Capital Asset Pricing Model (CAPM)

The theoretical rate of return of an investment with zero risk of financial loss. It represents the minimum return an investor expects for any investment, as they can earn this rate without taking any risk.

• Proxy: Typically represented by the yield on a 10-year government bond (e.g., US Treasury Note) for long-term analysis, or a 3-month T-bill for short-term models.
• Role in CAPM: Serves as the intercept in the Security Market Line (SML). It is the baseline to which the risk premium is added.
• Real-World Nuance: The true "risk-free" asset is a theoretical construct. Sovereign bonds carry minimal but non-zero inflation and default risk, making them the accepted practical proxy.

A measure of a security's volatility relative to the overall market portfolio. It quantifies the sensitivity of an asset's excess return to the excess return of the market.

• β = 1: The asset moves in perfect lockstep with the market.
• β > 1: The asset is more volatile than the market (e.g., high-growth tech stocks). Amplifies market moves.
• β < 1: The asset is less volatile (e.g., utilities). Dampens market moves.
• Calculation: Mathematically derived from the covariance of the asset's return and the market return, divided by the variance of the market return.

The additional return over the risk-free rate that investors require to hold the broad market portfolio instead of risk-free assets. It represents the price of systematic risk.

• Ex-Ante vs. Ex-Post: The expected premium is forward-looking and drives investment decisions, while the historical premium is calculated from past data.
• Equity Risk Premium (ERP): Often used synonymously when the "market" is defined as a broad equity index like the S&P 500.
• Magnitude: Historically, the long-run arithmetic average ERP in the US has hovered between 4% and 6%, though it varies significantly by country and economic cycle.

A graphical representation of the CAPM equation. It plots expected return on the Y-axis against beta on the X-axis.

• Intercept: The risk-free rate (Rf).
• Slope: The market risk premium (Rm - Rf).
• Fair Valuation: Assets plotting above the SML are undervalued (offering high return for their risk), while assets below the line are overvalued.
• Alpha (α): The vertical distance between an asset's actual return and the SML. A positive alpha indicates a return exceeding the CAPM prediction.

A theoretical value-weighted portfolio containing every available risky asset in the world, including stocks, bonds, real estate, and even human capital.

• Proxy Problem: In practice, the true market portfolio is unobservable. Analysts use broad-based indices like the S&P 500 or MSCI World Index as a proxy, a limitation known as the Roll's Critique.
• Efficiency Assumption: CAPM assumes this portfolio is mean-variance efficient, meaning it sits on the efficient frontier and offers the highest Sharpe ratio.

The risk specific to an individual asset or a small group of assets. It is distinct from systematic market risk.

• Examples: A management scandal, a product recall, or a regulatory change affecting a specific industry.
• Diversification: CAPM assumes investors hold fully diversified portfolios, thereby eliminating idiosyncratic risk. Because it can be diversified away for free, the model does not reward investors for bearing it.
• Contrast: The total risk of an asset is the sum of systematic risk (measured by beta) and idiosyncratic risk. CAPM only prices the former.

The sole explanatory variable in CAPM, beta measures an asset's sensitivity to systematic, non-diversifiable market risk. A beta of 1.0 implies the asset moves in lockstep with the market; a beta greater than 1.0 indicates amplified volatility.

• Calculation: Covariance(Asset, Market) / Variance(Market)
• Interpretation: A stock with a beta of 1.5 is theoretically 50% more volatile than the benchmark.
• Critique: Relies on historical data and assumes a stable linear relationship, which often breaks during regime shifts.

The Security Market Line is the graphical representation of the CAPM formula. It plots expected return against systematic risk (beta), with the risk-free rate as the intercept and the market risk premium as the slope.

• Fair Valuation: Assets plotting above the SML are undervalued (generating alpha); those below are overvalued.
• Market Equilibrium: In an efficient market, all assets should theoretically lie directly on the SML.
• Slope: Represents the market risk premium (E(Rm) - Rf).

A direct extension of CAPM that adds size (SMB) and value (HML) factors to explain anomalies the single-factor model misses. Eugene Fama and Kenneth French demonstrated that small-cap stocks and high book-to-market (value) stocks historically generate excess returns.

• Three-Factor Formula: E(R) = Rf + β_mkt(MRP) + β_smb(SMB) + β_hml(HML)
• Five-Factor Expansion: Later added profitability (RMW) and investment (CMA) factors.
• Empirical Superiority: Explains over 90% of diversified portfolio returns vs. ~70% for CAPM.

In the context of CAPM, alpha represents the excess return of an investment relative to the return predicted by its beta. It is the vertical distance of an asset from the Security Market Line and quantifies manager skill or market inefficiency.

• Jensen's Alpha: A specific regression-based measure of risk-adjusted performance.
• Zero-Sum Game: In a perfectly efficient CAPM world, alpha should be zero after costs.
• Statistical Significance: A positive alpha must be tested for robustness to ensure it isn't random noise.

Developed by Stephen Ross as a more flexible alternative to CAPM, Arbitrage Pricing Theory allows for multiple systematic risk factors rather than a single market factor. It assumes that arbitrageurs will eliminate mispricing instantly.

• Multi-Factor Structure: E(R) = Rf + β1(RP1) + β2(RP2) + ... + βn(RPn)
• Unspecified Factors: Unlike CAPM, APT does not specify which macroeconomic factors (inflation, GDP, interest rates) drive returns.
• No-Arbitrage Principle: The model is built on the law of one price, not mean-variance optimization.

The Efficient Frontier is the set of optimal portfolios that offer the highest expected return for a defined level of risk, as measured by standard deviation. CAPM's market portfolio is the tangency point on this frontier.

• Construction: Derived from Mean-Variance Optimization (MVO) using a covariance matrix of asset returns.
• Tangency Portfolio: The point where the Capital Market Line (CML) touches the Efficient Frontier; this is the CAPM market portfolio.
• Limitation: Highly sensitive to input estimates; small changes in expected returns can drastically shift the frontier.