Which of the following is the better investment? $$ 2.62 \% $$ compounded monthly OR The investments will earn the same. The $$ 2.62 \% $$ compounded monthly is better. The $$ 2.8 \% $$ compounded annually is better. A comparison cannot be made unless the amount invested is known.

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To determine which investment option is better, we need to compare their **Effective Annual Rates (EAR)**. The EAR allows us to compare different interest compounding periods on an equal footing.

### **1. Calculating EAR for 2.62% Compounded Monthly**

The formula for EAR is:

$$
\text{EAR} = \left(1 + \frac{r}{n}\right)^n - 1
$$

Where:
- $$ r $$ = annual nominal interest rate (as a decimal)
- $$ n $$ = number of compounding periods per year

For **2.62% compounded monthly**:
- $$ r = 0.0262 $$
- $$ n = 12 $$

$$
\text{EAR} = \left(1 + \frac{0.0262}{12}\right)^{12} - 1 \approx \left(1 + 0.0021833\right)^{12} - 1 \approx 1.0265 - 1 = 0.0265 \text{ or } 2.65\%
$$

### **2. Calculating EAR for 2.8% Compounded Annually**

For **2.8% compounded annually**:
- $$ r = 0.028 $$
- $$ n = 1 $$

$$
\text{EAR} = \left(1 + \frac{0.028}{1}\right)^1 - 1 = 1.028 - 1 = 0.028 \text{ or } 2.8\%
$$

### **3. Comparison**

- **2.62% compounded monthly** has an **EAR of approximately 2.65%**.
- **2.8% compounded annually** has an **EAR of 2.8%**.

Since **2.8% compounded annually** results in a higher EAR compared to **2.62% compounded monthly**, it is the better investment option.

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**Answer:** The 2.8 % compounded annually is better.
The 2.8% compounded annually is the better investment.

Which of the following is the better investment? \( 2.62 \% \) compounded monthly OR The investments will earn the same. The \( 2.62 \% \) compounded monthly is better. The \( 2.8 \% \) compounded annually is better. A comparison cannot be made unless the amount invested is known.

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