Questions: Suppose that you have 7820 to invest. Which investment yields the greater return over 7 years: 9.35% compounded quarterly or 9.45% compounded semiannually? The accumulated value of the 9.35% investment is 14,934.18 The accumulated value of the 9.45% investment is . CONCLUSION: The investment at 9.35% compounded quarterly yields a greater return.

Solution Steps

To determine which investment yields a greater return, we need to calculate the future value of each investment using the compound interest formula. The formula for compound interest is:

\[ A = P \left(1 + \frac{r}{n}\right)^{nt} \]

where:

• \( A \) is the amount of money accumulated after n years, including interest.
• \( P \) is the principal amount (initial investment).
• \( r \) is the annual interest rate (decimal).
• \( n \) is the number of times that interest is compounded per year.
• \( t \) is the time the money is invested for in years.

1. Calculate the future value for the 9.35% interest rate compounded quarterly.
2. Calculate the future value for the 9.45% interest rate compounded semiannually.
3. Compare the two future values to determine which investment yields a greater return.

Using the compound interest formula:

\[ A_1 = P \left(1 + \frac{r_1}{n_1}\right)^{n_1 t} \]

where:

• \( P = 7820 \)
• \( r_1 = 0.0935 \)
• \( n_1 = 4 \)
• \( t = 7 \)

Substituting the values:

\[ A_1 = 7820 \left(1 + \frac{0.0935}{4}\right)^{4 \times 7} = 7820 \left(1 + 0.023375\right)^{28} = 7820 \left(1.023375\right)^{28} \approx 14934.1754 \]

Using the same compound interest formula:

\[ A_2 = P \left(1 + \frac{r_2}{n_2}\right)^{n_2 t} \]

where:

• \( r_2 = 0.0945 \)
• \( n_2 = 2 \)

Substituting the values:

\[ A_2 = 7820 \left(1 + \frac{0.0945}{2}\right)^{2 \times 7} = 7820 \left(1 + 0.04725\right)^{14} = 7820 \left(1.04725\right)^{14} \approx 14924.9168 \]

Now we compare \( A_1 \) and \( A_2 \):

\[ A_1 \approx 14934.1754 \quad \text{and} \quad A_2 \approx 14924.9168 \]

Since \( A_1 > A_2 \), the investment at \( 9.35\% \) compounded quarterly yields a greater return.

Final Answer