To solve the problem, we need to understand two key financial ratios: Return on Assets (ROA) and Return on Equity (ROE).
1. Return on Assets (ROA): This ratio measures the efficiency of a company in using its assets to generate profit. It is given by the formula:
[tex]\[
\text{ROA} = \frac{\text{Net Income}}{\text{Total Assets}}
\][/tex]
2. Return on Equity (ROE): This ratio measures the profitability relative to the shareholders' equity and is given by the formula:
[tex]\[
\text{ROE} = \frac{\text{Net Income}}{\text{Equity}}
\][/tex]
Given:
- \( \text{Total Assets} = \$280 \) million
- \( \text{Initial Equity} = \$28 \) million
- \( \text{ROA} = 4\% \)
First, we calculate the initial ROE.
### Step-by-Step Solution:
1. Initial ROE Calculation:
[tex]\[
\text{ROE} = \left( \frac{\text{ROA}}{100} \right) \times \left( \frac{\text{Total Assets}}{\text{Initial Equity}} \right) \times 100
\][/tex]
Plugging in values:
[tex]\[
\text{ROE} = \left( \frac{4}{100} \right) \times \left( \frac{280}{28} \right) \times 100
\][/tex]
Simplifying,
[tex]\[
\text{ROE} = 0.04 \times 10 \times 100 = 40\%
\][/tex]
The initial ROE is \( 40.00\% \).
2. New ROE Calculation When Equity Declines:
Now, suppose the equity capital declines to \$14 million, while the assets and ROA remain unchanged.
- \( \text{New Equity} = \$14 \) million
The new ROE is calculated similarly:
[tex]\[
\text{New ROE} = \left( \frac{\text{ROA}}{100} \right) \times \left( \frac{\text{Total Assets}}{\text{New Equity}} \right) \times 100
\][/tex]
Plugging in the new values:
[tex]\[
\text{New ROE} = \left( \frac{4}{100} \right) \times \left( \frac{280}{14} \right) \times 100
\][/tex]
Simplifying,
[tex]\[
\text{New ROE} = 0.04 \times 20 \times 100 = 80\%
\][/tex]
The new ROE is \( 80.00\% \).
So, if First National's equity capital declines to \$14 million while its assets and ROA stay the same, the new ROE is [tex]\( 80.00\% \)[/tex].
