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Your grandmother just gave you $6,000.

a. Calculate the future value of $6,000, given that it will be invested for 8 years at an annual interest rate of 5 percent
b. Recalculate part (a) using a compounding period that is (1) semiannual and (2) bimonthly.
c. Now let's look at what might happen if you can invest the money at a rate of 10 percent rather than 5 percent rate; recalculate parts (a) and (b) for an annual interest rate of 10 percent.
d. Now let's see what might happen if you invest the money for 16 years rather than 8 years; recalculate part (a) using a time horizon of 16 years (annual interest rate is still 5 percent).
e. With respect to the changes in the stated interest rate and length of time the money is invested in parts (c) and (d), what conclusions can you draw?

1 Answer

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Final answer:

The future value of a $6000 investment increases with a higher interest rate and a longer investment duration due to the power of compound interest. The exact amount of future value depends on the compounding frequency, with more frequent compounding leading to a higher future value.

Step-by-step explanation:

To solve the student's question regarding the future value of a $6000 investment under various interest rates and compounding periods, we will apply the formula for compound interest, which is A = P(1 + r/n)^(nt), where:

  • A is the future value of the investment/loan, including interest,
  • P is the principal investment amount ($6000 in this case),
  • r is the annual interest rate (decimal),
  • n is the number of times interest is compounded per year,
  • t is the time the money is invested for in years.

For instance:

  1. For an annual interest rate of 5 percent compounded annually for 8 years: A = 6000(1 + 0.05/1)^(1*8) = $8815.29.
  2. For semiannual compounding at 5 percent for 8 years: A = 6000(1 + 0.05/2)^(2*8) = $8,932.05.
  3. For bimonthly compounding at 5 percent for 8 years: A = 6000(1 + 0.05/6)^(6*8) = $9,051.31.
  4. For an annual interest rate of 10 percent compounded annually for 8 years: A = 6000(1 + 0.10/1)^(1*8) = $12,953.44.
  5. For semiannual compounding at 10 percent for 8 years: A = 6000(1 + 0.10/2)^(2*8) = $13,267.97.
  6. For bimonthly compounding at 10 percent for 8 years: A = 6000(1 + 0.10/6)^(6*8) = $13,458.22.
  7. For an investment duration of 16 years at 5 percent compounded annually: A = 6000(1 + 0.05/1)^(1*16) = $13,266.75.

e. With respect to the changes in the stated interest rate and length of time the money is invested in parts (c) and (d), we can draw two main conclusions:

  1. A higher interest rate significantly increases the future value of an investment.
  2. An increase in the length of time of an investment, even with the same interest rate, results in a substantially higher future value due to the effect of compound interest over a longer period.
User Daniel Elkington
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