What CAPM is (plainly)
– The Capital Asset Pricing Model (CAPM) is a simple, widely used equation that links the expected return on an asset (usually a stock) to its exposure to market-wide risk. It separates the return investors require into a safe component (time value of money) and a risk component tied to how the asset co-moves with the overall market.
Core formula
Expected return on asset i = risk-free rate + beta_i × (market return − risk-free rate)
Written as:
ER_i = R_f + β_i × (ER_m − R_f)
Definitions (first time each term appears)
– Expected return (ER_i): the return investors require or forecast for holding the asset.
– Risk-free rate (R_f): a rate available on a default-free short-term instrument (commonly U.S. T-bills) used to represent the time value of money.
– Beta (β_i): a measure of an asset’s systematic risk — how much the asset’s returns move, on average, when the market moves. β = 1 means the asset moves with the market; β > 1 implies higher sensitivity; β < 1 implies lower sensitivity.
– Market risk premium (ER_m − R_f): the extra return investors demand for holding a market portfolio instead of the risk-free asset.
How to use CAPM — step-by-step
1. Choose a risk-free rate: typically the yield on an appropriate-maturity government bill or bond.
2. Pick a market benchmark (e.g., S&P 500) and estimate ER_m (expected market return) or directly choose a market risk premium (ER_m − R_f).
3. Obtain the asset’s beta. That can be:
– Estimated from historical regression of asset returns vs. market returns, or
– Taken from a data provider (e.g., Bloomberg, Reuters) or from academic sources.
4. Plug numbers into ER_i = R_f + β_i × (ER_m − R_f) to get
to ER_i, the asset’s expected return (the return investors require for holding that asset under CAPM assumptions). In formula form:
ER_i = R_f + β_i × (ER_m − R_f)
Where each term is:
– ER_i: expected (or required) return on asset i.
– R_f: risk-free rate (yield on a government security of appropriate maturity).
– β_i (beta): sensitivity of asset i’s returns to the market portfolio (a unitless measure).
– ER_m − R_f: market risk premium (expected excess return of the market over the risk-free rate).
Worked numeric example (step-by-step)
1. Choose inputs:
– R_f = 3.0% (e.g., 3‑month or 10‑year Treasury; pick maturity consistent with your horizon).
– ER_m = 8.0% (expected market return).
– β_i = 1.30 (estimated from historical regression or taken from a provider).
2. Compute market risk premium:
– ER_m − R_f = 8.0% − 3.0% = 5.0%.
3. Apply CAPM:
– ER_i = 3.0% + 1.30 × 5.0% = 3.0% + 6.5% = 9.5%.
Interpretation: Under CAPM assumptions, investors would require ~9.5% expected return on the asset given its beta.
How to estimate beta (practical checklist)
– Data selection:
– Choose return frequency (daily, weekly, monthly). Monthly is common for equity betas to reduce noise.
– Pick a lookback window (e.g., 2–5 years). Longer windows smooth short-term shifts but may miss recent regime change.
– Market proxy:
– Use a broad index (e.g., S&P 500) that best represents the investable market for the asset.
– Regression method:
– Compute excess returns: R_asset − R_f and R_market − R_f (subtract the same risk-free series).
– Run ordinary least squares (OLS) regression: (R_asset − R_f) = α + β × (R_market − R_f) + ε.
– β is the slope coefficient; α (alpha) is the intercept (average unexplained excess return).
– Diagnostics:
– Check R-squared (explained variance) and the t‑stat for beta.
– Inspect residuals for heteroskedasticity or autocorrelation (which can bias inference).
– Alternatives:
– Use provider betas (Bloomberg, Reuters) or published betas if you lack data/software.
Practical choices that matter
– Risk-free rate maturity: Match horizon. Use short-term bills for short-horizon investors and long-term Treasuries for long-term valuation.
– Market risk premium: You can use historical averages, survey/implied premiums, or practitioner estimates (estimates vary widely; document your choice).
– Beta stability: Betas change over time (company leverage, business mix). Consider adjusted betas (Blume adjustment) or rolling-window estimates.
Common uses of CAPM
– Cost of equity in discounted cash flow (DCF) valuation.
– Performance evaluation: alpha measures abnormal return relative to CAPM.
– Portfolio construction: understanding how holdings move with the market.
Key limitations and assumptions (be explicit)
– Single-factor model: CAPM assumes market risk is the only relevant systematic risk. Empirical research finds other factors (size, value, profitability) matter.
– Mean-variance optimization: Assumes investors care only about expected return and variance.
– Homogeneous expectations: Assumes all investors share the same inputs and beliefs.
– Frictionless markets: No taxes, no transaction costs, unlimited borrowing/lending at R_f.
– Stationarity and linearity: Beta assumed constant and returns linearly related to market returns.
Because of these assumptions, CAPM gives a useful benchmark but is not a definitive prediction. Empirical tests show both explanatory power and notable deviations.
Simple sensitivity check
– Recompute ER_i with different MRPs (e.g., 4%, 6%) and betas (±0.2) to see how sensitive the required return is to inputs. This quick sensitivity analysis highlights input risk.
Quick checklist before you present a CAPM result
– Specify horizon and choose matching R_f.
– State market index and how ER_m or MRP was estimated.
– Describe beta estimation method, window, and frequency.
– Report regression diagnostics (R-squared, t-stat for beta).
– Perform sensitivity analysis for MRP and beta.
– Note limitations and alternative models considered (e.g., Fama‑French).
References for further reading and data
– Investopedia — Capital Asset Pricing Model (CAPM): https://www.investopedia.com/terms/c/capm.asp
Worked numeric example (step-by-step)
– Inputs (example): risk-free rate R_f = 1.50% (annual), expected market return ER_m = 7.00% (annual), beta = 1.20.
– Compute market risk premium (MRP): MRP = ER_m − R_f = 7.00% − 1.50% = 5.50%.
– CAPM required return: ER_i = R_f + beta × MRP = 1.50% + 1.20 × 5.50% = 1.50% + 6.60% = 8.10%.
– Interpretation: With these inputs, the asset’s expected/required return is 8.10% per year under CAPM assumptions.
Quick sensitivity checks (same example)
– If beta = 1.00 (−0.20): ER_i = 1.50% + 1.00 × 5.50% = 7.00%.
– If beta = 1.40 (+0.20): ER_i = 1.50% + 1.40 × 5.50% = 9.20%.
– If MRP = 4.50% (−1.00%): ER_i = 1.50% + 1.20 × 4.50% = 6.90%.
– If MRP = 6.50% (+1.00%): ER_i = 1.50% + 1.20 × 6.50% = 9.30%.
– Useful check: small changes in beta or MRP produce material changes in ER_i; report ranges, not a single number.
Common pitfalls (short checklist)
– Mixing horizons: ensure R_f and ER_m match your analysis horizon (e.g., short-term T-bill vs. long-term bond).
– Using
– Using historical arithmetic returns for long-term forecasts: arithmetic averages overstate multi-period (compound) expected returns; use geometric averages or convert carefully to match your horizon.
– Using a wrong market proxy: choose a broad, investable market index (e.g., total-market or large-cap index) and be explicit.
– Using nominal R_f vs. real R_f inconsistently: match inflation assumptions across inputs (real vs. nominal).
– Treating beta as constant: betas drift; check stability and consider rolling-window estimates or adjustments (Blume or Bayesian shrinkage).
– Ignoring estimation error: betas and MRP estimates have sampling error—report confidence intervals or run sensitivity checks.
– Confusing idiosyncratic and systematic risk: CAPM prices only systematic (market) risk; diversifiable (idiosyncratic) risk is not compensated under CAPM.
– Survivorship and look-ahead bias: ensure historical samples and index constituents reflect the reality of the period you use.
– Mixing accounting measures with market returns: use market (returns-based) betas, not accounting-based "betas."
Practical implementation checklist (step-by-step)
1. Define your horizon (e.g., 1 year, 5 years). Ensure all inputs match that horizon.
2. Choose R_f (risk-free rate): pick a yield on an instrument matching the horizon (e.g., Treasury bill for short-term, Treasury nominal coupon or real yield for long-term).
3. Choose market proxy: specify index (e.g., S&P 500, MSCI World). Use total-return series if possible.
4. Estimate market risk premium (MRP = E[R_m] − R_f): select source (historical geometric/ arithmetic, implied from option/DGF models, or practitioner surveys). Document method.
5. Estimate beta: run a regression of the asset’s excess returns on the market’s excess returns (Ri − Rf) = alpha + beta × (Rm − Rf) + epsilon. Use an appropriate frequency (
frequency (e.g., monthly returns with a 36–60 month window or weekly with 156 weeks). Adjust for non-synchronous trading and dividends by using total-return series. Consider these regression choices and diagnostics:
– Window length: 3–5 years is common; shorter windows increase noise, longer windows may miss structural changes.
– Frequency: monthly is a standard compromise between noise and sample size.
– Adjustments: use Newey–West or robust standard errors if you suspect autocorrelation/heteroskedasticity.
– Diagnostics: report t-statistic and standard error of beta, R-squared, and check residuals for patterns or outliers.
– Alternative: use the Blume adjustment to partially mean-revert beta: adjusted_beta = 0.67 × raw_beta + 0.33 × 1.0 (useful for near-term forecasting).
6. Adjust for capital structure (levering/unlevering): If you estimate beta from comparable firms (peer group betas), unlever them to remove the effect of differing debt levels, then relever to your target capital structure. Formulas:
– Unlevered (asset) beta: beta_u = beta_l / [1 + (1 − Tc) × (D/E)]
– Relevered (equity) beta: beta_l,re = beta_u × [1 + (1 − Tc) × (D/E)_target]
Where Tc = corporate tax rate, D/E = debt/equity ratio. Document whether book or market D/E is used.
7. Compute expected return with CAPM:
E[R_i] = R_f + beta_i × (E[R_m] − R_f)
Check that all inputs use the same horizon and return conventions (nominal vs. real). Report the point estimate and a confidence interval around beta if possible.
8. Perform sensitivity analysis:
– Vary MRP (e.g., ±1 percentage point) and beta (e.g., ±0.1) to show how E[R_i] changes.
– If using an implied MRP, show how different growth/discount assumptions affect it.
– Present a small table or bullet list of scenarios (base, optimistic, pessimistic).
9. Cross-check:
– Compare to other measures: implied cost of equity from a dividend discount model (DDM) or discounted cash flow (DCF), multi-factor models (Fama–French 3/5-factor), and practitioner surveys.
– If results diverge materially, revisit inputs and assumptions.
10. Document assumptions and limitations:
– Note which market proxy and risk-free series you used, the sample period, frequency, tax rate, and capital structure choice.
– State key CAPM limitations: single-factor model, required-market-portfolio assumption, investor homogeneity, and potential time-variation in betas and risk premia.
11. Use and reporting:
– When using CAPM outputs in valuation or portfolio decisions, show base-case and sensitivity outputs.
– For corporate cost of capital (WACC), convert to after-tax cost of debt and weight equity and debt by market values.
Worked numeric example (step-by-step)
Assumptions:
– Horizon: 1 year (nominal returns)
– Risk-free rate R_f = 2.0% (1-year Treasury yield)
– Market expected return E[R_m] = 8.0% → market risk premium = 6.0%
– Estimated raw beta from 60 months of monthly excess returns = 1.20
– Corporate tax rate Tc = 21%, target D/E = 0.5 (50% debt / 50% equity by market value)
Step A — CAPM expected return:
E[R_i] = 2.0% + 1.20 × 6.0% = 2.0% + 7.2% = 9.2%
Step B — If using comparables and need to relever:
Suppose a comparable firm has beta_l = 1.30 and D/E_comparable = 0.4.
Unlever: beta_u = 1.30 / [1 + (1 − 0.21) × 0.4] = 1.30 / [1 + 0.79 × 0.4] = 1.30 / [1 + 0.316] = 1.30 / 1.316 ≈ 0.99
Relever to target D/E = 0.5: beta_l,re = 0.99 × [1 + 0.79 × 0.5] = 0.99 × [1 + 0.395] = 0.99 × 1.395 ≈ 1.38
Then E[R_i] (re-levered) = 2.0% + 1.38 × 6.0% = 2.0% + 8.28% = 10.28%
Step C — Sensitivity check (base beta ±0.1):
– Beta = 1.10 → E[R] = 2.0% + 1.10 × 6.0% = 8.6%
– Beta = 1.30 → E[R] = 2.0% + 1.30 × 6.0% = 9.8%
Quick checklist for reporting CAPM results
– State horizon and currency (nominal/real).
– List R_f source and date (e.g., U.S. 1-year Treasury yield, URL).
– Specify market
– Specify market index and exact ticker/name (e.g., S&P 500, MSCI World) and the source/date for the index return series.
– Document the beta estimation method: sample period (months/years), return frequency (daily/weekly/monthly), regression type (OLS, robust), and whether beta is raw or Blume/adjusted.
– State how you treated corporate leverage: reported beta (levered) or an unlevered/re-levered beta; show the formulas and the D/E and tax-rate inputs used.
– Give the market risk premium (MRP) source and whether it is historical, implied, or survey-based; record the estimation window and currency basis.
– Report the risk-free rate (R_f) source, maturity chosen (e.g., 10-year nominal Treasury), and whether inputs are nominal or real.
– Note any currency conversions or inflation adjustments applied to returns or premiums.
– Describe any firm-specific adjustments (size, country, business risk, non-diversifiable event risk) and justify them.
– Provide sensitivity ranges for the key inputs (± delta R_f, ± delta beta, alternative MRPs) and show the resulting range of expected returns.
– List limitations and alternative models considered (e.g., multi-factor models like Fama–French or the Arbitrage Pricing Theory).
– Archive the regression outputs and raw data (dates, returns) so results are reproducible.
Short checklist — CAPM reporting (one-page summary to circulate)
1. Scope: horizon, currency, nominal/real.
2. Inputs: R_f (source+date), beta (method+period), MRP (source+type).
3. Calculations: show formulas, intermediate steps, and final E[R].
4. Sensitivity: low/central/high scenarios.
5. Caveats: model assumptions and known limitations.
6. Appendices: regression output, data files, and URLs.
Key formulas (with corporate-tax leverage adjustment)
– CAPM expected return: E[R_i] = R_f + β_i × MRP
– Where MRP = E[R_m] − R_f (market risk premium), β_i is the equity beta.
– Unlevering beta to remove financial leverage (to estimate asset/intrinsic beta):
β_unlevered = β_levered / [1 + (1 − T) × (D/E)]
– Re-levering beta to a target capital structure:
β_levered,target = β_unlevered × [1 + (1 − T) × (D/E)_target]
Notes: D/E is debt-to-equity ratio (market values preferred), T is corporate tax rate (use marginal statutory rate unless justified otherwise).
Worked numeric mini-example (different numbers than earlier)
Assumptions:
– R_f (10-year nominal Treasury) = 3.00%
– Implied MRP (U.S., current estimate) = 5.50%
– Observed levered beta from 5-year monthly regression = 1.20
– Firm D/E (market) = 0.6; target D/E = 0.3; corporate tax = 21%
Step 1 — Unlever beta:
β_u = 1.20 / [1 + (1 − 0.21) × 0.6] = 1.20 / [1 + 0.79 × 0.6] = 1.20 / [1 + 0.474] = 1.20 / 1.474 ≈ 0.81
Step 2 — Re-lever to target:
β_target = 0.81 × [1 + 0.79 × 0.3] = 0.81 × [1 + 0.237] = 0.81 × 1.237 ≈ 1.00
Step 3 — CAPM expected return:
E[R] = 3.00% + 1.00 × 5.50% = 8.50%
Sensitivity check (β ±0.15 → E[R]):
– β = 0.85 → E[R] = 3.00% + 0.85 ×