CAPM (Security Market Line)

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E(R_i) = R_f + \beta_i\,[E(R_m) - R_f]

$E(R_i)$: the expected (fair) return on asset i — E(·) means 'expected value of'
$R_f$: the risk-free rate: what T-bills pay for waiting, with zero risk
$\beta_i$: the asset's dose of market risk (1 = moves with the market)
$E(R_m) - R_f$: the market risk premium: the market's reward over cash for one unit of risk
$\beta_i[E(R_m)-R_f]$: the asset's risk premium: its beta times the market's premium

Why it must be true

Why should any asset pay more than cash? Only as compensation for risk you cannot diversify away — private, company-specific wobbles are free to eliminate, so the market pays nothing for bearing them. The only priced risk is market exposure, and beta measures how much of it you hold.

So the fair expected return is built from two pieces: the risk-free rate (payment for time), plus the market's excess reward scaled by your dose of market risk (payment for beta). Double the beta, double the risk premium — a straight line in beta, which is why it's called the Security Market Line.

The derivation

Start from what an investor gives up and takes on. Holding cash pays $R_f$ with certainty. Holding the market portfolio pays a premium $E(R_m) - R_f$ for one full unit of market risk ( $\beta = 1$ ).

In a market where everyone can mix cash and the market freely, an asset with sensitivity $\beta_i$ delivers exactly $\beta_i$ units of that same undiversifiable risk — so arbitrage-free pricing must pay it $\beta_i$ times the same premium:

E(R_i) = R_f + \beta_i\,[E(R_m) - R_f]

Anything above the line is a bargain (everyone buys it, price rises, expected return falls back to the line); anything below is overpriced. The line IS the equilibrium.

When to reach for it

Asked for a required / expected return given a beta, a risk-free rate and a market return (or market premium) — or to judge whether a forecast return is above or below fair.

Listen for

“required rate of return”“beta of … / market risk premium”“according to the CAPM / SML”“is the stock overvalued or undervalued”

Back-of-the-envelope

Estimate it in your head first — then the calculator only confirms.

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Anchor at the endpoints: β = 0 must give exactly Rf, β = 1 exactly the market return. Slot your beta between (or beyond) those two rails and interpolate.
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Premium first: market 10%, cash 4% → premium 6. Then β × 6 is a one-digit multiply; add 4 back. Doing it in premium-space avoids the classic error.
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β > 1 must land ABOVE the market return, β < 1 below it (given a positive premium). A β = 1.4 answer under the market return is a distractor.

Traps in applying it

✗Multiplying beta by the TOTAL market return instead of the premium — the single most common CAPM error.
✗Forgetting to add the risk-free rate back after scaling the premium.
✗Feeding in total risk (σ) where beta belongs — CAPM prices only the undiversifiable part.
✗Mismatched horizons: a 10-year bond yield as Rf with a 1-year premium estimate.

Limits & criticisms

CAPM assumes a single common factor, frictionless markets and mean-variance investors — and the data disagree politely: measured returns are flatter in beta than the SML predicts, and size, value and momentum earn premiums beta cannot explain (hence multi-factor models). The inputs are estimates too: beta wobbles by sample, and the equity premium is one of the most argued-over numbers in finance. It survives as a benchmark, not a law.

Where it came from

William Sharpe (1964), with Lintner and Mossin independently, asked what Markowitz's diversification math implies for prices if everyone uses it — and the answer was a one-factor line that won Sharpe the 1990 Nobel Prize. CAPM became the backbone of corporate finance: it is how firms estimate their cost of equity for valuations and capital budgeting, how courts set allowed utility returns, and the benchmark against which "alpha" is defined. Empirically the line is flatter than theory says (Fama–French, 1992), yet no replacement has dislodged it from practice.

One identity, 2 questions

The exam can hide any variable. Each face below is the same equation solved for a different unknown — drill them separately.

Required return from beta

E(R_i) = R_f + \beta_i[E(R_m) - R_f]

The pricing face: what the asset MUST be expected to earn for its dose of market risk — the discount rate for equity valuations.

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Implied beta

\beta_i = \frac{E(R_i) - R_f}{E(R_m) - R_f}

The reverse-engineering face: an asset's excess return as a fraction of the market's excess reveals how much market risk it must carry.

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On the BA II Plus

Worked example: An analyst uses a risk-free rate of 4% and an expected market return of 10%. What return does the SML require of an asset with beta 0.4?

1.0.1 [−] 0.04 [=]the market risk premium
2.[×] 0.4 [=]scale it by beta
3.[+] 0.04 [=]add back the price of time (decimal)

→ 6.4%

Where it leads

Master this and the following come almost for free:

Weighted-Average Cost of Capital Jensen's Alpha

Drill this formula Back to the library