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Chapter 8: Problem 14

Find the accumulated value of an investment of \(\$ 5000\) for 10 years at an interest rate of \(6.5 \%\) if the money is a. compounded semiannually; b. compounded quarterly; c. compounded monthly; d. compounded continuously.

Short Answer

Expert verified
The accumulated values for the investment when compounded semiannually, quarterly, monthly, and continuously can calculated by plug the values into the respective formulas and evaluating them. The exact answers depend on the exact value of \(e\), Euler's number.

Step by step solution

01

Understand the Formula for Compounded Interest

To solve this problem, the formula for compound interest is essential. It is \(FV = P(1 + r/n)^{nt}\), where \(P\) is the principal amount, \(r\) is the annual interest rate, \(n\) is the number of times the interest is compounded per year, and \(t\) is the time in years.
02

Calculate for Semiannual Compounding

For semiannual compounding (twice a year), \(n = 2\). Also, \(P = \$5000\), \(r = 6.5\% = 0.065\), and \(t = 10\) years. Plug these values into the formula: \(FV = 5000 * (1 + 0.065/2) ^ {2 * 10}\)
03

Calculate for Quarterly Compounding

For quarterly compounding (four times a year), \(n = 4\). Keep the values of \(P, r\), and \(t\) the same. Plug these values into the formula: \(FV = 5000 * (1 + 0.065/4) ^ {4 * 10}\)
04

Calculate for Monthly Compounding

For monthly compounding (twelve times a year), \(n = 12\). Keep the values of \(P, r\), and \(t\) the same. Plug these values into the formula: \(FV = 5000 * (1 + 0.065/12) ^ {12 * 10}\)
05

Understand the Formula for Continuous Compounding

For continuous compounding, the formula is different: \(FV = Pe^{rt}\), where \(e\) is Euler's number (~2.71828). Apply the same values for \(P, r\), and \(t\) and get the future value: \(FV = 5000 * e^{0.065 * 10}\)

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