CAPM and WACC Concept Card Template (Beta, Cost of Capital)

CAPM and WACC Concept Card: [Your course, case or company]

A concept card for the cost of capital: how the capital asset pricing model (CAPM) prices equity risk, how to estimate and adjust beta, how to unlever and relever beta with Hamada (1972) and its alternatives, where the equity risk premium comes from, how to measure the after-tax cost of debt, and how to combine them into a weighted average cost of capital (WACC) at target market-value weights. It is for MBA students in the core finance course, students preparing valuation or banking interviews, and managers who set hurdle rates. Worked example A builds a WACC for a fictional growth company step by step, Worked example B (half solved) estimates beta from returns, and Exercise C is left for you, with a separate answer key. Source basis: Markowitz (1952), Sharpe (1964), Lintner (1965), Mossin (1966), Modigliani and Miller (1958, 1963), Hamada (1972), Miles and Ezzell (1980), Harris and Pringle (1985), Blume (1971) and Fama and French (1993, 2004), written up in original words. Every market input in the examples (risk-free rate, equity risk premium, betas) is an invented Example value, not a current market figure. Replace every [bracketed] value and every row marked Example. This is a learning aid, not investment advice.

Document control

Contents

• When to use it
• The idea in plain language
• Formulas and notation
• Assumptions and limits
• Estimating each input
• Worked example A (fully solved): Corvane Robotics WACC
• Worked example B (half solved): Northfield beta from returns
• Exercise C (unsolved): Northfield WACC
• Process diagram
• How managers use the cost of capital
• Connections to other cards
• Common mistakes and misconceptions
• Self-test questions
• My notes and next steps
• Sources and further reading
• Answer key
• Revision history

When to use it

One-sentence rule: discount free cash flows to the firm at the WACC of a business with the same operating risk and target financing, where the cost of equity comes from the CAPM and the cost of debt is the after-tax yield that lenders demand today.

The idea in plain language

Portfolio theory. Markowitz (1952) showed that an investor who holds many assets cares about how each asset moves with the rest of the portfolio, not about its risk in isolation. Much of a single stock's volatility is company-specific (a product recall, a lost contract) and cancels out in a diversified portfolio. What cannot be diversified away is the exposure to market-wide shocks: recessions, interest-rate moves, shifts in risk appetite.

CAPM. Sharpe (1964), Lintner (1965) and Mossin (1966) derived the consequence for prices. If investors hold diversified portfolios, an asset's expected return depends only on its beta: the sensitivity of its return to the market's return. A beta of 1 moves with the market; 1.5 amplifies market moves by half again; 0.5 dampens them. The expected return equals the risk-free rate plus beta times the market's excess return (the equity risk premium). Plotted against beta, this relationship is the security market line.

Leverage changes beta. Modigliani and Miller (1958) showed that, without taxes and frictions, borrowing does not change the value of the firm's assets, but it concentrates the asset risk on the shareholders. A company's equity beta therefore rises with debt. To compare companies, you strip out leverage (unlever to an asset beta), average the business risk across peers, and then relever at the target company's own capital structure (Hamada 1972).

WACC. A firm is financed by equity holders and lenders. The WACC is the return the firm must earn on its assets to pay both their required returns, weighted by their shares of the firm's market value, with the cost of debt reduced by the tax saving on interest (Modigliani and Miller 1963). It is an opportunity cost set by capital markets, not an accounting number and not a target chosen by management.

Formulas and notation

Unlevering and relevering beta (three common versions):

Which unlevering formula? Hamada assumes debt is fixed in amount, risk-free and its tax shields are as risky as the debt. Harris and Pringle assume the firm keeps a constant debt-to-value ratio, so the tax shields carry asset risk and the tax rate drops out. Miles and Ezzell (1980) derive the constant-ratio case with annual rebalancing; their adjustment sits between the two and, at normal interest rates, very close to Harris and Pringle. With a debt beta, the formula also works for firms with risky debt. Use one version consistently for unlevering and relevering; the choice usually moves beta by less than the estimation error in beta itself (Worked example A shows the difference).

Assumptions and limits

1. Investors hold diversified portfolios and care only about mean and variance of returns (Markowitz 1952).
2. They can borrow and lend at one risk-free rate, have the same expectations and face no taxes or transaction costs. None of this is literally true; the model is a benchmark.
3. The market portfolio is observable. In practice a broad equity index stands in for it.
4. Beta is stable over the estimation window. Mergers, new business lines and leverage changes break this.
5. Empirical record. Tests from the 1970s onward found the security market line flatter than the CAPM predicts, and Fama and French (1993) showed that size and value factors explain returns that beta does not. Fama and French (2004) review the evidence. The CAPM remains the most common cost-of-equity method in practice (Graham and Harvey 2001) because it is simple, transparent and hard to manipulate, but treat its output as an estimate with a range.
6. WACC assumes a constant target capital structure and that the project or firm has the risk embedded in the peers' asset betas.

Estimating each input

Risk-free rate. Use the yield on a government bond in the same currency as the cash flows, with a maturity close to the duration of the cash flows (long-term valuations commonly use a 10-year yield). Match inflation: nominal cash flows, nominal rate. Where the government itself carries default risk, analysts subtract a default spread; document the method.

Beta.

• Regression beta: regress the stock's returns on the index's returns; the slope is beta (Excel =SLOPE(stock, market)). Common choices are five years of monthly returns or two years of weekly returns. The standard error is large: a beta of 1.3 with a standard error of 0.3 has a 95% interval of roughly 0.7 to 1.9.
• Adjusted beta: raw betas tend to move toward 1 over time (Blume 1971; Vasicek 1973 gave a Bayesian shrinkage version). A widely used commercial convention is adjusted beta = 0.67 x raw beta + 0.33 x 1.
• Bottom-up (peer) beta: unlever the betas of comparable listed companies, take the mean or median asset beta, and relever at the target's capital structure. Averaging across peers reduces estimation error, and it is the only option for private companies, divisions and young firms such as Corvane.

Equity risk premium. There is no single correct number, and it is the most debated input. Three families of estimates exist: (1) historical averages of equity returns over government bonds, which depend heavily on the period, the country and arithmetic versus geometric averaging (Dimson, Marsh and Staunton 2002 compile long-run international data); (2) implied premiums backed out of current index prices and expected cash flows, which Aswath Damodaran (NYU Stern) publishes and updates regularly on his website together with an annual paper on equity risk premiums; (3) surveys of finance professors, analysts and CFOs. Name the source, the method and the date you used, and keep the ERP consistent with the risk-free rate it was measured against. This card does not state a current value: check the named sources at your valuation date. Country risk premiums for emerging-market operations are a further judgment call; Damodaran's published country tables are one named source.

Cost of debt. Use the yield to maturity that the firm would pay on new long-term debt today, not the coupon on old debt and not the book interest expense divided by book debt. Sources: yields on the firm's traded bonds, a synthetic rating from the interest coverage ratio plus the market spread for that rating, or recent bank term sheets. The yield is a promised return; for low-rated debt, the expected return is lower because of default risk, so using the yield overstates the cost of debt. Multiply by $(1 - t_c)$ only to the extent the firm can use the interest deduction (loss-making firms may not, at least for now).

Weights. Use target market-value weights: the capital structure the firm will hold over the life of the cash flows. Look at the firm's current market values, its stated policy and peers' structures. Book values of equity are irrelevant; book value of debt is an acceptable proxy for market value when rates and credit quality are close to issue conditions. Include lease liabilities in debt if the cash flows exclude lease payments (IFRS 16 presents most leases on the balance sheet).

Worked example A (fully solved): Corvane Robotics WACC

Situation (fictional). Corvane Robotics is venture-backed and has no traded shares, so its beta cannot be estimated by regression. The CFO asks for a WACC to evaluate the Atlas production line (mba/mba-npv-irr-card) and to support a future valuation. Corvane plans a target debt-to-equity ratio of 0.25 (debt 20% of value) once it adds term debt, and its marginal tax rate is 25%. Market inputs are Example values for teaching only: risk-free rate 4.2%, equity risk premium 5.0%.

Step 1: unlever peer betas. Three fictional listed warehouse-automation peers:

Hamada for Peer A: $\beta_U = 1.45 / [1 + (1 - 0.25) \times 0.30] = 1.45 / 1.225 = 1.1837$. Excel: =B2/(1+(1-D2)*C2).

Step 2: relever at Corvane's target. $\beta_L = 1.1747 \times [1 + 0.75 \times 0.25] = 1.1747 \times 1.1875 = 1.3950$. The Harris-Pringle version gives 1.3858 and the debt-beta version 1.3936: the spread (about 0.01) is far smaller than the standard error of any single regression beta.

Step 3: cost of equity. $r_E = 4.2\% + 1.3950 \times 5.0\% = 11.17\%$ (exactly 11.1748%). Excel: =rf + beta * erp.

Step 4: cost of debt. A fictional peer with a credit profile similar to Corvane's target has a 5-year bond with a 6.0% annual coupon trading at 96.50 per 100 face. Yield to maturity: =RATE(5, 6, -96.5, 100) = 6.85% (6.8502%). After tax: $6.85\% \times (1 - 0.25) = 5.14\%$ (5.1376%).

Step 5: weights. $D/E = 0.25$ gives $D/V = 0.25/1.25 = 20\%$ and $E/V = 80\%$.

Step 6: WACC.

Excel: =0.8*11.1748% + 0.2*6.8502%*(1-25%). With the Harris-Pringle beta the WACC is 9.93%; with the debt-beta version 9.96%. Two common errors move it more: using book weights (book D/E 0.60, same cost of equity) gives 8.91%, and using the old coupon of 6.0% as the cost of debt gives 9.84%.

Security market line. The chart places the peers and Corvane on the Example security market line ($r_f$ = 4.2%, ERP = 5.0%).

Conclusion. Corvane's Example WACC is about 10.0%, which is the rate used for the Atlas line. Report it with a range: a plus or minus 0.2 change in asset beta moves the cost of equity by about 1.2 percentage points and the WACC by about 0.95 of a point (9.02% to 10.92%).

Worked example B (half solved): Northfield beta from returns

Situation (fictional). Northfield Outdoor Co. is listed. You have 12 monthly returns (in %) for its shares and for a broad market index. (Twelve observations is far too few for real work; it keeps the arithmetic visible.)

1. Sample covariance of $R_i$ and $R_m$ = 3.9368 (=COVARIANCE.S(B2:M2, B3:M3)); sample variance of $R_m$ = [ ] (=VAR.S(B3:M3)).
2. Beta = covariance / variance = [ ] (=SLOPE(B2:M2, B3:M3)).
3. R-squared = 0.6527 (=RSQ(B2:M2, B3:M3)). What share of Northfield's return variance is market risk, and what share is diversifiable?
4. The standard error of the slope is 0.3112 and the t critical value for 10 degrees of freedom is 2.2281. The 95% confidence interval for beta is [ ] to [ ].
5. Adjusted beta with the 0.67 / 0.33 convention = [ ].
6. Cost of equity with the raw beta and with the adjusted beta (Example $r_f$ = 4.2%, ERP = 5.0%) = [ ] and [ ].
7. What would you do before using either number in a valuation?

Exercise C (unsolved): Northfield WACC

Situation (fictional). Northfield has 40.0 million shares trading at USD 18.50. The market value of its debt, including lease liabilities, is USD 185.0m; book equity is USD 310.0m. A bottom-up beta from outdoor-retail peers, relevered at Northfield's structure, is 1.10. New five-year borrowing would cost 5.8% before tax; the marginal tax rate is 24%. Use the Example market inputs $r_f$ = 4.2% and ERP = 5.0%.

1. Compute the market value of equity and the market-value weights.
2. Compute the cost of equity and the after-tax cost of debt.
3. Compute the WACC, with the Excel formula.
4. Recompute the WACC wrongly with book equity instead of market equity. Which direction does the error go, and why?
5. The bottom-up beta (1.10) differs from the regression beta in Worked example B. Which would you use, and why?

Process diagram

The flow below is the order in which an analyst builds the cost of capital, with the main judgment call at each step shown in the box.

How managers use the cost of capital

• Hurdle rates. Firms set a hurdle at or above the WACC for projects of average risk. Using one company-wide rate for every project causes systematic errors: safe projects are rejected and risky ones accepted. Divisional rates based on pure-play peers fix most of this.
• Valuation. In a DCF the WACC discounts free cash flow to the firm (mba/mba-dcf-valuation-card). Small changes matter: at a long-run growth rate of 3%, moving WACC from 10% to 9% raises the terminal value multiple from $1/(0.10 - 0.03) = 14.3$ to $1/(0.09 - 0.03) = 16.7$ times next year's cash flow.
• Performance measurement. Return on invested capital above WACC means the business creates value; economic profit (also sold as EVA) equals invested capital times the spread.
• Impairment tests and regulation. Accounting impairment tests discount at a pre-tax rate derived from a market-participant WACC (finance/impairment-test-memo), and regulators of utilities set allowed returns from a WACC built on the same logic.
• Capital structure decisions. The WACC shortcut assumes a fixed target; when financing changes the value materially, compare with APV (mba/mba-capital-structure-card).

Connections to other cards

• mba/mba-npv-irr-card uses the WACC as the discount rate for project decisions.
• mba/mba-dcf-valuation-card uses it for free cash flow to the firm and the terminal value.
• mba/mba-capital-structure-card explains why WACC falls and then rises with leverage, the trade-off and pecking-order views, and APV.
• mba/mba-regression-for-managers-card and academic/stats-linear-regression cover the regression behind beta, including standard errors and R-squared.
• mba/mba-corporate-finance-guide places this card in sessions 5 and 6.
• finance/comparable-companies-analysis and finance/valuation-report are working formats that need a documented WACC.

Common mistakes and misconceptions

1. Using book-value weights. Weights are market values at the target structure; book equity is a historical accounting figure.
2. Using the coupon rate or book interest as the cost of debt. Use today's yield for the firm's credit quality.
3. Forgetting the tax adjustment, or applying it when the firm pays no tax. The shield exists only if the deduction can be used.
4. Mismatching currencies or inflation. A USD risk-free rate with EUR cash flows, or a nominal rate with real cash flows, produces a wrong value of unknown sign.
5. Using the firm's WACC for a project of different risk. A low-risk utility unit and a venture arm need different rates.
6. Relevering at the peers' leverage instead of the target's. Unlever with each peer's D/E, relever with the target company's.
7. Treating the regression beta as precise. With a standard error of 0.3, a second decimal is noise; use peer averages and adjustment.
8. Adding arbitrary premia. Ad hoc "company-specific" premia for risks that are diversifiable, or already in the cash-flow forecast, double count. Put specific risks in scenarios.
9. Using the venture capital target return as the WACC. A 30% VC target reflects optimistic plans and illiquidity, not the cost of capital for expected cash flows.
10. Quoting an equity risk premium without a source and date. Name the source and method; ERP estimates differ by several percentage points across methods.
11. Mixing a short-term risk-free rate with long-term cash flows. Match maturity roughly to the cash-flow duration.
12. Believing more debt always lowers WACC. Cheaper debt raises the cost of equity; beyond some point distress costs raise both.

Self-test questions

1. Why does the CAPM reward only systematic risk? Use the word "diversification" in your answer.
2. Write the CAPM. A stock has beta 0.8, $r_f$ = 3.5% and ERP = 5.5% (Example values). What is its required return?
3. A peer has equity beta 1.6, D/E = 0.6 and tax rate 25%. Compute its Hamada asset beta.
4. Relever an asset beta of 0.9 at D/E = 1.0 and tax rate 20% using Hamada. Then do it with Harris-Pringle (zero debt beta). Why do the two differ?
5. A company's bonds have a 4% coupon and yield 7.5% today. Which rate goes into the WACC, and why?
6. Name three sources or methods for the equity risk premium and one weakness of each.
7. Why does using book weights usually understate the WACC of a growing, profitable company?
8. A conglomerate has a WACC of 8%. Its new biotech venture has peers with an average asset beta twice the conglomerate's. What rate should it use for the venture, in principle?
9. Give two reasons a regression beta from a 12-month window is unreliable.
10. Interleaving: which rate discounts each cash flow? (a) Free cash flow to the firm. (b) Dividends to shareholders. (c) A tax shield on debt that is fixed in amount and as safe as the debt.

My notes and next steps

Sources and further reading

• Markowitz, H. (1952). Portfolio selection. The Journal of Finance, 7(1), 77-91.
• Modigliani, F., and Miller, M. H. (1958). The cost of capital, corporation finance and the theory of investment. The American Economic Review, 48(3), 261-297.
• Modigliani, F., and Miller, M. H. (1963). Corporate income taxes and the cost of capital: a correction. The American Economic Review, 53(3), 433-443.
• Sharpe, W. F. (1964). Capital asset prices: a theory of market equilibrium under conditions of risk. The Journal of Finance, 19(3), 425-442.
• Lintner, J. (1965). The valuation of risk assets and the selection of risky investments in stock portfolios and capital budgets. The Review of Economics and Statistics, 47(1), 13-37.
• Mossin, J. (1966). Equilibrium in a capital asset market. Econometrica, 34(4), 768-783.
• Blume, M. E. (1971). On the assessment of risk. The Journal of Finance, 26(1), 1-10.
• Hamada, R. S. (1972). The effect of the firm's capital structure on the systematic risk of common stocks. The Journal of Finance, 27(2), 435-452.
• Vasicek, O. A. (1973). A note on using cross-sectional information in Bayesian estimation of security betas. The Journal of Finance, 28(5), 1233-1239.
• Miles, J. A., and Ezzell, J. R. (1980). The weighted average cost of capital, perfect capital markets, and project life: a clarification. Journal of Financial and Quantitative Analysis, 15(3), 719-730.
• Harris, R. S., and Pringle, J. J. (1985). Risk-adjusted discount rates: extensions from the average-risk case. Journal of Financial Research, 8(3), 237-244.
• Fama, E. F., and French, K. R. (1993). Common risk factors in the returns on stocks and bonds. Journal of Financial Economics, 33(1), 3-56.
• Fama, E. F., and French, K. R. (2004). The capital asset pricing model: theory and evidence. Journal of Economic Perspectives, 18(3), 25-46.
• Graham, J. R., and Harvey, C. R. (2001). The theory and practice of corporate finance: evidence from the field. Journal of Financial Economics, 60(2-3), 187-243.
• Dimson, E., Marsh, P., and Staunton, M. (2002). Triumph of the Optimists: 101 Years of Global Investment Returns. Princeton University Press.
• Damodaran, A. Equity risk premium data (implied premiums, country risk premiums) and the annually updated paper on equity risk premiums, pages.stern.nyu.edu/~adamodar (named as a source; no value quoted here; check the current update at your valuation date).
• Textbooks (by name only): Berk, J., and DeMarzo, P., Corporate Finance; Brealey, R. A., Myers, S. C., and Allen, F., Principles of Corporate Finance; Koller, T., Goedhart, M., and Wessels, D., Valuation: Measuring and Managing the Value of Companies.
• Open resource: OpenStax, Principles of Finance (CC BY-NC-SA 4.0), chapters on risk and return and on the cost of capital.

Answer key

Worked example B. (1) Variance of $R_m$ = 2.9184. (2) Beta = $3.9368 / 2.9184 = 1.3490$ (=SLOPE gives 1.348960). (3) About 65% of the variance is explained by the market; about 35% is company-specific and diversifiable. (4) $1.3490 \pm 2.2281 \times 0.3112$ gives 0.6557 to 2.0422: very wide. (5) $0.67 \times 1.3490 + 0.33 = 1.2338$. (6) Raw: $4.2\% + 1.3490 \times 5.0\% = 10.94\%$; adjusted: $4.2\% + 1.2338 \times 5.0\% = 10.37\%$. (7) Use a longer window (60 months or 104 weeks), compare with a peer-based bottom-up beta, and check that no event (acquisition, leverage change) distorts the window.

Exercise C. (1) $E = 40.0 \times 18.50 =$ USD 740.0m; $V = 740.0 + 185.0 = 925.0$; $E/V = 80\%$, $D/V = 20\%$. (2) $r_E = 4.2\% + 1.10 \times 5.0\% = 9.70\%$; after-tax $r_D = 5.8\% \times 0.76 = 4.408\%$. (3) $WACC = 0.8 \times 9.70\% + 0.2 \times 4.408\% = 8.64\%$ (8.6416%); Excel =0.8*9.7%+0.2*5.8%*(1-24%). (4) With book equity of 310.0, $E/V = 310/495 = 62.6\%$, which gives more weight to cheap debt and understates the WACC (about 7.72%). Market equity is above book for a profitable company, so book weights understate the equity share. (5) Prefer the bottom-up beta: it averages several firms, so its error is smaller, and it reflects the target structure; the 12-month regression has a 95% interval from about 0.66 to 2.04.

Self-test.

1. Company-specific risk can be eliminated at almost no cost by diversification, so the market does not pay a premium for bearing it; only risk that remains in a diversified portfolio (market risk, measured by beta) is priced.
2. $E[R] = r_f + \beta \times ERP = 3.5\% + 0.8 \times 5.5\% = 7.9\%$.
3. $\beta_U = 1.6 / (1 + 0.75 \times 0.6) = 1.6 / 1.45 = 1.1034$.
4. Hamada: $0.9 \times (1 + 0.8 \times 1.0) = 1.62$. Harris-Pringle: $0.9 \times (1 + 1.0) = 1.80$. Hamada assumes a fixed amount of debt whose tax shields are as safe as the debt; Harris-Pringle assumes debt rebalanced to a constant ratio, so the tax shields carry business risk and do not reduce equity risk.
5. The 7.5% yield: it is what lenders require today for this credit risk. The coupon reflects conditions at issue.
6. Historical (sensitive to period and averaging method), implied from prices (depends on the cash-flow and growth forecast), surveys (opinions, may follow recent returns). Any named source must be dated.
7. Market equity exceeds book equity for profitable, growing firms, so book weights overweight debt, which is the cheaper source after tax.
8. A rate based on the biotech peers' asset beta relevered at the venture's financing, not the conglomerate's 8%.
9. Twelve observations give a large standard error; one-off events in a short window can dominate the estimate (and beta may not be stable).
10. (a) WACC. (b) Cost of equity. (c) The cost of debt.

Revision history

Standard and licence note

Based on the capital asset pricing model and cost-of-capital theory in the primary works listed above (Markowitz 1952, Sharpe 1964, Lintner 1965, Mossin 1966, Modigliani and Miller 1958 and 1963, Hamada 1972, Miles and Ezzell 1980, Harris and Pringle 1985). The text, examples, numbers and questions are original; no textbook text is copied or paraphrased, and Corvane Robotics, Northfield Outdoor Co. and the peer companies are fictional. All market inputs are Example values, not current data; Damodaran's published estimates are named only as a source to check. The template is independent and not affiliated with or endorsed by any business school, publisher or data provider. Check current market data and the current official versions of any standard before you rely on a figure. Template licence: MIT.