Find the accumulated value of an investment of $ 20 comma 000 for 7 years at an interest rate of 1.65 % if the money is a. compounded semiannually; b. compounded quarterly; c. compounded monthly d. compounded continuously. Question content area bottom Part 1 a. What is the accumulated value if the money is compounded semiannually? $ enter your response here (Round to the nearest cent as needed
To calculate the accumulated value of an investment with compound interest, we can use the formula:
[ A = P \left(1 + \frac{r}{n}\right)^{nt} ]
Where:
Given:
For semiannual compounding, (n = 2).
[ A = 20000 \left(1 + \frac{0.0165}{2}\right)^{2 \times 7} ]
Calculating:
[ A = 20000 \left(1 + 0.00825\right)^{14} ] [ A = 20000 \left(1.00825\right)^{14} ] [ A = 20000 \times 1.125682 ] [ A \approx 22513.64 ]
So, the accumulated value if the money is compounded semiannually is approximately $22,513.64.
For quarterly compounding, (n = 4).
[ A = 20000 \left(1 + \frac{0.0165}{4}\right)^{4 \times 7} ]
Calculating:
[ A = 20000 \left(1 + 0.004125\right)^{28} ] [ A = 20000 \left(1.004125\right)^{28} ] [ A = 20000 \times 1.125949 ] [ A \approx 22518.98 ]
So, the accumulated value if the money is compounded quarterly is approximately $22,518.98.
For monthly compounding, (n = 12).
[ A = 20000 \left(1 + \frac{0.0165}{12}\right)^{12 \times 7} ]
Calculating:
[ A = 20000 \left(1 + 0.001375\right)^{84} ] [ A = 20000 \left(1.001375\right)^{84} ] [ A = 20000 \times 1.126825 ] [ A \approx 22536.50 ]
So, the accumulated value if the money is compounded monthly is approximately $22,536.50.
For continuous compounding, we use the formula:
[ A = Pe^{rt} ]
Where (e) is the base of the natural logarithm (approximately 2.71828).
Calculating:
[ A = 20000 \times e^{0.0165 \times 7} ] [ A = 20000 \times e^{0.1155} ] [ A \approx 20000 \times 1.1224 ] [ A \approx 22448.00 ]
So, the accumulated value if the money is compounded continuously is approximately $22,448.00.