Chapter 5: Introduction to Capital Market Theory

Capital Market Theory extends Markowitz's Modern Portfolio Theory by introducing a fundamental real-world asset: a risk-free asset ($R_f$) that yields a guaranteed return with zero variance ($\sigma_{Rf} = 0$).

The Risk-Free Asset and Capital Allocation Line (CAL)

When an investor combines a risk-free asset with a portfolio of risky assets $p$, the expected return and risk of the combined portfolio are linear:

$$E(R_c) = w_{Rf} R_f + (1 - w_{Rf}) E(R_p)$$

$$\sigma_c = (1 - w_{Rf}) \sigma_p$$

The line connecting the risk-free rate on the vertical axis to any risky portfolio is the Capital Allocation Line (CAL). Its slope represents the Sharpe Ratio of that portfolio:

$$\text{Slope of CAL} = \frac{E(R_p) - R_f}{\sigma_p}$$

Expected
Return
  ^                                           CAL / CML (Capital Market Line)
  |                                         /
  |                                       / * (Tangency Market Portfolio M)
  |                                     / *
  |                                   / * (Markowitz Risky Efficient Frontier)
  |                                 / *
  |                               / *
  |                             /*
  |  * (Risk-Free Rate Rf)     /
  |
  +----------------------------------------------------------------------------> Total Risk \sigma

The Capital Market Line (CML) and Market Portfolio

When all rational investors hold homogeneous expectations, they will seek the CAL with the steepest possible slope. This unique line is the Capital Market Line (CML), which is tangent to the Markowitz efficient frontier at exactly one point: the Market Portfolio ($M$).

• The Market Portfolio ($M$): A theoretical portfolio containing every risky asset in the global economy, held in exact proportion to its market capitalization value.
• Equation of the Capital Market Line:

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

Where:

• $\left[ \frac{E(R_m) - R_f}{\sigma_m} \right]$ is the Market Price of Risk (the Sharpe ratio of the market portfolio).
• $\sigma_p$ is the total standard deviation of the efficient portfolio.
• Critical Rule: The CML applies strictly and exclusively to efficient, fully diversified portfolios. It cannot be used to price individual securities or inefficient portfolios containing unsystematic risk.

Developed independently by William Sharpe, John Lintner, and Jan Mossin, the Capital Asset Pricing Model (CAPM) prices individual securities and inefficient portfolios by evaluating their systematic risk.

Derivation and Formula of CAPM

In an efficient market, investors are not rewarded for bearing diversifiable unsystematic risk. The required return on any individual asset $i$ is solely a function of its sensitivity to systematic market movements, measured by Beta ($\beta_i$):

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

Where Beta is mathematically defined as:

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

• $\beta_i = 1.0$: The asset moves in perfect lockstep with the broad market.
• $\beta_i > 1.0$: Aggressive asset; amplified sensitivity to market swings.
• $\beta_i < 1.0$: Defensive asset; dampened sensitivity to market swings.

+----------------------------------------------------------------------------------------------------+
|                                    CML vs. SML COMPARATIVE MATRIX                                  |
+--------------------------+-------------------------------------+-----------------------------------+
| ATTRIBUTE                | CAPITAL MARKET LINE (CML)           | SECURITY MARKET LINE (SML)        |
+--------------------------+-------------------------------------+-----------------------------------+
| Measure of Risk          | **Total Risk (\sigma_p)**            | **Systematic Market Risk (\beta_i)**|
| (Horizontal Axis)        | (Standard deviation).               | (Covariance with market).         |
+--------------------------+-------------------------------------+-----------------------------------+
| Scope of Applicability   | Only **fully diversified** efficient| **Every individual security**,    |
|                          | portfolios.                         | inefficient portfolio, or asset.  |
+--------------------------+-------------------------------------+-----------------------------------+
| Reward / Slope           | Sharpe Ratio: [E(R_m)-R_f]/\sigma_m | Equity Risk Premium: [E(R_m)-R_f] |
+--------------------------+-------------------------------------+-----------------------------------+

The Security Market Line (SML) and Jensen's Alpha ($\alpha$)

The graphical representation of the CAPM is the Security Market Line (SML):

• Underpriced (Undervalued) Asset: An asset plotting above the SML offers an expected return higher than its CAPM required return. It possesses Positive Jensen's Alpha ($\alpha > 0$) and is a BUY.
• Overpriced (Overvalued) Asset: An asset plotting below the SML offers an expected return lower than its CAPM required return. It possesses Negative Jensen's Alpha ($\alpha < 0$) and is a SELL.

While single-index CAPM provides elegant theoretical clarity, empirical studies reveal that market beta alone cannot fully explain the cross-section of realized equity returns.

1. Arbitrage Pricing Theory (APT)

Formulated by Stephen Ross, APT is a multi-factor asset pricing model built on the principle of the law of one price and absence of arbitrage:

$$E(R_i) = R_f + \beta_{i1} F_1 + \beta_{i2} F_2 + \dots + \beta_{ik} F_k$$

Where $F_k$ represents macroeconomic risk factors (unexpected changes in inflation, industrial production, credit default spreads, and sovereign yield curve shifts).

2. The Fama-French Three-Factor Model

Eugene Fama and Kenneth French demonstrated that two firm-specific variables—size and book-to-market equity—capture significant variation in stock returns unexplained by CAPM beta:

$$E(R_i) - R_f = \alpha_i + \beta_{i,MKT} [R_m - R_f] + \beta_{i,SMB} \cdot \text{SMB} + \beta_{i,HML} \cdot \text{HML}$$

Where:

• SMB (Small Minus Big): The historic return premium of small-cap companies over large-cap companies (Size factor).
• HML (High Minus Low): The historic return premium of high book-to-market value companies over low book-to-market growth companies (Value factor).

3. The Carhart Four-Factor Model

Mark Carhart extended the Fama-French model by incorporating the Momentum Factor (WML - Winners Minus Losers):

• Captures the tendency of past 12-month winning stocks to continue outperforming past losing stocks. Used extensively by Category III long/short and quantitative fund managers to isolate pure managerial alpha.