Module 19: Behavioral Finance & Market Anomalies

Introduction: When Humans Are the Data-Generating Process

Classical finance assumes rational, utility-maximizing agents who instantly incorporate all available information into prices. Behavioral finance starts from a different premise: humans are the data-generating process in financial markets, and humans are systematically biased. These biases are not random noise — they are structured, predictable deviations from rationality that leave measurable fingerprints in asset prices. For a statistician, behavioral finance is the study of systematic errors in the human estimation process.

1. Loss Aversion and Prospect Theory

1.1 The Core Finding

Kahneman and Tversky (1979) discovered that people do not evaluate outcomes in terms of final wealth levels (as expected utility theory assumes), but in terms of gains and losses relative to a reference point. Furthermore, losses are weighted approximately 2.5 times more heavily than equivalent gains.

1.2 Financial Implications of Loss Aversion

2. Overconfidence: Miscalibrated Confidence Intervals

2.1 The Calibration Problem

When people construct 90% confidence intervals for unknown quantities, the true value falls outside their intervals roughly 50% of the time. This is not a minor miscalibration — it is a massive, systematic failure of probabilistic reasoning.

2.2 Forms of Overconfidence

2.3 Trading Volume and Overconfidence

Barber and Odean (2000) showed that individual investors who trade the most earn the lowest net returns. Overconfident traders trade too frequently, incurring transaction costs that erode returns. The annual return shortfall attributable to excessive trading is approximately 3.7 percentage points for the most active traders versus the least active.

3. Anchoring: Biased Priors

3.1 The Mechanism

Anchoring is the tendency for initial values to disproportionately influence subsequent estimates, even when the initial values are arbitrary or irrelevant. In experiments, people who first see a random number give estimates biased toward that number.

3.2 Financial Examples of Anchoring

• Purchase price anchoring: Investors anchor to the price they paid, making sell decisions based on gains/losses rather than current valuation.
• 52-week high anchoring: Stocks near their 52-week high are perceived as “expensive” even if fundamentals have improved.
• Analyst forecast anchoring: Earnings forecasts cluster around the consensus and adjust insufficiently to new information.
• Round number anchoring: Prices cluster around round numbers ($50, $100), creating support/resistance levels.

4. Herding: Correlated Errors Create Bubbles

4.1 The Statistics of Herding

When investors make independent errors, those errors cancel in aggregate (by the law of large numbers), and prices remain efficient. But when investors herd — imitating each other rather than acting on private information — their errors become correlated, and the law of large numbers breaks down.

4.2 Herding Mechanisms

4.3 Bubbles as Statistical Phenomena

A price bubble occurs when the market price deviates significantly from fundamental value for an extended period. From a statistical perspective, bubbles are characterized by:

• Super-exponential growth: Prices grow faster than exponential, implying accelerating returns.
• Increasing positive autocorrelation: Momentum intensifies during the bubble.
• Decreasing diversity: Investors converge on the same narrative.
• Fat-tailed crash risk: The bubble’s end is a tail event in the return distribution.

5. The Momentum Anomaly

5.1 Definition and Evidence

The momentum anomaly, documented by Jegadeesh and Titman (1993), states that stocks that have performed well over the past 3–12 months tend to continue performing well, and stocks that have performed poorly tend to continue underperforming. This is one of the most robust and widely replicated findings in empirical finance.

5.2 Behavioral Explanations

• Underreaction: Investors are slow to update beliefs due to anchoring and conservatism. Prices adjust gradually to new information, creating trending behavior.
• Overreaction (delayed): Initial underreaction is followed by overreaction as herding and extrapolation take over, pushing prices beyond fair value.
• Disposition effect: Winners are sold too early (limiting upside response) while losers are held (limiting downside response).

5.3 Python: Testing Momentum

Python
import numpy as np
import pandas as pd
import yfinance as yf
from scipy import stats

# Download a universe of ETFs as proxies for asset classes
tickers = ["XLK", "XLF", "XLE", "XLV", "XLI",
           "XLP", "XLU", "XLB", "XLY", "XLRE"]
data = yf.download(tickers, start="2005-01-01", end="2023-12-31")
prices = data["Adj Close"]

# Monthly returns
monthly_prices = prices.resample("M").last()
monthly_returns = monthly_prices.pct_change().dropna()

# ──────────────────────────────────────────────
# Momentum strategy: buy top 3 performers over past 12 months
# ──────────────────────────────────────────────
formation_period = 12  # months
holding_period = 1    # month
n_long = 3
n_short = 3

# Compute trailing returns (skip most recent month to avoid reversal)
trailing_ret = monthly_prices.pct_change(formation_period).shift(1)

momentum_returns = []
for date in monthly_returns.index[formation_period + 1:]:
    if date not in trailing_ret.index:
        continue

    ranks = trailing_ret.loc[date].dropna().rank(ascending=False)

    # Long winners, short losers
    winners = ranks[ranks <= n_long].index
    losers = ranks[ranks > len(ranks) - n_short].index

    if date in monthly_returns.index:
        long_ret = monthly_returns.loc[date, winners].mean()
        short_ret = monthly_returns.loc[date, losers].mean()
        momentum_returns.append({
            "date": date,
            "long": long_ret,
            "short": short_ret,
            "long_short": long_ret - short_ret,
            "long_only": long_ret,
        })

mom_df = pd.DataFrame(momentum_returns).set_index("date")

# ──────────────────────────────────────────────
# Statistical analysis
# ──────────────────────────────────────────────
for col in ["long_short", "long_only"]:
    ret = mom_df[col]
    ann_ret = ret.mean() * 12
    ann_vol = ret.std() * np.sqrt(12)
    sharpe = ann_ret / ann_vol
    t_stat, p_val = stats.ttest_1samp(ret, 0)

    print(f"\n=== {col.replace('_', ' ').title()} ===")
    print(f"Ann. Return:  {ann_ret:+.2%}")
    print(f"Ann. Vol:     {ann_vol:.2%}")
    print(f"Sharpe Ratio: {sharpe:.3f}")
    print(f"t-statistic:  {t_stat:.3f}")
    print(f"p-value:      {p_val:.4f}")
    print(f"Skewness:     {ret.skew():.3f}")
    print(f"Kurtosis:     {ret.kurtosis():.3f}")

# Test for serial correlation in momentum returns
from statsmodels.stats.diagnostic import acorr_ljungbox
lb = acorr_ljungbox(mom_df["long_short"].dropna(), lags=[6, 12], return_df=True)
print(f"\nLjung-Box test for autocorrelation in momentum returns:")
print(lb)

6. Mean Reversion: The Long-Run Counterpart

6.1 The Evidence

While momentum operates at the 3–12 month horizon, mean reversion operates at longer horizons (3–5 years). DeBondt and Thaler (1985) showed that past losers over 3–5 year periods subsequently outperform past winners. This is negative serial correlation at long lags.

6.2 Autocorrelation Structure of Returns

7. The Value Premium

7.1 Definition

The value premium is the historical tendency for “cheap” stocks (low price relative to fundamentals) to outperform “expensive” stocks (high price relative to fundamentals). Common measures of value include:

7.2 Behavioral Explanation

Behavioral finance attributes the value premium to systematic overextrapolation:

• Investors extrapolate past growth rates too far into the future.
• “Glamour” stocks (high growth, high P/E) are bid up beyond fair value because investors are seduced by the narrative.
• “Value” stocks (low growth, low P/E) are overlooked or feared because their recent performance has been poor.
• When reality disappoints (growth slows) or surprises positively (value improves), prices correct.

7.3 Risk-Based Explanation

The alternative explanation is that value stocks are genuinely riskier: they are often distressed companies with high leverage and uncertain futures. The value premium is compensation for bearing this risk, not a free lunch from behavioral mispricing.

Python
import numpy as np
import pandas as pd
import yfinance as yf
from scipy import stats

# Proxy for value premium: compare value vs growth ETFs
etfs = {
    "Value": "IWD",      # iShares Russell 1000 Value
    "Growth": "IWF",     # iShares Russell 1000 Growth
    "Market": "SPY",     # S&P 500
}

data = yf.download(list(etfs.values()), start="2005-01-01", end="2023-12-31")
prices = data["Adj Close"]
prices.columns = list(etfs.values())
monthly = prices.resample("M").last().pct_change().dropna()

# Value minus Growth spread (HML proxy)
hml_proxy = monthly["IWD"] - monthly["IWF"]

# Test if value premium is statistically significant
t_stat, p_val = stats.ttest_1samp(hml_proxy, 0)
ann_premium = hml_proxy.mean() * 12
ann_vol = hml_proxy.std() * np.sqrt(12)

print("=== Value Premium (Value - Growth) ===")
print(f"Annualized premium: {ann_premium:+.2%}")
print(f"Annualized vol:     {ann_vol:.2%}")
print(f"Sharpe ratio:       {ann_premium/ann_vol:.3f}")
print(f"t-statistic:        {t_stat:.3f}")
print(f"p-value:            {p_val:.4f}")

# Rolling 3-year value premium (to see time variation)
rolling_premium = hml_proxy.rolling(36).mean() * 12
print(f"\nRecent 3Y ann. premium: {rolling_premium.iloc[-1]:+.2%}")
print(f"Historical avg:        {rolling_premium.mean():+.2%}")

# Compare performance by decade
for period_name, start, end in [
    ("2005-2009", "2005", "2009"),
    ("2010-2014", "2010", "2014"),
    ("2015-2019", "2015", "2019"),
    ("2020-2023", "2020", "2023"),
]:
    subset = hml_proxy[start:end]
    print(f"{period_name}: ann. premium = {subset.mean()*12:+.2%}, "
          f"t = {stats.ttest_1samp(subset, 0)[0]:.2f}")

8. Are Anomalies Real Alpha or Risk Premiums?

8.1 The Debate Framework

For every market anomaly, there are three possible explanations:

8.2 The Replication Crisis in Finance

Harvey, Liu, and Zhu (2016) documented over 400 published factors that allegedly explain the cross-section of stock returns. Many of these cannot be replicated out-of-sample. Applying multiple testing corrections, the majority of published factors have t-statistics that fail to reach the adjusted significance threshold.

9. Calendar Anomalies

9.1 The January Effect

Historically, stock returns in January have been higher than other months, especially for small-cap stocks. The most common explanation is tax-loss selling: investors sell losing positions in December to harvest tax losses, depressing prices. When selling pressure subsides in January, prices rebound.

9.2 Day-of-Week Effect

Returns on Mondays have historically been lower than other days (the “Monday effect”). Possible explanations include the accumulation of negative news over weekends and institutional trading patterns.

9.3 Other Calendar Patterns

9.4 Python: Testing Calendar Effects

Python
import numpy as np
import pandas as pd
import yfinance as yf
from scipy import stats

spy = yf.download("SPY", start="2000-01-01", end="2023-12-31")
returns = spy["Adj Close"].pct_change().dropna()

# ──────────────────────────────────────────────
# Test 1: January Effect
# ──────────────────────────────────────────────
monthly_returns = spy["Adj Close"].resample("M").last().pct_change().dropna()
monthly_returns.index = monthly_returns.index.month

jan_returns = monthly_returns[monthly_returns.index == 1]
other_returns = monthly_returns[monthly_returns.index != 1]

t_jan, p_jan = stats.ttest_ind(jan_returns, other_returns)
print("=== January Effect ===")
print(f"January mean return: {jan_returns.mean():+.4f}")
print(f"Other months mean:   {other_returns.mean():+.4f}")
print(f"t-statistic: {t_jan:.3f}, p-value: {p_jan:.4f}")

# Monthly return by month
monthly_returns_full = spy["Adj Close"].resample("M").last().pct_change().dropna()
by_month = monthly_returns_full.groupby(monthly_returns_full.index.month).agg(["mean", "std", "count"])
by_month.columns = ["Mean Return", "Std Dev", "Count"]
by_month["t-stat"] = by_month["Mean Return"] / (by_month["Std Dev"] / np.sqrt(by_month["Count"]))
by_month.index = ["Jan", "Feb", "Mar", "Apr", "May", "Jun",
                  "Jul", "Aug", "Sep", "Oct", "Nov", "Dec"]
print("\nMonthly return patterns:")
print(by_month.round(4))

# ──────────────────────────────────────────────
# Test 2: Day-of-Week Effect
# ──────────────────────────────────────────────
returns_dow = returns.copy()
returns_dow.index = returns_dow.index.dayofweek  # 0=Mon, 4=Fri

by_dow = returns.groupby(returns.index.dayofweek).agg(["mean", "std", "count"])
by_dow.columns = ["Mean Return", "Std Dev", "Count"]
by_dow["t-stat"] = by_dow["Mean Return"] / (by_dow["Std Dev"] / np.sqrt(by_dow["Count"]))
by_dow.index = ["Monday", "Tuesday", "Wednesday", "Thursday", "Friday"]

print("\n=== Day-of-Week Effect ===")
print(by_dow.round(6))

# ANOVA test: are day-of-week returns significantly different?
groups = [returns[returns.index.dayofweek == d] for d in range(5)]
f_stat, p_anova = stats.f_oneway(*groups)
print(f"\nANOVA F-statistic: {f_stat:.3f}, p-value: {p_anova:.4f}")

# ──────────────────────────────────────────────
# Test 3: Halloween Effect (Nov-Apr vs May-Oct)
# ──────────────────────────────────────────────
winter = monthly_returns_full[monthly_returns_full.index.month.isin([11,12,1,2,3,4])]
summer = monthly_returns_full[monthly_returns_full.index.month.isin([5,6,7,8,9,10])]

t_hw, p_hw = stats.ttest_ind(winter, summer)
print("\n=== Halloween Effect (Nov-Apr vs May-Oct) ===")
print(f"Winter mean: {winter.mean():+.4f} ({winter.mean()*12:+.2%} ann.)")
print(f"Summer mean: {summer.mean():+.4f} ({summer.mean()*12:+.2%} ann.)")
print(f"t-statistic: {t_hw:.3f}, p-value: {p_hw:.4f}")

10. Chapter Summary