Module 4: Numeric Questions

For students

Last updated: 23/09/2025 22:50

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 1 (until Midterm)

Module	Chapter	Slides	T/F	MCQ	Numeric	Long	Self-quiz
4	ch12	🎞️	✅	❓	🔢	📝	🧪

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Answer the following questions based on the discussions in class.

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Q1.

Unlevered Beta and CAPM (Using Business Risk Only)

A firm has levered beta = 1.68, D/E = 0.34, tax rate = 29%.
What is its unlevered (asset) beta? (Use the standard formula removing leverage.)

1. Unlevered beta (no units):

✅ Why this is the answer: Unlevered beta measures business risk only (no financial leverage).
We “strip out” leverage using: β_U = β_L / [1 + (1 – T) × (D/E)].
This lets us compare firms on operating risk, independent of capital structure.
Your numbers: β_U = 1.68 / [1 + (1 − 0.29) × 0.34] = 1.35.

If the risk-free rate is 2% and the expected market return is 9%, what is the expected return using CAPM with the unlevered beta?

2. Expected return (in %):

✅ CAPM intuition: Investors require compensation for time value (Rf) and for bearing market risk (β × MRP).
We use the unlevered beta to price the business risk without leverage:
E[R] = R_f + β_U × (R_m − R_f) = 11.47%.
This gives a baseline cost of equity for the project/asset before financing effects.

Q2.

WACC and Value Creation Threshold

A firm has Equity = $330m, Debt = $188m, Re = 15.1%, Rd = 8%, Tax = 30%.
Compute the WACC (after tax).

1. WACC (in %):

✅ Concept: WACC is the blended required return from shareholders and debtholders, after the interest tax shield.
Formula: WACC = (E/V)·R_e + (D/V)·R_d·(1 − T).
Your numbers: WACC = (330/518)×15.1% + (188/518)×8%×(1−30%) = 11.65%.

Suppose a project in this firm has expected return = 14.4%.
By how many percentage points does the project’s return exceed the WACC? (This gap signals value creation if positive.)

2. Value creation gap (pp):

✅ Decision rule: If Project Return > WACC, the project creates value (positive NPV under proportional scaling).
Your numbers: Gap = 14.4% − 11.65% = 2.75 pp. A positive gap indicates the project clears the firm’s opportunity cost of capital.

Q3.

Comparable Company Method: Unlever and Relever Beta

A comparable firm has β = 1.19, D/E = 0.46, tax = 23%.
(a) Compute the unlevered beta for the business.

1. Unlevered beta (no units):

✅ Step 1 (remove leverage): β_U = β_L / [1 + (1 − T) × (D/E)] → business-only risk.
Your numbers: 1.19 / [1 + (1 − 0.23) × 0.46] = 0.88.

(b) Your target firm will finance the project at D/E = 0.67. Compute the relevered beta to be used in CAPM.

2. Relevered beta (no units):

✅ Step 2 (apply target leverage): β_L = β_U × [1 + (1 − T) × (D/E)].
Your numbers: 0.8787 × [1 + (1 − 0.23) × 0.67] = 1.33.
This aligns the project’s risk with your capital structure.

Q4.

Portfolio Beta and Cost of Capital

A two-asset portfolio invests 54% in Asset A (β = 1.09) and 46% in Asset B (β = 0.9).
Compute the portfolio beta.

1. Portfolio beta (no units):

✅ Aggregation rule: Portfolio beta is the value-weighted average of component betas: β_p = Σ w_iβ_i.
Your numbers: 54%×1.09 + 46%×0.9 (scaled to weights) → 1.003.

With Rf = 3.2% and Rm = 9.2%, compute the portfolio’s cost of capital via CAPM.

2. Cost of capital (in %):

✅ CAPM logic: Required return rises with systematic risk (β).
E[R_p] = R_f + β_p(R_m − R_f) = 9.22%.
Interpretation: this is the portfolio’s opportunity cost of capital.

Q5.

Relevering Beta – Aurora Technologies

Aurora Technologies has an asset beta of 0.82.
Its debt-to-equity ratio is 1.3, and the corporate tax rate is 30.93%.

1. What is the equity beta of Aurora Technologies?

✅ Equity Beta = β_A × [1 + (1 – tax) × D/E] = 1.56.
This accounts for financial leverage amplifying the risk borne by equity holders.

Q6.

Cost of Debt Capital – CAPM and Default Approach

A company’s bond has a beta of 0.11. The risk-free rate is 4% and the market risk premium is 6%. What is the expected return on debt using the CAPM?

✅ Expected Return (CAPM) = r_f + β × MRP = 4.65%.

The bond has a Yield to Maturity (YTM) of 6.87%, probability of default of 0.5%, and expected loss given default of 59.8%. Estimate the expected return using the default method.

✅ Expected Return (Default Method) = YTM − Default Probability × Loss Rate = 6.57%.

Q7.

Portfolio Beta with Multiple Divisions

An institutional investor holds two divisions of Aurora Technologies:
- Division A weight = 59.94%, beta = 1.29
- Division B weight = 40.06%, beta = 0.77

1. What is the overall portfolio beta?

✅ Portfolio Beta = w_Aβ_A + w_Bβ_B = 1.08.
This shows how different risk exposures combine in a weighted portfolio.

Q8.

After-tax Cost of Debt

Aurora Technologies issues new bonds with a yield to maturity of 5.8%.
The corporate tax rate is 23.98%.

1. What is the after-tax cost of debt (in %)?

✅ After-tax cost of debt = r_D × (1 – tax) = 4.41%.
Interest is tax-deductible, so the effective cost of debt is lower than the nominal rate.

Q9.

Project-specific Discount Rate

Aurora Technologies considers investing in a high-risk digital platform.
- Risk-free rate = 2.98%
- Market risk premium = 4.81%
- Project beta = 1.43

1. What is the appropriate discount rate for this project (in %)?

✅ Discount Rate = r_f + β_project × MRP = 9.84%.
Projects with higher betas require higher discount rates to compensate investors for risk.