💼 Portfolio Management

CAPM Formula — Capital Asset Pricing Model Explained

The CAPM formula explained: expected return = risk-free rate + beta × market premium. Capital asset pricing model with examples and CFA Level 1 practice.

Key Concepts

Capital Market Line (CML)

CAL using the MARKET portfolio as the optimal risky portfolio. E(R) = R<sub>f</sub> + [(R<sub>m</sub>-R<sub>f</sub>)/σ<sub>m</sub>]σ<sub>p</sub>.

Systematic vs Unsystematic Risk

Systematic (market/non-diversifiable): β measures sensitivity. Unsystematic (company-specific): eliminated by diversification.

CAPM

E(R<sub>i</sub>) = R<sub>f</sub> + β<sub>i</sub>(E(R<sub>m</sub>) - R<sub>f</sub>). Only systematic risk (β) is rewarded. Security Market Line (SML) plots this.

Beta

β = Cov(R<sub>i</sub>, R<sub>m</sub>) / Var(R<sub>m</sub>). Market β=1. β>1: more volatile than market. β<1: less volatile.

Performance Measures

Sharpe (total risk), Treynor (systematic risk), Jensen's Alpha (excess return above CAPM), M² (risk-adjusted RAP).

Formulas

From this module

CAPM / SML

E(Ri) = Rf + βi[E(Rm) - Rf]

Where: R<sub>f</sub> = risk-free, β = beta, E(R<sub>m</sub>)-R<sub>f</sub> = market risk premium

Beta

βi = Cov(Ri, Rm) / σ²m

Where: σ²<sub>m</sub> = variance of market returns

Treynor Ratio

Treynor = (Rp - Rf) / βp

Where: Excess return per unit of systematic risk

Jensen's Alpha

α = Rp - [Rf + β(Rm - Rf)]

Where: Positive α = outperformance vs CAPM

M-squared

M² = (Sharpep - Sharpem) × σm

Where: Risk-adjusted performance in return units

Master Formula Sheet -- Portfolio Management

Utility Function

U = E(R) - ½ × A × σ²

A = risk aversion (A>0: risk-averse)

CAL Equation

E(Rᶜ) = Rᶠ + [E(Rₚ)-Rᶠ]/σₚ × σᶜ

Slope = Sharpe ratio

Portfolio Variance (2 assets)

σ²ₚ = w²ₐσ²ₐ + w²_bσ²_b + 2wₐw_bρσₐσ_b

Diversification when ρ < 1

CAPM

E(Rᵢ) = Rᶠ + βᵢ[E(R_m) - Rᶠ]

Only systematic risk rewarded

Beta

β = Cov(Rᵢ, R_m) / σ²_m

Systematic risk measure

Treynor Ratio

Treynor = (Rₚ - Rᶠ) / βₚ

Return per unit of beta risk

Jensen's Alpha

α = Rₚ - [Rᶠ + β(R_m - Rᶠ)]

Excess return above CAPM

M-squared

M² = (Sharpeₚ - Sharpe_m) × σ_m

Risk-adjusted in return units

Decision Frameworks

Which performance measure to use?

Use when:

  • Sharpe: for well-diversified portfolios (total risk matters)
  • Treynor: for portfolios that are part of a larger diversified portfolio (β matters)
  • Jensen’s Alpha: for measuring active manager skill

Avoid when:

  • Sharpe for poorly diversified portfolios (total risk includes diversifiable risk)
  • Treynor when portfolio is the investor’s entire holdings

Test Your Understanding

R<sub>f</sub>=2%, β=1.2, Market return=9%. CAPM expected return:

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