Q25. What is the yield to maturity (YTM) of MI's zero-coupon debt with a $125 million face value, given bankruptcy costs and asset values?

Background

Topic: Debt Valuation, Yield to Maturity, Bankruptcy Costs

This question tests your understanding of how to calculate the yield to maturity (YTM) for debt when there is a risk of default and bankruptcy costs. It involves expected cash flows, present value, and the concept of risk-free discounting.

Key Terms and Formulas

• Yield to Maturity (YTM): The effective annual return to debtholders, calculated as the percentage change from the present value to the face value of debt.

• Present Value (PV): The discounted expected value of the debt's payoff, using the risk-free rate.

• Bankruptcy Costs: In default, debtholders receive the asset value minus 20% bankruptcy costs.

Formula for Present Value of Debt:

Discounting to Present Value:

Yield to Maturity Formula:

Step-by-Step Guidance

1. Identify the possible asset values at maturity: $100 million, and $191\frac{1}{3}$.

2. Determine the payoff to debtholders in each scenario:

   • When assets are $100 due to bankruptcy costs.

   • When assets are $150 million: Debtholders receive the full $125$ million.

3. Calculate the expected payoff to debtholders by weighting each outcome by its probability:

4. Discount the expected payoff to present value using the risk-free rate ():

5. Set up the YTM calculation using the present value and face value:

Try solving on your own before revealing the answer!

Q26. What is the initial value of MI's equity, given bankruptcy costs and asset values?

Background

Topic: Equity Valuation, Bankruptcy Costs, Expected Value

This question tests your ability to calculate the value of equity in a firm when there is a risk of default and bankruptcy costs. It involves expected payoffs to equity holders and discounting to present value.

Key Terms and Formulas

• Equity Value: The expected payoff to shareholders, discounted to present value.

• Bankruptcy Costs: In default, equity holders receive nothing.

• Expected Payoff: Weighted average of possible equity outcomes.

Formula for Equity Value:

Step-by-Step Guidance

1. Identify the possible asset values at maturity: $100 million, and $191\frac{1}{3}$.

2. Determine the payoff to equity holders in each scenario:

   • When assets are $100.

   • When assets are $150 million.

   • When assets are $191 million.

3. Calculate the expected payoff to equity holders by weighting each outcome by its probability:

4. Discount the expected payoff to present value using the risk-free rate ():

Try solving on your own before revealing the answer!