Module 9: Numeric Questions

For students

Last updated: 14/11/2025 13:26

The questions are based on or inspired by the following references:

• Berk & DeMarzo, Corporate Finance, 5th ed. (2020)
  
• Brealey & Myers, Principles of Corporate Finance, 13th ed. (2020)

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💡 You can also press Ctrl + P (or Cmd + P on Mac) to print or save your responses as a .pdf file.

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⚠️ These exercises are powered by AI-assisted technologies and may contain occasional formatting or logic errors. Please report any issues you encounter so I can improve the experience.

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📙 Part 2 (Midterm to Final)

Module	Chapter	Slides	T/F	MCQ	Numeric	Long	Self-quiz
9	ch16	🎞️	✅	❓	🔢	📝	🧪

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Answer the following questions based on the discussions in class.

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Q1.

Timing of Equity Issuance Under Asymmetric Information

A firm currently has 11.33M shares. True firm value today is 222.16M, but the market believes it is only 190.12M. In six months, the true value will be 230.36M, but markets will believe it is 219.27M.

The firm needs to raise 27.22M and is considering issuing equity today or in six months.

Compute:

• 1. share price today
• 2. share price in six months
• 3. shares issued today
• 4. shares issued in six months

a. Price today ($):

✅ Price today = Market belief / shares = 190.12 / 11.33 = 16.78.

b. Price in six months ($):

✅ Price future = Market belief future / shares = 219.27 / 11.33 = 19.35.

c. Shares issued today (millions):

✅ Shares today = 27.22 / 16.78 = 1.62M.

d. Shares issued in six months (millions):

✅ Shares future = 27.22 / 19.35 = 1.41M. So issuing in the future seems better.

Q2.

Leverage, Bankruptcy Costs, and Equity Value

Nimbus Devices will be worth either 100.92M, 147.32M, or 185.59M in one year, with each outcome equally likely. Because the project risk is fully diversifiable, the discount rate is the risk-free rate of 7%.

Nimbus considers issuing zero-coupon debt with face value of 107.22M due in one year. If the firm defaults (which occurs only in the low state), bankruptcy costs destroy 25% of asset value before debtholders are paid. Compute:

• 1. the value of the firm today without leverage, \(V_U\) (millions)
• 2. the value of debt today, \(V_D\) (millions)
• 3. the value of equity today, \(V_E\) (millions)
• 4. the present value of financial distress costs, \(PV(\text{distress})\) (millions)

a. Unlevered firm value \(V_U\) (millions):

✅ \(V_U = \dfrac{E[V_1]}{1+r_f} = \dfrac{144.61}{1 + 0.07} =\) 135.15M.

b. Debt value today \(V_D\) (millions):

✅ Debt payoff is the expected amount debtholders receive across all states, discounted at the risk-free rate.
\(V_D = \dfrac{E[\text{Debt Payoff}]}{1+r_f} = \dfrac{96.71}{1 + 0.07} =\) 90.38M.

Here:
• Debt payoffs = (75.69, 107.22, 107.22)
• Expected payoff = (96.71)
• Discounting at 7% gives 90.38M.

c. Equity value today \(V_E\) (millions):

✅✅ Equity holders receive only residual value after debtholders are paid.
\(V_E = \dfrac{E[\text{Equity Payoff}]}{1+r_f} = \dfrac{39}{1 + 0.07} =\) 37M.

Here:
• Equity payoffs = (0, 40.1, 78.37)
• Expected equity payoff = 39
• Discounting at 7% gives 37M.

d. \(PV(\text{distress})\) (millions):

✅ \(PV(\text{distress}) = V_U - V_L = 135 - (90 + 37) =\) 8M.

Q3.

Equity Issuance Under Asymmetric Information

A firm has 10.07M shares outstanding. Managers know the true total value is 214.62M, but the market believes it is only 193.76M. The firm needs to raise 42.91M immediately and issues equity at the market price.

Compute:

1. the market price per share

2. the number of shares the firm must issue

3. the true value per share after the issue (millions)

a. Market price ($):

✅ Price = Market belief / shares = 193.76 / 10.07 = 19.24.

b. Shares issued (millions):

✅ Shares issued = Needed / Price = 42.91 / 19.24 = 2.23M.

c. True post-issue price ($):

✅ True post-issue price = True value / total shares = 214.62 / (10.07 + 2.23) = 17.45.

Q4.

Trade-off Theory: Taxes vs. Financial Distress

Aurora Technologies has an unlevered value of 329.2M and is considering issuing permanent debt of 118.26M. The corporate tax rate is 21%. If Aurora uses this debt level, the probability (p) of financial distress will be 28%, and the loss (L ) if distress occurs is estimated at 81.08M.

Assume a simple trade-off theory where: the present value of the interest tax shield is \(\tau_c D\), the present value of expected financial distress costs is \(p \times L\), compute (in millions):

• 1. the present value of the tax shield, \(PV(\text{tax shield})\)
• 2. the present value of expected financial distress costs, \(PV(\text{distress})\)
• 3. the value of the levered firm, \(V_L\)
• 4. the net effect of leverage on firm value, \(V_L - V_U\)

a. \(PV(\text{tax shield})\) (millions):

✅ \(PV(\text{tax shield}) = \tau_c D = 0.21 \times 118.26 =\) 24.83M.

b. \(PV(\text{distress})\) (millions):

✅ \(PV(\text{distress}) = p \times L = 0.28 \times 81.08 =\) 22.7M.

c. Levered firm value \(V_L\) (millions):

✅ \(V_L = V_U + PV(\text{tax shield}) - PV(\text{distress}) = 329.2 + 24.83 - 22.7 =\) 331.33M.

d. Net effect of leverage \(V_L - V_U\) (millions):

✅ \(V_L - V_U = PV(\text{tax shield}) - PV(\text{distress}) = 24.83 - 22.7 =\) 2.13M.

Q5.

Debt Overhang — Positive-NPV Project Rejected by Equity

A firm has assets worth 95.93M and outstanding zero-coupon debt with face value 116.42M due tomorrow. It can undertake a new project costing 10.71M that will pay 26.1M tomorrow. Compute:

• 1. the change in equity payoff if the firm invests
• 2. the NPV to equity of undertaking the project
• 3. whether equity will invest (1 = Yes, 0 = No)

a. Δ Equity payoff (millions):

✅ ΔEquity = max(95.93+26.1−116.42,0) − max(95.93−116.42,0) = 5.61M.

b. Equity NPV (millions):

✅ NPV_E = ΔEquity − cost = 5.61 − 10.71 = -5.1M.

c. Equity invests? (1 = Yes, 0 = No):

✅ Equity invests only if NPV_E > 0 → No.

Q6.

Asset Substitution — Risk Shifting by Equity Holders

A firm has assets worth 119.56M and owes 107.91M tomorrow. It can invest in one of two mutually exclusive projects: a safe project paying 16.06M for sure; or a risky project paying 57.01M with probability 0.4, and 0 otherwise.

Compute:

• 1. the equity value if the firm chooses the safe project
• 2. the expected equity value if it chooses the risky project
• 3. which project equity prefers (1 = risky, 0 = safe)
• 4. which project maximizes firm value (1 = risky, 0 = safe)

a. Equity (safe) (millions):

✅ Equity(safe) = max(119.56 + 16.06 − 107.91, 0) = 27.71M.

b. Equity (risky) (millions):

✅ Equity(risky) = 0.4·max(119.56 + 57.01 − 107.91, 0) + 0.6·0 = 27.46M.

c. Equity prefers risky? (1/0):

✅ Equity chooses the project with the higher expected equity payoff. \(E[E_{\text{risky}}] - E[E_{\text{safe}}] = -0.25\)M, equityholders prefer: safe.

d. Project that maximizes firm value? (1 = risky, 0 = safe):

✅ Firm value (safe) = 135.62M; firm value (risky) = 142.364M. Max Value: risky.