ 4 \times 7 } = 20000 (1 + 0.01125)^{28} = 20000 (1.01125)^{28} Calculating this, we find A ≈ 27357.03 .
Calculating Monthly Compounding c. Compounded monthly: Here, P = 20000 , r = 0.045 , n = 12 , and t = 7 . Plugging these values into the compound interest formula, we get: A = 20000 \tleft(1+\frac{ 0.045 }{ 12 }\right)^{ 12 \times 7 } = 20000 (1 + 0.00375)^{84} = 20000 (1.00375)^{84} Calculating this, we find A ≈ 27389.05 .
Calculating Continuous Compounding d. Compounded continuously: Here, P = 20000 , r = 0.045 , and t = 7 . Plugging these values into the continuous compounding formula, we get: A = 20000 e 0.045 × 7 = 20000 e 0.315 Calculating this, we find A ≈ 27405.19 .
Final Answer Therefore, the accumulated values for each compounding scenario are: a. Semiannually: $27,309.67 b. Quarterly: $27,357.03 c. Monthly: $27,389.05 d. Continuously: $27,405.19
Examples
Understanding compound interest is crucial for making informed financial decisions. For instance, when planning for retirement, knowing how different compounding frequencies affect your investment can significantly impact your savings over time. Suppose you invest in a bond with a fixed interest rate; the more frequently the interest is compounded (e.g., daily vs. annually), the faster your investment grows. This knowledge helps you choose the best investment options to reach your financial goals sooner.
