Investment Accumulation with Different Compounding Methods

Answered using GPT-4o mini by AI Answer Generator

Question:

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

Answer:

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:

  • (A) = the accumulated amount (future value)
  • (P) = the 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 or borrowed

Given:

  • (P = 20,000)
  • (r = 1.65% = 0.0165)
  • (t = 7) years

a. Compounded Semiannually

For semiannual compounding, (n = 2).

[ A = 20000 \left(1 + \frac{0.0165}{2}\right)^{2 \times 7} ]

Calculating:

  1. Calculate (\frac{0.0165}{2} = 0.00825)
  2. Calculate (2 \times 7 = 14)
  3. Calculate (A):

[ 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.

b. Compounded Quarterly

For quarterly compounding, (n = 4).

[ A = 20000 \left(1 + \frac{0.0165}{4}\right)^{4 \times 7} ]

Calculating:

  1. Calculate (\frac{0.0165}{4} = 0.004125)
  2. Calculate (4 \times 7 = 28)
  3. Calculate (A):

[ 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.

c. Compounded Monthly

For monthly compounding, (n = 12).

[ A = 20000 \left(1 + \frac{0.0165}{12}\right)^{12 \times 7} ]

Calculating:

  1. Calculate (\frac{0.0165}{12} = 0.001375)
  2. Calculate (12 \times 7 = 84)
  3. Calculate (A):

[ 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.

d. Compounded Continuously

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.

Summary of Results:

  • Compounded Semiannually: $22,513.64
  • Compounded Quarterly: $22,518.98
  • Compounded Monthly: $22,536.50
  • Compounded Continuously: **$22,448