4. You invest $2000 in an investment that earns 6% interest, compounded quarterly. How much will the investment be worth after 5 years?

5. You invest $3000 in an investment that earns 5% interest, compounded monthly. How much will the investment be worth after 8 years?

6. The formula that financial managers and accountants use to determine the value of investments that are subject to compounding interest is A = P(1 + r/n)^nt where A is the final balance, P is the initial deposit, r is the annual interest rate, n is the number of times the interest is compounded per year and t is the number of years. Redo Exercises 4 and 5 using this formula.

1 answer

To find the future value of the investments using the formula \( A = P(1 + \frac{r}{n})^{nt} \), we'll apply it to both cases, 4 and 5.

Exercise 4:

Investing $2000 at an annual interest rate of 6% compounded quarterly for 5 years.

  • P (initial investment) = $2000
  • r (annual interest rate) = 6% = 0.06
  • n (number of times compounded per year) = 4 (quarterly)
  • t (number of years) = 5

Using the formula: \[ A = 2000 \left(1 + \frac{0.06}{4}\right)^{4 \times 5} \]

Now, calculate step by step:

  1. Calculate \( \frac{0.06}{4} = 0.015 \)
  2. Calculate \( 4 \times 5 = 20 \)
  3. Calculate \( 1 + 0.015 = 1.015 \)
  4. Raise to the power of 20: \[ 1.015^{20} \approx 1.346855007 \]
  5. Now, multiply by $2000: \[ A \approx 2000 \times 1.346855007 \approx 2693.71 \]

So, the investment will be worth approximately $2693.71 after 5 years.

Exercise 5:

Investing $3000 at an annual interest rate of 5% compounded monthly for 8 years.

  • P (initial investment) = $3000
  • r (annual interest rate) = 5% = 0.05
  • n (number of times compounded per year) = 12 (monthly)
  • t (number of years) = 8

Using the formula: \[ A = 3000 \left(1 + \frac{0.05}{12}\right)^{12 \times 8} \]

Now, calculate step by step:

  1. Calculate \( \frac{0.05}{12} \approx 0.00416667 \)
  2. Calculate \( 12 \times 8 = 96 \)
  3. Calculate \( 1 + 0.00416667 \approx 1.00416667 \)
  4. Raise to the power of 96: \[ 1.00416667^{96} \approx 1.432364654 \]
  5. Now, multiply by $3000: \[ A \approx 3000 \times 1.432364654 \approx 4297.09 \]

So, the investment will be worth approximately $4297.09 after 8 years.

Summary:

  • After 5 years, the investment in exercise 4 will be worth approximately $2693.71.
  • After 8 years, the investment in exercise 5 will be worth approximately $4297.09.