Final answer:
To find the accumulated value of an investment at different compounding frequencies, we use the compound interest formula A = P(1 + r/n)^(nt). For an investment of $20,000 for 3 years at an interest rate of 4%: a. Compounded semiannually, the accumulated value is approximately $21,312.22. b. Compounded quarterly, the accumulated value is approximately $21,393.01. c. Compounded monthly, the accumulated value is approximately $21,449.36. d. Compounded continuously, the accumulated value is approximately $21,493.08.
Step-by-step explanation:
To find the accumulated value of an investment, we can use the compound interest formula A = P(1 + r/n)^(nt), where A is the accumulated value, P is the principal amount, r is the interest rate, n is the number of times interest is compounded per year, and t is the number of years.
a. Compounded semiannually:
- Interest rate = 4% = 0.04
- Number of times interest is compounded per year (n) = 2
- Principal amount (P) = $20,000
- Time (t) = 3 years
Using the formula, A = $20,000(1 + 0.04/2)^(2*3), we can calculate the accumulated value to be approximately $21,312.22.
b. Compounded quarterly:
- Interest rate = 4% = 0.04
- Number of times interest is compounded per year (n) = 4
- Principal amount (P) = $20,000
- Time (t) = 3 years
Using the formula, A = $20,000(1 + 0.04/4)^(4*3), we can calculate the accumulated value to be approximately $21,393.01.
c. Compounded monthly:
- Interest rate = 4% = 0.04
- Number of times interest is compounded per year (n) = 12
- Principal amount (P) = $20,000
- Time (t) = 3 years
Using the formula, A = $20,000(1 + 0.04/12)^(12*3), we can calculate the accumulated value to be approximately $21,449.36.
d. Compounded continuously:
- Interest rate = 4% = 0.04
- Principal amount (P) = $20,000
- Time (t) = 3 years
Using the formula, A = $20,000*e^(0.04*3), where e is Euler's number (approximately 2.71828), we can calculate the accumulated value to be approximately $21,493.08.