Asset Pricing Theory
Paris Dauphine University-PSL
Mathematical Foundations of Market Efficiency and Information Processing
Informationally efficient market definition: \[E[\tilde{p}_{t+1} | \Phi_t] = [1 + E(\tilde{r}_{t+1} | \Phi_t)] p_t\]
Key assumptions: 1. No transaction costs 2. Information freely available 3. Rational expectations: \(E[\tilde{r}_{t+1} | \Phi_t] = E[\tilde{r}_{t+1} | \Phi_t^*]\) where \(\Phi_t^*\) is true information set
Three forms of efficiency:
Weak form: \(\Phi_t\) contains only historical prices \(\{p_s\}_{s<t}\) - Implication: \(E[\tilde{r}_{t+1} | \{r_s\}_{s<t}] = E[\tilde{r}_{t+1}]\) - Testable: No predictability from past returns
Semi-strong form: \(\Phi_t\) contains all public information - Includes: Financial statements, news, analyst reports - Implication: Prices adjust immediately to public news
Strong form: \(\Phi_t\) contains all information (public + private) - Includes: Insider information, private signals - Implication: No abnormal returns to any trader
Arithmetic random walk: \[p_t = p_{t-1} + \epsilon_t, \quad \epsilon_t \sim iid(0,\sigma^2)\]
Properties: - \(E[p_t | p_{t-1}] = p_{t-1}\) - \(\text{Var}(p_t | p_{t-1}) = \sigma^2\) - Independent increments
Geometric random walk (log prices): \[\ln p_t = \ln p_{t-1} + \mu + \epsilon_t, \quad \epsilon_t \sim iid(0,\sigma^2)\]
Equivalent to multiplicative model: \[p_t = p_{t-1} \cdot e^{\mu + \epsilon_t}\]
Martingale property derivation: - For arithmetic RW: \(E[p_t | \mathcal{F}_{t-1}] = p_{t-1}\) - For geometric RW: \(E[\ln p_t | \mathcal{F}_{t-1}] = \ln p_{t-1} + \mu\)
Submartingale (with drift): \[E[p_t | \mathcal{F}_{t-1}] = (1 + \mu) p_{t-1} > p_{t-1}\]
Fair game property: \[E[\tilde{r}_t | \mathcal{F}_{t-1}] = 0\]
Fundamental testing challenge: - Cannot test EMH in isolation - Must test joint hypothesis: EMH + Equilibrium model
Mathematical formulation: \[H_0: \text{EMH holds} \land \text{Model } M \text{ is correct}\]
Rejection possibilities: 1. Type I error: False rejection of correct EMH + M 2. EMH violation: EMH false, M correct 3. Model misspecification: EMH true, M false 4. Both wrong: EMH false, M false
Example with CAPM: - Reject: \(E[r_i] \neq \beta_i E[r_M]\) - Could be: EMH violation OR CAPM misspecified OR both
Market model estimation (pre-event period): \[R_{it} = \alpha_i + \beta_i R_{mt} + \epsilon_{it}, \quad \epsilon_{it} \sim iid(0,\sigma_i^2)\]
Parameter estimation: \[\hat{\alpha}_i = \bar{R}_i - \hat{\beta}_i \bar{R}_m\] \[\hat{\beta}_i = \frac{\sum (R_{it} - \bar{R}_i)(R_{mt} - \bar{R}_m)}{\sum (R_{mt} - \bar{R}_m)^2}\]
Abnormal return on event day t: \[AR_{it} = R_{it} - (\hat{\alpha}_i + \hat{\beta}_i R_{mt})\]
Standardized abnormal return: \[SAR_{it} = \frac{AR_{it}}{\hat{\sigma}_{AR,i}}\]
Where residual standard deviation: \[\hat{\sigma}_{AR,i} = \sqrt{\frac{\sum_{t \in \text{estimation}} AR_{it}^2}{T-2}}\]
CAR over window \((\tau_1, \tau_2)\): \[CAR_i(\tau_1, \tau_2) = \sum_{t=\tau_1}^{\tau_2} AR_{it}\]
Statistical test for single security: - Under null: \(AR_{it} \sim iid(0, \sigma_i^2)\) - Test statistic: \(t_i = \frac{CAR_i(\tau_1, \tau_2)}{\hat{\sigma}_{CAR,i}} \sqrt{|\tau_2 - \tau_1| + 1}\) - Where: \(\hat{\sigma}_{CAR,i} = \hat{\sigma}_{AR,i} \sqrt{|\tau_2 - \tau_1| + 1}\)
Cross-sectional test for N securities: \[t = \frac{\overline{CAR}}{\hat{\sigma}(\overline{CAR})} \sqrt{N}\]
Assumptions: 1. Normality: \(AR_{it} \sim N(0, \sigma_i^2)\) 2. Independence: Across securities and time 3. Stationarity: Parameters constant over estimation period
Autocorrelation function for returns: \[\rho_k = \frac{\sum_{t=k+1}^T (r_t - \bar{r})(r_{t-k} - \bar{r})}{\sum_{t=1}^T (r_t - \bar{r})^2}\]
Sample autocorrelation properties: - Under independence: \(E[\rho_k] = 0\) - Variance: \(\text{Var}(\rho_k) \approx \frac{1}{T}\) for large T - Asymptotic normality: \(\sqrt{T} \rho_k \sim N(0,1)\)
Ljung-Box portmanteau test: \[Q = T(T+2) \sum_{k=1}^K \frac{\rho_k^2}{T-k}\]
Derivation: - Under null: \(Q \sim \chi^2_K\) - Tests joint hypothesis: \(\rho_1 = \rho_2 = \cdots = \rho_K = 0\) - More powerful than individual tests
Box-Pierce test (simpler version): \[Q_{BP} = T \sum_{k=1}^K \rho_k^2 \sim \chi^2_K\]
Lo-MacKinlay (1988) variance ratio:
Single-period variance: \[\hat{\sigma}^2(1) = \frac{1}{T-1} \sum_{t=2}^T (r_t - \bar{r})^2\]
q-period variance: \[\hat{\sigma}^2(q) = \frac{1}{q(T-q+1)} \sum_{t=q}^T (p_t - p_{t-q})^2\]
Variance ratio: \[VR(q) = \frac{\hat{\sigma}^2(q)}{\hat{\sigma}^2(1)}\]
Random walk hypothesis: - For RW: \(VR(q) = 1\) - For mean-reverting: \(VR(q) > 1\) - For momentum: \(VR(q) < 1\)
Test statistic: \[z(q) = \frac{VR(q) - 1}{\sqrt{\phi(q)}}\]
Where asymptotic variance: \[\phi(q) = \frac{2(2q-1)(q-1)}{3q(T-q+1)}\]
Intuition: Variance ratio compares actual variability to RW prediction
Definition of a run: - Sequence of consecutive positive or negative returns - Direction change defines run boundary
Number of runs in sequence of T returns: \[R = 1 + \sum_{t=2}^T I(r_t r_{t-1} < 0)\]
Under independence null hypothesis: - Let \(n_+ =\) number of positive returns - Let \(n_- =\) number of negative returns - Total: \(n_+ + n_- = T\)
Expected runs: \[E[R] = \frac{2n_+ n_-}{n_+ + n_-} + 1 = \frac{2n_+ n_-}{T} + 1\]
Variance: \[\text{Var}(R) = \frac{2n_+ n_- (2n_+ n_- - n_+ - n_-)}{(n_+ + n_-)^2 (n_+ + n_- - 1)}\]
Test statistic: \[z = \frac{R - E[R]}{\sqrt{\text{Var}(R)}} \sim N(0,1)\]
Intuition: Too few runs suggest positive dependence, too many suggest negative dependence
Standardized unexpected earnings (SUE): \[SUE_t = \frac{X_t - E[X_t | \Phi_{t-1}]}{\hat{\sigma}_X}\]
Where: - \(X_t =\) actual earnings per share - \(E[X_t | \Phi_{t-1}] =\) analyst consensus forecast - \(\hat{\sigma}_X =\) standard deviation of forecast errors
Post-earnings announcement drift (PEAD): \[AR_{t+k} = \gamma_k \cdot SUE_t + \epsilon_{t+k}\]
Ball and Brown (1968) findings: - Immediate price adjustment: \(AR_t \propto SUE_t\) - Subsequent drift: Positive (negative) SUE predicts positive (negative) AR for months
Theoretical explanations: 1. Risk premium: High SUE firms have higher risk 2. Behavioral: Underreaction to earnings surprise 3. Institutional constraints: Limits to arbitrage
Signaling hypothesis (Bhattacharya 1979): - Dividends signal future earnings - Managers use dividends to convey private information
Price reaction model: \[\Delta P = \gamma \cdot \Delta D + \epsilon\]
Where: - \(\Delta P =\) price change around announcement - \(\Delta D =\) dividend change - \(\gamma > 0\) if signaling valuable
Miller-Modigliani irrelevance theorem: - In perfect markets: Dividends irrelevant for firm value - Assumptions: No taxes, no transaction costs, perfect information
Signaling restores relevance: - Asymmetric information creates signaling role - Dividend increases signal positive private information - Market rationally responds to signal
Abnormal return calculation: \[AR_{insider} = R_{insider} - R_{benchmark}\]
Where benchmark could be: - Market index - Size-matched portfolio - Industry-matched portfolio
Jaffe (1974) methodology: - Sample: SEC filings of insider trades - Time period: 1960s data - Result: Significant positive AR for insider purchases
Seyhun (1986) extensions: - Insider sales show negative AR - Returns persist for months after trade - Stronger for smaller firms
Jensen’s alpha (risk-adjusted returns): \[\alpha_i = E[R_i] - R_f - \beta_i (E[R_M] - R_f)\]
CAPM benchmark: - If \(\alpha_i > 0\): Outperformance - If \(\alpha_i < 0\): Underperformance - If \(\alpha_i = 0\): Fair performance
Persistence tests: \[P(\alpha_{i,t+1} > 0 | \alpha_{i,t} > 0)\]
Carhart (1997) four-factor model: \[R_{it} - R_{ft} = \alpha_i + \beta_{i,M} (R_{Mt} - R_{ft}) + \beta_{i,SMB} SMB_t + \beta_{i,HML} HML_t + \beta_{i,MOM} MOM_t\]
Factor definitions: - SMB (Small Minus Big): Size effect - HML (High Minus Low): Value effect
- MOM (Momentum): Momentum effect
Effective spread: \[S_{effective} = 2 \cdot \frac{|P - P_{mid}|}{P_{mid}}\]
Where: - \(P =\) transaction price - \(P_{mid} =\) midpoint of bid-ask spread
Realized spread: \[S_{realized} = 2 \cdot \frac{|P - P_{VWAP}|}{P_{VWAP}}\]
Where VWAP is volume-weighted average price over subsequent period
Price impact regression: \[\Delta P = \lambda \cdot \frac{Q}{ADV} + \epsilon\]
Where: - \(Q =\) trade size - \(ADV =\) average daily volume - \(\lambda =\) price impact coefficient
Easley-KO-O’Hara PIN model:
Model assumptions: - Informed traders: Probability α, trade μ shares - Uninformed traders: Trade ε shares (buy or sell)
Probability of informed trading: \[PIN = \frac{\alpha \mu}{\alpha \mu + 2\epsilon}\]
Likelihood function: \[L = \prod_{t=1}^T \left[ \alpha \mu e^{-\mu} \frac{(\mu)^B}{B!} \cdot \epsilon e^{-\epsilon} \frac{(\epsilon)^S}{S!} \right]\]
Maximum likelihood estimation of (α, μ, ε)
Adverse selection component: \[AS = \frac{PIN}{2} \cdot \sigma_V^2\]
Kyle’s lambda (price impact): \[\lambda = \frac{\sqrt{2}}{\sigma_u} \cdot \frac{\sigma_V}{\sqrt{T}}\]
Derivation from Kyle model: - Informed trade: \(x = \beta (V - \mu)\) - Market maker: \(P = \mu + \lambda (x + u)\) - Equilibrium: \(\lambda = \frac{\sigma_V}{\beta \sigma_u}\)
Banz (1981) finding: \[E[R_{small}] > E[R_{large}] + \beta_{small} (E[R_M] - R_f)\]
Evidence: - Small firms earn 1-2% higher monthly returns - Persists after controlling for beta - Stronger in January (January effect)
Fama-French three-factor model: \[R_{it} - R_{ft} = \alpha_i + \beta_{i,M} (R_{Mt} - R_{ft}) + \beta_{i,SMB} SMB_t + \beta_{i,HML} HML_t\]
SMB factor construction: \[SMB_t = \frac{1}{3} (Small_{value} + Small_{neutral} + Small_{growth} - Big_{value} - Big_{neutral} - Big_{growth})\]
Value premium discovery: - Basu (1977): High E/P ratios predict higher returns - Rosenberg et al. (1985): High B/M ratios predict higher returns
Value premium: \[E[R_{value}] - E[R_{growth}] \approx 4-5\% \text{ annually}\]
Book-to-market factor (HML): \[HML_t = \frac{1}{3} (Small_{value} + Big_{value} - Small_{growth} - Big_{growth})\]
Theoretical explanations: 1. Risk-based: Value firms have higher distress risk 2. Behavioral: Investor over-optimism for growth stocks 3. Institutional: Value investing requires contrarian behavior
Jegadeesh-Titman (1993) strategy: - Buy past 6-month winners - Sell past 6-month losers - Hold for 6 months - Result: 1% monthly abnormal returns
Momentum factor (MOM): \[MOM_t = \frac{1}{2} (Small_{up} + Big_{up} - Small_{down} - Big_{down})\]
Where: - \(Up =\) past 12-month winners (skip most recent month) - \(Down =\) past 12-month losers (skip most recent month)
Reversal at longer horizons: - Momentum profits reverse after 3-5 years - Suggests overreaction followed by correction
Value function: \[v(x) = \begin{cases} x^\alpha & x \geq 0 \\ -\lambda (-x)^\beta & x < 0 \end{cases}\]
Parameters: - \(\alpha \approx 0.88\) (concave for gains) - \(\beta \approx 0.88\) (convex for losses) - \(\lambda \approx 2.25\) (loss aversion)
Probability weighting function: \[w(p) = \frac{p^\gamma}{(p^\gamma + (1-p)^\gamma)^{1/\gamma}}\]
With \(\gamma \approx 0.61\) (overweight small probabilities)
Asset pricing implications: - Disposition effect: Investors hold losers too long, sell winners too soon - Narrow framing: Evaluate investments in isolation - Mental accounting: Separate treatment of gains/losses
Shleifer-Vishny (1997) model:
Arbitrageur capital dynamics: \[\frac{dC}{C} = \mu dt + \sigma dW - \delta dt\]
Performance-based capital withdrawal: - Good performance: Capital inflow (μ > 0) - Poor performance: Capital outflow (δ > 0)
Procyclical arbitrage: \[\text{Arbitrage activity} \propto C_t\]
Key insight: Capital flows exacerbate mispricings exactly when arbitrage is most needed
De Long et al. (1990) noise trader model:
Noise trader demand: \[D_{NT,t} = \gamma P_{t-1} + \epsilon_t\]
Arbitrageur response: \[D_{A,t} = -\phi (P_t - P^*)\]
Equilibrium price: \[P_t = P^* + \frac{\gamma}{1 + \phi} P_{t-1} + \frac{1}{1 + \phi} \epsilon_t\]
Persistent mispricing when arbitrage limited
Bayesian updating: \[p(\theta | y) = \frac{p(y | \theta) p(\theta)}{p(y)}\]
Price as sufficient statistic: - In rational expectations equilibrium - \(p_t = E[V | \Phi_t]\) - All relevant information impounded in price
Aggregation theorem (Grossman-Stiglitz 1980): \[\frac{1}{N} \sum_{i=1}^N p_i \to E[V | \Phi_{aggregate}] \text{ as } N \to \infty\]
No-trade theorem (Milgrom-Stokey 1982): - If information is common knowledge - And preferences are common knowledge - Then no trade occurs in equilibrium
Recursive least squares (RLS): \[\hat{\theta}_{t+1} = \hat{\theta}_t + \frac{P_t x_t (y_t - x_t' \hat{\theta}_t)}{1 + x_t' P_t x_t}\]
Where: - \(P_t =\) precision matrix (inverse covariance) - \(x_t =\) regressors - \(y_t =\) dependent variable
Kalman filter for state estimation:
State equation: \[\theta_{t+1} = \theta_t + w_t, \quad w_t \sim N(0, Q)\]
Measurement equation: \[y_t = x_t' \theta_t + v_t, \quad v_t \sim N(0, R)\]
Kalman update: - Prediction: \(\hat{\theta}_{t|t-1}, P_{t|t-1}\) - Innovation: \(\nu_t = y_t - x_t' \hat{\theta}_{t|t-1}\) - Kalman gain: \(K_t = P_{t|t-1} x_t (x_t' P_{t|t-1} x_t + R)^{-1}\) - Update: \(\hat{\theta}_{t|t} = \hat{\theta}_{t|t-1} + K_t \nu_t\)
Applications to asset pricing: - Learning about expected returns - Updating beliefs about firm fundamentals - Adaptive expectations formation
EMH Contributions: - Prices aggregate information efficiently under rational expectations - Random walk as null hypothesis for weak-form tests - Event studies methodology for semi-strong form tests - Joint hypothesis problem highlights testing challenges
Challenges and Extensions: - Anomalies require explanation: Size, value, momentum effects - Behavioral foundations: Prospect theory, limits to rationality - Limits to arbitrage: Capital constraints, noise trader risk - Market microstructure effects: Bid-ask spreads, information asymmetry
Mathematical Rigor: - Statistical testing of efficiency: Serial correlation, variance ratios, runs tests - Information processing models: Bayesian updating, Kalman filtering - Equilibrium with frictions: Adverse selection, transaction costs - Learning frameworks: Recursive estimation, adaptive expectations
Policy Implications: - Market design affects efficiency: Trading mechanisms, disclosure rules - Understanding frictions crucial: For asset pricing theory and practice - Behavioral insights inform: Investment strategies and risk management
Asset Pricing Theory - Day 6