The price of risk is beta — and only the risk you cannot diversify away is paid for. Sharpe's one-factor equilibrium says every asset's expected return is Rf + β(E[Rm] − Rf), so every asset plots on a single straight line, the security market line. Idiosyncratic wobble earns nothing. Down the center, data flows: return series go in, the engine prices, the expected return comes out. The blue team builds and defends it; the red team tries to break it. This is a model, not investment advice.
source Sharpe, W. F., Capital Asset Prices: A Theory of Market Equilibrium under Conditions of Risk, J. Finance 19 (1964) 425–442 — doi:10.1111/j.1540-6261.1964.tb02865.x. A one-factor equilibrium model AMBER. Rendered, not quoted.
Beta is not a mood; it falls out of two moments of the return series:
βi = Cov(Ri, Rm) / Var(Rm) → E[Ri] = Rf + βi(E[Rm] − Rf)
Expected return is linear in β. The market has β=1, the risk-free asset β=0, and every asset built from real returns sits exactly on the line below — computed live, not looked up.
| asset | β = Cov/Var | total σ | E[R] (CAPM) |
|---|
Sharpe stands on the-markowitz-portfolio. Markowitz (1952) showed only the covariance structure of a portfolio matters; the efficient frontier is its output.
Sharpe adds an equilibrium: let everyone hold that frontier and borrow/lend at Rf, and in equilibrium the single risky portfolio they all hold is the market. What gets priced is then only an asset's covariance with that market — its beta. The min-variance portfolio becomes the one point every line passes through. AMBER a one-factor equilibrium, true by assumption, not by measurement.
The blue team's live check: rebuild the whole roster from its return series and confirm every asset prices onto the SML, the market has β=1, and two equal-β assets get the same price. If red tampers, this badge is where it shows.
The raw material is returns across states of the world: a market series Rm and, for each asset, its own series Ri. Two numbers are extracted from them — a covariance with the market and the market's own variance. Their ratio is beta.
Each asset here is built honestly as Ri = Rf(1−β) + βRm + ε, where the idiosyncratic term ε is constructed orthogonal to the market (zero covariance, zero mean) by Gram–Schmidt. So the sample beta equals its target to machine precision, and the realized average return equals the SML price — nothing is fitted after the fact.
Assumed AMBER one period, constant variances, everyone optimizes mean–variance with the same beliefs, frictionless markets, borrow/lend at Rf.
Drag idiosyncratic σ: total risk moves, expected return does not — only β is priced. That is the whole model.
Every quantity is computed from the constructed series on the spot — the beta is Cov/Var, the price is Rf+β·premium, never a stored answer.
What the machine produces, proven: for any asset built from real returns, its realized average return equals Rf + β(E[Rm] − Rf) to 1e−6 — it lies on the security market line. Two assets with the same β but wildly different total variance earn the same expected return: diversifiable risk is not paid for.
The blue team's witness (left) confirms these numbers live; the red team (right) tries to make them wrong.
Roll's critique (1977): the true market portfolio is unobservable, so CAPM is arguably untestable — any test is a joint test of the model and your proxy for the market. CAPM is not a law of prices; it is the first clean statement that only shared risk should be paid for. It is a model, and no line here is advice.
"CAPM tells you which stocks to buy." Cut. It is an equilibrium theory of prices, not a stock-picking rule and not personal advice.
"Higher risk always earns higher return." Cut. Only systematic (beta) risk is compensated; idiosyncratic risk earns nothing — the panel proves it live.
"Beta measures how risky a stock is." Kept, corrected. Beta measures only co-movement with the market (Cov/Var), not total volatility — a stock can be violent and low-beta at once.
The red team's move: price the wrong risk. Swap beta for the asset's total variance (σi/σm) so a violent, low-beta stock is handed a fat expected return it never earned. The blue team's witness (window 7) is watching.
Under the tamper, the two equal-beta assets get different prices, every point lifts off the security market line, and the witness recomputes, disagrees with the realized means, and turns red. Nothing is faked; the attack is real and it is caught.