Questions: Would you prefer a savings account that paid 7% interest compounded quarterly, 6.8% compounded monthly, 7.2% compounded weekly, or an account that paid 7.5% with annual compounding?

Would you prefer a savings account that paid 7% interest compounded quarterly, 6.8% compounded monthly, 7.2% compounded weekly, or an account that paid 7.5% with annual compounding?

Transcript text: Would you prefer a savings account that paid $7 \%$ interest compounded quarterly, $6.8 \%$ compounded monthly, $7.2 \%$ compounded weekly, or an account that paid $7.5 \%$ with annual compounding?

Solution

Solution Steps

To determine which savings account is preferable, we need to calculate the effective annual rate (EAR) for each option. The EAR can be calculated using the formula: \[ \text{EAR} = \left(1 + \frac{r}{n}\right)^n - 1 \] where $ r $ is the nominal interest rate and $ n $ is the number of compounding periods per year.

Step 1: Calculate Effective Annual Rate (EAR) for Quarterly Compounding

For the account with $ 7\% $ interest compounded quarterly: \[ \text{EAR}_{\text{quarterly}} = \left(1 + \frac{0.07}{4}\right)^4 - 1 \approx 0.0719 \]

Step 2: Calculate Effective Annual Rate (EAR) for Monthly Compounding

For the account with $ 6.8\% $ interest compounded monthly: \[ \text{EAR}_{\text{monthly}} = \left(1 + \frac{0.068}{12}\right)^{12} - 1 \approx 0.0702 \]

Step 3: Calculate Effective Annual Rate (EAR) for Weekly Compounding

For the account with $ 7.2\% $ interest compounded weekly: \[ \text{EAR}_{\text{weekly}} = \left(1 + \frac{0.072}{52}\right)^{52} - 1 \approx 0.0746 \]

Step 4: Calculate Effective Annual Rate (EAR) for Annual Compounding

For the account with $ 7.5\% $ interest compounded annually: \[ \text{EAR}_{\text{annually}} = \left(1 + \frac{0.075}{1}\right)^{1} - 1 \approx 0.0750 \]

Step 5: Compare the Effective Annual Rates

The calculated EARs are: