Module 4: Numeric Questions

For students

Last updated: 09/01/2026 13:06

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
4	ch12

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Answer the following questions based on the discussions in class.

Q1.

Unlevered Beta and CAPM (Using Business Risk Only)

A firm has levered beta = 1.54, D/E = 0.7, tax rate = 25%.
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 = 1.54 / [1 + (1 − 0.25) × 0.7] = 1.01.

If the risk-free rate is 3% and the expected market return is 10%, 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) = 10.07%.
This gives a baseline cost of equity for the project/asset before financing effects.

Q2.

WACC and Value Creation Threshold

A firm has Equity = $355m, Debt = $180m, Re = 12.5%, Rd = 8.1%, Tax = 35%.
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 = (355/535)×12.5% + (180/535)×8.1%×(1−35%) = 10.07%.

Suppose a project in this firm has expected return = 11.2%.
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 = 11.2% − 10.07% = 1.13 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 = 1.12, D/E = 0.46, tax = 30%.
(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: 1.12 / [1 + (1 − 0.3) × 0.46] = 0.85.

(b) Your target firm will finance the project at D/E = 0.53. 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: 0.8472 × [1 + (1 − 0.3) × 0.53] = 1.16.
This aligns the project’s risk with your capital structure.

Q4.

Portfolio Beta and Cost of Capital

A two-asset portfolio invests 61% in Asset A (β = 1.05) and 39% in Asset B (β = 0.76).
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: 61%×1.05 + 39%×0.76 (scaled to weights) → 0.937.

With Rf = 3.8% and Rm = 9.9%, 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) = 9.52%.
Interpretation: this is the portfolio’s opportunity cost of capital.

Q5.

Relevering Beta – Aurora Technologies

Aurora Technologies has an asset beta of 0.76.
Its debt-to-equity ratio is 1.21, and the corporate tax rate is 33.69%.

1. What is the equity beta of Aurora Technologies?

✅ Equity Beta = β_A × [1 + (1 – tax) × D/E] = 1.36.
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 0.1. The risk-free rate is 4% and the market risk premium is 5.99%. What is the expected return on debt using the CAPM?

✅ Expected Return (CAPM) = r_f + β × MRP = 4.62%.

The bond has a Yield to Maturity (YTM) of 7.06%, probability of default of 0.49%, and expected loss given default of 58.5%. Estimate the expected return using the default method.

✅ Expected Return (Default Method) = YTM − Default Probability × Loss Rate = 6.77%.

Q7.

Portfolio Beta with Multiple Divisions

An institutional investor holds two divisions of Aurora Technologies:
- Division A weight = 64.56%, beta = 1.3
- Division B weight = 35.44%, beta = 0.62

1. What is the overall portfolio beta?

✅ Portfolio Beta = w_Aβ_A + w_Bβ_B = 1.06.
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 5.9%.
The corporate tax rate is 25.87%.

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

✅ After-tax cost of debt = r_D × (1 – tax) = 4.37%.
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 = 3.32%
- Market risk premium = 5.95%
- Project beta = 1.5

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

✅ Discount Rate = r_f + β_project × MRP = 12.24%.
Projects with higher betas require higher discount rates to compensate investors for risk.

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