Chapter 16: Introduction to Capital Market Theory
16.1 Risk-Free Asset (R_f), Capital Allocation Line (CAL), and Market Portfolio (M)
Introduction of the Risk-Free Asset ($R_f$)
Modern Portfolio Theory operates purely in a world of risky assets. Capital Market Theory, pioneered by William Sharpe, John Lintner, and Jan Mossin, extends Markowitz's framework by introducing a Risk-Free Asset ($R_f$)—an instrument with zero variance of return ($\sigma_f = 0$) and zero covariance with all risky assets ($\text{Cov}(R_f, R_i) = 0$). In practice, short-term sovereign debt (e.g., 91-day Government of India Treasury Bills) proxies the risk-free rate.
The Capital Allocation Line (CAL)
When an investor combines the risk-free asset with any risky portfolio $P$, the resulting complete portfolio has expected return and standard deviation: $$E(R_c) = w_f R_f + (1 - w_f) E(R_p) = R_f + w_p [E(R_p) - R_f]$$ $$\sigma_c = \sqrt{w_p^2 \sigma_p^2 + 0 + 0} = w_p \sigma_p \implies w_p = \frac{\sigma_c}{\sigma_p}$$ Substituting $w_p$ into the return equation yields the straight-line equation of the Capital Allocation Line (CAL): $$E(R_c) = R_f + \left[ \frac{E(R_p) - R_f}{\sigma_p} \right] \sigma_c$$
Expected Return E(R)
^
| / CAL (Tangency Line)
| /
| / Market Portfolio (M)
| *
| / \ Markowitz Efficient Frontier
| / \
| / \
| /
| /
| Risk-Free Rate (Rf) *
+-------------------------------------------> Total Risk (sigma)
The slope of the Capital Allocation Line is the Sharpe Ratio of risky portfolio $P$: $$\text{Slope of CAL} = \frac{E(R_p) - R_f}{\sigma_p}$$
The Tangency Portfolio and The Market Portfolio ($M$)
To maximize risk-adjusted returns, an investor selects the specific risky portfolio that produces the steepest possible CAL slope. This unique portfolio is the Tangency Portfolio, where the CAL is exactly tangent to the Markowitz Efficient Frontier.
Under the Capital Market Theory assumption of homogeneous expectations, all investors identify the identical tangency portfolio. In equilibrium, this portfolio must hold all available risky assets in proportion to their relative market values. This unique tangency portfolio is designated as the Market Portfolio ($M$).
16.2 Capital Market Line (CML) and Capital Asset Pricing Model (CAPM) Derivation
The Capital Market Line (CML)
The Capital Market Line is the specific Capital Allocation Line resulting from combining the Risk-Free Asset ($R_f$) with the broad Market Portfolio ($M$):
$$E(R_p) = R_f + \left[ \frac{E(R_m) - R_f}{\sigma_m} \right] \sigma_p$$ Where:
- $E(R_p)$ = Expected return on any efficient portfolio
- $R_f$ = Risk-free rate of return
- $E(R_m)$ = Expected return on the market portfolio
- $\sigma_m$ = Standard deviation of the market portfolio
- $\frac{E(R_m) - R_f}{\sigma_m}$ = Market price of risk (equilibrium Sharpe ratio)
- $\sigma_p$ = Standard deviation of the efficient portfolio
+---------------------------------------------------------------------------------------------------+
| CRITICAL DISTINCTION: CML VS. SML |
+-----------------------------------+---------------------------------------------------------------+
| Capital Market Line (CML) | Security Market Line (SML) |
+-----------------------------------+---------------------------------------------------------------+
| Measure of Risk: TOTAL RISK | Measure of Risk: SYSTEMATIC RISK ONLY |
| (Standard Deviation sigma) | (Beta beta) |
| Applies ONLY to EFFICIENT | Applies to ALL SECURITIES & PORTFOLIOS |
| fully-diversified portfolios | (Individual stocks, inefficient portfolios, efficient baskets)|
+-----------------------------------+---------------------------------------------------------------+
The CML cannot price individual securities because individual securities contain unsystematic risk that can be eliminated through diversification.
Derivation of the Capital Asset Pricing Model (CAPM)
In an efficient market, rational investors will not be compensated with risk premiums for holding unsystematic (idiosyncratic) risk, since such risk can be diversified away at zero cost. Therefore, the only risk rewarded by the market is Systematic Market Risk, quantified by Beta ($\beta$):
$$\beta_i = \frac{\text{Cov}(R_i, R_m)}{\sigma_m^2} = \frac{\rho_{im} \sigma_i \sigma_m}{\sigma_m^2} = \rho_{im} \left( \frac{\sigma_i}{\sigma_m} \right)$$
The fundamental equation of the Capital Asset Pricing Model (CAPM) expresses the expected return of any asset $i$ as: $$E(R_i) = R_f + \beta_i \left[ E(R_m) - R_f \right]$$ Where:
- $\beta_i$ = Systematic risk coefficient of security $i$
- $[E(R_m) - R_f]$ = Market Risk Premium (MRP)
- $\beta_i [E(R_m) - R_f]$ = Equity Risk Premium demanded for holding security $i$
16.3 Security Market Line (SML), Jensen's Alpha, Roll's Critique, and Multi-Factor Models
The Security Market Line (SML)
The graphical representation of the CAPM is the Security Market Line (SML), plotting expected return $E(R)$ on the vertical axis against systematic risk (Beta $\beta$) on the horizontal axis:
Expected Return E(R)
^
| / SML
| * A / (Underpriced Stock: Alpha > 0 -> BUY!)
| /
| * M / E(Rm)
| /
| / * B (Overpriced Stock: Alpha < 0 -> SELL!)
| /
| /
| Rf *
+-------------------------+---------------------> Systematic Risk (Beta)
Beta = 1.0
Valuation via the SML: Jensen's Alpha ($\alpha$)
The SML provides an objective benchmark for determining whether an asset is fairly priced, undervalued, or overvalued:
- Undervalued Asset (Point A): Lies ABOVE the SML. Its estimated expected return exceeds the return required by the CAPM ($E(R_A) > R_f + \beta_A [E(R_m) - R_f]$). The security generates Positive Jensen's Alpha ($\alpha > 0$). Investors buy the security, bidding up its price until its expected return falls back onto the SML.
- Overvalued Asset (Point B): Lies BELOW the SML. Its expected return is insufficient given its systematic risk. The security generates Negative Jensen's Alpha ($\alpha < 0$). Investors sell/short the security until its price falls to equilibrium.
$$\alpha_i = R_i - \left[ R_f + \beta_i (R_m - R_f) \right]$$
Richard Roll's Critique of CAPM
In 1977, Richard Roll published a famous critique challenging the empirical testability of the CAPM:
- Unobservability of the True Market Portfolio: The CAPM requires that the Market Portfolio ($M$) contain all risky assets worldwide (including real estate, private businesses, commodities, human capital, art).
- Proxy Problem: In empirical tests, researchers use broad stock indices (like the Nifty 50 or S&P 500) as proxies for $M$. If the proxy is mean-variance inefficient, the CAPM can appear invalid even if the true model holds.
Multi-Factor Capital Market Models
To overcome single-factor CAPM limitations, financial economists developed multi-factor asset pricing frameworks:
- Arbitrage Pricing Theory (APT): Developed by Stephen Ross (1976), APT assumes asset returns are driven by multiple macroeconomic factors (GDP growth, inflation, yield curve spreads, industrial production): $$E(R_i) = R_f + \beta_{i1} F_1 + \beta_{i2} F_2 + \dots + \beta_{ik} F_k$$
- Fama-French Three-Factor Model (1993): Captures market risk plus two fundamental empirical size and value anomalies:
$$E(R_i) = R_f + \beta_i [E(R_m) - R_f] + s_i \text{SMB} + h_i \text{HML}$$
- SMB (Small Minus Big): Size risk premium (small-cap outperformance)
- HML (High Minus Low): Value risk premium (high book-to-market outperformance)
- Carhart Four-Factor Model (1997): Adds a fourth factor: WML (Winners Minus Losers) to capture the empirical Momentum anomaly.