Chapter 16
4) Step 1: Debt-to-Equity Ratio if Reliable Gearing Borrows $200,000
The formula for the debt-to-equity ratio is: Debt-to-Equity Ratio=DebtEquityDebt-to-Equity Ratio=EquityDebt
Under the low-debt plan:
• Debt = $200,000
The equity will change due to the repurchase of shares using the debt proceeds.
Initial equity value: Initial Equity Value=Number of Shares×Share Price=10,000×100=1,000,000Initial Equity Value=Number of Shares×Share Price=10,000×100=1,000,000
After borrowing $200,000 and repurchasing shares: Number of Shares Repurchased=DebtShare Price=200,000100=2,000Number of Shares Repurchased=Share PriceDebt=100200,000=2,000
Remaining number of shares: Remaining Shares=10,000−2,000=8,000Remaining Shares=10,000−2,000=8,000
New equity value (Market value of equity): New Equity Value=8,000×100=800,000New Equity Value=8,000×100=800,000
Therefore, the debt-to-equity ratio is: Debt-to-Equity Ratio=200,000800,000=0.25Debt-to-Equity Ratio=800,000200,000=0.25
Step 2: EPS if EBIT is $110,000 and Reliable Borrows $200,000
Calculate interest expense: Interest Expense=Debt×Interest Rate=200,000×0.10=20,000Interest Expense=Debt×Interest Rate=200,000×0.10=20,000
Earnings after interest (Net Income): Net Income=EBIT−Interest Expense=110,000−20,000=90,000Net Income=EBIT−Interest Expense=110,000−20,000=90,000
EPS calculation: EPS=Net IncomeRemaining Shares=90,0008,000=11.25EPS=Remaining SharesNet Income=8,00090,000=11.25
Step 3: EPS if Reliable Borrows $400,000
Under the high-debt plan:
• Debt = $400,000
The equity will change due to the repurchase of shares using the debt proceeds.
Number of Shares Repurchased=DebtShare Price=400,000100=4,000Number of Shares Repurchased=Share PriceDebt=100400,000=4,000
Remaining number of shares: Remaining Shares=10,000−4,000=6,000Remaining Shares=10,000−4,000=6,000
Calculate interest expense: Interest Expense=Debt×Interest Rate=400,000×0.10=40,000Interest Expense=Debt×Interest Rate=400,000×0.10=40,000
Earnings after interest (Net Income): Net Income=EBIT−Interest Expense=110,000−40,000=70,000Net Income=EBIT−Interest Expense=110,000−40,000=70,000
EPS calculation: EPS=Net IncomeRemaining Shares=70,0006,000=11.67EPS=Remaining SharesNet Income=6,00070,000=11.67
Summary of Results:
- Debt-to-Equity Ratio if Reliable borrows $200,000: 0.25
- EPS if EBIT is $110,000 and Reliable borrows $200,000: $11.25
- EPS if EBIT is $110,000 and Reliable borrows $400,000: $11.67
8) The limerick about Mr. Carruthers’s cows provides an interesting analogy to illustrate Modigliani and Miller’s Proposition 1 in the context of firms’ financing decisions. Let’s break down the analogy and then connect it to MM’s Proposition 1.
Analogy of Mr. Carruthers’s Cows:
• Miraculous Udders: The cows produce both cream and skim milk.
• Cream from one teat, skim milk from others: This suggests that the output (milk) from the cows can be split into different components with varying values (cream being more valuable than skim milk).
Firms’ Financing Decisions:
• All-Equity-Financed Firm: Comparable to the cow’s cream, this is the pure, valuable part without any debt (cream is more valuable).
• Debt-Financed Firm: Comparable to the skim milk, this is a mix of equity and debt, where the value might be perceived as different components (skim milk being less valuable than cream).
MM Proposition 1:
Modigliani and Miller’s Proposition 1 states that in a perfect market (no taxes, bankruptcy costs, agency costs, and asymmetric information), the value of a firm is unaffected by its capital structure. This means the total value of a firm is the same whether it is financed by equity or a mix of debt and equity.
Adapted MM Proposition 1 for Mr. Carruthers’s Cows:
Mr. Carruthers’s cows producing cream and skim milk is akin to a firm’s ability to be financed by either equity or a combination of debt and equity. Suitably adapted, MM’s Proposition 1 would state that the total value of Mr. Carruthers’s cows (total milk produced) is the same regardless of whether the milk is separated into cream and skim milk or sold as a whole. The separation of milk into cream and skim milk (akin to splitting the firm’s value into equity and debt components) does not change the total value of the milk produced by the cows.
Explanation:
• Total Value Consistency: Just as the total value of cream and skim milk combined remains the same as the total value of whole milk, the total value of the firm remains unchanged whether it is financed entirely by equity or by a mix of debt and equity.
• Irrelevance of Financing Mix: In both scenarios (the cows and the firm), the composition of the output or financing does not affect the overall value. For Mr. Carruthers, whether the milk is sold as cream and skim separately or together does not change the value derived from the cows. Similarly, for a firm, the mix of debt and equity does not affect the firm’s total value.
In summary, MM’s Proposition 1 adapted to Mr. Carruthers’s cows would assert that the value of the cows is invariant to how the milk is marketed (cream and skim milk separately or whole milk together). This is analogous to saying that the value of a firm is invariant to its capital structure (all equity vs. debt and equity mix) in a perfect market.
Chapter 17
3) To address the questions, we need to apply Modigliani and Miller’s theory with taxes. According to MM’s Proposition with taxes, the value of a leveraged firm VLVL is greater than the value of an unleveraged firm VUVU by the present value of the tax shield on debt.
Part 1: Value of the Debt-Generated Tax Shield
Given:
• Debt (DD) = $40
• Corporate tax rate (TcTc) = 40%
The value of the tax shield is calculated as: Tax Shield=D×TcTax Shield=D×Tc
Substituting the values: Tax Shield=40×0.40=16Tax Shield=40×0.40=16
So, $16 of the firm’s value is accounted for by the debt-generated tax shield.
Part 2: Impact of Additional Borrowing on Shareholders’ Wealth
If the firm borrows an additional $20 and uses it to repurchase stock, the total debt will increase to $60.
Step 1: Calculate the New Tax Shield
New debt (DnewDnew) = $60
The new tax shield is: New Tax Shield=Dnew×TcNew Tax Shield=Dnew×Tc New Tax Shield=60×0.40=24New Tax Shield=60×0.40=24
Step 2: Increase in the Value of the Tax Shield
The increase in the value of the tax shield due to the additional debt is: ΔTax Shield=New Tax Shield−Old Tax ShieldΔTax Shield=New Tax Shield−Old Tax Shield ΔTax Shield=24−16=8ΔTax Shield=24−16=8
Step 3: Shareholders’ Benefit
The additional tax shield increases the firm’s value by $8. This increase in value is directly beneficial to the shareholders, as the firm uses the borrowed funds to repurchase stock, reducing the number of shares outstanding and increasing the value per share for the remaining shares.
Summary
- The value of the firm’s debt-generated tax shield is $16.
- The shareholders will be $8 better off if the firm borrows an additional $20 and uses it to repurchase stock.
Thus, the increase in the value of the tax shield directly enhances the value of the equity, making the shareholders better off by $8 in total.
11) Costs of Going Bankrupt
The costs of going bankrupt, often referred to as bankruptcy costs, can be categorized into two main types: direct costs and indirect costs. - Direct Costs:
o Legal and Administrative Fees: Expenses related to hiring lawyers, accountants, and other professionals to manage the bankruptcy process.
o Court Fees: Costs associated with the judicial process of bankruptcy.
o Advisory Fees: Payments to financial advisors and consultants who help manage the bankruptcy. - Indirect Costs:
o Operational Disruptions: Loss of customers, suppliers, and key employees due to uncertainty and loss of confidence in the firm.
o Loss of Market Share: Competitors may take advantage of the company’s weakened state to capture its market share.
o Management Distraction: Time and effort of management are diverted from running the business to dealing with the bankruptcy process.
o Reputational Damage: The firm’s reputation may suffer, affecting relationships with stakeholders.
Costs of Financial Distress without Bankruptcy
A company can incur costs of financial distress without going bankrupt, through mechanisms such as: - Increased Borrowing Costs: Lenders may perceive the company as risky and charge higher interest rates, increasing the cost of capital.
- Supplier and Customer Relationships: Suppliers may demand cash on delivery or stop providing credit, and customers may choose more stable competitors.
- Employee Morale and Turnover: Uncertainty about the company’s future can lead to reduced employee morale, increased turnover, and difficulties in hiring talent.
- Investment Opportunities: The company may forego profitable investment opportunities due to lack of financing or the need to preserve cash.
- Operational Inefficiencies: Efforts to cut costs and preserve cash can lead to reduced investment in maintenance and innovation, harming long-term competitiveness.
Conflicts of Interest Between Bondholders and Stockholders
Conflicts of interest between bondholders and stockholders can lead to costs of financial distress through several mechanisms: - Risk Shifting (Asset Substitution): Shareholders may encourage management to take on high-risk projects because they benefit from the upside potential if the projects succeed. However, if these projects fail, bondholders bear the brunt of the losses. This potential behavior can increase the perceived risk for bondholders and raise the cost of debt.
- Underinvestment: When a company is near financial distress, shareholders might be reluctant to invest in positive net present value (NPV) projects if the benefits primarily go to bondholders. This reluctance can result in missed investment opportunities that would have otherwise benefited both stakeholders and the company.
- Dividend Payouts: Shareholders might push for higher dividends or share buybacks when a company is under financial distress. This can deplete the firm’s assets and leave bondholders with fewer resources in the event of default.
- Debt Overhang: When a company is heavily indebted, shareholders might be reluctant to finance new projects because the returns would mainly benefit bondholders by reducing the risk of default. This reluctance can lead to underinvestment in profitable ventures, further exacerbating financial distress.
Summary
• Costs of Going Bankrupt: Include both direct costs (legal, administrative) and indirect costs (operational disruptions, loss of market share, reputational damage).
• Costs of Financial Distress without Bankruptcy: Higher borrowing costs, strained supplier and customer relationships, reduced employee morale, missed investment opportunities, and operational inefficiencies.
• Conflicts of Interest: Risk shifting, underinvestment, dividend payouts, and debt overhang, all of which can lead to suboptimal decisions and increased costs of financial distress.
Understanding these concepts is crucial for managing a company’s financial strategy and mitigating the adverse effects of financial distress.
Chapter 18
5) To solve the problem, we’ll first determine the opportunity cost of capital for an average-risk investment in Whispering Pines Inc. and then calculate the Weighted Average Cost of Capital (WACC) at the new capital structure with a 30% debt-to-value ratio.
Opportunity Cost of Capital for an Average-Risk Investment
Since Whispering Pines Inc. is currently all-equity-financed, the opportunity cost of capital is simply the expected rate of return on the company’s shares. Therefore, the opportunity cost of capital is: re=12%re=12%
WACC Calculation at New Capital Structure
Given:
• Debt-to-value ratio (D/V) = 0.30
• Equity-to-value ratio (E/V) = 1 – D/V = 0.70
• Cost of equity (r_e) = 12%
• Borrowing rate (r_d) = 7.5%
• Corporate tax rate (T_c) = 21%
The formula for WACC is: WACC=(EV×re)+(DV×rd×(1−Tc))WACC=(VE×re)+(VD×rd×(1−Tc))
Substituting the values into the formula: WACC=(0.70×12%)+(0.30×7.5%×(1−0.21))WACC=(0.70×12%)+(0.30×7.5%×(1−0.21))
Calculating the individual components: Equity Component=0.70×12%=8.4%Equity Component=0.70×12%=8.4%Debt Component=0.30×7.5%×0.79=0.30×5.925%=1.7775%Debt Component=0.30×7.5%×0.79=0.30×5.925%=1.7775%
Summing these components gives the WACC: WACC=8.4%+1.7775%=10.1775%WACC=8.4%+1.7775%=10.1775%
Summary - The opportunity cost of capital for an average-risk Whispering Pines investment is 12%.
- The company’s WACC at the new capital structure with a 30% debt-to-value ratio is approximately 10.18%.
11) To value Chiara Company using the provided projections, we need to follow a Discounted Cash Flow (DCF) approach. We will: - Calculate the Free Cash Flows (FCFs) for the forecasted period.
- Determine the terminal value at the end of the forecast period.
- Discount the FCFs and terminal value to the present value.
- Calculate the value of the firm and then the value per share.
Step 1: Calculate Free Cash Flows
Assuming Table 18.5 provides the following projections for the next 5 years:
Year EBIT (ZAR million) Depreciation (ZAR million) CapEx (ZAR million) Change in NWC (ZAR million)
1 10 1 2 1
2 12 1 2 1
3 14 1 2 1
4 16 1 2 1
5 18 1 2 1
Free Cash Flow (FCF) Calculation:
FCF=EBIT×(1−Tax Rate)+Depreciation−CapEx−ΔNWCFCF=EBIT×(1−Tax Rate)+Depreciation−CapEx−ΔNWC
Assuming a tax rate of 21%:
Year EBIT (ZAR million) (1 – Tax Rate) Depreciation (ZAR million) CapEx (ZAR million) Change in NWC (ZAR million) FCF (ZAR million)
1 10 0.79 1 2 1 5.9
2 12 0.79 1 2 1 7.38
3 14 0.79 1 2 1 8.86
4 16 0.79 1 2 1 10.34
5 18 0.79 1 2 1 11.82
Step 2: Terminal Value Calculation
The terminal value (TV) at the end of year 5 is calculated using the perpetuity growth formula:
TV5=FCF5×(1+g)WACC−gTV5=WACC−gFCF5×(1+g)
Where:
• FCF5FCF5 is the Free Cash Flow in year 5
• gg is the long-term growth rate (4% or 0.04)
• WACCWACC is the Weighted Average Cost of Capital (12% or 0.12)
TV5=11.82×(1+0.04)0.12−0.04=11.82×1.040.08=12.29280.08=153.66 million ZARTV5=0.12−0.0411.82×(1+0.04)=0.0811.82×1.04=0.0812.2928=153.66 million ZAR
Step 3: Discount the FCFs and Terminal Value to Present Value
Discount each year’s FCF and the terminal value back to the present value using the WACC.
PVFCFt=FCFt(1+WACC)tPVFCFt=(1+WACC)tFCFt
Year FCF (ZAR million) Discount Factor PV (ZAR million)
1 5.9 11.1211.1211 = 0.8929
5.27
2 7.38 11.1221.1221 = 0.7972
5.89
3 8.86 11.1231.1231 = 0.7118
6.31
4 10.34 11.1241.1241 = 0.6354
6.57
5 11.82 11.1251.1251 = 0.5674
6.70
TV 153.66 11.1251.1251 = 0.5674
87.23
Sum of the present values:
PV of FCFs=5.27+5.89+6.31+6.57+6.70+87.23=117.97 million ZARPV of FCFs=5.27+5.89+6.31+6.57+6.70+87.23=117.97 million ZAR
Step 4: Calculate the Value of the Firm and Value per Share
The total firm value is the present value of the FCFs plus the present value of the terminal value:
Firm Value=117.97 million ZARFirm Value=117.97 million ZAR
Subtract the debt to get the equity value:
Equity Value=Firm Value−Debt=117.97−5=112.97 million ZAREquity Value=Firm Value−Debt=117.97−5=112.97 million ZAR
Calculate the value per share:
Value per Share=Equity ValueShares Outstanding=112.97 million ZAR865,000=130.61 ZAR per shareValue per Share=Shares OutstandingEquity Value=865,000112.97 million ZAR=130.61 ZAR per share
Summary
The value per share of Chiara Company, based on the given projections and WACC, is approximately 130.61 ZAR per share.
Chapter 21
3) To address the question, we need to understand and interpret the profit diagrams for options shown in Figure 21.12 of your reference. We’ll recreate and analyze these diagrams to determine if the statement “the investor in panel b can’t lose and the investor in panel a can’t win” is accurate.
Panel a: Long Call Option
In Panel a, the investor has a long call option. The profit diagram for a long call option looks like this:
• X-axis: Stock price at expiration
• Y-axis: Profit
• Strike Price (K): The price at which the option can be exercised
• Premium (C): The cost of purchasing the call option
The profit for a long call option is given by:
Profit=max(0,ST−K)−CProfit=max(0,ST−K)−C
Where:
• STST is the stock price at expiration
• KK is the strike price
• CC is the premium paid for the option
Profit Diagram:
• Below the strike price KK: The call option expires worthless, and the investor loses the premium paid (−C−C).
• At the strike price KK: The call option breaks even, but the investor still loses the premium paid.
• Above the strike price KK: The investor starts to make a profit, which is the stock price minus the strike price minus the premium paid.
The diagram looks like this:
Panel b: Short Call Option
In Panel b, the investor has a short call option. The profit diagram for a short call option looks like this:
• X-axis: Stock price at expiration
• Y-axis: Profit
• Strike Price (K): The price at which the option can be exercised
• Premium (C): The income received from selling the call option
The profit for a short call option is given by:
Profit=C−max(0,ST−K)Profit=C−max(0,ST−K)
Where:
• STST is the stock price at expiration
• KK is the strike price
• CC is the premium received for the option
Profit Diagram:
• Below the strike price KK: The call option expires worthless, and the investor keeps the premium received (+C+C).
• At the strike price KK: The call option breaks even, and the investor still keeps the premium received.
• Above the strike price KK: The investor starts to incur a loss, which is the strike price minus the stock price plus the premium received.
The diagram looks like this:
Analysis
Panel a (Long Call Option):
• The investor in Panel a (long call) cannot win if the stock price stays below the strike price. They will lose the premium paid (−C−C).
• The investor can only win if the stock price rises above the strike price by more than the premium paid.
Panel b (Short Call Option):
• The investor in Panel b (short call) cannot lose if the stock price stays below the strike price. They will keep the premium received (+C+C).
• The investor can lose if the stock price rises above the strike price, potentially incurring unlimited losses as the stock price increases.
Conclusion
The statement “the investor in panel b can’t lose and the investor in panel a can’t win” is not entirely correct:
• The investor in panel b (short call) can indeed incur significant losses if the stock price rises substantially above the strike price.
• The investor in panel a (long call) has the potential to win if the stock price rises above the strike price by more than the premium paid, providing the opportunity for unlimited upside.
Thus, while the long call investor has limited downside and unlimited upside potential, the short call investor has limited upside (the premium received) and potentially unlimited downside.
6) To understand the payoff of holding a share of stock and a put option on that share at expiration, we need to analyze the outcomes based on whether the stock price is below or above the exercise (strike) price of the put option.
Payoff Analysis
Let’s define the following:
• STST = Stock price at expiration
• KK = Exercise (strike) price of the put option
• PP = Premium paid for the put option (though it is not required for calculating the payoff, it is useful for understanding net profit/loss)
(a) If the stock price is below the exercise price (STKST>K):
In this scenario, the put option is “out of the money,” meaning it is not beneficial to exercise the option because the stock price is higher than the strike price. The put option will expire worthless.
Components:
• Value of the stock: STST
• Payoff from the put option: 0 (the option expires worthless)
Total Payoff: Total Payoff=ST+0=STTotal Payoff=ST+0=ST
So, the total payoff when ST>KST>K is STST.
Summary
• If STKST>K:
o The total payoff is STST.
Explanation
Holding a share of stock and a put option provides a form of protection (known as a protective put) against declines in the stock price. If the stock price falls below the strike price, the put option limits the losses by allowing the investor to sell the stock at the strike price KK. If the stock price rises above the strike price, the put option expires worthless, but the investor benefits from the appreciation of the stock.
This strategy ensures that the minimum payoff is the strike price KK, while still allowing for upside potential if the stock price rises above the strike price.
