# Equity Financing and ROE Analysis This structured solution follows the steps for evaluating the impact of a new equity-financed investment on a company's share price, EPS, and ROE. --- ## Given Data | Item | Value | |--------------|------------| | Stock price (\(P_\)) | $40 | | Number of shares | ,000 | | Total assets | $9,800,000 | | Total liabilities | $4,700,000 | | Net income | $420,000 | | Cost of new investment| $640,000 | | Financing | New equity issue | | Required ROE on investment | 35% | --- ## Step 1: Compute Current Earnings Per Share (EPS) \[ \text{EPS} = \frac{\text{Net income}}{\text{Number of shares}} = \frac{420{,}000}{30{,}000} = \$14.00 \] --- ## Step 2: Compute Current Price-to-Earnings (P/E) Ratio \[ \text{P/E} = \frac{P_}{\text{EPS}} = \frac{40}{14} = 2.8571 \] --- ## Step 3: Compute Expected Earnings from New Investment \[ \text{New income} = \text{Required ROE} \times \text{Cost of investment} = .35 \times 640{,}000 = \$224{,}000 \] --- ## Step 4: Determine Number of New Shares Issued \[ \text{New shares} = \frac{\text{Cost of investment}}{\text{Stock price}} = \frac{640{,}000}{40} = 16{,}000 \] --- ## Step 5: Compute New Total Earnings and Shares - **Total earnings:** \(420,000 + 224,000 = 644,000\) - **Total shares:** \(30,000 + 16,000 = 46,000\) --- ## Step 6: Compute New EPS \[ \text{New EPS} = \frac{644,000}{46,000} = \$14.00 \] --- ## Step 7: Compute New Stock Price (If P/E Remains Constant) \[ \text{New price} = \text{P/E} \times \text{New EPS} = 2.8571 \times 14 = \$40.00 \] - **Result:** The stock price remains unchanged. - **Interpretation:** The project’s ROE equals the required return, so NPV = . --- ## Step 8: Targeting a Higher Stock Price (\$50/share) ### Required EPS for \$50/share (with same P/E): \[ \text{New EPS}_{\text{target}} = \frac{50}{2.8571} = 17.50 \] ### Required Total Earnings: \[ \text{Total earnings required} = 17.50 \times 46,000 = 805,000 \] \[ \text{Incremental project earnings needed} = 805,000 - 420,000 = 385,000 \] \[ \text{Required ROE} = \frac{385,000}{640,000} = .6016 = 60.16\% \] - **So, to reach \$50/share, project must earn 60.16% ROE.** --- ## Step 9: Dilution Analysis - **Accounting dilution:** Occurs (more shares issued). - **Market value dilution:** Does **not** occur (share price unchanged at $40). --- ## Final Answers Summary | Item | Answer | |----------------------------------------|---------------| | ROE on investment (for zero NPV) | 35.00% | | New stock price | $40.00 | | NPV of investment | $ | | ROE required for $50/share | 60.16% | | Accounting dilution | Occurs | | Market value dilution | Does not occur| --- **Note:** - If the project's ROE equals the required return (implied by current P/E), there is no value creation or destruction for existing shareholders. - Only projects with ROE above this threshold increase share price (positive NPV). ---  *Stock price versus project ROE; price only rises if ROE > required return*