Cost of Capital Estimation
Estimating the Cost of Equity and the Weighted Average Cost of Capital (WACC)
1 Introduction
When a firm considers undertaking a new investment—whether building a factory, launching a product, or acquiring another company—it needs a benchmark to evaluate whether that investment is worthwhile. This benchmark is the cost of capital: the minimum return the firm must earn to satisfy its investors. If a project cannot clear this hurdle, the firm destroys value by pursuing it.
The cost of capital matters because it represents the opportunity cost of the funds being deployed. Shareholders could invest their money elsewhere—in other stocks, bonds, or projects with similar risk profiles. Bondholders lent money expecting a certain return. If the firm cannot generate returns that compensate all these investors appropriately, capital will flow elsewhere, and the firm’s value will decline.
This lecture focuses on estimating two critical components of the cost of capital. First, we examine how to estimate the cost of equity—what shareholders require to compensate them for bearing the firm’s risk. We’ll use both the Capital Asset Pricing Model (CAPM) and the Fama-French three-factor model to derive these estimates. Second, we’ll learn how to combine the cost of equity with the cost of debt to calculate the weighted average cost of capital (WACC), which represents the overall return a firm must earn on its assets.
Along the way, we’ll confront a challenge that often gets overlooked: the inputs to these models—particularly the equity risk premium and factor premia—are notoriously difficult to estimate precisely. Small differences in these estimates can dramatically change our conclusions about whether a project creates or destroys value.
2 Cost of Equity Using the CAPM
The Capital Asset Pricing Model provides a theoretically elegant framework for estimating the cost of equity. Under CAPM, the expected return on any asset depends on just one factor: its sensitivity to the overall market portfolio. The model states:
\[E[R_{i}] = R_{f} + \beta_i \times (E[R_{m}] - R_{f})\]
where:
- \(E[R_i]\) = the expected return on asset \(i\) (this is the cost of equity we’re trying to estimate)
- \(R_f\) = the risk-free rate
- \(\beta_i\) = the firm’s market beta, measuring its sensitivity to market movements
- \(E[R_m]\) = the expected return on the market portfolio
- \(E[R_m] - R_f\) = the equity risk premium (also called the market risk premium)
The intuition behind this formula is straightforward. Investors start with the risk-free rate as their baseline—this is what they could earn with zero risk. The second term compensates them for bearing systematic risk. If a stock has a beta of 1.2, it amplifies market movements by 20%, making it riskier than the average stock. Investors demand a proportionally higher premium.
2.1 Estimating the Risk-Free Rate
The risk-free rate should come from Treasury securities, as these carry essentially no default risk. The key decision is matching the horizon of your analysis with the maturity of the Treasury instrument. If you’re estimating the cost of equity for a project with a 10-year life, use the yield on 10-year Treasury notes. For shorter-horizon analyses, use Treasury bills of corresponding maturity.
This matching principle matters because the yield curve is typically upward sloping—longer-maturity Treasuries offer higher yields to compensate for interest rate risk. Using a short-term rate when evaluating a long-term project would understate the appropriate risk-free benchmark.
2.2 Estimating Beta
Beta measures how a stock’s returns co-move with the market. A beta of 1.0 means the stock moves in lockstep with the market. A beta greater than 1.0 indicates the stock amplifies market movements (more volatile, more systematic risk), while a beta less than 1.0 indicates dampened movements.
To estimate beta, we regress the firm’s historical excess returns on the market’s historical excess returns:
\[R_{i,t} - R_{f,t} = \alpha_i + \beta_i \times (R_{m,t} - R_{f,t}) + \varepsilon_{i,t}\]
The slope coefficient from this regression is our beta estimate. In practice, analysts typically use 3-5 years of monthly returns, though choices vary. The S&P 500 is commonly used as a proxy for the market portfolio.
One important consideration: beta estimates are noisy and can change over time as firms evolve. A technology startup that matures into a stable business will likely see its beta decline. For this reason, many analysts adjust raw beta estimates toward 1.0 (the market average) using formulas like the Bloomberg adjustment: \(\beta_{adjusted} = 0.67 \times \beta_{raw} + 0.33 \times 1.0\).
3 Cost of Equity Using the Fama-French Three-Factor Model
The CAPM’s simplicity is both its strength and its weakness. While theoretically elegant, decades of empirical research have documented that beta alone doesn’t fully explain cross-sectional differences in stock returns. Smaller firms and firms with high book-to-market ratios (value stocks) have historically earned higher returns than the CAPM predicts.
The Fama-French three-factor model addresses these anomalies by adding two additional risk factors:
\[E[R_i] = R_f + \beta_{i,m} \times (E[R_m] - R_f) + \beta_{i,SMB} \times E[R_{SMB}] + \beta_{i,HML} \times E[R_{HML}]\]
where:
- \(\beta_{i,m}\) = the firm’s sensitivity to the market factor (same concept as CAPM beta)
- \(\beta_{i,SMB}\) = the firm’s sensitivity to the size factor (Small Minus Big)
- \(\beta_{i,HML}\) = the firm’s sensitivity to the value factor (High Minus Low book-to-market)
- \(E[R_{SMB}]\) = the expected premium on the SMB factor
- \(E[R_{HML}]\) = the expected premium on the HML factor
The SMB factor captures the historical tendency of small-cap stocks to outperform large-cap stocks. The HML factor captures the tendency of value stocks (high book-to-market) to outperform growth stocks (low book-to-market). By including these factors, the model provides a richer description of risk and return.
3.1 Estimating the Factor Betas
To use the Fama-French model, we run a multiple regression of the firm’s excess returns on all three factors simultaneously:
\[R_{i,t} - R_{f,t} = \alpha_i + \beta_{i,m}(R_{m,t} - R_{f,t}) + \beta_{i,SMB} \times R_{SMB,t} + \beta_{i,HML} \times R_{HML,t} + \varepsilon_{i,t}\]
The coefficients from this regression give us the three factor loadings. Kenneth French’s website provides freely downloadable time series of the SMB and HML factors, making this estimation straightforward.
3.2 Estimating the Factor Premia
Just as with the equity risk premium, we must estimate the expected returns on the SMB and HML factors. The standard approach is historical averaging over long periods—typically 20 years or more to capture multiple economic cycles.
Historical averages suggest an SMB premium of roughly 2-3% and an HML premium of roughly 3-5% annually, though these estimates vary considerably depending on the sample period and geography. Recent decades have seen much weaker performance from both factors in U.S. markets, leading some to question whether these premia will persist.
The SMB and HML premia are even more uncertain than the equity risk premium. Consider the range of estimates:
| Factor | Long-Run Historical Average (1926-present) | Recent 20-Year Average | Difference |
|---|---|---|---|
| SMB | ~2.5% | ~0.5% | 2.0% |
| HML | ~4.0% | ~-1.5% | 5.5% |
The HML factor has actually been negative in recent decades—value stocks have underperformed growth stocks. Should we use long historical averages, assuming mean reversion? Or should we weight recent data more heavily, thinking the world has changed?
This uncertainty means Fama-French cost of equity estimates can differ substantially depending on which factor premia you assume. For a firm with high SMB and HML loadings, using historical versus recent premia could change the cost of equity estimate by several percentage points.
Consider a small biotechnology company. After running the three-factor regression, you estimate the following factor loadings:
- \(\beta_{m}\) = 1.35
- \(\beta_{SMB}\) = 0.80 (positive exposure to size factor—behaves like a small-cap stock)
- \(\beta_{HML}\) = -0.25 (negative exposure to value factor—behaves like a growth stock)
You also gather these inputs:
- Risk-free rate: 4.0%
- Equity risk premium: 6.5%
- SMB premium: 2.5%
- HML premium: 4.0%
What is the cost of equity according to the Fama-French model?
Using the Fama-French formula:
\[E[R_i] = R_f + \beta_{i,m}(E[R_m] - R_f) + \beta_{i,SMB} \times E[R_{SMB}] + \beta_{i,HML} \times E[R_{HML}]\]
Substituting our values:
\[E[R_i] = 4.0\% + 1.35 \times 6.5\% + 0.80 \times 2.5\% + (-0.25) \times 4.0\%\]
Breaking this down:
- Risk-free component: \(4.0\%\)
- Market premium component: \(1.35 \times 6.5\% = 8.775\%\)
- SMB component: \(0.80 \times 2.5\% = 2.0\%\)
- HML component: \(-0.25 \times 4.0\% = -1.0\%\)
\[E[R_i] = 4.0\% + 8.775\% + 2.0\% - 1.0\% = 13.775\%\]
The Fama-French cost of equity is approximately 13.8%.
Notice several things: First, the market beta of 1.35 substantially increases the required return above the market. Second, the positive SMB loading adds to the cost of equity—this small firm bears size-related risk. Third, the negative HML loading actually reduces the cost of equity—this growth stock has negative exposure to value risk. Compare this to what CAPM alone would give: \(4.0\% + 1.35 \times 6.5\% = 12.775\%\). The Fama-French model yields a higher estimate because the size factor effect dominates the value factor effect.
4 The Weighted Average Cost of Capital (WACC)
So far, we’ve focused on the cost of equity—what shareholders require. But most firms finance themselves with a mix of equity and debt. The weighted average cost of capital combines the costs of different financing sources, weighted by their proportions in the firm’s capital structure.
The standard WACC formula is:
\[WACC = \frac{E}{V} \times r_E + \frac{D}{V} \times r_D \times (1 - T_c)\]
where:
- \(E\) = market value of equity
- \(D\) = market value of debt
- \(V = E + D\) = total firm value
- \(r_E\) = cost of equity (estimated using CAPM or Fama-French)
- \(r_D\) = cost of debt (yield to maturity on the firm’s bonds)
- \(T_c\) = corporate tax rate
The \((1 - T_c)\) term reflects the tax deductibility of interest payments. Because interest expense reduces taxable income, the after-tax cost of debt is lower than the stated interest rate. This tax shield is one reason firms use debt financing.
4.1 Why WACC Matters
WACC serves as the discount rate for valuing projects with risk similar to the firm’s overall operations. When a firm evaluates a new investment, it should compare the project’s expected return to its WACC. Projects earning more than the WACC create value; projects earning less destroy it.
This makes intuitive sense. The WACC represents what the firm pays to obtain capital from all sources. If a project can’t clear this hurdle, the firm would be better off returning the money to investors, who could earn their required returns elsewhere.
4.2 Estimating the Component Costs
Cost of equity (\(r_E\)): Use CAPM or Fama-French as discussed above.
Cost of debt (\(r_D\)): For firms with publicly traded bonds, use the yield to maturity on those bonds. For firms without traded debt, you can estimate the cost of debt by adding a credit spread to the risk-free rate based on the firm’s credit rating.
| Credit Rating | Typical Spread Over Treasuries |
|---|---|
| AAA | 0.5% - 0.8% |
| AA | 0.8% - 1.2% |
| A | 1.2% - 1.8% |
| BBB | 1.8% - 2.5% |
| BB | 2.5% - 4.0% |
| B | 4.0% - 6.0% |
Capital structure weights: Use market values, not book values. For equity, this is straightforward—multiply shares outstanding by the stock price. For debt, the market value of traded bonds is ideal, though book value is often used as an approximation if bonds don’t trade frequently.
A manufacturing company has the following capital structure and costs:
- Market value of equity: $800 million
- Market value of debt: $200 million
- Cost of equity (estimated via CAPM): 11.0%
- Yield to maturity on the company’s bonds: 6.0%
- Corporate tax rate: 25%
Calculate the company’s WACC.
First, calculate the total firm value and capital structure weights:
\[V = E + D = 800 + 200 = 1{,}000 \text{ million}\]
\[\frac{E}{V} = \frac{800}{1{,}000} = 0.80 = 80\%\]
\[\frac{D}{V} = \frac{200}{1{,}000} = 0.20 = 20\%\]
Now apply the WACC formula:
\[WACC = \frac{E}{V} \times r_E + \frac{D}{V} \times r_D \times (1 - T_c)\]
\[WACC = 0.80 \times 11.0\% + 0.20 \times 6.0\% \times (1 - 0.25)\]
\[WACC = 8.8\% + 0.20 \times 6.0\% \times 0.75\]
\[WACC = 8.8\% + 0.9\%\]
\[WACC = 9.7\%\]
The company’s weighted average cost of capital is 9.7%. This is the minimum return the firm should require from new investments with similar risk to its existing operations. Notice how the WACC (9.7%) is lower than the cost of equity (11.0%) because the firm benefits from cheaper debt financing and the interest tax shield.
Using the same company from above, how would WACC change if the firm shifted to 50% debt and 50% equity financing? Assume the cost of equity rises to 12.5% and the cost of debt rises to 7.0% due to higher financial risk.
With the new capital structure:
\[\frac{E}{V} = 0.50 \quad \text{and} \quad \frac{D}{V} = 0.50\]
Applying the WACC formula with updated costs:
\[WACC = 0.50 \times 12.5\% + 0.50 \times 7.0\% \times (1 - 0.25)\]
\[WACC = 6.25\% + 0.50 \times 7.0\% \times 0.75\]
\[WACC = 6.25\% + 2.625\%\]
\[WACC = 8.875\%\]
The WACC decreases to approximately 8.9% with higher leverage. Even though both the cost of equity and cost of debt increased (reflecting higher financial risk), the greater weight on tax-advantaged debt more than compensates. This illustrates why firms use debt—it can reduce WACC up to a point. However, too much debt eventually increases financial distress costs, and both \(r_E\) and \(r_D\) rise sharply, eventually increasing WACC.
5 Key Takeaways
Estimating the cost of capital is both essential and inherently uncertain. The CAPM provides a simple, theoretically grounded approach: the cost of equity equals the risk-free rate plus a premium for bearing market risk, scaled by the firm’s beta. This framework makes clear that only systematic risk—risk correlated with the market—commands a premium, because diversified investors can eliminate idiosyncratic risk.
The Fama-French three-factor model extends this logic by recognizing that market beta doesn’t capture all sources of systematic risk. By adding size and value factors, the model better explains observed patterns in stock returns. For many firms—particularly small caps and value stocks—the Fama-French model yields meaningfully different cost of equity estimates than CAPM alone.
Throughout these calculations, we confront estimation uncertainty. The equity risk premium, which drives the largest component of the cost of equity, varies significantly depending on methodology and sample period. Estimates range from 4% to 8%, and each percentage point flows directly into the cost of equity. The SMB and HML premia add further uncertainty, with recent performance diverging sharply from historical averages. Thoughtful analysts recognize these limitations and use sensitivity analysis to understand how their conclusions depend on these assumptions.
For most corporate finance applications, we need the weighted average cost of capital, not just the cost of equity. WACC blends the costs of equity and debt according to their proportions in the firm’s capital structure, with debt receiving a tax benefit due to interest deductibility. WACC serves as the hurdle rate for evaluating new investments: projects must exceed this benchmark to create shareholder value. When applying WACC, remember that it’s only appropriate for projects with risk profiles similar to the firm’s existing operations. Significantly riskier or safer projects require adjusted discount rates.
6 Key Formulas Summary
| Concept | Formula | When to Use |
|---|---|---|
| CAPM Cost of Equity | \(E[R_i] = R_f + \beta_i(E[R_m] - R_f)\) | Estimating required return for any asset when you have an estimate of market beta |
| Fama-French Cost of Equity | \(E[R_i] = R_f + \beta_m(E[R_m] - R_f) + \beta_{SMB} \cdot E[R_{SMB}] + \beta_{HML} \cdot E[R_{HML}]\) | Estimating required return when you want to account for size and value risk exposures |
| Beta Regression | \(R_{i,t} - R_{f,t} = \alpha + \beta(R_{m,t} - R_{f,t}) + \varepsilon_t\) | Estimating market beta from historical returns data |
| Weighted Average Cost of Capital | \(WACC = \frac{E}{V}r_E + \frac{D}{V}r_D(1-T_c)\) | Calculating the firm’s overall cost of capital for project evaluation |
| Adjusted Beta | \(\beta_{adj} = 0.67 \times \beta_{raw} + 0.33 \times 1.0\) | Adjusting raw beta estimates to account for mean reversion |
7 Practice Problems
A large pharmaceutical company has a beta of 0.75, reflecting its relatively stable cash flows. The current yield on 10-year Treasury bonds is 4.8%, and you estimate the equity risk premium at 5.5%. Calculate the cost of equity using CAPM.
Using the CAPM formula:
\[E[R_i] = R_f + \beta_i \times (E[R_m] - R_f)\]
\[E[R_i] = 4.8\% + 0.75 \times 5.5\%\]
\[E[R_i] = 4.8\% + 4.125\%\]
\[E[R_i] = 8.925\%\]
The pharmaceutical company’s cost of equity is approximately 8.9%. The below-market beta reflects that pharmaceutical companies tend to be less cyclical than the overall economy—people need medications regardless of economic conditions.
A real estate investment trust (REIT) has the following factor loadings estimated from a three-factor regression:
- \(\beta_m\) = 0.90
- \(\beta_{SMB}\) = -0.15 (negative—behaves like a large-cap stock)
- \(\beta_{HML}\) = 0.45 (positive—behaves like a value stock)
Assume a risk-free rate of 3.5%, equity risk premium of 6.0%, SMB premium of 2.0%, and HML premium of 3.5%. Calculate the cost of equity using the Fama-French model.
Using the Fama-French formula:
\[E[R_i] = R_f + \beta_m(E[R_m] - R_f) + \beta_{SMB} \times E[R_{SMB}] + \beta_{HML} \times E[R_{HML}]\]
\[E[R_i] = 3.5\% + 0.90 \times 6.0\% + (-0.15) \times 2.0\% + 0.45 \times 3.5\%\]
Breaking down each component:
- Risk-free: \(3.5\%\)
- Market: \(0.90 \times 6.0\% = 5.4\%\)
- SMB: \(-0.15 \times 2.0\% = -0.3\%\)
- HML: \(0.45 \times 3.5\% = 1.575\%\)
\[E[R_i] = 3.5\% + 5.4\% - 0.3\% + 1.575\% = 10.175\%\]
The REIT’s cost of equity is approximately 10.2%. The positive value loading (HML) increases the cost of equity because REITs tend to have high book-to-market ratios and thus bear value-related risk. Interestingly, the negative size loading slightly offsets this because the REIT behaves like a large-cap stock.
A technology company has the following characteristics:
- 50 million shares outstanding at $60 per share
- $500 million in bonds trading at par (book value equals market value)
- Cost of equity: 13.0%
- Yield to maturity on bonds: 5.5%
- Corporate tax rate: 21%
Calculate the company’s WACC.
First, calculate market values and weights:
Market value of equity: \[E = 50 \text{ million shares} \times \$60 = \$3{,}000 \text{ million}\]
Market value of debt: \[D = \$500 \text{ million}\]
Total firm value: \[V = E + D = 3{,}000 + 500 = \$3{,}500 \text{ million}\]
Weights: \[\frac{E}{V} = \frac{3{,}000}{3{,}500} = 0.857 = 85.7\%\] \[\frac{D}{V} = \frac{500}{3{,}500} = 0.143 = 14.3\%\]
WACC calculation: \[WACC = \frac{E}{V} \times r_E + \frac{D}{V} \times r_D \times (1 - T_c)\]
\[WACC = 0.857 \times 13.0\% + 0.143 \times 5.5\% \times (1 - 0.21)\]
\[WACC = 11.14\% + 0.143 \times 5.5\% \times 0.79\]
\[WACC = 11.14\% + 0.62\%\]
\[WACC = 11.76\%\]
The technology company’s WACC is approximately 11.8%. Notice that because this firm uses relatively little debt (only 14.3% of capital), the WACC is close to the cost of equity. The tax shield provides only modest benefit given the low debt weight.
Using the pharmaceutical company from Practice Problem 1 (beta = 0.75, risk-free rate = 4.8%), calculate how much the cost of equity changes if the equity risk premium estimate is 7.0% instead of 5.5%.
With ERP = 5.5%: \[E[R_i] = 4.8\% + 0.75 \times 5.5\% = 4.8\% + 4.125\% = 8.925\%\]
With ERP = 7.0%: \[E[R_i] = 4.8\% + 0.75 \times 7.0\% = 4.8\% + 5.25\% = 10.05\%\]
Change in cost of equity: \[10.05\% - 8.925\% = 1.125\%\]
The cost of equity increases by 1.125 percentage points when the equity risk premium estimate increases by 1.5 percentage points. Note that the change in cost of equity equals \(\beta \times \Delta ERP = 0.75 \times 1.5\% = 1.125\%\). For higher-beta stocks, the sensitivity to ERP estimates would be even greater.
8 Ask an LLM
Here are three questions you might ask an AI assistant to deepen your understanding of these concepts:
- “Can you explain the economic intuition for why small stocks and value stocks might earn higher returns? Is this compensation for risk, or could it be market inefficiency?”
- “What are the main criticisms of using historical averages to estimate the equity risk premium, and what alternative approaches do practitioners use?”
- “If I’m evaluating a project that is significantly riskier than my firm’s typical operations, how should I adjust the WACC to account for this?”
- “How do I estimate beta for a private company that doesn’t have publicly traded stock?”
- “The Fama-French model was developed using U.S. data. Does it work in other countries, and what factors tend to explain returns globally?”