Let's solve each part step by step using the **Dividend-Growth Model** formula: \[ V = \frac{D_0(1 + g)}{k - g} \] Where: - \( V \) = Value of the stock - \( D_0 \) = Current dividend - \( k \) = Required rate of return - \( g \) = Dividend growth rate --- ## **a. Value of stock with: \( D_0 = \$3.60 \), \( k = 9\% \), \( g = 4\% \)** ### **Step 1: Write the formula with values** - \( D_0 = 3.60 \) - \( k = 9\% = 0.09 \) - \( g = 4\% = 0.04 \) \[ V = \frac{3.60(1 + 0.04)}{0.09 - 0.04} \] ### **Step 2: Calculate numerator** \[ 3.60 \times (1 + 0.04) = 3.60 \times 1.04 = 3.744 \] ### **Step 3: Calculate denominator** \[ 0.09 - 0.04 = 0.05 \] ### **Step 4: Divide numerator by denominator** \[ V = \frac{3.744}{0.05} = 74.88 \] **Final Answer (a):** \[ \boxed{\$74.88} \] --- ## **b. If dividend increases to \$5.00 (other variables constant)** - \( D_0 = 5.00 \) - \( k = 0.09 \) - \( g = 0.04 \) \[ V = \frac{5.00 \times 1.04}{0.09 - 0.04} \] **Numerator:** \( 5.00 \times 1.04 = 5.20 \) **Denominator:** \( 0.05 \) \[ V = \frac{5.20}{0.05} = 104.00 \] **Final Answer (b):** \[ \boxed{\$104.00} \] --- ## **c. If required return declines to 5.5% (\( k = 0.055 \)), other variables constant** - \( D_0 = 3.60 \) - \( k = 0.055 \) - \( g = 0.04 \) \[ V = \frac{3.60 \times 1.04}{0.055 - 0.04} \] **Numerator:** \( 3.60 \times 1.04 = 3.744 \) **Denominator:** \( 0.055 - 0.04 = 0.015 \) \[ V = \frac{3.744}{0.015} = 249.60 \] **Final Answer (c):** \[ \boxed{\$249.60} \] --- ## **d. If growth rate declines to 3% (\( g = 0.03 \)), other variables constant** - \( D_0 = 3.60 \) - \( k = 0.09 \) - \( g = 0.03 \) \[ V = \frac{3.60 \times 1.03}{0.09 - 0.03} \] **Numerator:** \( 3.60 \times 1.03 = 3.708 \) **Denominator:** \( 0.09 - 0.03 = 0.06 \) \[ V = \frac{3.708}{0.06} = 61.80 \] **Final Answer (d):** \[ \boxed{\$61.80} \] --- ## **e. If dividend increases to \$4.20, growth rate declines to 3%, required return remains 9%** - \( D_0 = 4.20 \) - \( k = 0.09 \) - \( g = 0.03 \) \[ V = \frac{4.20 \times 1.03}{0.09 - 0.03} \] **Numerator:** \( 4.20 \times 1.03 = 4.326 \) **Denominator:** \( 0.09 - 0.03 = 0.06 \) \[ V = \frac{4.326}{0.06} = 72.10 \] **Final Answer (e):** \[ \boxed{\$72.10} \] --- ### **Summary Table** | Part | Dividend (\(D_0\)) | Growth (\(g\)) | Return (\(k\)) | Value (\(V\)) | |------|-------------------|---------------|---------------|--------------| | a | \$3.60 | 4% | 9% | \$74.88 | | b | \$5.00 | 4% | 9% | \$104.00 | | c | \$3.60 | 4% | 5.5% | \$249.60 | | d | \$3.60 | 3% | 9% | \$61.80 | | e | \$4.20 | 3% | 9% | \$72.10 | --- **Explanation:** - The formula calculates the present value of an infinite series of dividends growing at a constant rate. - Increasing the dividend or decreasing the required return increases the stock value. - Increasing the growth rate also increases the value, while decreasing it reduces the value. - Each step substitutes the new values as specified by the question. Let me know if you need more explanation on any step!

Let's carefully analyze the problem step-by-step to find the **net new long-term debt** for Raines Umbrella Corporation. --- ### **Given Data:** | Item | Amount | |--------|---------| | Sales | $730,000 | | Cost of Goods Sold (COGS) | $500,000 | | Administrative & Selling Expenses | $125,000 | | Depreciation Expenses | $180,000 | | Interest Expense | $56,000 | | Tax Rate | 21% | | Cash Dividends Paid | $64,000 | | Net Capital Spending | $0 | | Net Working Capital (NWC) | $0 | | No new stock issued | Yes | --- ### **Step 1: Calculate Earnings Before Interest and Taxes (EBIT)** **Formula:** \[ \text{EBIT} = \text{Sales} - \text{COGS} - \text{Admin & Selling Expenses} - \text{Depreciation} \] **Calculation:** \[ EBIT = 730,000 - 500,000 - 125,000 - 180,000 \] \[ EBIT = 730,000 - 805,000 = -75,000 \] **Result:** \[ \boxed{\text{EBIT} = -\$75,000} \] *Note:* EBIT is negative, indicating an operating loss. --- ### **Step 2: Calculate Earnings Before Taxes (EBT)** **Formula:** \[ EBT = EBIT - \text{Interest Expense} \] **Calculation:** \[ EBT = -75,000 - 56,000 = -\$131,000 \] **Result:** \[ \boxed{\text{EBT} = -\$131,000} \] --- ### **Step 3: Calculate Net Income** **Tax calculation:** Since EBT is negative, the company has a tax loss, and taxes paid are zero (no tax on negative income). \[ \text{Taxes} = EBT \times \text{Tax Rate} = -\$131,000 \times 0.21 \] But taxes are only paid on positive EBT, and with a negative EBT, taxes are zero. **Net Income:** \[ \text{Net Income} = EBT - \text{Taxes} \] Since taxes on negative EBT are zero, \[ \text{Net Income} = -\$131,000 \] ### **Note:** - Tax savings due to losses are not considered here explicitly, as the problem states to ignore tax loss carryforwards. --- ### **Step 4: Calculate the amount of retained earnings** **Retained earnings** are typically the net income minus dividends paid. \[ \text{Retained Earnings} = \text{Net Income} - \text{Dividends} \] **Calculation:** \[ \text{Retained Earnings} = -\$131,000 - \$64,000 = -\$195,000 \] *This is a net reduction in retained earnings.* --- ### **Step 5: Determine the change in net debt** Since **net capital spending** and **NWC** are both zero, and **no new stock was issued**, the change in the firm's **net debt** is driven by the difference between **funds needed** and **funds generated**. - The **funds needed** for the firm (cash flow to support dividends, debt repayment, etc.) is reflected by the net change in debt. - The firm's **cash flow from operations** is often approximated by **net income plus depreciation** (since depreciation is a non-cash expense). --- ### **Step 6: Calculate the **cash flow from operations** (CFO)** **Formula:** \[ CFO = \text{Net Income} + \text{Depreciation} \] **Calculation:** \[ CFO = -\$131,000 + \$180,000 = \$49,000 \] --- ### **Step 7: Connect cash flows to changes in debt** - The **cash paid in dividends** was \$64,000. - The **cash flow from operations** is \$49,000. - Since **net capital spending** and **NWC** are zero, the change in debt (net new debt) must account for the difference between dividends paid and cash flow, considering the overall cash flow. **Note:** - If the company paid out \$64,000 in dividends, and its cash flow from operations is only \$49,000, it must have financed the difference by **borrowing**. **Calculation:** \[ \text{Net New Long-Term Debt} = \text{Dividends} - \text{Cash Flow from Operations} \] \[ = 64,000 - 49,000 = 15,000 \] --- ### **Final Answer:** \[ \boxed{\text{Net new long-term debt} = \mathbf{\$15,000}} \] This means Raines Umbrella Corporation borrowed **\$15,000** in new long-term debt during 2024 to fund dividends after accounting for its cash flow. --- ### **Summary:** - The company had a net operating loss but still paid dividends. - The shortfall between cash flow and dividends was financed by new debt. - **Net new long-term debt = \$15,000**. Let me know if you'd like further clarification!