📖 WACC 计算：股权成本（CAPM）

Corporate Finance

I. Lesson Focus

This lesson explains how to estimate the cost of equity using the Capital Asset Pricing Model (CAPM) and how to incorporate it correctly into the weighted average cost of capital (WACC). Candidates must master the CAPM formula, the economic rationale behind each input, common estimation techniques, and the most frequent exam traps regarding beta adjustment, risk-free rate selection, and capital structure weights.

II. The Problem

A manufacturing firm is evaluating a new production line with risk similar to its existing operations. The firm’s target capital structure is 60% equity and 40% debt. Management needs to know the minimum return that investors require to compensate for the risk of the project. This minimum return is the WACC, and the most difficult component to estimate is the cost of equity. This lesson focuses on using the CAPM to calculate the cost of equity and demonstrates how to integrate it properly into the WACC calculation for project appraisal and valuation.

III. Fundamental Concept of Cost of Equity

The cost of equity ($r_e$) is the rate of return that equity investors require to compensate them for the risk they bear by owning the company’s shares. It is an opportunity cost: the expected return investors could earn by investing in securities of comparable risk. In the WACC formula, the cost of equity usually carries the largest weight; therefore, even small errors in $r_e$ can materially affect the final discount rate and investment decisions.

IV. Core Logic of the CAPM

The Capital Asset Pricing Model (CAPM) is the primary method required at CFA Level I for estimating the cost of equity. Its central idea is that the expected return on any asset equals the risk-free rate plus compensation for systematic (non-diversifiable) risk only.

The CAPM formula is: $$ r_e = r_f + \beta_i \times (r_m - r_f) $$ where: - $r_f$ = risk-free rate - $\beta_i$ = beta of the security - $r_m - r_f$ = equity market risk premium (ERP)

Only systematic risk, measured by beta, is rewarded. Unsystematic risk can be eliminated through diversification and therefore commands no risk premium.

V. Practical Estimation of CAPM Inputs

1. Risk-Free Rate ($r_f$)

The risk-free rate should match the duration of the cash flows being discounted. In CFA exams, the 10-year or 20-year government bond yield is most commonly used. The key principle is term matching: the risk-free rate and the equity risk premium must have consistent time horizons.

2. Beta ($\beta$)

Beta measures the sensitivity of a stock’s returns to market returns. It is calculated as: $$ \beta_i = \frac{\text{Cov}(R_i, R_m)}{\text{Var}(R_m)} $$ In practice, beta is obtained by regressing the stock’s historical excess returns against the market’s excess returns. Raw historical betas are often adjusted toward 1.0 using the Blume adjustment: $$ \text{Adjusted }\beta = \frac{2}{3} \times \text{Historical }\beta + \frac{1}{3} \times 1.0 $$ This adjustment reflects the empirical tendency of betas to revert to the market mean of 1.0 over time.

3. Equity Market Risk Premium (ERP)

Common estimation approaches include: - Historical method (arithmetic or geometric average of past excess returns) - Supply-side models - Survey methods

For discounting purposes, the arithmetic historical average is generally preferred because it better represents the expected value of returns. CFA exam vignettes usually provide the ERP directly or require candidates to choose the appropriate average.

VI. Integrating Cost of Equity into WACC

The after-tax WACC formula is: $$ \text{WACC} = w_e \times r_e + w_d \times r_d \times (1 - t) $$ where $r_e$ is obtained from the CAPM, $w_e$ and $w_d$ are the target capital structure weights (not book or current market weights unless they equal the target), $r_d$ is the before-tax cost of debt, and $t$ is the marginal tax rate. Debt cost must be adjusted for the tax shield provided by interest deductibility.

Worked Cases

Case 1: Basic CAPM Calculation

A company has a historical beta of 1.2. The 10-year government bond yield is 3.5% and the equity risk premium is 6.0%. Calculate the cost of equity.

Solution: $$ r_e = 3.5\% + 1.2 \times 6.0\% = 3.5\% + 7.2\% = 10.7\% $$

Case 2: Adjusted Beta and Full WACC

A firm’s target capital structure is 55% equity and 45% debt. Historical beta is 1.35, risk-free rate is 4.0%, ERP is 5.5%, before-tax cost of debt is 7%, and tax rate is 25%. First adjust beta, then compute WACC.

Adjusted beta: $$ \text{Adjusted }\beta = \frac{2}{3} \times 1.35 + \frac{1}{3} \times 1.0 = 0.900 + 0.333 = 1.233 $$

Cost of equity: $$ r_e = 4.0\% + 1.233 \times 5.5\% = 4.0\% + 6.78\% = 10.78\% $$

WACC: $$ \text{WACC} = 0.55 \times 10.78\% + 0.45 \times 7\% \times (1 - 0.25) = 5.929\% + 2.3625\% = 8.2915\% \approx 8.29\% $$

Case 3: Parameter Selection Trap

Analyst A uses the 3-month T-bill rate of 2.1% as $r_f$ and a geometric-average ERP of 4.8%. Analyst B uses the 10-year Treasury yield of 3.8% and an arithmetic-average ERP of 6.2%. The stock beta is 1.1. Which analyst’s cost of equity estimate is more appropriate for valuing a long-term project?

Calculations: - Analyst A: $2.1\% + 1.1 \times 4.8\% = 7.38\%$ - Analyst B: $3.8\% + 1.1 \times 6.2\% = 10.62\%$

Analyst B is more appropriate. Long-term projects require a long-term risk-free rate paired with a consistent long-term ERP. The arithmetic average is also the preferred measure for discount-rate estimation.

Traps

Key Formulas

• CAPM: $r_e = r_f + \beta \times (r_m - r_f)$
• Blume-adjusted beta: $\text{Adjusted }\beta = \frac{2}{3}\beta_h + \frac{1}{3} \times 1$
• WACC: $w_e \times r_e + w_d \times r_d \times (1-t)$
• Beta definition: $\beta_i = \frac{\text{Cov}(R_i,R_m)}{\text{Var}(R_m)}$
• Levered beta relation (introduced later): $\beta_L = \beta_U \times [1 + (1-t)\frac{D}{E}]$

Practice Questions

Q1. When using the CAPM to calculate the cost of equity, the most appropriate risk-free rate is:
A. Always the 3-month T-bill rate
B. The government bond yield that matches the duration of the cash flows being valued
C. Always the 30-year government bond yield
D. The company’s own long-term bond yield

Q2. A stock has a historical beta of 1.4. Using the Blume adjustment, the adjusted beta is closest to:
A. 1.27
B. 1.40
C. 1.00
D. 1.60

Q3. Which of the following is NOT an assumption of the CAPM?
A. Investors are risk averse
B. All investors have homogeneous expectations
C. Unlimited risk-free borrowing and lending is possible
D. The market risk premium equals zero

Q4. A company’s target capital structure is 40% equity and 60% debt. Given $r_e = 12\%$, before-tax $r_d = 8\%$, and a tax rate of 30%, the WACC is closest to:
A. 7.68%
B. 8.00%
C. 9.12%
D. 10.40%

Q5. With a beta of 0.8, risk-free rate of 4%, and equity risk premium of 7%, the cost of equity is:
A. 9.6%
B. 5.6%
C. 11.2%
D. 4.0%

Q6. For estimating the equity risk premium to be used in discount rates, CFA curriculum most commonly recommends:
A. Geometric average only
B. Arithmetic average
C. Next year’s expected excess return only
D. Corporate credit spreads

Q7. If a stock has a beta of –0.3, its cost of equity according to the CAPM:
A. Must be higher than the risk-free rate
B. May be lower than the risk-free rate
C. Equals the risk-free rate
D. Cannot be calculated using CAPM

Q8. Which statement about capital structure weights in WACC is correct?
A. Current market-value weights should always be used
B. Weights should reflect management’s long-term target capital structure
C. Book-value weights are superior to market-value weights
D. Weights are irrelevant to the WACC calculation

Answers

Takeaways

• The CAPM formula $r_e = r_f + \beta(r_m - r_f)$ must be memorized and applied precisely.
• Always match the maturity of the risk-free rate and the equity risk premium.
• Apply the Blume adjustment to historical beta unless the vignette states otherwise.
• Use target capital structure weights and after-tax cost of debt in the WACC formula.
• Arithmetic historical ERP is generally more appropriate than geometric for discounting.
• A negative beta is theoretically acceptable and results in $r_e < r_f$.