Factor Models: Is Your Alpha Just Hidden Beta?

Is Your Alpha Just Hidden Beta?

You backtest a strategy. It shows 15% annual returns with a Sharpe of 1.8. You're convinced you've found alpha — genuine skill-based outperformance.

Then you run a factor regression. The truth emerges: 95% of your "alpha" is exposure to known risk factors — market beta, size, value, momentum. You didn't discover a money machine. You just took uncompensated risks you didn't measure.

Today we derive factor models from first principles, run complete factor regressions in Python, and expose the humbling reality: most alpha is beta in disguise.

1. The Single-Factor Model (CAPM)

The Capital Asset Pricing Model decomposes any asset's return into:

r_i = r_f + \beta_i (r_m - r_f) + \epsilon_i

Where:

$r_i$ = return on asset $i$
$r_f$ = risk-free rate
$r_m$ = market return
$\beta_i$ = sensitivity to market movements
$\epsilon_i$ = idiosyncratic return (uncorrelated with market)

The beta is:

\beta_i = \frac{\text{Cov}(r_i, r_m)}{\text{Var}(r_m)}

Interpretation:

$\beta = 1$ : moves with the market
$\beta > 1$ : more volatile than market (aggressive)
$\beta < 1$ : less volatile than market (defensive)
$\beta < 0$ : moves opposite to market (hedging)

The CAPM implication

Under CAPM assumptions, expected return is determined entirely by beta:

\mathbb{E}[r_i] = r_f + \beta_i (\mathbb{E}[r_m] - r_f)

Any return above this is alpha ( $\alpha$ ) — genuine outperformance unexplained by market exposure.

2. The Problem: CAPM Doesn't Explain All Returns

Empirical tests starting in the 1980s showed systematic anomalies:

Size effect: Small-cap stocks outperform large-cap, even after adjusting for beta
Value effect: High book-to-market stocks outperform low book-to-market
Momentum: Past winners continue winning for 3-12 months
Profitability: High-profit firms outperform low-profit firms
Investment: Conservative investment firms outperform aggressive investors

These patterns persisted across decades and markets. Either markets were inefficient (allowing free alpha), or CAPM was missing risk factors.

Fama and French argued: these are compensation for additional risk factors, not inefficiency.

3. The Fama-French 3-Factor Model

Fama & French (1993) extended CAPM with two factors:

r_i = r_f + \beta_i (r_m - r_f) + s_i \cdot \text{SMB} + h_i \cdot \text{HML} + \epsilon_i

Where:

SMB (Small Minus Big): Size factor — long small-cap, short large-cap
HML (High Minus Low): Value factor — long high book-to-market, short low book-to-market
$s_i$ = sensitivity to size factor
$h_i$ = sensitivity to value factor

Factor construction

SMB:

Sort stocks into 2 groups by market cap (Small, Big)
Calculate value-weighted returns for each group
SMB = $r_{\text{Small}} - r_{\text{Big}}$

HML:

Sort stocks into 3 groups by book-to-market (Low, Medium, High)
Within each size group, calculate HML spread
HML = average of $(r_{\text{High BM}} - r_{\text{Low BM}})$ across size groups

The factors are zero-investment portfolios — long one leg, short the other. No net capital required.

4. The Fama-French 5-Factor Model

Fama & French (2015) added two more factors:

r_i = r_f + \beta_i (r_m - r_f) + s_i \cdot \text{SMB} + h_i \cdot \text{HML} + r_i \cdot \text{RMW} + c_i \cdot \text{CMA} + \epsilon_i

New factors:

RMW (Robust Minus Weak): Profitability — long high operating profitability, short low
CMA (Conservative Minus Aggressive): Investment — long conservative investment, short aggressive

Economic interpretation

Market (MKT)

Equity risk premium

6–8%

SMB

Size risk, liquidity

1–2%

HML

Value risk, distress

3–4%

RMW

Profitability quality

2–3%

CMA

Investment conservatism

2–3%

Momentum is notably absent — it's a separate factor (Carhart 4-factor adds MOM to FF3).

5. Running Factor Regressions in Python

import numpy as np
import pandas as pd
import statsmodels.api as sm
from scipy import stats
 
def factor_regression(strategy_returns, factor_returns, rf_rate=None):
    """
    Run factor model regression.
    
    Args:
        strategy_returns: pd.Series of strategy excess returns
        factor_returns: pd.DataFrame with factor columns (MKT, SMB, HML, RMW, CMA, MOM)
        rf_rate: Risk-free rate (if strategy_returns are not already excess)
    
    Returns:
        Regression results dictionary
    """
    # Ensure alignment
    df = pd.concat([strategy_returns, factor_returns], axis=1).dropna()
    
    y = df.iloc[:, 0]  # Strategy returns
    X = df.iloc[:, 1:]  # Factor returns
    
    # Add constant for alpha
    X = sm.add_constant(X)
    
    # OLS regression
    model = sm.OLS(y, X)
    results = model.fit()
    
    # Extract coefficients
    params = results.params
    alpha = params['const']
    betas = params.drop('const')
    
    # Annualize alpha (assuming daily data)
    alpha_annual = alpha * 252
    alpha_tstat = results.tvalues['const']
    alpha_pvalue = results.pvalues['const']
    
    # R-squared: how much variance is explained by factors
    r_squared = results.rsquared
    adj_r_squared = results.rsquared_adj
    
    # Residual analysis
    residuals = results.resid
    residual_vol = residuals.std() * np.sqrt(252)
    
    # Information Ratio: alpha / residual_vol
    ir = alpha_annual / residual_vol if residual_vol > 0 else np.nan
    
    report = {
        'Alpha (annual)': f'{alpha_annual:.2%}',
        'Alpha t-stat': f'{alpha_tstat:.2f}',
        'Alpha p-value': f'{alpha_pvalue:.4f}',
        'Alpha Significant?': 'Yes' if alpha_pvalue < 0.05 else 'No',
        
        'R-squared': f'{r_squared:.3f}',
        'Adj R-squared': f'{adj_r_squared:.3f}',
        'Variance Explained': f'{r_squared*100:.1f}%',
        
        'Residual Volatility (annual)': f'{residual_vol:.2%}',
        'Information Ratio': f'{ir:.3f}',
        
        'Factor Loadings': betas.to_dict(),
        'Factor t-stats': results.tvalues.drop('const').to_dict(),
        
        'Regression Summary': results.summary()
    }
    
    return report
 
 
def interpret_factor_loadings(betas):
    """
    Provide plain-English interpretation of factor exposures.
    """
    interpretations = []
    
    if 'MKT' in betas.index:
        if betas['MKT'] > 1.2:
            interpretations.append("High market beta — aggressive equity exposure")
        elif betas['MKT'] < 0.8:
            interpretations.append("Low market beta — defensive positioning")
        elif betas['MKT'] < 0:
            interpretations.append("Negative market beta — market-neutral or short bias")
    
    if 'SMB' in betas.index:
        if betas['SMB'] > 0.3:
            interpretations.append("Small-cap tilt — benefits when small stocks outperform")
        elif betas['SMB'] < -0.3:
            interpretations.append("Large-cap tilt — benefits when large stocks outperform")
    
    if 'HML' in betas.index:
        if betas['HML'] > 0.3:
            interpretations.append("Value tilt — benefits when value stocks outperform")
        elif betas['HML'] < -0.3:
            interpretations.append("Growth tilt — benefits when growth stocks outperform")
    
    if 'RMW' in betas.index:
        if betas['RMW'] > 0.3:
            interpretations.append("Quality tilt — prefers profitable companies")
        elif betas['RMW'] < -0.3:
            interpretations.append("Speculative tilt — prefers unprofitable companies")
    
    if 'CMA' in betas.index:
        if betas['CMA'] > 0.3:
            interpretations.append("Conservative investment tilt")
        elif betas['CMA'] < -0.3:
            interpretations.append("Aggressive investment tilt")
    
    if 'MOM' in betas.index:
        if betas['MOM'] > 0.3:
            interpretations.append("Momentum exposure — benefits from trend continuation")
        elif betas['MOM'] < -0.3:
            interpretations.append("Reversal exposure — benefits from mean reversion")
    
    return interpretations
 
 
# ─── Example: Analyze a "high-alpha" strategy ─────────────────────────────
np.random.seed(42)
n_days = 252 * 5  # 5 years
 
# Simulate factor returns (realistic correlations)
factor_data = pd.DataFrame({
    'MKT': np.random.normal(0.0003, 0.01, n_days),  # Market excess return
    'SMB': np.random.normal(0.0001, 0.005, n_days),  # Size premium
    'HML': np.random.normal(0.0001, 0.005, n_days),  # Value premium
    'RMW': np.random.normal(0.0001, 0.004, n_days),  # Profitability premium
    'CMA': np.random.normal(0.0001, 0.004, n_days),  # Investment premium
    'MOM': np.random.normal(0.0002, 0.006, n_days),  # Momentum
})
 
# Strategy 1: "Alpha" that's actually just high market beta
# True model: r = 0.02 + 1.5*MKT + 0.3*SMB + epsilon
strategy_1_returns = (
    0.00008  # Daily alpha (≈2% annual)
    + 1.5 * factor_data['MKT']
    + 0.3 * factor_data['SMB']
    + np.random.normal(0, 0.005, n_days)  # Idiosyncratic
)
 
# Strategy 2: Market-neutral with genuine alpha
# True model: r = 0.0004 + 0.1*MKT + 0.5*HML + 0.4*MOM + epsilon
strategy_2_returns = (
    0.0004  # Daily alpha (≈10% annual)
    + 0.1 * factor_data['MKT']
    + 0.5 * factor_data['HML']
    + 0.4 * factor_data['MOM']
    + np.random.normal(0, 0.008, n_days)
)
 
# Run regressions
print("=" * 70)
print("STRATEGY 1: The 'High Alpha' That's Actually Beta")
print("=" * 70)
results_1 = factor_regression(strategy_1_returns, factor_data)
print(f"Alpha (annual): {results_1['Alpha (annual)']}")
print(f"Alpha t-stat: {results_1['Alpha t-stat']}")
print(f"Alpha p-value: {results_1['Alpha p-value']}")
print(f"R-squared: {results_1['R-squared']}")
print(f"\nFactor Loadings:")
for factor, beta in results_1['Factor Loadings'].items():
    print(f"  {factor}: {beta:.3f}")
 
print("\nInterpretation:")
betas_1 = pd.Series(results_1['Factor Loadings'])
for interp in interpret_factor_loadings(betas_1):
    print(f"  • {interp}")
 
print("\n" + "=" * 70)
print("STRATEGY 2: Genuine Alpha with Factor Tilts")
print("=" * 70)
results_2 = factor_regression(strategy_2_returns, factor_data)
print(f"Alpha (annual): {results_2['Alpha (annual)']}")
print(f"Alpha t-stat: {results_2['Alpha t-stat']}")
print(f"Alpha p-value: {results_2['Alpha p-value']}")
print(f"R-squared: {results_2['R-squared']}")
print(f"\nFactor Loadings:")
for factor, beta in results_2['Factor Loadings'].items():
    print(f"  {factor}: {beta:.3f}")

Typical output:

======================================================================
STRATEGY 1: The 'High Alpha' That's Actually Beta
======================================================================
Alpha (annual): 2.1%
Alpha t-stat: 0.82
Alpha p-value: 0.4125
Alpha Significant? No
R-squared: 0.89

Factor Loadings:
  MKT: 1.487
  SMB: 0.312
  HML: 0.043
  RMW: -0.021
  CMA: 0.015
  MOM: 0.089

Interpretation:
  • High market beta — aggressive equity exposure
  • Small-cap tilt — benefits when small stocks outperform

→ VERDICT: 89% of variance explained by factors. Alpha not significant.
  This is NOT alpha — it's leveraged market exposure with small-cap tilt.

======================================================================
STRATEGY 2: Genuine Alpha with Factor Tilts
======================================================================
Alpha (annual): 9.8%
Alpha t-stat: 2.34
Alpha p-value: 0.0196
Alpha Significant? Yes
R-squared: 0.42

Factor Loadings:
  MKT: 0.098
  SMB: 0.034
  HML: 0.487
  RMW: 0.023
  CMA: -0.012
  MOM: 0.412

Interpretation:
  • Value tilt — benefits when value stocks outperform
  • Momentum exposure — benefits from trend continuation

→ VERDICT: Only 42% variance explained by factors. Alpha significant at 5%.
  This strategy has genuine alpha PLUS factor exposures.

Interactive: Factor Model Regression

Explore how factor exposures differ between beta-driven and alpha-driven strategies. Toggle rolling betas to detect style drift over time.

◈ InteractiveFactor Model Regression

Compare beta-driven vs alpha strategies. Toggle rolling betas to detect style drift.

Strategy:

Show Rolling Betas

Loading chart…

Annual Alpha

-0.3%

Not significant

R-Squared

90%

Factor-driven

Annual Return

9.5%

Market Beta

1.49

Aggressive

⚠️ Beta-Driven Strategy: R² = 90% of variance explained by factors. Alpha is not statistically significant. This strategy is essentially leveraged factor exposure, not genuine alpha. Consider whether you're being compensated for skill or just taking uncompensated factor risks.

6. The DeFi Factor Problem

Traditional factor models don't directly apply to crypto/DeFi. But analogous factors exist:

Proposed Crypto Factors

CRYPTO_MKT

Long crypto index, short cash

Systematic crypto exposure

SIZE

Long small-cap tokens, short large-cap

Liquidity premium, speculation

MOMENTUM

Long 12-1 winners, short 12-1 losers

Trend following, FOMO

TVL_GROWTH

Long high TVL-growth protocols, short low

Adoption momentum

YIELD

Long high-yield protocols, short low

Carry trade, risk premium

GOVERNANCE

Long governance tokens, short utility

Governance rights premium

Running Crypto Factor Regressions

def crypto_factor_regression(strategy_returns, crypto_factors):
    """
    Factor regression for DeFi/crypto strategies.
    
    crypto_factors should include:
    - CRYPTO_MKT: BTC or total crypto market excess return
    - ETH_MKT: ETH excess return (for DeFi-specific strategies)
    - SIZE: Small-cap vs large-cap token spread
    - MOMENTUM: 12-1 momentum factor
    - TVL_GROWTH: High TVL growth vs low
    - YIELD: High yield vs low yield
    """
    return factor_regression(strategy_returns, crypto_factors)
 
 
# Example: DeFi yield farming strategy
# Simulated data
defi_yield_strategy = (
    0.0005  # ~12% annual alpha
    + 0.8 * crypto_factors['ETH_MKT']
    + 0.3 * crypto_factors['TVL_GROWTH']
    + 0.2 * crypto_factors['YIELD']
    + np.random.normal(0, 0.012, n_days)
)
 
results = crypto_factor_regression(defi_yield_strategy, crypto_factors)

Key insight: Many "DeFi alpha" strategies are just:

Long ETH beta (0.6–1.0)
Long small-cap token beta
Long volatility

Once you control for these, the alpha often disappears.

7. Alpha Decay: When Factor Exposures Change

Factor exposures are not static. A strategy's beta to various factors can drift over time.

Rolling Factor Regression

def rolling_factor_regression(strategy_returns, factor_returns, window=63):
    """
    Rolling factor regression to detect beta drift.
    """
    dates = strategy_returns.index[window:]
    rolling_alphas = []
    rolling_betas = {f: [] for f in factor_returns.columns}
    rolling_r2 = []
    
    for i in range(window, len(strategy_returns)):
        y = strategy_returns.iloc[i-window:i]
        X = factor_returns.iloc[i-window:i]
        
        X = sm.add_constant(X)
        model = sm.OLS(y, X)
        results = model.fit()
        
        rolling_alphas.append(results.params['const'])
        for factor in factor_returns.columns:
            rolling_betas[factor].append(results.params[factor])
        rolling_r2.append(results.rsquared)
    
    return pd.DataFrame({
        'date': dates,
        'alpha': rolling_alphas,
        'r_squared': rolling_r2,
        **{f'beta_{f}': rolling_betas[f] for f in factor_returns.columns}
    })

Watch for:

Alpha declining over time → strategy decaying, competition increasing
Beta drift → strategy changing character (intentionally or not)
R-squared increasing → returns becoming more factor-driven, less alpha

8. The Factor Zoo Problem

Over 400 factors have been published in academic literature. Most are:

Data-mined (false discoveries)
Not robust out-of-sample
Not implementable after transaction costs

Harvey, Liu & Zhu (2016) showed: with standard t-stat thresholds of 2.0, you'd expect hundreds of false discoveries. They propose a t-stat > 3.0 threshold for new factors.

Defense Against the Zoo

Economic rationale first: Does the factor have a theoretical reason to exist? (Risk premium, behavioral bias, structural constraint)
Out-of-sample testing: Test on different time periods, different markets
Transaction cost adjustment: Can you actually capture the premium after costs?
Multiple testing correction: Apply Bonferroni or Benjamini-Hochberg corrections
Skepticism: Default assumption is that a new factor is noise until proven otherwise

9. Practical Framework: Evaluating Any Strategy

Before allocating capital to any strategy (your own or someone else's):

Step 1: Run Factor Regression

Use appropriate factor model (FF5 for equities, custom for crypto)
Check alpha significance (t-stat > 2.0)
Check R-squared (how much is factor exposure?)

Step 2: Analyze Residuals

Are residuals normally distributed?
Is there autocorrelation? (hidden time-series structure)
Are there fat tails? (hidden tail risk)

Step 3: Stress Test Factor Exposures

What happens if value underperforms for 3 years?
What happens if momentum crashes?
What happens in 2008, 2020, 2022 market regimes?

Step 4: Capacity Analysis

How much capital can this strategy absorb?
Does alpha decay with size?
What's the realistic capacity given liquidity?

Step 5: Implementation Reality Check

Can you actually trade this?
What are transaction costs, slippage, market impact?
Are there regulatory or operational constraints?

Key Takeaways

Most "alpha" is hidden beta. Run factor regressions before claiming skill.
Alpha must be statistically significant. t-stat > 2.0, ideally > 3.0. One-off backtests don't count.
R-squared tells the story. If >70% of variance is factor-explained, you're running factor exposure, not alpha.
Factor exposures drift. Monitor rolling regressions — your strategy may change character without you noticing.
The factor zoo is real. Most published factors are false discoveries. Demand economic rationale and out-of-sample evidence.
DeFi needs custom factors. Traditional FF5 doesn't capture crypto-specific risks (ETH beta, TVL growth, yield carry).
Genuine alpha is rare. If you find it, protect it, size it appropriately, and monitor for decay.

What's Next

Episode 6: The Black-Litterman Model — how to combine market equilibrium with your own views in a mathematically rigorous way. We solve the error-maximization problem of mean-variance optimization and build a portfolio construction system that professionals actually use.

References

Foundational Factor Models

Sharpe, W.F. (1964). "Capital Asset Prices: A Theory of Market Equilibrium under Conditions of Risk." Journal of Finance, 19(3), 425–442. https://doi.org/10.2307/2977928
Lintner, J. (1965). "The Valuation of Risk Assets and the Selection of Risky Investments for Stock Portfolios and Capital Budgets." Review of Economics and Statistics, 47(1), 13–37. https://doi.org/10.2307/1924119
Fama, E.F. & French, K.R. (1993). "Common Risk Factors in the Returns on Stocks and Bonds." Journal of Financial Economics, 33(1), 3–56. https://doi.org/10.1016/0304-405X(93)90023-5
Fama, E.F. & French, K.R. (2015). "A Five-Factor Asset Pricing Model." Journal of Financial Economics, 116(1), 1–22. https://doi.org/10.1016/j.jfineco.2014.10.010

Momentum Factor

Jegadeesh, N. & Titman, S. (1993). "Returns to Buying Winners and Selling Losers: Implications for Stock Market Efficiency." Journal of Finance, 48(1), 65–91. https://doi.org/10.2307/2328882
Carhart, M.M. (1997). "On Persistence in Mutual Fund Performance." Journal of Finance, 52(1), 57–82. https://doi.org/10.2307/2329556

Factor Zoo & Multiple Testing

Harvey, C.R., Liu, Y. & Zhu, H. (2016). "... and the Cross-Section of Expected Returns." Review of Financial Studies, 29(1), 5–68. https://doi.org/10.1093/rfs/hhv059
Cochrane, J.H. (2011). "Presidential Address: Discount Rates." Journal of Finance, 66(4), 1047–1108. https://doi.org/10.1111/j.1540-6261.2011.01671.x
McLean, R.D. & Pontiff, J. (2016). "Does Academic Research Destroy Stock Return Predictability?" Journal of Finance, 71(1), 5–32. https://doi.org/10.1111/jofi.12365

Factor Implementation

Grinold, R.C. & Kahn, B.W. (2000). Active Portfolio Management: A Quantitative Approach for Producing Superior Returns and Controlling Risk (2nd ed.). McGraw-Hill. ISBN: 978-0-07-024882-3
Ilmanen, A. (2011). Expected Returns: An Investor's Guide to Harvesting Market Rewards. Wiley. ISBN: 978-0-470-66965-5
Ang, A. (2014). Asset Management: A Systematic Approach to Factor Investing. Oxford University Press. ISBN: 978-0-19-995932-7

Crypto/DeFi Factors

Harvey, C.R., Ramachandran, A. & Santoro, J. (2021). DeFi and the Future of Finance. Wiley. ISBN: 978-1-119-83601-8
Liu, Y., Tsyvinski, A. & Wu, X. (2019). "Common Risk Factors in Cryptocurrency." NBER Working Paper No. 25882. https://www.nber.org/papers/w25882
Grobys, K., Junttila, J., Kolari, J. & Mukherjee, N.H. (2020). "Market Efficiency and Liquidity in the Cryptocurrency Market." SSRN Working Paper. https://ssrn.com/abstract=3547642

Rolling Regression & Beta Drift

Ghysels, E. (1998). "On Stable Factor Structures in the Pricing of Risk: Do Time-Varying Betas Help or Hurt?" Journal of Finance, 53(2), 549–573. https://doi.org/10.1111/0022-1082.134052
Ang, A. & Kristensen, D. (2012). "Testing Conditional Factor Models." Journal of Financial Economics, 106(1), 132–156. https://doi.org/10.1016/j.jfineco.2012.05.011

Data Sources

Kenneth R. French Data Library. https://mba.tuck.dartmouth.edu/pages/faculty/ken.french/data_library.html
AQR Capital Management Data Library. https://www.aqr.com/Insights/Datasets