Weighted Average Cost of Capital (WACC) Detailed Study Notes

Understanding the Discount Rate and CAPM

• Setting the Discount Rate     - Cash flows must be discounted to the present using a discount rate that accurately reflects the risk associated with those cash flows.     - For decision rules like NPV and IRR, the discount rate was previously provided as a given variable. In practice, this rate must be derived.     - Example Scenario:         - Period 1 (Now): $(100.00)$         - Period 2 (Later): $112.00$         - Interest rate ($i$): $10\%$         - $NPV = 1.82$         - $IRR = 12\%$

• Three Primary Approaches to Setting the Discount Rate     - Past Experience or Comparables: Setting the rate based on historical data or similar projects/firms.     - Direct use of CAPM:         - Determining the CAPM of the project itself.         - This requires a project beta ($\beta$).         - Challenges include knowing the covariance of project returns and market returns ($COV(r_p, r_m)$).         - Practitioners may proxy the beta by looking at similar assets, using experience, or inferring it, while remaining cautious about the capital structure.     - Weighted Average Cost of Capital (WACC):         - This approach assumes the project carries the same risk profile as the firm's existing operations.         - The firm's current cost of debt ($k_d$) and cost of equity ($k_e$) are used to reflect appropriate return levels.         - These costs are weighted according to the firm's capital structure.

• Calculating the Discount Rate with CAPM     - In instances where a project beta is available, the risk-adjusted discount rate can be calculated.     - CAPM Formula: $E[R] = R_f + \beta(E[R_m] - R_f)$     - Example Calculation:         - Risk-free rate ($R_f$): $0.04$         - Expected Market return ($E[R_m]$): $0.10$         - Project beta ($\beta$): $1.2$         - Calculation: $0.04 + 1.2\times(0.10 - 0.04) = 0.112$         - Resulting Risk-adjusted discount rate: $11.2\%$     - Problem: In practice, a specific "project beta" is usually not readily available.

Weighted Average Cost of Capital (WACC) Fundamentals

• Definition and Purpose     - Firms raise capital through debt and equity and must pay for these funds.     - Debt costs are paid explicitly via interest.     - Equity costs are paid implicitly via opportunity cost.     - WACC is the minimum rate an investment or project must return to satisfy the required returns of those who supplied the firm's capital (debt and equity holders).     - It represents an average of costs from all sources ($k_d$ and $k_e$).

• WACC as a Risk Measure     - WACC reflects the overall risk of the firm.     - Positive NPV: Occurs if project cash flows are more than sufficient to cover the WACC.     - Negative NPV: Occurs if cash flows do not sufficiently cover the WACC.     - Accuracy Requirement: if the WACC does not appropriately reflect the project's specific risk, the resulting NPV calculation will be incorrect.

• WACC Formula (No Taxes)     - Formula: $WACC = \frac{E}{V}k_e + \frac{D}{V}k_d$     - Variables:         - $k_e$: Cost of equity         - $D$: Market value of debt         - $E$: Market value of equity         - $k_d$: Cost of debt         - $V = E + D$ (Total Asset Value)     - Note: $k_e$ is generally greater than $k_d$. Also, $\frac{E}{V} + \frac{D}{V} = 1$. If $\frac{E}{V} = 1$, then $WACC = k_e$. If $\frac{E}{V} = 0.5$, then weights are equal.     - Example (No Taxes):         - $V = 100$         - $E = 50$ (with $k_e = 0.15$)         - $D = 50$ (with $k_d = 0.10$)         - calculation: $\frac{50}{100}\times0.15 + \frac{50}{100}\times0.10 = 0.125$ or $12.5\%$

• WACC Formula (With Taxes)     - Formula: $WACC = \frac{E}{V}k_e + \frac{D}{V}k_d(1 - t)$     - Variable:         - $t$: Corporate tax rate.     - Tax Considerations: Interest payments on debt are tax-deductible, whereas dividends (equity payments) are not. The term $(1-t)$ represents the effective after-tax cost of debt.     - Example (With Taxes):         - $V = 100, E = 50, D = 50$         - $k_e = 0.15, k_d = 0.10, t = 0.3$         - Calculation: $\frac{50}{100}\times0.15 + \frac{50}{100}\times0.10\times(1 - 0.30) = 0.11$ or $11\%$

Elements of WACC: Debt, Equity, and Weights

• Market Values (E and D)     - Market values should always be used when available rather than book values, as they reflect the true economic claim of each funding source.

• Calculating Cost of Equity ($k_e$)     - Method 1: CAPM         - Formula: $k_e = R_f + \beta(E[R_m] - R_f)$         - Beta reflects specific types of risk: $\beta_L$ (Low risk), $\beta_m$ (Market risk), $\beta_H$ (High risk).         - Calculation for Beta: $\beta_i = \frac{COV(r_i, r_m)}{VAR(r_m)}$     - Method 2: Dividend Pricing Formula         - Basic Form: $P_0 = \sum_{t=1}^{\infty} \frac{D_t}{(1 + k_e)^t}$         - Constant Dividend Model: $P_0 = \frac{D}{k_e}$, rearrange for equity cost: $k_e = \frac{D}{P_0}$. This applies to perpetual bonds and non-redeemable preference shares but fails to capture growth.         - Constant Growth Model: $P_0 = \frac{D_1}{k_e - g}$, rearrange for equity cost: $k_e = \frac{D_1}{P_0} + g$. This requires assumptions about the growth rate ($g$).         - Non-Constant Growth (Standard Model): Solve for $k_e$ via trial-and-error (similar to finding IRR).         - $P_0 = \sum_{t=1}^n \frac{D_t}{(1 + k_e)^t} + \frac{P_n}{(1 + k_e)^n}$, where $P_n = \frac{D_{n+1}}{k_e - g}$.

• Comprehensive Cost of Equity Example     - Firm Data:         - Expected dividend ($D_1$): $\$2.20$         - Dividend growth rate ($g$): $10.20\%$         - Share price ($P_0$): $\$46.00$         - Beta ($\beta$): $1.28$     - Market Data:         - Market risk premium ($E[R_m] - R_f$): $7\%$         - Risk-free rate ($R_f$): $6\%$     - Calculations:         - Dividend Growth Approach: $k_e = \frac{2.20}{46.00} + 0.1020 = 0.1498$         - CAPM Approach: $k_e = 0.06 + 1.28\times(0.07) = 0.1496$         - both approaches converge at approximately $15.00\%$.

• Calculating Cost of Debt ($k_d$)     - The relevant cost of debt is the current implicit after-tax interest rate based on market values.     - Valuation model for fixed interest and face value repayment: $P_0 = \sum_{t=1}^n \frac{C}{(1 + k_d)^t} + \frac{FV}{(1 + k_d)^n}$     - The Preferred Approach (Practiced in EFB210):         1. Solve for before-tax cost of debt ($k_d$) using the bond pricing formula and trial-and-error.         2. Calculate the after-tax component using $(1 - t)$.         - Example: $P_0 = 113.420$, Coupon ($C$) = $10$, Years ($n$) = $10$, Face Value ($FV$) = $100$.         - $113.420 = 10\times[\frac{1 - (1 + k_d)^{-10}}{k_d}] + 100\times(1 + k_d)^{-10}$         - By trial and error, before-tax $k_d = 0.08$.         - If tax ($t$) = $0.30$, after-tax cost is $0.08\times(1 - 0.3) = 0.056$ or $5.6\%$.     - The Other Approach:         - Use after-tax coupon payments: $C\times(1 - t) = 10\times(1 - 0.30) = 7.00$.         - Solve for after-tax $k_d$ directly from the price: $113.420 = 7\times[\frac{1 - (1 + k_d)^{-10}}{k_d}] + 100\times(1 + k_d)^{-10}$.         - Solution yields after-tax $k_d = 0.0525$.

WACC Case Study and Application

• Comprehensive WACC Calculation Example     - Inputs:         - Shares: $1,000,000$ units at $\$4$ market price.         - Bonds: $10,000$ units at $\$113.42$ market price.         - Bond terms: $10\%$ coupon, $10$ years maturity.         - Equity attributes: $\beta = 1.75$, most recent dividend ($D_0$) = $\$0.28$, growth ($g$) = $7\%$.         - Market attributes: $E[r_m] = 10\%$, $R_f = 4\%$.         - Tax rate: $30\%$.     - Step 1: Calculate $k_e$         - Dividend Model: $k_e = \frac{0.28\times(1.07)}{4.00} + 0.07 = 0.1449$         - CAPM: $k_e = 0.04 + 1.75\times(0.10 - 0.04) = 0.1450$     - Step 2: Solve for $k_d$         - Bond Price formula: $113.41 = 10\times[\frac{1 - (1 + k_d)^{-10}}{k_d}] + 100\times(1 + k_d)^{-10}$         - $k_d = 0.08$     - Step 3: Calculate Market Values (E and D)         - $E = 1,000,000 \times 4.00 = 4,000,000$         - $D = 10,000 \times 113.420 = 1,134,200$         - Total Value ($V$) = $4,000,000 + 1,134,200 = 5,134,200$     - Step 4: Final WACC Calculation         - $WACC = \frac{4,000,000}{5,134,200}\times0.145 + \frac{1,134,200}{5,134,200}\times0.08\times(1 - 0.30)$         - Result: $0.1254$ or $12.54\%$

• Multiple Capital Components     - When there are more than two sources of capital (e.g., ordinary equity, preference shares, debentures, overdrafts), the formula expands:     - Formula: $WACC = \sum_{s=1}^n w_s k_s$     - Example Table Approach:         - Ordinary Equity ($E$): Market Value $50M$, Weight $0.5000$, After-Tax Cost $0.1500$, Weighted Cost $0.0750$         - Preference Shares ($P$): Market Value $20M$, Weight $0.2000$, After-Tax Cost $0.1000$, Weighted Cost $0.0200$         - Debentures ($D_1$): Market Value $20M$, Weight $0.2000$, After-Tax Cost $0.0800$, Weighted Cost $0.0160$         - Overdraft ($D_2$): Market Value $10M$, Weight $0.1000$, After-Tax Cost $0.0900$, Weighted Cost $0.0090$         - Total Market Value: $\$100,000,000$         - Total WACC: $0.1200$ or $12\%$

Capital Structure Theory

• Modigliani and Miller (MM)     - Assumptions for Neutrality: Perfect capital markets, no taxes, risk-free borrowing/lending, no bankruptcy costs, fixed investment policies.     - Proposition 1: Two firms with identical assets and operations have the same total value regardless of capital structure (debt/equity mix).     - Proposition 2: Leverage does not change firm value, but it does change the costs of debt and equity. Specifically, as the Debt-to-Equity ($D/E$) ratio increases, $k_e$ increases.

• Risk Components     - Business Risk: Risk inherent in the firm's operations.     - Finance Risk: As leverage ($D/E$) increases, business risk is concentrated into a smaller equity base, leading to the increase in $k_e$.     - With Taxes: The WACC curve typically dips because of the tax shield provided by debt $(1-t)k_d$.

Practical Application and Recap

• When to Use Firm WACC     - Using a firm's WACC for a project is only appropriate if the project has the same risk as the overall firm (e.g., a simple expansion).     - If a project is different (e.g., a takeover in a foreign industry), using the firm's WACC may result in a discount rate that is either too high or too low relative to the project's real risk.     - The discount rate must always reflect the risk of the project, not necessarily the source of funds.

• WACC Summary for EFB210     - Key formula: $WACC = \frac{E}{V}k_e + \frac{D}{V}k_d(1 - t)$.     - Cash flows used in DCF must be net cash flows after tax but before interest and interest tax savings.     - Use market values for $E$ and $D$.     - Weights ($\frac{E}{V}$, $\frac{D}{V}$) represent the target capital structure.     - Assumes project risk equals firm risk.