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:
- For an annual interest rate of 5 percent compounded annually for 8 years: A = 6000(1 + 0.05/1)^(1*8) = $8815.29.
- For semiannual compounding at 5 percent for 8 years: A = 6000(1 + 0.05/2)^(2*8) = $8,932.05.
- For bimonthly compounding at 5 percent for 8 years: A = 6000(1 + 0.05/6)^(6*8) = $9,051.31.
- For an annual interest rate of 10 percent compounded annually for 8 years: A = 6000(1 + 0.10/1)^(1*8) = $12,953.44.
- For semiannual compounding at 10 percent for 8 years: A = 6000(1 + 0.10/2)^(2*8) = $13,267.97.
- For bimonthly compounding at 10 percent for 8 years: A = 6000(1 + 0.10/6)^(6*8) = $13,458.22.
- 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:
- A higher interest rate significantly increases the future value of an investment.
- 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.