Question

8. Darlene wants to invest $15000 into a savings account. One bank offers 3.9% compounded quarterly and another bank offers 3% compounded monthly.
a. Which should she choose if she wants to maximize her money? Explain.
b. Does the answer to part a depend on how much she invests? Does the answer to part a depend on how many years the money is in the account?

Answer 1

To determine which bank Darlene should choose, we will calculate the future value of her investment for both banks given the interest rates and compounding frequencies.

### a. Future Value Calculation

The future value \( FV \) of an investment can be calculated using the formula:

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

Where:
- \( P \) = principal amount (the initial amount of money)
- \( r \) = annual interest rate (decimal)
- \( n \) = number of times that interest is compounded per year
- \( t \) = number of years the money is invested

**Bank 1: 3.9% compounded quarterly**

Given:
- \( P = 15000 \)
- \( r = 0.039 \)
- \( n = 4 \) (quarterly)
- Assuming \( t = 1 \) year initially for comparison

\[
FV_1 = 15000 \left(1 + \frac{0.039}{4}\right)^{4 \times 1}
\]
\[
FV_1 = 15000 \left(1 + 0.00975\right)^{4}
\]
\[
FV_1 = 15000 \left(1.00975\right)^{4}
\]
\[
FV_1 \approx 15000 \times 1.039428
\]
\[
FV_1 \approx 15591.42
\]

**Bank 2: 3% compounded monthly**

Given:
- \( P = 15000 \)
- \( r = 0.03 \)
- \( n = 12 \) (monthly)

\[
FV_2 = 15000 \left(1 + \frac{0.03}{12}\right)^{12 \times 1}
\]
\[
FV_2 = 15000 \left(1 + 0.0025\right)^{12}
\]
\[
FV_2 = 15000 \left(1.0025\right)^{12}
\]
\[
FV_2 \approx 15000 \times 1.030439
\]
\[
FV_2 \approx 15456.59
\]

### Comparison of Future Values
- **Bank 1 Future Value**: \( FV_1 \approx 15591.42 \)
- **Bank 2 Future Value**: \( FV_2 \approx 15456.59 \)

Since \( 15591.42 > 15456.59 \), Darlene should choose the bank offering **3.9% compounded quarterly** to maximize her money.

### b. Does the answer depend on the investment amount or time?

1. **Investment Amount**: The answer does not depend on the amount Darlene invests. The calculations for future value are proportional to the principal investment amount; if she invests more or less, the relative advantage of the bank with the higher interest rate and better compounding will still hold.

2. **Time in the Account**: The answer does depend on how long the money is in the account. Generally, as the number of years \( t \) increases, the effects of compounding become more significant. However, with both banks maintaining the same interest rate relationships (more frequent compounding vs less), the bank with the higher rate (3.9% compounded quarterly) would still perform better over any duration of time, although the extent of the difference could vary based on the length of time. Generally, longer durations enhance the advantages of compounding.

In conclusion, while the choice of bank is independent of how much Darlene invests, it is dependent on how long the money remains in the account, specifically in favor of the bank with the higher percent rate compounded more frequently.