- Identify the continuous compounding formula \(A(t) = Pe^{rt}\) and the effective rate formula \(r_{\text{eff}} = e^r - 1\).
- Calculate the effective annual rate \(r_{\text{eff}} = e^{0.108} - 1 \approx 0.1140477\).
- Calculate the accumulated value \(A(5) = 71900 \cdot e^{0.108 \times 5} \approx 123,380.89\).
- Final Answer: \(r_{\text{eff}} = \boxed{0.1140477}\) and \(A(5) = \boxed{\$123,380.89}\).
Explanation
- Problem Analysis and Formulas
To determine the effective annual rate and the accumulated value of an investment with continuous compounding, we use the given parameters: the principal \(P = \$71,900\), the annual interest rate \(r = 10.8\% = 0.108\), and the duration of the investment \(t = 5\) years. Continuous compounding assumes interest is added constantly, following the formula \(A(t) = Pe^{rt}\), where \(e\) is Euler's number (approximately \(2.71828\)). The effective annual rate \(r_{\text{eff}}\), which represents the interest earned in one year, is calculated by the formula \(r_{\text{eff}} = e^r - 1\).
- Calculating the Effective Annual Rate
The effective annual rate is found using the formula \(r_{\text{eff}} = e^r - 1\). With \(r = 0.108\):
\[r_{\text{eff}} = e^{0.108} - 1\]
Using the value of \(e^{0.108} \approx 1.114047745\), we find:
\[r_{\text{eff}} \approx 1.114047745 - 1 = 0.114047745\]
Rounding to 7 decimal places as required, we obtain \(r_{\text{eff}} = 0.1140477\).
- Calculating the Accumulated Value
The accumulated value \(A(5)\) after 5 years is calculated using the formula \(A(t) = Pe^{rt}\):
\[A(5) = 71,900 \times e^{0.108 \times 5}\]
First, calculate the exponent: \(0.108 \times 5 = 0.54\).
Next, compute \(e^{0.54} \approx 1.71600686\):
\[A(5) = 71,900 \times 1.71600686 \approx 123,380.89339\]
Rounding the accumulated value to the nearest cent, we get \(A(5) = \$123,380.89\).
- Final Results
The effective annual rate is \(0.1140477\), and the accumulated value of the investment after 5 years is \(\$123,380.89\).
Examples
Understanding continuous compounding is essential for financial planning and investment analysis. For example, a bank might offer a savings account that compounds interest continuously, and calculating the effective annual rate allows you to compare this account's performance directly against other investment options that compound on an annual or monthly basis. By applying these formulas, you can accurately project your future savings and make informed decisions about where to invest your money to maximize growth over time.