Derivation — CAPM from Portfolio Theory

The main CAPM lesson derived the Security Market Line using an intuitive no-arbitrage argument: assets lying off the SML would be arbitraged back onto it. That argument is compelling but circular — it assumes beta is the relevant risk measure and then confirms the SML. This note provides the rigorous derivation from first-order optimality conditions, where beta emerges as a consequence rather than an assumption.

The derivation proceeds in three steps:

Write the first-order conditions for the tangency portfolio (Sharpe-ratio-maximising portfolio with a riskless asset).
Impose market clearing: at equilibrium, the tangency portfolio must be the market portfolio.
Identify the Lagrange multiplier $\lambda$ by applying the optimality condition to the market portfolio itself — yielding the SML.

Learning objectives. By the end of this note you should be able to:

state the first-order conditions for the tangency portfolio;
show that the left-hand side of the first-order condition equals the covariance of stock $k$ with the market portfolio;
solve for the Lagrange multiplier $\lambda$ using the market portfolio self-consistency;
derive the Security Market Line $\bar{R}_k - R_F = \beta_k(\bar{R}_M - R_F)$ from the optimality conditions without pre-assuming beta.

In this lesson

Step 1: Tangency portfolio optimality conditions
Step 2: Identifying the left-hand side as a covariance
Step 3: Solving for $\lambda$ and deriving the SML
What this derivation establishes
Connection to the stochastic discount factor

Before you start

CAPM: The Security Market Line and Beta Pricing

The full lesson, with worked examples and exercise solutions, is for members.

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