13.3 The Capital Asset Pricing Model, CML versus SML & Beta Estimation

13.3 The Capital Asset Pricing Model, CML versus SML & Beta Estimation

Asset pricing theory seeks to determine the equilibrium relationship between an asset's expected return and its fundamental risk exposure. While Markowitz Modern Portfolio Theory provides the normative framework for how investors should construct optimal portfolios, the Capital Asset Pricing Model (CAPM), Arbitrage Pricing Theory (APT), and multifactor models describe how asset prices and required returns are determined in competitive capital markets.

                         Asset Pricing & Factor Risk Architecture
                                            │
         ┌──────────────────────────────────┼──────────────────────────────────┐
         │                                  │                                  │
Single-Factor Equilibrium (CAPM)      Arbitrage Pricing Theory (APT)     Empirical Multifactor Models
  • Systematic vs. Unsystematic Risk    • Law of One Price & No-Arbitrage  • Fama-French 3-Factor (Size, Value)
  • CML (Total Risk σ) vs. SML (Beta β) • Macro Factors (Chen-Roll-Ross)   • Carhart 4-Factor (Momentum)
  • Beta Estimation & SCL Regression    • Industrial Production, Inflation • Fama-French 5-Factor (Profit, Inv)
  • Jensen's Alpha Mispricing Metric    • Fundamental & Statistical Models • Barra Risk Factor Exposures

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1. The Capital Asset Pricing Model (CAPM): Foundations & Risk Decomposition

Theoretical Origins and Core Assumptions

Developed independently by William Sharpe (1964), John Lintner (1965), and Jan Mossin (1966), the Capital Asset Pricing Model (CAPM) extends Markowitz MVO to general market equilibrium under the following assumptions:

1. Homogeneous Expectations: All investors share identical economic forecasts regarding asset expected returns, variances, and covariances over a single holding period.
2. Mean-Variance Optimizers: All investors construct portfolios along the Markowitz Efficient Frontier.
3. Risk-Free Borrowing and Lending: All investors can borrow and lend unlimited amounts at the identical risk-free interest rate ($R_f$).
4. Frictionless, Competitive Markets: Zero transaction costs, no capital gains taxes, complete asset divisibility, and no individual investor possesses market power to influence prices (all are price takers).
5. All Assets are Marketable: All assets, including human capital and private businesses, are publicly traded and infinitely divisible.

The Equilibrium Market Portfolio ($M$)

Under these homogeneous assumptions, every investor identifies the exact same tangency portfolio on the efficient frontier. In equilibrium, the prices of all assets clear the market, meaning the optimal tangency portfolio must contain every available risky asset in proportion to its market capitalization weight ($w_i = \text{Market Cap}_i / \sum \text{Market Cap}$). This is the Market Portfolio ($M$).

Total Risk Decomposition: Systematic vs. Unsystematic Risk

The total variance of return for any individual security $i$ decomposes into two distinct, orthogonal risk components:

$\sigma_i^2 = \underbrace{\beta_i^2 \sigma_m^2}_{\text{Systematic (Market) Risk}} + \underbrace{\sigma^2(e_i)}_{\text{Unsystematic (Idiosyncratic) Risk}}$

   Portfolio Standard Deviation (σ)
     ▲
     │  
     │  Total Risk
     │   \ 
     │    \  Unsystematic / Idiosyncratic Risk
     │     \   (Diversifiable: Firm-Specific Events, Lawsuits, CEO Turnover)
     │      \   ─────────────────────────────────────────────────────────────
     │       ` - - - - - - - - - - - - - - - - - - - - - - - - - - - - - 
     │                                                Systematic Risk (Market / Non-Diversifiable)
     │ ══════════════════════════════════════════════ (GDP, Inflation, Interest Rates)
     │
     └────────────────────────────────────────────────────────► Number of Assets (N)
                                                   N ≈ 30-40

1. Unsystematic (Idiosyncratic / Firm-Specific) Risk ($\sigma^2(e_i)$):

   • Arises from events unique to an individual company or industry (e.g., product recalls, regulatory rulings, labor disputes, clinical trial results).
   • Can be completely eliminated at zero cost by assembling a well-diversified portfolio of approximately 30 to 40 uncorrelated securities.
   • Pricing Implication: Because diversification is costless, capital markets do not compensate investors for bearing unsystematic risk ($E(R)$ does not increase with idiosyncratic risk).

2. Systematic (Market / Non-Diversifiable) Risk ($\beta_i^2 \sigma_m^2$):

   • Arises from macroeconomic forces that simultaneously affect all operating businesses (e.g., changes in GDP growth, monetary policy interest rate adjustments, sovereign geopolitical crises, systemic inflation).
   • Cannot be eliminated through diversification.
   • Pricing Implication: In equilibrium, systematic risk is the only risk that commands an expected return premium in the capital markets.

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2. Capital Market Line (CML) vs. Security Market Line (SML)

Understanding the precise distinction between the Capital Market Line and the Security Market Line is one of the most critical conceptual foundations for CIMA candidates.

   Capital Market Line (CML)                         Security Market Line (SML)
   Expected Return E(R)                              Expected Return E(R)
     ▲                                                 ▲
     │               / CML (Slope = Sharpe Ratio)      │               / SML (Slope = Market Risk Premium)
     │             /                                   │             /
     │           .● Market Portfolio (M)               │           .● Market Portfolio (β = 1.0)
     │         .'                                      │         .'
     │       .'                                        │       .'
     │     .'                                          │     .'
     │   .'                                            │   .'
     │  ● Risk-Free Rate (Rf)                          │  ● Risk-Free Rate (Rf)
     └──────────────────────────────► Total Risk (σ)   └──────────────────────────────► Systematic Risk (β)

1. Capital Market Line (CML)

The Capital Market Line is the Capital Allocation Line that connects the risk-free asset ($R_f$) directly to the aggregate Market Portfolio ($M$):

$E(R_p) = R_f + \left[ \frac{E(R_m) - R_f}{\sigma_m} \right] \sigma_p$

• Horizontal Axis: Total Risk measured by standard deviation ($\sigma$).
• Slope: The Sharpe Ratio of the Market Portfolio: $S_m = \frac{E(R_m) - R_f}{\sigma_m}$.
• Applicability: Efficient, fully diversified portfolios only.
• Individual Assets: Individual stocks and inefficient portfolios possess uncompensated unsystematic risk and therefore must plot strictly below the CML.

2. Security Market Line (SML)

The Security Market Line is the graphical representation of the CAPM pricing equation:

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

• Horizontal Axis: Systematic Risk measured by Beta ($\beta$).
• Slope: The Market Risk Premium ($E(R_m) - R_f$).
• Applicability: All individual securities, inefficient portfolios, and efficient portfolios.
• Equilibrium Condition: In theoretical equilibrium, all correctly priced assets plot directly on the SML.

Direct Comparison: CML vs. SML

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3. Beta Estimation, Security Characteristic Line (SCL) & Jensen's Alpha

Mathematical Formulation of Beta ($\beta$)

Beta measures the sensitivity of an asset's return to fluctuations in the overall market portfolio:

$\beta_i = \frac{\text{Cov}(R_i, R_m)}{\text{Var}(R_m)} = \frac{\sigma_{im}}{\sigma_m^2} = \rho_{im} \left( \frac{\sigma_i}{\sigma_m} \right)$

• $\beta = 1.0$: Average market risk exposure (moves in tandem with market).
• $\beta > 1.0$: Aggressive / cyclical asset (e.g., technology, consumer discretionary; amplifies market swings).
• $\beta < 1.0$: Defensive asset (e.g., utilities, consumer staples, healthcare; dampened sensitivity).
• $\beta = 0.0$: Uncorrelated with market (e.g., risk-free Treasury bills).
• $\beta < 0.0$: Inverse market exposure (e.g., short funds, dedicated put strategies).

The Security Characteristic Line (SCL)

Beta and Jensen's alpha are estimated empirically via ordinary least squares (OLS) linear regression of excess asset returns against excess market returns over $T$ historical periods:

$(R_{i,t} - R_{f,t}) = \alpha_i + \beta_i (R_{m,t} - R_{f,t}) + e_{i,t}$

• Slope of SCL: Asset Beta ($\beta_i$).
• Intercept of SCL: Jensen's Alpha ($\alpha_i$).
• Residual Variance: $\sigma^2(e_i)$, representing the asset's idiosyncratic risk.
• Coefficient of Determination ($R^2$): Quantifies the percentage of the asset's total return variance explained by market systematic risk ($R^2 = \rho_{im}^2$).

   Excess Asset Return (Ri - Rf)
     ▲
     │                                              / Security Characteristic Line (SCL)
     │                                            /   Slope = Beta (β_i)
     │                                          / 
     │                                        / 
     │                                      / 
     │                                    / 
     │   Positive Alpha (+α) ──●        / 
     │                         │      / 
     │                         │    / 
     │                         │  / 
     │  Alpha Intercept (α) ──● 
     └────────────────────────┴────────────────────────────────► Excess Market Return (Rm - Rf)
                              0

Jensen's Alpha and SML Mispricing

Jensen's Alpha ($\alpha_i$) measures the realized or expected risk-adjusted return of an asset over and above the return required by the CAPM:

$\alpha_i = E(R_i) - \left[ R_f + \beta_i (E(R_m) - R_f) \right]$

1. Positive Alpha ($\alpha_i > 0$): The asset yields a higher return than warranted by its systematic risk. The asset is undervalued (underpriced) and plots above the SML (a "Buy" signal for active managers).
2. Zero Alpha ($\alpha_i = 0$): The asset is fairly priced in equilibrium and plots directly on the SML.
3. Negative Alpha ($\alpha_i < 0$): The asset generates less return than required for its systematic risk. The asset is overvalued (overpriced) and plots below the SML (a "Sell" or short signal).

Worked Example: Beta, SML Expected Return, and Alpha Analysis

An analyst evaluates a cyclical industrial stock with standard deviation $\sigma_i = 30.0%$. The broad market index has standard deviation $\sigma_m = 15.0%$ and expected return $E(R_m) = 9.0%$. The risk-free rate is $R_f = 3.0%$. The correlation between the stock and the market is $\rho_{im} = 0.70$. A consensus Wall Street forecast projects the stock will return $13.50%$.

1. Calculate Beta ($\beta_i$): $\beta_i = \rho_{im} \left( \frac{\sigma_i}{\sigma_m} \right) = 0.70 \times \left( \frac{0.30}{0.15} \right) = 0.70 \times 2.0 = \mathbf{1.40}$
2. Calculate CAPM Required Return on SML: $E(R_i)_{\text{SML}} = 3.0\% + 1.40 \times (9.0\% - 3.0\%) = 3.0\% + (1.40 \times 6.0\%) = 3.0\% + 8.40\% = \mathbf{11.40\%}$
3. Calculate Jensen's Alpha ($\alpha_i$): $\alpha_i = 13.50\% - 11.40\% = \mathbf{+2.10\%}$
4. Investment Conclusion: Because $\alpha_i = +2.10% > 0$, the stock plots above the SML and is undervalued in the market, offering superior risk-adjusted return potential.

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