Public Offerings, Underwriting, IPOs, Rights Offerings, and Dilution

Public Offerings and Underwriting

Public Offering: The process where a company offers its shares to the general public for the first time or issues new shares if it's already publicly traded.
Seasoned Equity Offering: A new issue by a company that already has shares trading in the market. They issue additional shares.
Underwriters: Financial institutions (often akin to banks) that help companies raise capital by selling newly issued shares in the market. Companies cannot directly sell shares to the public; they give shares to underwriters, who then sell them and take a small commission.
Syndicate: A group of underwriters working together to sell a large issue of securities. This term is important for multiple-choice questions (MCQs).
Spread: The difference between the underwriter's buying price (from the company) and the consumer's buying price (in the market). For example, if an underwriter buys shares for $10$ and sells them for $12$ , their spread is $2$ .

General Problem Solving Strategy (Exam Prep)

The instructor emphasizes the importance of understanding the steps to solve a problem rather than just remembering how to do a specific part of a broken-down question. Exams will present problems as paragraphs, requiring students to identify and execute all necessary steps independently.

Initial Public Offering (IPO) Valuation - The Winner's Curse

Scenario: Company A and Company B announce IPOs at $25 per share. One is undervalued by $6, and the other is overvalued by $4. An investor plans to buy $500$ shares of each issue without knowing which is which.
Concept: This scenario illustrates the "Winner's Curse," which is the tendency for the winning bid in an auction to exceed the intrinsic or true value of an item.
Profit/Loss Calculation: The investor gains from the undervalued stock and loses from the overvalued stock.
- Gain from Undervalued: 500 \text{ shares} \times \6/\text{share} = \ $3,000$
- Loss from Overvalued: 500 \text{ shares} \times \4/\text{share} = \ $2,000$
- Net Profit/Loss: $\$3,000 - \$2,000 = \$1,000 \text{ profit}$
Rationing of Underpriced Issues: If an issue is underpriced (undervalued), it is often rationed. This means only a portion of an order might be filled.
- Example: If only half the order for an undervalued issue is filled, the investor gets $250$ shares of the undervalued stock but still $500$ of the overvalued stock.
  - Gain (undervalued): 250 \text{ shares} \times \6/\text{share} = \ $1,500$
  - Loss (overvalued): 500 \text{ shares} \times \4/\text{share} = \ $2,000$
  - Net Profit/Loss: $\$1,500 - \$2,000 = -\$500 \text{ (net loss)}$

Calculating Shares to Be Sold for Capital Raising

Problem: A company needs to raise $15,000,000 via a general cash offering. The offer price is $20 per share, and underwriters charge a $10\%$ spread.
Underwriter Spread Calculation: The company receives the offer price minus the underwriter's percentage spread.
- Offer Price: $\$20$
- Underwriter Spread: $10\% \text{ of } \$20 = \$2$
- Net Price per Share to Company: $\$20 - \$2 = \$18$
Formula for Number of Shares to be Sold: This is a crucial formula not typically provided on formula sheets but used frequently.
- $\text{Number of Shares to be Sold} = \frac{\text{Capital to be Raised}}{\text{Net Price per Share to Company}}$
- $\text{Number of Shares} = \frac{\$15,000,000}{\$18} \approx 833,333.33 \text{ shares}$
Including Other Fees/Expenses: If there are additional administrative expenses (e.g., $500,000), these must be added to the capital needed.
- Total Capital Needed: $\$15,000,000 + \$500,000 = \$15,500,000$
- New Number of Shares: $\frac{\$15,500,000}{\$18} \approx 861,111.11 \text{ shares}$

Rights Offerings (Most Important Chapter Part)

Definition: An issue of common stock offered exclusively to existing shareholders, typically at a discounted price, for a limited time, before it is offered to the general public.
Shareholder Options: Existing shareholders can either:
1. Exercise their right: Buy the new shares at the discounted subscription price.
2. Sell their rights: Sell their rights to buy shares to another investor.
Example Problem Setup: A company has $200,000$ shares outstanding at $35 each and proposes a rights offering of $25,000$ new shares at $30 per share.
- Old Market Value (of the company):
  - $\text{Old Shares Outstanding} \times \text{Old Price} = 200,000 \times \$35 = \$7,000,000$
- New Market Value (of the company) / Cum-Rights Value: This includes the value of existing shares plus the value of the new shares being offered.
  - \text{New Market Value} = (200,000 \times \35) + (25,000 \times \ $30) = \$7,000,000 + \$750,000 = \$7,750,000$
Number of Rights to Buy One New Share ( $n$ ):
- This formula indicates how many existing shares (and thus rights) are needed to purchase one new share.
- $n = \frac{\text{Old Shares Outstanding}}{\text{New Shares Offered}}$
- $n = \frac{200,000}{25,000} = 8 \text{ rights per new share}$
X-Rights Date & X-Rights Stock Price ( $M_E$ ):
- Rights-on Stock Price: The stock price when the rights option is available.
- X-Rights Date: The date when the offer ends, and the value of the right typically declines.
- X-Rights Stock Price ( $M_E$ ): The stock price after the rights offering.
- $ME = \frac{n \times M0 + S}{n+1}$
 - Where: $n$ = number of rights; $M_0$ = old price per share (rights-on price); $S$ = subscription price for new shares.
 - M_E = \frac{8 \times \$35 + \$30}{8+1} = \frac{\$280 + \$30}{9} = \frac{\310}{9} \approx \ $34.44$
- Alternative Method for $M_E$ : Divide the new market value by the total number of shares after the offering.
 - $\text{Total Shares} = 200,000 + 25,000 = 225,000$
 - M_E = \frac{\7,750,000}{225,000} \approx \ $34.44$
Value of a Right ( $R_E$ ):
- This represents the independent market value of owning the right to buy a new share.
- Formula 1 (using $M_E$ ):
 - $RE = \frac{ME - S}{n}$
 - R_E = \frac{\$34.44 - \$30}{8} = \frac{\4.44}{8} \approx \ $0.55$
- Formula 2 (using $M_0$ ):
 - $RE = \frac{M0 - S}{n+1}$
 - R_E = \frac{\$35 - \$30}{8+1} = \frac{\5}{9} \approx \ $0.55$
- Formula 3 (difference in stock prices):
 - $RE = M0 - M_E$
 - $R_E = \$35 - \$34.44 = \$0.56$
- The presenter notes that minor discrepancies (like $0.55$ vs $0.56$ ) might occur due to rounding in intermediate steps.
Minimum and Maximum Subscription Prices (S):
- Maximum: The subscription price cannot be higher than the current market price ( $M_0$ ). If it were higher, no one would buy the new shares when existing shares are cheaper.
- Minimum: The minimum subscription price can technically be any value below the current market price, even close to zero.
Example: Shareholder Impact (Dilution for a single shareholder):
- A shareholder owns $5,000$ shares, and the current stock price ( $M_0$ ) is $50.
- Initial Portfolio Value: 5,000 \text{ shares} \times \50/\text{share} = \ $250,000$
- $n = 8$ (rights per new share), $S = \$40$ , $ME = \$48.89$ , $RE = \$1.11$
- Scenario 1: Shareholder does not exercise rights (sells them)
 - Portfolio value from shares: 5,000 \text{ shares} \times \48.89/\text{share} = \ $244,450$
 - Value from selling rights: 5,000 \text{ rights} \times \1.11/\text{right} = \ $5,550$
 - Total Portfolio Value: $\$244,450 + \$5,550 = \$250,000$
- Scenario 2: Shareholder does exercise rights
 - Number of new shares purchasable: $5,000 \text{ rights} \div 8 \text{ rights/share} = 625 \text{ new shares}$
 - Cost of new shares: 625 \text{ shares} \times \40/\text{share} = \ $25,000$
 - Portfolio value (existing shares): 5,000 \text{ shares} \times \48.89/\text{share} = \ $244,450$
 - Portfolio value (new shares): 625 \text{ shares} \times \48.89/\text{share} = \ $30,556.25$
 - Total Portfolio Value (after investing additional cash): $\$244,450 + \$30,556.25 = \$275,006.25$
 - Initial Value plus Investment: $\$250,000 + \$25,000 = \$275,000$
- Conclusion: Whether a shareholder exercises or sells their rights, their wealth is theoretically unchanged, assuming the market prices are efficient.
Working Backward in Rights Offering Problems: Sometimes, the question asks for old shares outstanding given final prices and the capital to be raised.
- Strategy: Use available information (like $M0, ME, S$ ) to calculate $RE$ . Then use $RE$ to find $n$ . Finally, use $n$ and newly issued shares (calculated from capital needed and subscription price) to find old shares.
Underwriter Spread in Rights Offerings: When calculating the number of new shares to be issued to raise a specific amount of capital, explicitly account for underwriter spreads (e.g., if the company needs $6,000,000 and the subscription price is $15 with a $4\%$ spread, the effective price received per share is \15 \times (1-0.04) = \ $14.4$ ). However, when calculating $ME$ or $RE$ using market prices and subscription price, the underwriter spread is typically not directly included in those formulas, as it's an expense to the company, not a component of the stock's market value or the subscription offer price to shareholders.
Arbitrage Opportunity (Incorrectly Priced Stock/Rights):
- Scenario: $n=5$ , $S=\$50$ , $M0=\$75$ . After rights offering, actual market price ( $ME$ ) is $70, and rights sell for $2 each ( $R_E=\$2$ ).
- Check Correct Pricing: Calculate theoretical $ME$ and $RE$
 - Theoretical M_E = \frac{5 \times \$75 + \$50}{5+1} = \frac{\$375 + \$50}{6} = \frac{\425}{6} \approx \ $70.83$
 - Since actual $ME=\$70$ (lower than theoretical) and actual $RE=\$2$ (lower than theoretical $RE = M0 - M_E = \$75 - \$70.83 = \$4.17$ ), prices are incorrectly priced.
- Creating Immediate Profit (Arbitrage Strategy):
 - Buy $n$ rights to buy one share: Buy $5$ rights for \2 \text{ each} = \ $10$
 - Buy one new share at subscription price: Buy $1 \text{ share for } \$50$
 - Total Cost: $\$10 + \$50 = \$60$
 - Sell one existing share (after rights offering) at the actual current market price: Sell $1 \text{ share for } \$70$
 - Profit: $\$70 (selling price) - \$60 (total cost) = \$10$
- This strategy exploits the mispricing to guarantee a profit.

Dilution

Definition: A reduction in earnings per share (EPS) or book value per share (BVPS) or market price per share, caused by the issuance of new shares without a corresponding increase in true value per share.
Key Information for Dilution Analysis: Often involves: shares outstanding, net income, investment cost, increase in net income from investment, stock price ( $M_0$ ), book value per share (BVPS), and constant price-to-earnings (PE) ratio.
Book Value vs. Market Price: Book value is the accounting value per share (e.g., $20$ as recorded), while market price is the current trading price based on market sentiment (e.g., $10$ due to market fluctuations).
- Market to Book Ratio: $\frac{\text{Market Price}}{\text{Book Value}} = \frac{\$10}{\$20} = 0.5$
New Shares Outstanding Calculation (for investment):
- To find the number of new shares to issue for an investment, use the current market price, not the book value.
- Investment Cost: $\$3,000,000$
- Market Price per Share: $\$10$
- New Shares Issued: $\frac{\$3,000,000}{\$10} = 300,000 \text{ shares}$
- Total Shares Outstanding (after investment): $2,000,000 \text{ (old)} + 300,000 \text{ (new)} = 2,300,000 \text{ shares}$
Accounting Dilution (based on Book Value):
- Book Value Per Share (BVPS) after Investment: (Old BV + New Capital) / Total Shares Outstanding
  - Old Book Value of Company: 2,000,000 \text{ shares} \times \20/\text{share} = \ $40,000,000$
  - New Capital (Investment Cost): $\$3,000,000$
  - Total Book Value: $\$40,000,000 + \$3,000,000 = \$43,000,000$
  - New BVPS: \frac{\43,000,000}{2,300,000 \text{ shares}} \approx \ $18.70$
- Conclusion for Accounting Dilution: If the new BVPS ( $\$18.70$ ) is less than the old BVPS ( $\$20$ ), accounting dilution takes place.
Market Dilution (based on Market Value / PE Ratio):
- Earnings Per Share (EPS) Before Investment:
  - Net Income: $\$4,000,000$
  - Old Shares: $2,000,000$
  - Old EPS: $\frac{\$4,000,000}{2,000,000} = \$2.00$
- Price-to-Earnings (PE) Ratio (assumed constant):
  - Old Market Price: $\$10$
  - Old EPS: $\$2.00$
  - PE Ratio: $\frac{\$10}{\$2.00} = 5$
- EPS After Investment:
  - New Net Income: \4,000,000 \text{ (old)} + \ $900,000 \text{ (increase)} = \$4,900,000$
  - Total Shares: $2,300,000$
  - New EPS: \frac{\4,900,000}{2,300,000 \text{ shares}} \approx \ $2.13$
- New Market Price Per Share (using constant PE ratio):
  - PE Ratio (5) = New Market Price / New EPS ( $\$2.13$ )
  - New Market Price: $5 \times \$2.13 = \$10.65$
- Conclusion for Market Dilution: If the new market price ( $\$10.65$ ) is greater than the old market price ( $\$10$ ), market dilution does not occur.
- Overall Conclusion: According to accounting books, the company dilutes. However, according to market perception (using the constant PE ratio principle), the company will not dilute. Thus, careful consideration of both is needed.

General Advice for the Course

Practice is Paramount: This course (especially Chapter 15) requires extensive practice to understand the steps and nuances of each problem. Allocate at least $4-5$ hours per week for practice.
Start with Slide Questions: Focus on mastering all problems discussed in class slides before moving to textbook questions.
Midterm Focus: Chapters $15$ and $16$ are heavily weighted and crucial for the midterm. Chapters $17$ and $18$ are generally easier.
Assignments: Complete assignments on time as they are relatively easy $15\%$ of the grade.
Final Exam: The final exam will be cumulative, requiring knowledge from all chapters.