Week 2: Cost of Capital Section B Core

Master WACC, Beta calculations, and M&A valuation for the 70-mark problem-solving section.

      🎯 Exam Relevance: Cost of capital is central to Section B valuation questions. You must be able to: calculate WACC with market values, unlever/relever betas for comparables, handle M&A beta calculations, and convert between EV and equity value. Always show formula → substitution → calculation → interpretation.
    

Essential Formulas for Cost of Capital

1. WACC (Weighted Average Cost of Capital)

Section B Core

WACC Formula

WACC = r_e × (E / V) + r_d × (1 - t) × (D / V)

Where:

r_e = Cost of equity
r_d = Cost of debt (YTM)
E = Market value of equity
D = Market value of debt
V = E + D (Total firm value)
t = Corporate tax rate

Always use MARKET values, not book values. The tax shield only applies to debt!

2. Cost of Equity (CAPM)

CAPM

CAPM Formula

r_e = R_f + β_E × (R_m - R_f)

Where:

R_f = Risk-free rate
β_E = Equity beta (levered beta)
R_m = Expected market return
(R_m - R_f) = Market risk premium

3. Unlevering & Relevering Beta

Frequently Tested

Unlevering Beta (No Tax - Simplified)

β_A = β_E / (1 + D/E)

Relevering Beta (No Tax - Simplified)

β_E = β_A × (1 + D/E)

With Debt Beta (More Accurate)

β_E = β_A × (1 + D/E) - β_D × (D/E)

Bottom-Up Beta Process:
1. Find comparable firms' levered betas
2. Unlever each using their D/E ratios
3. Average the unlevered betas
4. Relever using TARGET firm's D/E ratio

4. Cost of Debt (YTM Calculation)

Bond Math

Zero-Coupon Bond YTM

Price = Face Value / (1 + YTM/n)^n×T

Example from Tutorial:

→

Price = $252.5725, Face = $1000, T = 20 years, Semi-annual (n=2)

252.5725 = 1000 / (1 + YTM/2)⁴⁰

Solving: YTM = 7% (annual rate = semi-annual × 2)

5. Merger Beta (Portfolio Approach)

M&A Problems

Combined Asset Beta

β_combined = (V_A/V_total) × β_A + (V_T/V_total) × β_T

Key Steps for M&A Beta:
1. Calculate combined unlevered beta (weighted average)
2. Calculate post-acquisition debt = Old debt + New debt issued
3. Calculate post-acquisition equity = Old equity + New equity issued
4. Relever: New β_E = Combined β_A × (1 + New D/E)

If acquisition is financed with DEBT, equity stays same but debt increases → higher leverage → higher beta. If financed with EQUITY, both equity increases and target equity is absorbed.

6. Enterprise Value vs Equity Value

Common Trap

EV to Equity Conversion

Equity Value = Enterprise Value - Net Debt

Net Debt = Total Debt - Cash

DCF gives you Enterprise Value. To get share price, subtract debt, add cash, then divide by shares outstanding!

Practice Problems (Tutorial Style)

Work through these step-by-step. Click to reveal each step.

Problem 1: Bottom-Up Beta & WACC

Full Solution

Problem: Estimate WACC for RWP Corp given:

Market value of debt: $600 million
Market value of equity: $4 billion
Zero-coupon bond: Price = $252.5725, Face = $1000, 20 years, semi-annual
Risk-free rate: 3%, Market return: 7%, Tax rate: 40%

Company	Levered Beta	D/E Ratio
Genoxys Inc	1.6	0.5
Lovarmaceutical Inc	1.9	1.0
Fanepill Corp	1.5	0.4
Udo Laboratory Inc	1.3	0.2
Refillobos Corp	1.5	0.3

Step 1: Unlever each comparable's beta

Formula: β_A = β_E / (1 + D/E)

Company	β_E	D/E	β_A = β_E/(1+D/E)
Genoxys	1.6	0.5	1.6/1.5 = 1.067
Lovarmaceutical	1.9	1.0	1.9/2.0 = 0.950
Fanepill	1.5	0.4	1.5/1.4 = 1.071
Udo Laboratory	1.3	0.2	1.3/1.2 = 1.083
Refillobos	1.5	0.3	1.5/1.3 = 1.154

Average unlevered beta = 1.065

Step 2: Relever for RWP's capital structure

RWP's D/E = 600/4000 = 0.15

β_E = β_A × (1 + D/E) = 1.065 × (1 + 0.15) = 1.225

Step 3: Calculate Cost of Equity

r_e = R_f + β_E(R_m - R_f)
r_e = 3% + 1.225 × (7% - 3%) = 3% + 4.9% = 7.9%

Step 4: Calculate Cost of Debt (YTM)

Zero-coupon bond: 252.5725 = 1000 / (1 + r/2)⁴⁰

(1 + r/2)⁴⁰ = 1000/252.5725 = 3.9593

1 + r/2 = 3.9593^1/40 = 1.035

r/2 = 0.035 → r_d = 7% (annual)

Step 5: Calculate WACC

E/(D+E) = 4000/4600 = 0.8696

D/(D+E) = 600/4600 = 0.1304

WACC = 7.9% × 0.8696 + 7% × (1-0.40) × 0.1304
WACC = 6.87% + 0.55% = 7.42%

Problem 2: Merger Beta

M&A Focus

Problem: Novell (MV equity = $2B, β = 1.5) acquires WordPerfect (MV equity = $1B, β = 1.3). Neither has debt.

a) Acquisition financed with equity
b) Acquisition financed with debt

Part (a): Equity-Financed Acquisition

Combined unlevered beta (weighted average):

β_combined = (2/3) × 1.5 + (1/3) × 1.3 = 1.0 + 0.433 = 1.43

Post-deal: D = $0, E = $2B + $1B = $3B

D/E = 0 → Levered beta = 1.43 × (1 + 0) = 1.43

Part (b): Debt-Financed Acquisition

Combined unlevered beta = 1.43 (same as above)

Post-deal: D = $1B (new borrowing), E = $2B (unchanged)

D/E = 1/2 = 0.5

β_E = 1.43 × (1 + 0.5) = 1.43 × 1.5 = 2.15

Interpretation: Debt financing increases leverage, which amplifies equity risk and beta. Cost of equity rises significantly!

Problem 3: Beta with Debt Beta

Advanced

Problem: Using department store data, relever for ABC Corp (D/E = 27%, BBB-rated debt → β_D = 0.1)

Industry average asset beta = 1.16 (from solution)

Solution with Debt Beta

β_E = β_A × (1 + D/E) - β_D × (D/E)

β_E = 1.16 × (1 + 0.27) - 0.1 × (0.27)
β_E = 1.16 × 1.27 - 0.027
β_E = 1.473 - 0.027 = 1.45

When debt beta is non-zero, the formula accounts for the fact that debt holders also bear some systematic risk. This slightly reduces equity beta compared to assuming β_D = 0.

🎯 Your Turn: Quick Calculation

Test Yourself

A firm has β_E = 2.0 and D/E = 0.6. What is its unlevered (asset) beta?

Your Answer (β_A):

Quick Concept Check

Test your understanding of cost of capital concepts.

Question 1 of 8

Interactive Calculators

Use these tools to check your work and build intuition.

📊 WACC Calculator

Market Value of Equity ($M)

Market Value of Debt ($M)

Cost of Equity (%)

Cost of Debt (%)

Tax Rate (%)

WACC = r_e × (E/V) + r_d × (1-t) × (D/V)

WACC = 7.42%

Breakdown:
E/V = 86.96% | D/V = 13.04% | After-tax r_d = 4.2%

🔄 Beta Lever/Unlever Calculator

Levered Beta (β_E)

D/E Ratio

β_A = β_E / (1 + D/E)

Unlevered β = 1.000

Relever for New D/E:

New D/E Ratio

β_E = β_A × (1 + D/E)

New β_E = 1.150

📈 Cost of Equity (CAPM) Calculator

Risk-Free Rate (%)

Equity Beta

Market Return (%)

r_e = R_f + β_E × (R_m - R_f)

Cost of Equity = 7.90%

Market Risk Premium = 4.00% | Risk Premium = 4.90%

⚠️ Common Exam Traps & How to Avoid Them

These are the mistakes that cost students marks. Don't make them!

Trap #1: Using Book Values Instead of Market Values

Wrong: Using book value of equity from balance sheet

Right: Always use market value of equity (shares × price) and market value of debt (present value at YTM)

WACC reflects opportunity cost of capital for investors. Investors care about market values!

Trap #2: Forgetting Tax Shield on Debt

Wrong: WACC = r_e(E/V) + r_d(D/V)

Right: WACC = r_e(E/V) + r_d(1-t)(D/V)

Interest is tax-deductible! The (1-t) captures this benefit.

Trap #3: Wrong D/E vs D/V Conversions

If given D/V (debt to value), convert to D/E:

If D/V = 0.3, then E/V = 0.7
D/E = (D/V) / (E/V) = 0.3/0.7 = 0.429

Beta formulas use D/E. WACC uses D/V and E/V weights!

Trap #4: Semi-Annual Compounding on YTM

Wrong: Using the semi-annual rate directly as cost of debt

Right: Multiply semi-annual rate by 2 for annual YTM

→

Example: (1 + r/2)⁴⁰ = 3.9593 → r/2 = 3.5% → r_d = 7%

Trap #5: Merger Beta - Wrong Weights

Wrong: Equal-weighting acquirer and target betas

Right: Value-weight by firm values

Combined β = (V_acquirer/V_total) × β_acquirer + (V_target/V_total) × β_target

Trap #6: EV vs Equity Value

Wrong: DCF result = Share price × Shares

Right: DCF gives Enterprise Value. Must adjust!

Equity Value = EV - Net Debt = EV - (Debt - Cash)
Share Price = Equity Value / Shares Outstanding

This is explicitly called out as a "common exam trap" in your exam instructions!

🎯 Interpretation Marks (40-50% of Section B!)

Your exam notes say interpretation is a "major portion of marks." After calculations, always state:

What does the result mean? (e.g., "Higher leverage increased equity beta from 1.43 to 2.15")
What are the implications? (e.g., "Higher cost of equity will increase WACC, potentially reducing NPV of projects")
What decision follows? (e.g., "Debt financing increases financial risk; management should consider if tax benefits outweigh distress costs")

📋 Week 2 Exam Checklist

Before moving on, make sure you can:

☐ Calculate WACC using market values and after-tax cost of debt
☐ Apply CAPM to find cost of equity
☐ Unlever comparable betas and relever for target capital structure
☐ Calculate YTM for zero-coupon and coupon bonds
☐ Handle M&A beta calculations for both debt and equity financing
☐ Convert between D/E and D/V ratios
☐ Convert Enterprise Value to Equity Value to Share Price
☐ Always show: Formula → Substitution → Calculation → Interpretation