The future value of $1,000 invested at an APR of 8% for 7 years varies depending on the compounding frequency: $1,718.19 if compounded annually, $1,747.26 if compounded semiannually, $1,765.87 if compounded monthly, and $1,793.47 if compounded continuously, all rounded to two decimal places.
To calculate the future value of an investment, we use the formula for compound interest, which is A = P(1 + r/n)^(nt), where A is the amount of money accumulated after n years, including interest, P is the principal amount (the initial amount of money), r is the annual interest rate (decimal), n is the number of times the interest is compounded per year, and t is the time the money is invested for in years.
a) With an APR of 8% compounded annually, the future value of $1,000 after seven years is calculated as follows:
A = $1,000 ×(1 + 0.08/1)⁷= $1,000 × (1.08)⁷ = $1,000 × 1.718186 = $1,718.19 (rounded to two decimal places).
b) For the future value with an APR of 8% compounded semiannually, the calculation is:
A = $1,000 × (1 + 0.08/2)¹⁴ = $1,000 × (1.04)¹⁴ = $1,000 × 1.747257 = $1,747.26 (rounded to two decimal places).
c) When it's compounded monthly, the future value is:
A = $1,000 ×(1 + 0.08/12)⁵⁶ = $1,000 ×(1.006666)⁸⁴ = $1,000 × 1.765869 = $1,765.87 (rounded to two decimal places).
d) If the interest is compounded continuously, we use the formula A = Pe^(rt), where e is Euler's number (approx. 2.71828). The calculation will be:
A = $1,000 × e^(0.08 ×7) = $1,000 ×2.71828^(0.56) = $1,000 ×1.793467 = $1,793.47 (rounded to two decimal places).