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Chapter 5: Problem 57

Investment A offers a \(10 \%\) return compounded semiannually, and investment \(\mathrm{B}\) offers a \(9.75 \%\) return compounded continuously. Which investment has a higher rate of return over a 4-yr period?

Short Answer

Expert verified
Investment A has a higher rate of return over a 4-year period, with a future value of approximately \(1.477\) compared to Investment B's future value of approximately \(1.474\).

Step by step solution

01

Write down the formulas for both types of compound interest

For the semiannually compounded investment A, we will use the compound interest formula: \(A = P(1 + \frac{r}{n})^{nt}\). For the continuously compounded investment B, we will use the continuous compound interest formula: \(A = Pe^{rt}\).
02

Set up the formulas for both investments with the given information

For Investment A, the annual interest rate is \(10\%\) or \(0.1\), and the compounding period is semiannual, so \(n = 2\). The time period is 4 years. Therefore, the formula for Investment A is: \(A_A = P(1 + \frac{0.1}{2})^{2 * 4}\). For Investment B, the annual interest rate is \(9.75\%\) or \(0.0975\). Since it's compounded continuously, the formula for Investment B is: \(A_B = Pe^{0.0975 * 4}\).
03

Compare the rates of return

To determine which investment has a higher rate of return, we compare their formulas using the future amounts of a hypothetical $1 investment. For Investment A: \(A_A = 1(1 + \frac{0.1}{2})^{2 * 4} = (1 + 0.05)^{8}\) \(A_A \approx 1.477\) For Investment B: \(A_B = 1e^{0.0975 * 4} \approx 1.474\)
04

Determine which investment has a higher rate of return

Since the future value of Investment A (\(1.477\)) is greater than the future value of Investment B (\(1.474\)), Investment A has a higher rate of return over a 4-year period.

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