Capital Asset Pricing Model (CAPM)

The formula in one line

The CAPM expected return for asset i is:

E[Rᵢ] = R_f + βᵢ × (E[R_m] − R_f)

Where R_f is the risk-free rate (typically the 10-year Treasury yield), E[R_m] is the expected return on the market portfolio (typically proxied by the S&P 500), and βᵢ is the asset's beta — the slope coefficient from a regression of Rᵢ − R_f on R_m − R_f.

The bracket term (E[R_m] − R_f) is the market risk premium: the extra return investors demand for holding the diversified market over a riskless bond. Multiply it by beta and add the risk-free rate, and you have the price-of-risk-adjusted expected return for any tradable asset.

Worked example: β = 1.5 stock

Take a hypothetical stock with beta 1.5, in an environment where the 10-year Treasury yields 4% and the expected market return is 10%.

R_f = 4%
E[R_m] = 10%
β = 1.5

Market risk premium = 10% − 4% = 6%
Risk premium for this stock = 1.5 × 6% = 9%
Expected return = 4% + 9% = 13%

So CAPM says: this stock should be priced today such that you expect to earn 13% per year going forward. If the company's projected cash flows imply a 15% return at today's price, the stock is undervalued — buy it and earn 2 percentage points of alpha (in expectation, with risk-adjusted discount rate held constant). If the implied return is 11%, you'd be taking on extra risk for less compensation than the market offers — sell or avoid.

This is exactly how corporate finance teams use CAPM. To value a project, they discount projected cash flows at R_f + β × (E[R_m] − R_f) using the firm's equity beta (or industry average, if the firm is private or pre-revenue). A 13% required return means a $100 cash flow next year is worth $100/1.13 = $88.50 today.

The intuition: diversifiable risk is free

Total return variance for any stock decomposes into two pieces: market-correlated (systematic) and idiosyncratic (firm-specific). Hold one stock and you eat both. Hold thirty uncorrelated stocks and the firm-specific pieces average out — by the law of large numbers, idiosyncratic risk approaches zero in a portfolio. The systematic piece survives diversification because every stock moves with the market to some degree.

The market doesn't pay you for risks anyone can shed for free. That's the entire conceptual content of CAPM: the only risk that earns a premium is the risk you can't diversify away. Beta measures exactly that piece — covariance with the market portfolio, scaled by market variance:

βᵢ = Cov(Rᵢ, R_m) / Var(R_m)

A stock with beta 2 has twice the systematic exposure of the market — it earns twice the risk premium, but it also carries twice the unavoidable risk. Risk and return move together, but only along the dimension that can't be diversified.

CAPM vs other asset pricing models

Model	Risk factors	Year	Typical R² (cross-section)	Strength
CAPM	1 (market beta)	1964 (Sharpe)	~25%	Simplicity, theoretical purity
Fama-French 3-factor	3 (market, size, value)	1992	~70%	Captures size and value premia
Carhart 4-factor	4 (+ momentum)	1997	~75%	Adds Jegadeesh-Titman momentum effect
Fama-French 5-factor	5 (+ profitability, investment)	2015	~75-90%	Captures quality factors; subsumes value in some samples
APT (Ross 1976)	k (any priced factors, derived from arbitrage)	1976	varies by specification	No reliance on market portfolio; allows unknown factor structure
Consumption CAPM (CCAPM)	1 (consumption beta)	1978-1979 (Breeden, Lucas)	poor (~10%)	Microfounded in marginal utility — but the empirical equity premium puzzle is severe
q-factor model (Hou-Xue-Zhang)	4 (market, size, investment, profitability)	2015	~80%	Investment-based, often beats Fama-French 5-factor on anomalies

Each successor adds factors to capture cross-sectional return patterns CAPM misses. Fama-French (1992) showed that small-cap stocks and high book-to-market (value) stocks earn extra returns that CAPM beta can't explain — those are the SMB ("small minus big") and HML ("high minus low") factors. Adding them roughly tripled the cross-sectional R² versus CAPM alone.

The Security Market Line

If you plot expected return on the y-axis and beta on the x-axis, CAPM predicts every fairly priced asset lies on a single line — the Security Market Line (SML) — passing through (0, R_f) with slope equal to the market risk premium.

Assets above the line are mispriced cheap (alpha > 0); assets below are mispriced rich. The empirical fact that small caps historically plotted above the SML — earning more than their betas predicted — was the original "anomaly" that motivated Fama and French to add the size factor.

Estimating beta in practice

import numpy as np
import statsmodels.api as sm

# Weekly returns over 2 years
stock_excess = stock_returns - rf_weekly       # n x 1
market_excess = market_returns - rf_weekly     # n x 1

X = sm.add_constant(market_excess)
model = sm.OLS(stock_excess, X).fit()
beta = model.params[1]
alpha = model.params[0]
r_squared = model.rsquared

Three implementation choices that materially affect beta:

• Frequency. Daily returns add noise from non-synchronous trading; monthly returns reduce the sample size. Most practitioners settle on weekly.
• Window length. Bloomberg uses 2 years (104 weeks). Damodaran prefers 5 years for stable firms. Tesla's beta computed over 2020-2022 is roughly 2.0; over 2023-2025 it's roughly 1.5. Same firm, different number.
• Market proxy. S&P 500 is the U.S. default. The "true" market portfolio includes bonds, real estate, human capital — none of which is investable as an index. This is Roll's critique (1977): CAPM is essentially untestable because the true market portfolio is unobservable.

The Blume adjustment shrinks raw beta toward 1: adjusted β = (2/3) × raw β + (1/3) × 1. Bloomberg reports both raw and adjusted by default. The shrinkage reflects mean-reversion in beta over time and reduces estimation noise — Marshall Blume showed in 1971 that betas computed in one period regress toward 1 in the next period.

Why CAPM fails empirically

• Roll's critique (1977). The true market portfolio is unobservable. Any test of CAPM is jointly a test of the model and of whether the chosen proxy is mean-variance efficient. Failing the test could just mean the proxy is bad.
• Low-beta anomaly. Low-beta stocks have historically earned higher Sharpe ratios than high-beta stocks — the opposite of the prediction. This is the "betting against beta" trade documented by Frazzini and Pedersen (2014).
• Size and value premia. Fama-French (1992) showed that small-cap and high-book/market stocks earn returns CAPM cannot explain — leading directly to the multi-factor models above.
• Momentum. Past winners outperform past losers for 6-12 months, with no help from beta exposure. Carhart's 1997 paper added a UMD ("up minus down") factor.
• Beta instability. Estimated betas are noisy and time-varying. Microsoft's beta has ranged from 0.7 to 1.5 over different five-year windows. Using one number to summarize systematic exposure presumes a stationarity that doesn't hold in messy data.

Why it survives anyway

CAPM is wrong but useful. Three reasons it persists:

• Pedagogy. It cleanly separates diversifiable from systematic risk, which is the most important conceptual lesson in investments.
• Tractability. Discount rate = R_f + β × ERP. One regression, one number. Multi-factor models require more inputs and produce ranges no one wants to defend in a board meeting.
• Equilibrium grounding. Multi-factor models are largely empirical; CAPM is derived from utility maximization under specific assumptions. When you need a theory-grounded discount rate, CAPM is the cleanest available.

Damodaran's surveys consistently find that 70-80% of corporate finance teams use CAPM as their primary cost-of-capital model, despite knowing its limits. Where they deviate, they typically use industry-average betas with simple Blume shrinkage rather than switching to Fama-French.

Common pitfalls

• Using historical risk premium as forward-looking. The 6.7% historical U.S. premium reflects realized returns; expected forward returns may be lower. Damodaran's implied premium estimate from S&P 500 dividend yields and earnings forecasts has fluctuated 4-6% in recent decades.
• Using one beta forever. Beta drifts. Re-estimate at least annually for active applications.
• Confusing levered and unlevered beta. Equity beta includes financial leverage. To compare firms with different capital structures, unlever first: β_unlevered = β_equity / (1 + (1 − tax) × D/E), then re-lever to the target structure.
• Plugging in negative expected returns. CAPM produces negative expected returns when β < 0 and the market premium is positive. That's correct — gold and inverse ETFs do offer insurance value — but check that your application can sensibly use a negative discount rate.
• Treating beta as causal. Beta is a statistical association. A stock with a high estimated beta isn't caused to be more risky; it's measured to have moved more with the market historically.