Raising Capital Problems

Solution

a. What is the new market value of the company?

The new market value of the company, or the firm's total value after the rights offering, is the initial market value plus the funds raised from the new issue.

• Initial Market Value = 435,000 shares $\times$ \$71/share = \$30,885,000
• Funds Raised = 50,000 new shares $\times$ \$64/share = \$3,200,000
• New Market Value = \$30,885,000 + \$3,200,000 = \$34,085,000

b. How many rights are associated with one of the new shares?

This asks how many old shares a person must own to buy one new share. This is the definition of the number of rights needed to buy a new share.

• Number of Rights per Share = (Number of Old Shares) / (Number of New Shares) = 435,000 / 50,000 = 8.7 rights/share

c. What is the ex-rights price?

The ex-rights price is the new share price after the rights offering, calculated by dividing the new market value by the total number of shares outstanding after the offering.

• Total New Shares Outstanding = 435,000 (old) + 50,000 (new) = 485,000 shares
• Ex-rights Price = (New Market Value) / (Total New Shares) = \$34,085,000 / 485,000 = \$70.28

d. What is the value of a right?

The value of a right is the difference between the stock's price before the offering (rights-on price) and its price after the offering (ex-rights price).

• Value of a Right = Rights-on Price - Ex-rights Price = \$71.00 - \$70.28 = \$0.72

e. Why might a company have a rights offering rather than a general cash offer?

A rights offering is generally cheaper because it avoids the underwriting fees associated with a general cash offer. It also allows existing shareholders to maintain their proportionate ownership, avoiding dilution of control.

Solution

a. What is the maximum possible subscription price? What is the minimum?

The subscription price must be below the current market price for the rights to have value and for shareholders to have an incentive to exercise them. The maximum possible price is therefore just under the current market price, effectively \$48. The minimum is anything greater than zero, but typically a low, round number. A practical minimum is often a nominal value like \$1.

• Maximum Subscription Price: \$48
• Minimum Subscription Price: A nominal value greater than \$0 (e.g., \$1)

b. If the subscription price is set at \$41 per share, how many shares must be sold? How many rights will it take to buy one share?

We use the formula from the theoretical foundation:

• New Shares Sold = (Funds to be Raised) / (Subscription Price) = \$25,000,000 / \$41 = 609,756 shares
• Rights per Share = (Old Shares Outstanding) / (New Shares Sold) = 2,700,000 / 609,756 $\approx$ 4.43 rights per share

c. What is the ex-rights price? What is the value of a right?

We first find the new total market value and then divide by the new total shares outstanding to find the ex-rights price.

• Initial Market Value = 2,700,000 shares $\times$ \$48/share = \$129,600,000
• New Market Value = \$129,600,000 + \$25,000,000 = \$154,600,000
• Total New Shares = 2,700,000 + 609,756 = 3,309,756 shares
• Ex-rights Price = \$154,600,000 / 3,309,756 $\approx$ \$46.71
• Value of a Right = \$48.00 - \$46.71 = \$1.29

d. Show how a shareholder with 1,000 shares before the offering and no desire (or money) to buy additional shares is not harmed by the rights offer.

A shareholder's wealth is unaffected. They can sell their rights to get the cash value of the price drop. We can show this with a simple calculation:

• Initial Holding Value = 1,000 shares $\times$ \$48/share = \$48,000
• Shares Held After Selling Rights = 1,000 shares (at the new ex-rights price) = 1,000 $\times$ \$46.71 = \$46,710
• Cash from Selling Rights = 1,000 rights $\times$ \$1.29/right = \$1,290
• Total Wealth = \$46,710 + \$1,290 = \$48,000

The total wealth remains the same, proving the shareholder is not harmed.

Solution

We need to work backward from the given prices to find the number of shares. The key is to find the number of new shares that will be issued, and then use the formula for the ex-rights price.

• Value of a Right = Rights-on Price ($P_{RO}$) - Ex-rights Price ($P_X$) = \$56.00 - \$54.30 = \$1.70
• Number of Rights per Share (N) = (Rights-on Price - Subscription Price) / Value of a Right = (\$56 - \$41) / \$1.70 = 15 / 1.70 $\approx$ 8.824 rights/share
• New Shares to be Sold = (Funds to be Raised) / (Subscription Price) = \$17,500,000 / \$41 = 426,829 shares
• Existing Shares = (New Shares Sold) $\times$ (Rights per Share) = 426,829 $\times$ 8.824 = 3,767,114 shares

Solution

a. What would your profit be if you got all 1,000 shares of each?

• Profit on Woods (underpriced) = 1,000 shares $\times$ \$12/share = \$12,000
• Loss on Koepka (overpriced) = 1,000 shares $\times$ \$5/share = -\$5,000
• Total Profit = \$12,000 - \$5,000 = \$7,000

b. What profit do you actually expect?

Due to rationing (a common result of underpricing), you only get half of your order for the underpriced stock.

• Actual Shares of Woods = 1,000 / 2 = 500 shares
• Actual Profit on Woods = 500 shares $\times$ \$12/share = \$6,000
• Actual Shares of Koepka = 1,000 shares (no rationing for overpriced stock)
• Actual Loss on Koepka = 1,000 shares $\times$ \$5/share = -\$5,000
• Total Actual Profit = \$6,000 - \$5,000 = \$1,000

Solution

The flotation costs (the spread) are paid from the gross proceeds of the offering. The net proceeds must equal the funds needed.

• Net Proceeds = Funds Needed = \$75,000,000
• Gross Proceeds = Net Proceeds / (1 - Spread) = \$75,000,000 / (1 - 0.07) = \$75,000,000 / 0.93 = \$80,645,161.29
• Number of Shares to be Sold = Gross Proceeds / Offer Price = \$80,645,161.29 / \$23 = 3,506,311.36 shares

Since you can't sell a fraction of a share, the company would need to sell 3,506,312 shares to raise the necessary funds.

Solution

This problem is an extension of Problem 5, adding other direct expenses. The total amount the company needs to raise from the offering must cover both the funds for the project and the fixed expenses.

• Total Amount Needed from Offering = Funds Needed + Other Direct Expenses = \$75,000,000 + \$1,900,000 = \$76,900,000
• Gross Proceeds = Total Amount Needed / (1 - Spread) = \$76,900,000 / 0.93 = \$82,688,172.04
• Number of Shares to be Sold = Gross Proceeds / Offer Price = \$82,688,172.04 / \$23 = 3,595,137.91 shares

The company would need to sell 3,595,138 shares to cover all costs and raise the required funds.

Solution

Flotation costs include both direct and indirect expenses. The funds raised are the net proceeds received by the company.

• Funds Raised (Net Proceeds) = \$25.11/share $\times$ 30,000,000 shares = \$753,300,000
• Gross Spread (Direct Cost to the firm) = (\$27.00 - \$25.11) $\times$ 30,000,000 = \$1.89 $\times$ 30,000,000 = \$56,700,000
• Other Direct Costs = \$1,400,000
• Indirect Cost (Underpricing) = (\$32.49 - \$27.00) $\times$ 30,000,000 = \$5.49 $\times$ 30,000,000 = \$164,700,000
• Total Flotation Costs = Gross Spread + Other Direct Costs + Indirect Costs + Indirect Expenses = \$56,700,000 + \$1,400,000 + \$164,700,000 + \$375,000 = \$223,175,000
• Flotation Cost as a Percentage = (Total Flotation Costs) / (Funds Raised + Total Flotation Costs) = \$223,175,000 / (\$753,300,000 + \$223,175,000) = \$223,175,000 / \$976,475,000 = 22.86\%

Alternatively, the flotation cost as a percentage of the funds raised by the company would be:

Total Flotation Costs / Funds Raised = \$223,175,000 / \$753,300,000 = 29.62%.

Solution

This problem relates to the concept of market value dilution. The price of the stock will drop only if the Net Present Value (NPV) of the new project is negative. The offering price itself does not cause dilution. We assume in this problem that the NPV of the project is zero since no information about a new project is provided. If the NPV is zero, the stock price should not change, regardless of the offering price.

In a rights offering (which this effectively is, as the existing shareholders' wealth is considered), the ex-rights price would be calculated as:

\[ P_X = \frac{(P_{RO} \times N_{old}) + (P_S \times N_{new})}{N_{old} + N_{new}} \]

where $P_{RO}$ is the rights-on price (\$81), $N_{old}$ is the number of old shares (250,000), $N_{new}$ is the number of new shares (40,000), and $P_S$ is the subscription price (or offering price).

• Case 1: Offering Price = \$81

  This is a cash offer at the current market price. The new price should remain at \$81, as the new funds raised (\$81 $\times$ 40,000 = \$3,240,000) are assumed to be used for a project with a zero NPV, which adds exactly that much value to the firm. The new firm value is \$20,250,000 + \$3,240,000 = \$23,490,000. The new price per share is \$23,490,000 / 290,000 = \$81.

• Case 2: Offering Price = \$76

  Funds Raised = \$76 $\times$ 40,000 = \$3,040,000. New Firm Value = \$20,250,000 + \$3,040,000 = \$23,290,000. New Price per Share = \$23,290,000 / 290,000 = \$80.31. The stock price falls slightly due to the value "given away" by selling at a discount. This is a form of dilution.

• Case 3: Offering Price = \$68

  Funds Raised = \$68 $\times$ 40,000 = \$2,720,000. New Firm Value = \$20,250,000 + \$2,720,000 = \$22,970,000. New Price per Share = \$22,970,000 / 290,000 = \$79.21. The price falls even more.

Solution

a. Assuming a constant price-earnings ratio, what will the effect be of issuing new equity to finance the investment?

• Price-Earnings Ratio (P/E) = Price / EPS = \$64 / (\$12.2M / 5M) = \$64 / \$2.44 = 26.23
• New Total Net Income = \$12.2M + \$0.775M = \$12.975M
• New Shares Outstanding = 5M shares + (\$28M / \$64) = 5M + 0.4375M = 5.4375M shares
• New Earnings Per Share (EPS) = \$12.975M / 5.4375M = \$2.386
• New Stock Price = New EPS $\times$ P/E = \$2.386 $\times$ 26.23 = \$62.58
• Initial Book Value = 5M shares $\times$ \$19/share = \$95M
• New Total Book Value = \$95M + \$28M = \$123M
• New Book Value per Share = \$123M / 5.4375M = \$22.62
• New Market-to-Book Ratio = \$62.58 / \$22.62 = 2.76

What is going on here?

The market value per share falls from \$64 to \$62.58, which represents a true dilution of value. This occurs because the Net Present Value (NPV) of the new project is negative. The new project costs \$28M but only increases the firm's market value by the change in stock price multiplied by the new shares outstanding: $(\$62.58 - \$64) \times 5.4375M = -\$7.72M$. However, the value added is the total new value minus the old value and the funds raised: $(5.4375M \times \$62.58) - (5M \times \$64) - \$28M = \$340.35M - \$320M - \$28M = -\$7.65M$. The project's NPV is negative, causing the stock price to drop. The book value per share rises, which is a classic sign of accounting dilution, but it is not what matters to investors.

b. What would the new net income for the company have to be for the stock price to remain unchanged?

• For the price to remain unchanged at \$64, the new EPS must also be unchanged: New EPS = \$64 / 26.23 = \$2.44
• Required New Total Net Income = New EPS $\times$ New Shares Outstanding = \$2.44 $\times$ 5.4375M = \$13.268M
• Required Increase in Net Income = \$13.268M - \$12.2M = \$1.068M

Solution

First, let's find the current financial metrics.

• Book Value of Equity = Total Assets - Total Liabilities = \$9,400,000 - \$4,100,000 = \$5,300,000
• Book Value per Share = \$5,300,000 / 64,000 = \$82.81
• Earnings per Share (EPS) = \$980,000 / 64,000 = \$15.31
• Return on Equity (ROE) = Net Income / Book Value of Equity = \$980,000 / \$5,300,000 = 18.49%
• Price-Earnings (P/E) Ratio = Price / EPS = \$75 / \$15.31 = 4.90

Now, let's analyze the new investment. The return on the investment (ROI) is equal to the current ROE (18.49%). The cost of the investment is \$1.5M.

• Increase in Net Income = \$1,500,000 $\times$ 18.49% = \$277,358
• New Total Net Income = \$980,000 + \$277,358 = \$1,257,358
• New Shares Issued = \$1,500,000 / \$75 = 20,000 shares
• New Total Shares Outstanding = 64,000 + 20,000 = 84,000 shares
• New EPS = \$1,257,358 / 84,000 = \$14.97
• New Stock Price (assuming constant P/E) = New EPS $\times$ P/E = \$14.97 $\times$ 4.90 = \$73.35
• New Book Value = \$5,300,000 + \$1,500,000 = \$6,800,000
• New Book Value per Share = \$6,800,000 / 84,000 = \$80.95
• NPV = (New Total Market Value) - (Initial Market Value + Funds Raised) = (\$73.35 $\times$ 84,000) - (\$75 $\times$ 64,000) - \$1,500,000 = \$6,161,400 - \$4,800,000 - \$1,500,000 = -\$138,600

Both EPS and market value per share decrease, indicating true dilution. The book value per share also decreases, which is an accounting dilution. The NPV is negative, confirming that this is a value-destroying project that harms existing shareholders.

Solution

For the stock price to remain unchanged at \$75 (no market value dilution), the EPS must also remain unchanged. We can work backward from there.

• Required New EPS = Initial EPS = \$15.31
• Required New Total Net Income = Required New EPS $\times$ New Shares Outstanding = \$15.31 $\times$ 84,000 = \$1,286,040
• Required Increase in Net Income = \$1,286,040 - \$980,000 = \$306,040
• Required ROE on New Investment = (Required Increase in Net Income) / (Investment Cost) = \$306,040 / \$1,500,000 = 20.40%

The NPV of this investment would be zero since the stock price does not change. There is no true dilution of market value. There is still an accounting dilution of book value per share, but this is irrelevant to the firm's value.

Solution

We know the rights-on price and the ex-rights price, so we can find the value of a right. We can then use the value of a right to find the subscription price.

• Value of a Right = $P_{RO} - P_X$ = \$76 - \$71 = \$5
• Number of Old Shares = 29,000,000
• Total Value of Firm = 29,000,000 $\times$ \$76 = \$2,204,000,000
• New Total Value = \$2,204,000,000 + \$95,000,000 = \$2,299,000,000
• Total New Shares = New Total Value / Ex-rights Price = \$2,299,000,000 / \$71 = 32,380,281.7 shares
• New Shares Issued = Total New Shares - Old Shares = 32,380,281.7 - 29,000,000 = 3,380,281.7 shares
• Subscription Price = Funds Raised / New Shares Issued = \$95,000,000 / 3,380,281.7 = \$28.10

Solution

Part 1: The Ex-rights Price ($P_X$)

The ex-rights price is the total value of the firm after the offering, divided by the total number of shares. The total value of the firm after the offering is the initial value plus the funds raised. The number of shares outstanding after the offering is the initial number of shares plus the new shares issued.

• Initial Firm Value = $P_{RO} \times N_{old}$
• New Shares Issued = (Funds Raised) / $P_S$
• Funds Raised = (New Shares Issued) $\times$ $P_S$
• Number of Rights per Share (N) = $N_{old}$ / $N_{new}$ $\implies$ $N_{old} = N \times N_{new}$

Combining these, the ex-rights price is: \[ P_X = \frac{(P_{RO} \times N_{old}) + (N_{new} \times P_S)}{N_{old} + N_{new}} \] Substitute $N_{old} = N \times N_{new}$ into the equation: \[ P_X = \frac{(P_{RO} \times N \times N_{new}) + (N_{new} \times P_S)}{ (N \times N_{new}) + N_{new}} \] We can cancel out $N_{new}$ from the numerator and denominator: \[ P_X = \frac{(P_{RO} \times N) + P_S}{N+1} \]

Part 2: The Value of a Right

The value of a right is the difference between the rights-on price and the ex-rights price: \[ \text{Value of a Right} = P_{RO} - P_X \] Substitute the expression for $P_X$ from Part 1: \[ \text{Value of a Right} = P_{RO} - \frac{(P_{RO} \times N) + P_S}{N+1} \] Find a common denominator: \[ \text{Value of a Right} = \frac{P_{RO}(N+1) - (P_{RO} \times N) - P_S}{N+1} \] Simplify the numerator: \[ \text{Value of a Right} = \frac{P_{RO}N + P_{RO} - P_{RO}N - P_S}{N+1} \] \[ \text{Value of a Right} = \frac{P_{RO} - P_S}{N+1} \] This derivation proves the second part of the equation.

Solution

The problem is slightly tricky because the funds raised are the net proceeds after the underwriter's spread. The gross proceeds are what is used to calculate the number of new shares.

• Gross Proceeds = Net Proceeds / (1 - Spread) = \$5,100,000 / (1 - 0.06) = \$5,425,531.91
• New Shares Issued = Gross Proceeds / Subscription Price = \$5,425,531.91 / \$30 = 180,851 shares
• Rights per New Share (N) = Old Shares / New Shares = 530,000 / 180,851 = 2.93 rights per new share
• Value of a Right = ($P_{RO} - P_S$) / (N+1) = (\$55 - \$30) / (2.93 + 1) = \$25 / 3.93 = \$6.36
• Money from Selling Rights = Number of Rights Owned $\times$ Value of a Right = 5,000 $\times$ \$6.36 = \$31,800

Solution

First, let's determine the theoretical correct prices on the ex-rights day using the provided information.

• Theoretical Ex-rights Price ($P_X$) = ($P_{RO} \times N + P_S$) / (N+1) = (\$60 \times 4 + \$35) / (4+1) = (\$240 + \$35) / 5 = \$275 / 5 = \$55
• Theoretical Value of a Right = $P_{RO} - P_X$ = \$60 - \$55 = \$5

The stock and rights are NOT correctly priced. The stock is selling for \$53 (undervalued by \$2), and the rights are selling for \$3 (undervalued by \$2).

Arbitrage Transaction:

To make an immediate profit, you must buy the assets that are undervalued and sell the ones that are overvalued. The combined value of the stock and rights must be correct.

• Buy the stock for \$53.
• Buy 4 rights for 4 $\times$ \$3 = \$12.
• Total cost to acquire the shares and rights = \$53 + \$12 = \$65.
• At the same time, you can create a "synthetic" rights-on share. The combined value of one share ex-rights and the number of rights needed to buy one new share must equal the rights-on price: $P_X + N \times V_{right} = P_{RO}$. In this case: \$53 + 4 $\times$ \$3 = \$65. This is consistent with the theoretical rights-on price of \$60 + \$5 = \$65. Since the theoretical rights-on price is \$65, you are paying \$65 for a package that should be worth \$60. Wait, this transaction is not correct. I need to re-think it.