Use the compound interest formulas to solve Exercises 10–11. Suppose that you have \$5000 to invest. Which investment yields the greater return over 5 years: 1.5% compounded semiannually or 1.45% compounded monthly?

Verified step by step guidance

1. Identify the compound interest formula: \(A = P \left(1 + \frac{r}{n}\right)^{nt}\), where \(A\) is the amount after \(t\) years, \(P\) is the principal, \(r\) is the annual interest rate (in decimal), \(n\) is the number of compounding periods per year, and \(t\) is the time in years.
2. For the first investment (1.5% compounded semiannually): set \(P = 5000\), \(r = 0.015\), \(n = 2\), and \(t = 5\). Substitute these values into the formula to express the amount \(A_1\).
3. For the second investment (1.45% compounded monthly): set \(P = 5000\), \(r = 0.0145\), \(n = 12\), and \(t = 5\). Substitute these values into the formula to express the amount \(A_2\).
4. Calculate the expressions for \(A_1\) and \(A_2\) separately by evaluating the powers and multiplications (do not compute the final numerical values here, just set up the expressions).
5. Compare the two amounts \(A_1\) and \(A_2\) to determine which investment yields the greater return over 5 years.

Key Concepts

Compound Interest Formula

The compound interest formula calculates the amount of money accumulated over time with interest added periodically. It is given by A = P(1 + r/n)^(nt), where P is the principal, r is the annual interest rate, n is the number of compounding periods per year, and t is the time in years.

Compounding Frequency

Compounding frequency refers to how often interest is added to the principal balance within a year. Common frequencies include annually, semiannually, quarterly, and monthly. More frequent compounding results in interest being calculated on previously earned interest more often, increasing the total return.

Comparing Investment Returns

To determine which investment yields a greater return, calculate the final amount for each option using their respective interest rates and compounding frequencies. Comparing these amounts after the same time period shows which investment is more profitable.