📘 Multi-Factor Models

13. 📘 Multi-Factor Models#

13.1. 🎯 Learning Objectives#

By the end of this notebook, you will be able to:

  1. Understand why investors move beyond CAPM — Articulate the limitations of a single-market factor

  2. Estimate multi-factor models — Regress asset/fund returns on factor portfolios; interpret loadings, alphas, and \(R^2\)

  3. Evaluate fund performance — Distinguish genuine alpha from factor exposure using factor regressions

  4. Decompose portfolio risk — Use both top-down and bottom-up approaches to separate systematic from idiosyncratic components

  5. Estimate factor premia — Run Fama–MacBeth cross-sectional regressions and interpret characteristic-based risk prices

  6. Construct characteristic-adjusted returns — Separate skill from style for any portfolio

13.2. 📋 Table of Contents#

  1. Why Multi-Factor Models?

  2. The Time-Series Approach

  3. Performance Attribution: Cathie Wood

  4. Warren Buffett: Does He Beat the Market?

  5. Bottom-Up vs Top-Down Decomposition

  6. The Cross-Sectional Approach

  7. Exercises

  8. Key Takeaways

13.3. 🛠️ Setup#

#@title 🛠️ Setup: Run this cell first (click to expand)

#!pip install wrds

import wrds

import numpy as np
import pandas as pd
%matplotlib inline
import matplotlib.pyplot as plt
import statsmodels.api as sm
import pandas_datareader.data as web

plt.style.use('seaborn-v0_8-whitegrid')
plt.rcParams['figure.figsize'] = [10, 6]
plt.rcParams['font.size'] = 12

import warnings
warnings.filterwarnings('ignore')

def get_factors(factors='CAPM', freq='daily'):
    if freq == 'monthly':
        freq_label = ''
    else:
        freq_label = '_' + freq

    if factors == 'CAPM':
        fama_french = web.DataReader("F-F_Research_Data_Factors" + freq_label, "famafrench", start="1921-01-01")
        df_factor = fama_french[0][['RF', 'Mkt-RF']]
    elif factors == 'FF3':
        fama_french = web.DataReader("F-F_Research_Data_Factors" + freq_label, "famafrench", start="1921-01-01")
        df_factor = fama_french[0][['RF', 'Mkt-RF', 'SMB', 'HML']]
    elif factors == 'FF5':
        fama_french = web.DataReader("F-F_Research_Data_Factors" + freq_label, "famafrench", start="1921-01-01")
        df_factor = fama_french[0][['RF', 'Mkt-RF', 'SMB', 'HML']]
        fama_french2 = web.DataReader("F-F_Research_Data_5_Factors_2x3" + freq_label, "famafrench", start="1921-01-01")
        df_factor = df_factor.merge(fama_french2[0][['RMW', 'CMA']], on='Date', how='outer')
    else:
        fama_french = web.DataReader("F-F_Research_Data_Factors" + freq_label, "famafrench", start="1921-01-01")
        df_factor = fama_french[0][['RF', 'Mkt-RF', 'SMB', 'HML']]
        fama_french2 = web.DataReader("F-F_Research_Data_5_Factors_2x3" + freq_label, "famafrench", start="1921-01-01")
        df_factor = df_factor.merge(fama_french2[0][['RMW', 'CMA']], on='Date', how='outer')
        fama_french3 = web.DataReader("F-F_Momentum_Factor" + freq_label, "famafrench", start="1921-01-01")
        df_factor = df_factor.merge(fama_french3[0], on='Date')
        df_factor.columns = ['RF', 'Mkt-RF', 'SMB', 'HML', 'RMW', 'CMA', 'MOM']

    if freq == 'monthly':
        df_factor.index = pd.to_datetime(df_factor.index.to_timestamp())
    else:
        df_factor.index = pd.to_datetime(df_factor.index)

    return df_factor / 100

def get_daily_wrds_multiple_ticker(tickers, conn):
    permnos = conn.get_table(library='crsp', table='stocknames',
                             columns=['permno', 'ticker', 'namedt', 'nameenddt'])
    permnos['nameenddt'] = pd.to_datetime(permnos['nameenddt'])
    permnos = permnos[(permnos['ticker'].isin(tickers)) &
                      (permnos['nameenddt'] == permnos['nameenddt'].max())]
    permno_list = permnos['permno'].unique().tolist()
    print(f"Found PERMNOs: {permno_list}")

    query = f"""
        SELECT permno, date, ret, retx, prc
        FROM crsp.dsf
        WHERE permno IN ({','.join(map(str, permno_list))})
        ORDER BY date
    """
    daily_returns = conn.raw_sql(query, date_cols=['date'])
    daily_returns = daily_returns.merge(permnos[['permno', 'ticker']], on='permno', how='left')
    daily_returns = daily_returns.pivot(index='date', columns='ticker', values='ret')
    daily_returns = daily_returns[tickers]
    return daily_returns

def get_permnos(tickers, conn):
    permnos = conn.get_table(library='crsp', table='stocknames',
                             columns=['permno', 'ticker', 'namedt', 'nameenddt'])
    permnos['nameenddt'] = pd.to_datetime(permnos['nameenddt'])
    permnos = permnos[permnos['ticker'].isin(tickers)]
    return permnos

13.4. Why Multi-Factor Models? #

So far we have focused on the market as our single factor. In practice, it is standard to use models with many factors. Additional factors:

  • Soak up risk — making measures of alpha more precise

  • Difference out other sources of expected excess returns that are easy to access

  • Allow for better risk management across multiple dimensions

We extend the single-factor model by adding more regressors. With \(m\) factors:

\[r_t^i = b_{i,1} f_t^1 + b_{i,2} f_t^2 + \cdots + b_{i,m} f_t^m + \epsilon_{i,t}\]

In matrix notation, stacking all \(n\) assets:

\[R_t = B \cdot F_t + U_t\]

where \(B\) is \(n \times m\) (each row = one asset’s exposures), \(F_t\) is \(m \times 1\) (factor returns), and \(U_t\) is the vector of idiosyncratic residuals.

13.4.1. “Endogenous” Benchmarking#

Large allocators often set benchmarks for managers. The most common is the S&P 500 (≈ market return), but you can also construct endogenous benchmarks:

\[r^b_t = \sum_j \beta_j F_{j,t}\]

Use the multi-factor combination that best replicates the portfolio as the benchmark. This is typically done implicitly: you allocate to funds based on their alpha (hard to get) rather than their beta exposure (cheap to replicate).

💡 Key Insight:

Alpha is scarce; beta is plentiful. You should pay different prices for each. The gains from beta are in implementation (low cost); the gains from alpha are in selection (finding skill).

13.5. Estimating Multi-Factor Models: The Time-Series Approach #

We start with known factors and estimate betas using time-series regressions. This works especially well when factors are excess returns themselves.

For each asset, regress its excess returns on the factor excess returns:

\[r_t^{e,i} = \alpha_i + \beta_{i,1} f_t^1 + \cdots + \beta_{i,m} f_t^m + \epsilon_{i,t}\]

13.5.1. Application: What Do Momentum ETFs Actually Deliver?#

We’ll take the largest ETFs claiming to implement momentum and see what factor exposures they actually have.

tickers = ["MTUM", "SPMO", "XMMO", "IMTM", "XSMO", "PDP", "JMOM", "DWAS", "VFMO", "XSVM", "QMOM"]
conn = wrds.Connection()

# Get daily returns and factor data
df_ETF = get_daily_wrds_multiple_ticker(tickers, conn)
df_factor = get_factors('FF6', 'daily')

# Align and compute excess returns
df_ETF, df_factor = df_ETF.align(df_factor, join='inner', axis=0)
df_ETF = df_ETF.subtract(df_factor['RF'], axis=0)

Loading library list...
Done
Found PERMNOs: [13512, 13851, 15161, 15725, 17085, 17392, 17622, 90621, 90622, 90623, 91876]

# Example: full regression for QMOM
X = sm.add_constant(df_factor.drop(columns=['RF']))
y = df_ETF["QMOM"].dropna()
X = X.loc[y.index]
model = sm.OLS(y, X).fit()
print(model.summary())

                            OLS Regression Results                            
==============================================================================
Dep. Variable:                   QMOM   R-squared:                       0.762
Model:                            OLS   Adj. R-squared:                  0.762
Method:                 Least Squares   F-statistic:                     1217.
Date:                Mon, 13 Apr 2026   Prob (F-statistic):               0.00
Time:                        15:15:29   Log-Likelihood:                 7764.5
No. Observations:                2284   AIC:                        -1.552e+04
Df Residuals:                    2277   BIC:                        -1.547e+04
Df Model:                           6                                         
Covariance Type:            nonrobust                                         
==============================================================================
                 coef    std err          t      P>|t|      [0.025      0.975]
------------------------------------------------------------------------------
const       2.253e-05      0.000      0.133      0.894      -0.000       0.000
Mkt-RF         1.0687      0.016     68.221      0.000       1.038       1.099
SMB            0.4718      0.029     16.195      0.000       0.415       0.529
HML            0.1421      0.025      5.581      0.000       0.092       0.192
RMW           -0.3528      0.037     -9.491      0.000      -0.426      -0.280
CMA           -0.1667      0.046     -3.618      0.000      -0.257      -0.076
MOM            0.5716      0.017     33.695      0.000       0.538       0.605
==============================================================================
Omnibus:                      197.078   Durbin-Watson:                   2.333
Prob(Omnibus):                  0.000   Jarque-Bera (JB):             1242.962
Skew:                           0.005   Prob(JB):                    1.24e-270
Kurtosis:                       6.614   Cond. No.                         292.
==============================================================================

Notes:
[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.

# Run the regression for all momentum ETFs
Results = pd.DataFrame([], index=tickers, columns=X.columns)
for ticker in tickers:
    y = df_ETF[ticker]
    X = sm.add_constant(df_factor.drop(columns=['RF']))
    X = X[y.isna() == False]
    y = y[y.isna() == False]
    model = sm.OLS(y, X).fit()
    Results.loc[ticker, :] = model.params
    Results.at[ticker, 't_alpha'] = model.tvalues['const']
    Results.at[ticker, 'ivol'] = model.resid.std() * 252**0.5
    Results.at[ticker, 'Sample size'] = y.shape[0] / 252

Results['const'] = Results['const'].astype(float) * 252
Results.rename(columns={'const': 'alpha'}, inplace=True)
Results = Results[['alpha', 't_alpha', 'Mkt-RF', 'SMB', 'HML', 'RMW', 'CMA', 'MOM', 'ivol', 'Sample size']]
Results
alpha t_alpha Mkt-RF SMB HML RMW CMA MOM ivol Sample size
MTUM -0.007184 -0.441952 1.024871 -0.105213 -0.057124 -0.127846 -0.027354 0.325909 0.055400 11.690476
SPMO 0.018246 0.776076 1.013657 -0.158928 -0.049394 -0.02647 0.076656 0.25509 0.071106 9.210317
XMMO 0.009431 0.496884 1.040259 0.337397 0.041132 0.030845 -0.105733 0.199555 0.084279 19.805556
IMTM -0.033974 -1.061349 0.802455 -0.013844 0.057041 -0.133071 0.12565 0.118228 0.100624 9.944444
XSMO -0.012768 -0.639363 0.984281 0.860455 0.191119 0.100718 -0.09243 0.165786 0.088671 19.805556
PDP -0.015862 -1.055797 1.054885 0.142881 -0.00835 -0.083504 -0.184164 0.252145 0.063254 17.817460
JMOM 0.004130 0.203645 0.963249 0.012772 -0.06464 -0.07863 -0.066677 0.105358 0.053914 7.123016
DWAS -0.010285 -0.495609 1.088441 1.075939 0.23163 -0.226312 -0.078938 0.400114 0.072943 12.428571
VFMO 0.011316 0.563876 1.03016 0.449136 0.185386 -0.221027 -0.061281 0.396359 0.052364 6.861111
XSVM -0.004102 -0.220702 0.932729 0.983862 0.528043 0.329306 0.194897 -0.056401 0.082522 19.805556
QMOM 0.005677 0.132772 1.068744 0.471783 0.142069 -0.352846 -0.166669 0.571551 0.128281 9.063492

🤔 Think and Code:

  1. Which fund is “better”? Is it all about alpha in this case?

  2. What other things should you look at beyond the alpha column?

  3. Is this table providing a fair comparison, given different sample sizes?

13.6. Performance Attribution: Cathie Wood #

Factor models let us decompose a manager’s strategy: what explains their returns? What tilts do they have? What kind of stocks do they like?

13.6.1. Application: What Does Cathie Wood Like?#

Cathie Wood

Cathie Wood is the founder of ARK Invest (~$60B AUM), investing in disruptive technologies — self-driving cars, genomics, AI. She gained fame for spectacular returns and unconventional stock picks.

df = pd.read_pickle('https://raw.githubusercontent.com/amoreira2/Fin418/main/assets/data/df_WarrenBAndCathieW_monthly.pkl')
_temp = df.drop(['BRK'], axis=1).dropna()

Factors = _temp.drop(['RF', 'ARKK'], axis=1)
ArK = _temp.ARKK - _temp.RF

(ArK + 1).cumprod().plot(title='ARKK Cumulative Excess Return', figsize=(10, 5))
plt.ylabel('Growth of $1')
plt.tight_layout()
plt.show()

print(f"Annualized mean excess return: {ArK.mean()*252:.1%}")
../../_images/114971d50d95c2d4d1817531fd277ce9de4e109941595ec89a95db35c75dad02.png 
Annualized mean excess return: 644.7%

The Fama-French factors capture different investment styles:

Factor

Strategy

HML

Buy high book-to-market (value), sell low (growth)

SMB

Buy small caps, sell large caps

RMW

Buy high profitability, sell low profitability

CMA

Buy low investment (conservative), sell high investment (aggressive)

MOM

Buy recent winners, sell recent losers

For now, think of these as important trading strategies that practitioners know well. We’ll discuss their economics in detail later.

# Multi-factor regression (annualized)
x = sm.add_constant(Factors * 252)
y = ArK * 252
results = sm.OLS(y, x).fit()
results.summary()
OLS Regression Results
Dep. Variable: y R-squared: 0.838
Model: OLS Adj. R-squared: 0.820
Method: Least Squares F-statistic: 44.90
Date: Mon, 13 Apr 2026 Prob (F-statistic): 7.32e-19
Time: 15:15:29 Log-Likelihood: -215.14
No. Observations: 59 AIC: 444.3
Df Residuals: 52 BIC: 458.8
Df Model: 6
Covariance Type: nonrobust
coef std err t P>|t| [0.025 0.975]
const 1.7109 1.396 1.225 0.226 -1.091 4.513
Mkt-RF 1.5432 0.155 9.931 0.000 1.231 1.855
SMB 0.3449 0.249 1.387 0.171 -0.154 0.844
HML -0.9504 0.204 -4.667 0.000 -1.359 -0.542
RMW -0.8065 0.306 -2.636 0.011 -1.420 -0.193
CMA -0.5312 0.379 -1.403 0.167 -1.291 0.229
Mom -0.2441 0.176 -1.390 0.170 -0.596 0.108
Omnibus: 11.073 Durbin-Watson: 1.753
Prob(Omnibus): 0.004 Jarque-Bera (JB): 11.234
Skew: 0.896 Prob(JB): 0.00364
Kurtosis: 4.165 Cond. No. 16.1


Notes:
[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.

🤔 Think and Code:

  1. How much of ARKK’s return behavior can we explain with factors?

  2. What kind of stocks does Cathie Wood like? (Look at the factor loadings)

  3. How much portfolio variance comes from market exposure alone vs. being anti-value?

  4. What would the volatility of the hedged (residual) portfolio be?

  5. When did she earn her alpha? Is it smooth or concentrated in a few periods?

13.7. Warren Buffett: Does He Beat the Market? #

Warren Buffett

Warren Buffett is the chairman and CEO of Berkshire Hathaway. His top holdings include Apple, Bank of America, Chevron, Coca-Cola, and American Express. He’s known for a long-term, value-oriented approach — large, blue-chip companies with strong balance sheets and attractive valuations.

Let’s apply the same factor regression framework to Berkshire Hathaway.

# Single-factor CAPM regression
BrK = df.BRK - df.RF
x = sm.add_constant(df['Mkt-RF'])
results = sm.OLS(BrK, x).fit()
results.summary()
OLS Regression Results
Dep. Variable: y R-squared: 0.223
Model: OLS Adj. R-squared: 0.220
Method: Least Squares F-statistic: 79.32
Date: Mon, 13 Apr 2026 Prob (F-statistic): 7.17e-17
Time: 15:15:29 Log-Likelihood: 442.23
No. Observations: 279 AIC: -880.5
Df Residuals: 277 BIC: -873.2
Df Model: 1
Covariance Type: nonrobust
coef std err t P>|t| [0.025 0.975]
const 0.0054 0.003 1.797 0.073 -0.001 0.011
Mkt-RF 0.5919 0.066 8.906 0.000 0.461 0.723
Omnibus: 51.668 Durbin-Watson: 1.989
Prob(Omnibus): 0.000 Jarque-Bera (JB): 198.575
Skew: 0.710 Prob(JB): 7.59e-44
Kurtosis: 6.882 Cond. No. 22.3


Notes:
[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.

  • What do we learn? Is the alpha large economically? Statistically?

  • How should we think about this alpha?

Now let’s use the full multi-factor model:

# Multi-factor regression: FF5 + Momentum
Factors = df.drop(['BRK', 'RF', 'ARKK'], axis=1)
x = sm.add_constant(Factors)
y = df.BRK - df.RF
results = sm.OLS(y, x).fit()
results.summary()
OLS Regression Results
Dep. Variable: y R-squared: 0.405
Model: OLS Adj. R-squared: 0.392
Method: Least Squares F-statistic: 30.81
Date: Mon, 13 Apr 2026 Prob (F-statistic): 3.71e-28
Time: 15:15:29 Log-Likelihood: 479.45
No. Observations: 279 AIC: -944.9
Df Residuals: 272 BIC: -919.5
Df Model: 6
Covariance Type: nonrobust
coef std err t P>|t| [0.025 0.975]
const 0.0037 0.003 1.307 0.192 -0.002 0.009
Mkt-RF 0.6938 0.070 9.907 0.000 0.556 0.832
SMB -0.3087 0.097 -3.193 0.002 -0.499 -0.118
HML 0.5732 0.130 4.398 0.000 0.317 0.830
RMW 0.3486 0.123 2.827 0.005 0.106 0.591
CMA -0.4156 0.191 -2.171 0.031 -0.792 -0.039
Mom -0.0152 0.059 -0.255 0.799 -0.132 0.102
Omnibus: 34.823 Durbin-Watson: 1.948
Prob(Omnibus): 0.000 Jarque-Bera (JB): 71.498
Skew: 0.647 Prob(JB): 2.98e-16
Kurtosis: 5.116 Cond. No. 82.1


Notes:
[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.

🤔 Think and Code:

  1. Did adding factors change the alpha? By how much?

  2. What kind of stocks does Warren like? (Look at the factor loadings)

  3. What does this tell us about his investment style vs. his stock-picking skill?

  4. How does his profile compare to Cathie Wood’s?

13.8. Bottom-Up vs Top-Down Decomposition #

So far we estimated fund factor exposures by looking at how the fund’s returns co-move with factors (top-down). An alternative: look through the fund at individual holdings (bottom-up).

If a portfolio with weights \(X\) earns excess returns \(r = X'R\), and each asset satisfies:

\[R = A + B \cdot F + U\]

then the portfolio satisfies:

\[r = X'A + X'B \cdot F + X'U\]

So the portfolio’s exposure to factor \(j\) is the dollar-weighted average of the asset betas:

\[\beta_{p,j} = \sum_i x_i \, \beta_{i,j}\]

💡 Key Insight:

For high-turnover portfolios, the bottom-up approach tracks exposures much better because it refreshes at the holding level. For stable portfolios, top-down regressions are simpler and avoid the noise of estimating individual-stock betas.

13.8.1. Sample Portfolio: Tech → Retail Rotation#

import pandas as pd

date1, date2, date3 = '2014-12-31', '2015-12-31', '2016-12-31'

# Portfolio 1: Tech (2014-2015)
portfolio_data1 = {
    'date': [date1]*5,
    'ticker': ['AAPL', 'GOOGL', 'MSFT', 'NVDA', 'AMZN'],
    'weight': [0.2, 0.2, 0.2, 0.2, 0.2]
}
# Portfolio 2: Retail (2015-2016)
portfolio_data2 = {
    'date': [date2]*4,
    'ticker': ['COST', 'WMT', 'TGT', 'KR'],
    'weight': [0.25, 0.25, 0.25, 0.25]
}

portfolio_df1 = pd.DataFrame(portfolio_data1)
portfolio_df2 = pd.DataFrame(portfolio_data2)

# Expand to daily holdings
date_range1 = pd.date_range(start=date1, end=date2, freq='B')
date_range2 = pd.date_range(start=date2, end=date3, freq='B')

monthly_portfolio1 = pd.DataFrame(
    [(d, t, w) for d in date_range1 for t, w in zip(portfolio_df1['ticker'], portfolio_df1['weight'])],
    columns=['date', 'ticker', 'weight'])
monthly_portfolio2 = pd.DataFrame(
    [(d, t, w) for d in date_range2 for t, w in zip(portfolio_df2['ticker'], portfolio_df2['weight'])],
    columns=['date', 'ticker', 'weight'])

final_portfolio_df = pd.concat([monthly_portfolio1, monthly_portfolio2], ignore_index=True)
final_portfolio_df
date ticker weight
0 2014-12-31 AAPL 0.20
1 2014-12-31 GOOGL 0.20
2 2014-12-31 MSFT 0.20
3 2014-12-31 NVDA 0.20
4 2014-12-31 AMZN 0.20
... ... ... ...
2353 2016-12-29 KR 0.25
2354 2016-12-30 COST 0.25
2355 2016-12-30 WMT 0.25
2356 2016-12-30 TGT 0.25
2357 2016-12-30 KR 0.25

2358 rows × 3 columns

# Get stock returns and factors
tickers = final_portfolio_df.ticker.unique().tolist()
df_stocks = get_daily_wrds_multiple_ticker(tickers, conn)
df_factor = get_factors('FF6', 'daily').dropna()
df_stocks = df_stocks.subtract(df_factor['RF'], axis=0)

Found PERMNOs: [10107, 14593, 16678, 49154, 55976, 84788, 86580, 87055, 90319]

# Merge portfolio weights with stock returns
df_merged = df_stocks.stack()
df_merged.name = 'eret'
df_merged = final_portfolio_df.merge(df_merged, left_on=['date', 'ticker'], right_index=True, how='left')
df_merged.head()
date ticker weight eret
0 2014-12-31 AAPL 0.2 -0.019019
1 2014-12-31 GOOGL 0.2 -0.008631
2 2014-12-31 MSFT 0.2 -0.012123
3 2014-12-31 NVDA 0.2 -0.01571
4 2014-12-31 AMZN 0.2 0.000161

13.8.2. Top-Down Approach#

Construct the portfolio return first, then run the multi-factor regression:

fund_return = df_merged.groupby('date').apply(lambda x: (x['eret'] * x['weight']).sum())
df_factor, fund_return = df_factor.align(fund_return, join='inner', axis=0)

# Full-sample regression
y = fund_return.dropna()
X = sm.add_constant(df_factor.drop(columns=['RF']).loc[y.index])
model = sm.OLS(y, X).fit()
model.summary()
OLS Regression Results
Dep. Variable: y R-squared: 0.539
Model: OLS Adj. R-squared: 0.533
Method: Least Squares F-statistic: 97.05
Date: Mon, 13 Apr 2026 Prob (F-statistic): 1.66e-80
Time: 15:15:34 Log-Likelihood: 1715.3
No. Observations: 505 AIC: -3417.
Df Residuals: 498 BIC: -3387.
Df Model: 6
Covariance Type: nonrobust
coef std err t P>|t| [0.025 0.975]
const 0.0005 0.000 1.408 0.160 -0.000 0.001
Mkt-RF 0.9482 0.044 21.464 0.000 0.861 1.035
SMB -0.1250 0.080 -1.573 0.116 -0.281 0.031
HML 0.0096 0.101 0.096 0.924 -0.188 0.207
RMW 0.7211 0.117 6.160 0.000 0.491 0.951
CMA -0.5525 0.147 -3.770 0.000 -0.840 -0.265
MOM 0.1502 0.050 3.008 0.003 0.052 0.248
Omnibus: 94.691 Durbin-Watson: 1.942
Prob(Omnibus): 0.000 Jarque-Bera (JB): 342.514
Skew: 0.819 Prob(JB): 4.21e-75
Kurtosis: 6.687 Cond. No. 461.


Notes:
[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.

Now suppose you know the portfolio changed at end-2015. You can break the regression into two windows — but what do you lose in precision?

# Period 1: tech portfolio (2014-2015)
y1 = fund_return[:'2015-12-31'].dropna()
X1 = sm.add_constant(df_factor.drop(columns=['RF']).loc[y1.index])
model1 = sm.OLS(y1, X1).fit()
display(model1.summary())

# Period 2: retail portfolio (2016)
y2 = fund_return['2015-12-31':].dropna()
X2 = sm.add_constant(df_factor.drop(columns=['RF']).loc[y2.index])
model2 = sm.OLS(y2, X2).fit()
model2.summary()
OLS Regression Results
Dep. Variable: y R-squared: 0.771
Model: OLS Adj. R-squared: 0.765
Method: Least Squares F-statistic: 137.9
Date: Mon, 13 Apr 2026 Prob (F-statistic): 9.55e-76
Time: 15:15:34 Log-Likelihood: 907.78
No. Observations: 253 AIC: -1802.
Df Residuals: 246 BIC: -1777.
Df Model: 6
Covariance Type: nonrobust
coef std err t P>|t| [0.025 0.975]
const 0.0009 0.000 1.995 0.047 1.08e-05 0.002
Mkt-RF 1.0123 0.048 21.135 0.000 0.918 1.107
SMB -0.2453 0.101 -2.439 0.015 -0.443 -0.047
HML 0.2501 0.135 1.859 0.064 -0.015 0.515
RMW 0.7186 0.170 4.231 0.000 0.384 1.053
CMA -2.0396 0.214 -9.511 0.000 -2.462 -1.617
MOM -0.0418 0.063 -0.659 0.511 -0.167 0.083
Omnibus: 86.620 Durbin-Watson: 1.814
Prob(Omnibus): 0.000 Jarque-Bera (JB): 377.103
Skew: 1.336 Prob(JB): 1.30e-82
Kurtosis: 8.351 Cond. No. 568.


Notes:
[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.
OLS Regression Results
Dep. Variable: y R-squared: 0.335
Model: OLS Adj. R-squared: 0.319
Method: Least Squares F-statistic: 20.63
Date: Mon, 13 Apr 2026 Prob (F-statistic): 1.51e-19
Time: 15:15:34 Log-Likelihood: 870.23
No. Observations: 253 AIC: -1726.
Df Residuals: 246 BIC: -1702.
Df Model: 6
Covariance Type: nonrobust
coef std err t P>|t| [0.025 0.975]
const -0.0004 0.001 -0.705 0.481 -0.001 0.001
Mkt-RF 0.7193 0.070 10.242 0.000 0.581 0.858
SMB 0.1228 0.108 1.135 0.258 -0.090 0.336
HML -0.0331 0.125 -0.265 0.791 -0.279 0.213
RMW 0.7591 0.144 5.289 0.000 0.476 1.042
CMA 0.0837 0.175 0.479 0.633 -0.261 0.428
MOM 0.1594 0.069 2.314 0.022 0.024 0.295
Omnibus: 8.063 Durbin-Watson: 2.104
Prob(Omnibus): 0.018 Jarque-Bera (JB): 13.473
Skew: -0.109 Prob(JB): 0.00119
Kurtosis: 4.109 Cond. No. 408.


Notes:
[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.

13.8.3. Strategy Abnormal Returns#

Armed with betas, we construct abnormal returns by stripping out factor-explained performance:

\[\text{Abnormal}_t = R_t - \sum_j \beta_j \, f_t^j\]
abnormal_return = fund_return - df_factor.drop(columns=['RF']) @ model.params[1:]

fig, ax = plt.subplots(figsize=(10, 5))
fund_return.cumsum().plot(ax=ax, label='Fund return')
abnormal_return.cumsum().plot(ax=ax, label='Abnormal return')
ax.set_title('Fund vs. Abnormal Cumulative Returns')
ax.legend()
plt.tight_layout()
plt.show()
../../_images/db7717ea6987f895b9d9ad46f61ac5861622bb1dfb2abfd0c97abb3fb9e0ad89.png

🤔 Think and Code:

  1. How can you compute abnormal returns more easily from regression outputs? Hint: which regression statistic equals the average abnormal return?

  2. What does the pattern of abnormal returns tell you about the fund’s skill?

13.8.4. Bottom-Up Approach#

Now we estimate factor betas for each stock, then use portfolio weights to compute fund exposures date-by-date:

# Estimate factor betas for each stock
df_factor, df_stocks = df_factor.align(df_stocks, join='inner', axis=0)
Xf = df_factor.drop(columns=['RF'])

B = pd.DataFrame([], index=tickers, columns=Xf.columns)
for ticker in df_stocks.columns:
    y = df_stocks[ticker].dropna()
    X = sm.add_constant(Xf.loc[y.index])
    model = sm.OLS(y, X).fit()
    B.loc[ticker, :] = model.params[1:]

B
Mkt-RF SMB HML RMW CMA MOM
AAPL 1.023921 -0.099693 0.098624 0.821751 -1.516863 -0.029002
GOOGL 0.944916 -0.445416 -0.053068 -0.052756 -1.334679 0.161329
MSFT 1.227477 -0.309171 0.164822 0.690796 -1.156395 0.097671
NVDA 1.269493 0.688041 -0.293219 0.298152 -0.264944 0.168911
AMZN 0.986571 -0.399886 0.24572 -0.143481 -2.160607 0.257187
COST 0.788138 -0.030709 0.042829 0.740032 -0.038182 0.210871
WMT 0.777156 -0.142485 -0.255371 0.858924 0.442454 0.105975
TGT 0.862613 0.297714 -0.103554 1.250017 0.49437 0.102883
KR 0.734953 0.055092 0.023944 0.298958 -0.165853 0.313963

With individual betas in hand, we can compute fund-level exposures date by date using current portfolio weights. This matters a lot for funds that trade frequently:

_temp = final_portfolio_df.merge(B, left_on='ticker', right_index=True, how='left')
Fund_B = _temp.groupby('date').apply(
    lambda x: pd.Series((x[Xf.columns].values * x['weight'].values.reshape(-1, 1)).sum(axis=0), index=Xf.columns))

Fund_B.plot(title='Fund Factor Exposures Over Time', figsize=(10, 5))
plt.ylabel('Beta')
plt.tight_layout()
plt.show()

Fund_B
../../_images/58be42cf8391313c48eb74baf0574607ae382dbc58f38f0f658fd652a84be028.png
Mkt-RF SMB HML RMW CMA MOM
date
2014-12-31 1.090475 -0.113225 0.032576 0.322892 -1.286698 0.131219
2015-01-01 1.090475 -0.113225 0.032576 0.322892 -1.286698 0.131219
2015-01-02 1.090475 -0.113225 0.032576 0.322892 -1.286698 0.131219
2015-01-05 1.090475 -0.113225 0.032576 0.322892 -1.286698 0.131219
2015-01-06 1.090475 -0.113225 0.032576 0.322892 -1.286698 0.131219
... ... ... ... ... ... ...
2016-12-26 0.790715 0.044903 -0.073038 0.786983 0.183197 0.183423
2016-12-27 0.790715 0.044903 -0.073038 0.786983 0.183197 0.183423
2016-12-28 0.790715 0.044903 -0.073038 0.786983 0.183197 0.183423
2016-12-29 0.790715 0.044903 -0.073038 0.786983 0.183197 0.183423
2016-12-30 0.790715 0.044903 -0.073038 0.786983 0.183197 0.183423

523 rows × 6 columns

📌 Remember:

There is no reason to believe asset betas are stable over time. The general recipe:

  • Daily data: 1–2 year estimation windows

  • Monthly data: ~5 year windows

Long samples give precision if betas are constant; short samples capture time-variation.

13.9. The Cross-Sectional Approach #

In the time-series approach, we start from factors and estimate betas. Now we flip this: start from characteristics (which are the betas) and estimate the returns associated with each characteristic.

13.9.1. Time-Series vs. Cross-Sectional#

Time-Series

Cross-Sectional

Starts from

Factor returns

Asset characteristics

Estimates

Betas (loadings)

Factor premia (returns to characteristics)

Requires

Traded factors

Large cross-section of stocks

Best for

Small number of well-defined factors

Many characteristics simultaneously

13.9.2. The Recipe#

  1. Get excess returns \(R\) for all stocks at date \(t\)

  2. Get characteristics \(X\) for those stocks as of date \(t-1\) (to avoid look-ahead bias!)

  3. Normalize characteristics cross-sectionally (z-scores)

  4. Run the cross-sectional regression: \(R = X \beta + \epsilon\)

From OLS: \(\beta = (X'X)^{-1}X'R\)

💡 Key Insight:

The \(\beta\) coefficients are excess returns themselves — they are returns on “pure play” portfolios designed to have a loading of 1 on one characteristic and zero on all others. The weights \((X'X)^{-1}X'\) are the portfolio weights.

# Load characteristics data
url = "https://github.com/amoreira2/Fin418/blob/main/assets/data/characteristics_raw.pkl?raw=true"
df_X = pd.read_pickle(url)
# Shift dates to end-of-month basis
df_X.set_index(['date', 'permno'], inplace=True)
df_X.head()
re rf rme size value prof fscore debtiss repurch nissa ... momrev valuem nissm strev ivol betaarb indrrev price age shvol
date permno
2006-01-31 10085 0.025224 0.0035 0.0304 14.132980 -0.775040 -2.223152 7 0 1 0.691947 ... 0.527791 -0.711504 0.697500 -0.003088 0.003396 1.030378 -0.003491 3.569814 5.480639 0.723779
10104 0.025984 0.0035 0.0304 18.034086 -2.186115 -0.458025 6 1 1 0.690818 ... 0.111133 -1.633254 0.687115 -0.030952 0.012757 1.473739 -0.005108 2.502255 5.480639 1.007820
10107 0.072982 0.0035 0.0304 19.399144 -1.357207 -1.094087 4 1 1 0.686126 ... 0.133546 -1.725634 0.667695 -0.055275 0.006959 1.166726 -0.029431 3.263849 5.480639 0.856907
10137 0.095710 0.0035 0.0304 15.226304 -0.256102 -2.418484 7 0 0 0.824237 ... 0.295023 -0.743060 0.782290 0.137262 0.012228 0.834982 0.124839 3.454738 6.349139 0.952114
10138 0.057586 0.0035 0.0304 15.913684 -1.553967 -1.227315 7 1 1 0.704786 ... 0.213203 -1.661273 0.701101 0.005004 0.006970 1.263471 -0.001327 4.277083 5.476464 0.605547

5 rows × 32 columns

# Standardize characteristics cross-sectionally (z-scores by date)
X_std = (df_X.drop(columns=['re', 'rf', 'rme'])
         .groupby('date')
         .transform(lambda x: (x - x.mean()) / x.std()))

# Run the cross-sectional regression for a single month
date = '2006-09'
X = X_std.loc[date]
R = df_X.loc[date, 're']

# Multiply by 100 for percentage returns
model = sm.OLS(100 * R, X).fit()
print(model.summary())

                                 OLS Regression Results                                
=======================================================================================
Dep. Variable:                     re   R-squared (uncentered):                   0.158
Model:                            OLS   Adj. R-squared (uncentered):              0.131
Method:                 Least Squares   F-statistic:                              6.017
Date:                Mon, 13 Apr 2026   Prob (F-statistic):                    1.81e-20
Time:                        15:15:36   Log-Likelihood:                         -3128.4
No. Observations:                 962   AIC:                                      6315.
Df Residuals:                     933   BIC:                                      6456.
Df Model:                          29                                                  
Covariance Type:            nonrobust                                                  
==============================================================================
                 coef    std err          t      P>|t|      [0.025      0.975]
------------------------------------------------------------------------------
size           0.3851      0.243      1.585      0.113      -0.092       0.862
value         -2.7649      0.815     -3.392      0.001      -4.364      -1.165
prof           2.2067      0.928      2.377      0.018       0.385       4.028
fscore        -0.1319      0.228     -0.579      0.563      -0.579       0.315
debtiss        0.6979      0.233      3.001      0.003       0.241       1.154
repurch        0.2854      0.231      1.234      0.217      -0.168       0.739
nissa         -0.3563      0.361     -0.987      0.324      -1.065       0.352
growth         0.5318      0.255      2.085      0.037       0.031       1.032
aturnover     -0.2630      1.170     -0.225      0.822      -2.560       2.034
gmargins      -0.2840      0.640     -0.444      0.657      -1.540       0.972
ep            -0.3085      0.263     -1.172      0.242      -0.825       0.208
sgrowth       -0.1833      0.218     -0.842      0.400      -0.611       0.244
lev            3.3657      0.614      5.485      0.000       2.161       4.570
roaa           0.8461      0.327      2.586      0.010       0.204       1.488
roea          -0.3104      0.267     -1.160      0.246      -0.835       0.215
sp            -0.2633      0.401     -0.656      0.512      -1.051       0.525
mom            0.2888      0.384      0.753      0.452      -0.464       1.042
indmom        -1.1630      0.247     -4.706      0.000      -1.648      -0.678
mom12         -0.0708      0.358     -0.198      0.843      -0.773       0.631
momrev        -0.2125      0.230     -0.923      0.356      -0.664       0.239
valuem         1.4420      0.761      1.895      0.058      -0.051       2.935
nissm          0.1557      0.350      0.445      0.657      -0.532       0.843
strev          1.9517      0.585      3.334      0.001       0.803       3.101
ivol          -0.1956      0.318     -0.615      0.539      -0.819       0.428
betaarb        0.4195      0.297      1.412      0.158      -0.163       1.002
indrrev       -2.2913      0.561     -4.084      0.000      -3.392      -1.190
price         -0.2100      0.245     -0.856      0.392      -0.691       0.271
age           -0.4385      0.234     -1.872      0.061      -0.898       0.021
shvol         -0.3185      0.334     -0.955      0.340      -0.973       0.336
==============================================================================
Omnibus:                       52.178   Durbin-Watson:                   1.910
Prob(Omnibus):                  0.000   Jarque-Bera (JB):              137.505
Skew:                          -0.253   Prob(JB):                     1.38e-30
Kurtosis:                       4.782   Cond. No.                         16.9
==============================================================================

Notes:
[1] R² is computed without centering (uncentered) since the model does not contain a constant.
[2] Standard Errors assume that the covariance matrix of the errors is correctly specified.

What does this mean?

  • The size coefficient means a portfolio with one standard deviation of size exposure (and zero of everything else) earned that return in this month

  • Because we normalized, “one unit” means one cross-sectional standard deviation above the mean

What are the portfolios behind these coefficients?

# Portfolio weights for each characteristic "pure play"
# Rows = characteristics, columns = stocks
Characteristic_portfolio_weights = np.linalg.inv(X.T @ X) @ X.T
Characteristic_portfolio_weights.index = X.columns
Characteristic_portfolio_weights
date 2006-09-30
permno 10104 10107 10137 10138 10143 10145 10147 10182 10225 10299 ... 89702 89753 89757 89805 89813 90352 90609 90756 91556 92655
size 0.002726 0.004360 -0.000686 -0.000294 -0.001032 0.001538 0.001587 -0.000573 0.000838 -0.000320 ... -0.000651 -0.000082 0.002007 0.000279 0.001528 0.000292 0.000142 -0.000617 -0.000251 0.002837
value -0.002632 0.006965 0.000676 0.000402 0.001094 0.000918 0.000914 0.001651 0.000402 0.000781 ... 0.002959 -0.001872 -0.017175 -0.008553 -0.000069 0.007953 0.001755 -0.002131 -0.004422 -0.000381
prof -0.004929 -0.005258 0.002034 0.002285 -0.008617 0.002159 -0.000689 -0.001997 0.003331 -0.006068 ... -0.019083 0.002454 0.003818 0.001534 -0.017922 -0.005647 -0.000107 -0.002436 0.000742 0.002495
fscore -0.000836 0.000893 -0.000555 -0.000177 0.002782 0.001506 0.002053 -0.000181 -0.001518 0.001342 ... 0.000608 -0.000435 -0.001723 -0.000495 0.000966 -0.001019 -0.000310 0.001647 -0.000310 -0.000057
debtiss -0.001236 0.001006 -0.000597 0.000775 -0.000740 0.002011 -0.001833 0.001377 -0.000026 -0.000127 ... 0.000919 0.000659 0.001524 -0.000626 -0.001255 -0.000257 0.001134 0.001364 0.001422 -0.000213
repurch 0.000264 -0.001071 0.001303 0.000584 0.002529 0.000491 0.000340 -0.001383 -0.001643 -0.000055 ... -0.000987 -0.001089 -0.002098 -0.001532 0.001370 0.000295 -0.002262 0.001152 0.000432 0.000759
nissa -0.001617 0.000601 -0.000104 -0.000428 -0.002482 -0.000482 -0.000480 0.000256 -0.001693 0.000070 ... 0.001202 0.000048 -0.002312 0.000341 0.000154 0.000719 -0.000753 -0.000134 -0.000160 -0.000975
growth 0.002627 -0.002042 -0.001180 0.000641 0.006734 0.001257 0.000534 0.000382 0.002447 -0.000216 ... -0.000920 -0.000798 0.000595 0.000434 0.001237 -0.000225 0.000504 0.001170 0.000455 0.002122
aturnover 0.004871 0.004464 -0.003664 -0.007132 0.007850 -0.000981 0.000484 -0.008215 -0.003439 0.002336 ... 0.023270 -0.004813 -0.000239 -0.002821 0.022379 0.015255 0.000800 0.004665 0.001854 -0.001814
gmargins 0.003855 0.004819 -0.002342 -0.003515 0.005676 -0.002311 0.000605 0.001060 -0.001483 0.003727 ... 0.011010 -0.002337 -0.002309 -0.000579 0.010097 0.003309 0.001513 0.000325 -0.000946 -0.002444
ep 0.000271 -0.000069 -0.000293 0.000143 -0.000826 -0.000250 -0.000066 0.006725 0.000147 0.000138 ... -0.000719 0.000504 0.000279 0.000525 -0.000788 -0.011986 0.000181 -0.001006 -0.000486 -0.000425
sgrowth -0.000287 0.000121 0.000095 -0.000050 0.001933 0.000055 0.000081 -0.000231 -0.000377 0.000371 ... -0.000251 -0.000266 -0.000869 -0.000080 -0.000567 -0.001799 -0.000175 0.000012 0.000174 -0.000431
lev -0.001423 -0.001301 -0.001663 -0.005339 -0.005967 0.000626 -0.000846 -0.012928 -0.000303 -0.003485 ... 0.002164 -0.002672 0.003537 0.000205 0.000187 0.009137 0.002843 0.002029 0.002139 0.000065
roaa 0.001069 0.001812 -0.000686 0.002124 -0.008606 -0.001630 -0.000170 -0.002489 -0.001095 0.002394 ... 0.000023 0.000402 0.000735 0.001093 -0.001175 0.002575 0.002885 -0.000064 0.000134 -0.001065
roea -0.000707 -0.000298 0.000059 -0.000892 0.002531 0.000550 0.000388 0.000520 0.000250 -0.000938 ... 0.000191 -0.000201 -0.001692 -0.000749 0.000008 0.000998 -0.000370 -0.000188 -0.000506 0.000377
sp 0.000065 0.000296 -0.000223 0.002129 -0.000102 -0.000631 -0.000166 0.017161 -0.000159 0.001461 ... -0.002150 0.000303 -0.001075 0.000744 -0.001675 -0.011035 -0.000667 -0.001549 -0.001146 -0.000327
mom 0.003275 -0.001444 0.000414 0.000814 0.004850 0.000500 -0.000680 -0.001967 0.001638 0.001538 ... -0.000963 0.006294 0.006965 0.000683 0.000819 -0.003108 -0.005419 -0.001557 -0.001139 -0.002296
indmom -0.001036 -0.000790 0.001130 -0.000069 -0.000559 0.000744 0.001156 -0.000728 -0.003179 -0.000662 ... -0.000162 0.000246 0.000676 0.000207 -0.000602 0.001851 0.000070 0.001938 -0.000137 -0.000500
mom12 -0.000448 -0.000538 0.000771 0.000531 -0.003858 -0.001458 -0.000226 0.000581 -0.001932 -0.001952 ... -0.000535 -0.002585 -0.002551 0.002501 0.000042 0.001954 0.004019 0.001748 0.001367 0.001481
momrev -0.000637 -0.000340 0.002424 -0.000383 -0.004906 -0.000384 -0.000023 -0.001467 0.000014 -0.000539 ... 0.000848 -0.000252 0.002038 -0.002305 -0.000394 0.000808 -0.000372 -0.000406 -0.001321 0.000021
valuem 0.002454 -0.006450 -0.000259 0.000282 -0.000720 -0.001481 0.000044 -0.000981 -0.000074 -0.000625 ... -0.002613 0.003338 0.017058 0.007713 0.001418 -0.005177 -0.001665 0.001393 0.002914 -0.000042
nissm 0.001268 -0.001002 0.000261 0.000340 0.002121 0.000014 -0.000332 -0.001073 0.001485 -0.000074 ... -0.000455 0.000915 0.001289 0.002882 0.000152 0.001970 -0.001322 -0.000142 -0.000059 0.001160
strev 0.001438 0.000856 0.001983 -0.003842 -0.001651 -0.005159 0.008296 -0.001921 -0.000597 0.003292 ... 0.000462 0.001209 0.000576 -0.000857 -0.000587 -0.007146 0.001954 0.002193 -0.002015 0.001781
ivol -0.000693 -0.000813 -0.000575 -0.000243 -0.000003 -0.000482 0.000029 -0.001962 0.000155 -0.002279 ... -0.001825 0.002788 -0.000676 -0.001814 0.000046 -0.000871 -0.001244 0.000096 -0.001520 0.000576
betaarb 0.001033 -0.001787 -0.000101 0.002654 0.001072 0.002249 0.000876 0.000410 -0.000529 0.001736 ... 0.000448 0.001858 0.002971 -0.000773 0.000476 0.002359 -0.000862 -0.000513 -0.000372 -0.002154
indrrev -0.000593 -0.000945 -0.001676 0.004517 0.003180 0.004748 -0.007188 -0.000257 0.000174 -0.003453 ... -0.000215 0.000768 0.001334 0.001912 0.001349 0.008045 -0.002434 0.000882 0.001426 -0.000810
price -0.002615 -0.001900 -0.000078 0.000075 0.001128 -0.000128 -0.001855 0.000754 0.001165 0.000277 ... 0.001324 0.001663 0.003084 -0.001295 0.000703 -0.002491 -0.003785 0.000367 -0.001120 -0.000556
age 0.000211 -0.000759 0.001319 0.000274 0.002850 0.000730 -0.000111 -0.000180 0.001711 0.000712 ... -0.002721 -0.001921 -0.003869 -0.002800 -0.003422 -0.002200 0.000486 0.000351 -0.000007 -0.000230
shvol -0.000060 0.001666 0.000600 -0.002817 0.002260 -0.001090 0.000591 -0.000607 -0.000647 0.001927 ... 0.000396 -0.000141 -0.000070 -0.000177 -0.001354 -0.000996 0.000648 -0.000197 0.000323 -0.000739

29 rows × 962 columns

13.9.3. Applications#

With these cross-sectional regressions we can:

  1. Compute characteristic-adjusted returns for any portfolio — just subtract the returns implied by its characteristics

  2. Construct factor return time-series — splice together the regression coefficients across dates to get \([\beta_t, \beta_{t+1}, \ldots]\)

13.9.4. Constructing Characteristic-Adjusted Returns#

We can get a portfolio’s characteristics and compute the returns implied by those characteristics. Subtracting these from actual returns gives the characteristic-adjusted return — the equivalent of “hedging” but using characteristics instead of time-series betas.

# Step 1: Define two sample portfolios (tech and retail)
portfolio_data1 = {'port': [1]*5,
    'ticker': ['AAPL', 'GOOG', 'MSFT', 'NVDA', 'AMZN'],
    'weight': [0.2, 0.2, 0.2, 0.2, 0.2]}

portfolio_data2 = {'port': [2]*4,
    'ticker': ['COST', 'WMT', 'TGT', 'KR'],
    'weight': [0.25, 0.25, 0.25, 0.25]}

portfolio_df = pd.concat([pd.DataFrame(portfolio_data1), pd.DataFrame(portfolio_data2)], ignore_index=True)
print(portfolio_df)

   port ticker  weight
0     1   AAPL    0.20
1     1   GOOG    0.20
2     1   MSFT    0.20
3     1   NVDA    0.20
4     1   AMZN    0.20
5     2   COST    0.25
6     2    WMT    0.25
7     2    TGT    0.25
8     2     KR    0.25

# Step 2: Get PERMNOs for ticker matching (our data uses PERMNOs, not tickers)
permno = get_permnos(portfolio_df.ticker.unique(), conn)
permno['namedt'] = pd.to_datetime(permno['namedt'])
permno['nameenddt'] = pd.to_datetime(permno['nameenddt'])

date = '2008-03'
d = pd.to_datetime(date)
# Get PERMNOs valid at this date (they can change over time!)
permno_d = permno[(permno['nameenddt'] >= d) & (permno['namedt'] <= d)]
portfolio_df = portfolio_df.merge(permno_d[['permno', 'ticker']], on='ticker', how='left')
portfolio_df
port ticker weight permno
0 1 AAPL 0.20 14593
1 1 GOOG 0.20 90319
2 1 MSFT 0.20 10107
3 1 NVDA 0.20 86580
4 1 AMZN 0.20 84788
5 2 COST 0.25 87055
6 2 WMT 0.25 55976
7 2 TGT 0.25 49154
8 2 KR 0.25 16678 
# Step 3: Merge portfolio with characteristics data
# Here we do it for one date; for multiple dates, add 'date' as a second identifier
X = X_std.loc[date].reset_index()
port_stocks_X = portfolio_df.merge(X, left_on='permno', right_on='permno', how='left')
port_stocks_X
port ticker weight permno date size value prof fscore debtiss ... momrev valuem nissm strev ivol betaarb indrrev price age shvol
0 1 AAPL 0.20 14593 2008-03-31 2.410205 -1.315480 0.455224 -0.091077 1.303219 ... 0.575718 -1.421058 0.147848 -0.580727 0.330686 0.752957 -0.868990 1.888114 0.398208 2.904418
1 1 GOOG 0.20 90319 2008-03-31 2.528013 -1.118659 0.564933 -0.091077 1.303219 ... 0.755154 -1.005315 0.229125 -1.402481 0.158169 -0.694430 -1.224557 3.959159 -2.339689 1.693268
2 1 MSFT 0.20 10107 2008-03-31 3.257090 -1.363239 0.985259 0.755457 1.303219 ... 0.726842 -1.531135 -0.438239 -1.377024 -0.637930 -0.456596 -1.195484 -0.492776 0.105671 -0.399737
3 1 NVDA 0.20 86580 2008-03-31 0.680226 -1.634531 0.817117 0.755457 1.303219 ... 1.172971 -0.842731 0.223491 -1.079041 1.603828 2.812533 -1.183732 -0.867861 -1.084778 1.545659
4 1 AMZN 0.20 84788 2008-03-31 1.240524 -3.728875 1.118980 -0.091077 1.303219 ... 1.249185 -3.528315 0.034269 -1.451173 0.544068 1.264655 -1.341326 0.854323 -0.858657 1.347997
5 2 COST 0.25 87055 2008-03-31 1.154301 0.138992 0.785716 -0.091077 -0.766462 ... -0.499439 -0.268807 -0.301382 -0.674229 -0.579135 -0.414050 -0.454017 0.791327 0.126212 0.223698
6 2 WMT 0.25 55976 2008-03-31 2.960595 -0.322603 1.029022 -0.937610 -0.766462 ... -0.424544 -0.260600 -0.348921 -0.082608 -1.065065 -0.802397 0.221643 0.444707 0.753804 -1.140483
7 2 TGT 0.25 49154 2008-03-31 1.815796 -0.209013 0.909948 1.601991 -0.766462 ... 0.974779 -0.055104 -0.423087 -0.319158 0.450388 0.077535 -0.048508 0.536987 0.871398 0.443988
8 2 KR 0.25 16678 2008-03-31 0.934108 -0.149013 1.326149 0.755457 -0.766462 ... -0.172023 -0.295956 -0.406029 -0.282320 -0.487468 -0.728124 -0.006437 -0.671970 1.220560 -0.344465

9 rows × 34 columns

# Step 4: Compute portfolio-level characteristics (weighted average)
X_names = X.drop(columns=['permno', 'date']).columns
port_X = port_stocks_X.groupby('port').apply(lambda x: x['weight'] @ x[X_names])
port_X
weight size value prof fscore debtiss repurch nissa growth aturnover gmargins ... momrev valuem nissm strev ivol betaarb indrrev price age shvol
port
1 2.023212 -1.832157 0.788303 0.247537 1.303219 0.202595 -0.021355 0.558931 0.525197 0.320116 ... 0.895974 -1.665711 0.039299 -1.178089 0.399764 0.735824 -1.162818 1.068192 -0.755849 1.418321
2 1.716200 -0.135409 1.012709 0.332190 -0.766462 0.642131 -0.272052 -0.261772 1.427599 -0.806162 ... -0.030307 -0.220117 -0.369855 -0.339578 -0.420320 -0.466759 -0.071830 0.275263 0.742994 -0.204316

2 rows × 29 columns

# Step 5: Estimate returns associated with each characteristic (full universe)
X = X_std.loc[date]
R = df_X.loc[date, 're']
model = sm.OLS(R, X).fit()
R_X = model.params
R_X

size        -0.008685
value        0.010048
prof         0.014476
fscore      -0.000212
debtiss     -0.009435
repurch      0.008764
nissa        0.007878
growth       0.001369
aturnover   -0.026765
gmargins    -0.011491
ep          -0.001229
sgrowth     -0.007586
lev         -0.026462
roaa        -0.008852
roea         0.006974
sp           0.007249
mom          0.005713
indmom      -0.008373
mom12        0.003797
momrev       0.000066
valuem      -0.006634
nissm       -0.005530
strev        0.013887
ivol        -0.007777
betaarb      0.003666
indrrev     -0.011385
price       -0.003223
age          0.004363
shvol       -0.007341
dtype: float64

# Step 6: Characteristic-implied returns
# This is the equivalent of sum(beta_j * f_j), but using characteristics as "betas"
# and the cross-sectional regression coefficients as "factors"
port_characteristic_returns = port_X[X_names] @ R_X
print("Characteristic-implied returns:")
print(port_characteristic_returns)

Characteristic-implied returns:
port
1   -0.027497
2    0.006339
dtype: float64

# Step 7: Characteristic-adjusted returns = actual - implied
_temp = portfolio_df.merge(R.reset_index(), left_on='permno', right_on='permno')
R_port = _temp.groupby('port').apply(lambda x: x['weight'] @ x['re'])

print("Raw excess returns:")
print(R_port)
print("\nCharacteristic-implied returns:")
print(port_characteristic_returns)
print("\nCharacteristic-adjusted returns:")
print(R_port - port_characteristic_returns)

Raw excess returns:
port
1    0.029733
2    0.030074
dtype: float64

Characteristic-implied returns:
port
1   -0.027497
2    0.006339
dtype: float64

Characteristic-adjusted returns:
port
1    0.057230
2    0.023735
dtype: float64

13.9.5. Why Practitioners Like This#

  • No time-series betas needed — avoids all the issues with sample length and beta instability

  • Characteristics can change freely — we estimate date-by-date, so the model adapts instantly

  • Scales to many factors — just add columns to the regression (sector, country, currency, etc.)

13.9.6. What Are the Issues?#

  • Ignores covariances — characteristic-neutral ≠ factor-neutral. A stock classified as “retail” might co-move with tech

  • Loads on small stocks — OLS treats all observations equally, and most stocks are tiny. Fixes: weighted least squares (by market cap), or restrict to the largest 20% of stocks

⚠️ Caution:

The characteristic and factor-based approaches are complements, not substitutes. Characteristics are observable and easy to work with, but factors capture the actual return co-movement structure. Use both.

13.10. 📝 Exercises #

13.10.1. Exercise 1: Factor Attribution#

🔧 Exercise:

Pick a fund or ETF of your choice (e.g., QQQ, XLF, ARKW).

  1. Download its daily returns from WRDS

  2. Run a multi-factor regression (FF5 + Momentum)

  3. Report: alpha, t-stat, \(R^2\), and the dominant factor exposures

  4. In 2-3 sentences: what is this fund actually giving you?

# Your code here

13.10.2. Exercise 2: Bottom-Up vs Top-Down#

🤔 Think and Code:

Using the Tech → Retail portfolio from above:

  1. Compare the fund betas from the top-down regression (full sample) to the bottom-up approach

  2. Where do the biggest discrepancies appear? Why?

  3. Which approach would you trust more for a high-turnover hedge fund?

# Your code here

13.11. 🧠 Key Takeaways #

  • Multi-factor models are the industry workhorse. They capture multiple rewarded risks simultaneously, delivering more realistic benchmarks and richer performance attribution.

  • Alpha is scarce; beta is plentiful. Time-series regressions reveal that most “smart-beta” ETFs provide factor exposure, not outperformance — true skill shows up only in the intercept.

  • Bottom-up attribution excels for high-turnover managers. Refreshing exposures at the holding level avoids the lag and instability that afflict purely return-based estimates.

  • Characteristic models broaden the toolkit but ignore covariances. They neutralize portfolios on observed attributes quickly and at scale, yet leave hidden co-movement risks untouched — factor and characteristic views are complements, not substitutes.

13.12. 📎 Solutions#

13.12.1. ETF Evaluation (Think and Code)#

💡 Click to see answer

Alpha alone is insufficient. You also need to consider:

  1. t-statistic — is the alpha statistically significant, or could it be zero?

  2. Idiosyncratic volatility — higher ivol means more tracking error and noisier alpha estimates

  3. Sample size — some ETFs are newer with less data; shorter samples produce less reliable estimates

  4. Factor loadings — a fund with high MOM loading is delivering factor exposure you could get cheaply from an index. That is not skill.

The comparison is only fair if sample periods overlap. Different start dates mean different market conditions, which can bias the results.

13.12.2. Cathie Wood Factor Profile (Think and Code)#

💡 Click to see answer

ARKK typically has \(R^2\) around 0.4–0.6 with the FF5+MOM model. The loadings reveal:

  • High market beta (~1.3+) — aggressive, amplifies market moves

  • Strongly negative HML — anti-value / growth tilt (buys expensive, innovative firms)

  • Positive SMB — tilts toward smaller firms

  • Negative CMA — likes firms investing heavily (high capex)

Market exposure dominates variance, but the anti-value tilt contributes significantly. You can compute this as \(\beta_{HML}^2*\text{Var}(HML)\).

The residual volatility is the regression’s \(\sigma(\epsilon)\) — this is what you’d bear if you hedged all factor exposures.

Her alpha is likely concentrated in 2020 (pandemic tech/innovation boom), not smoothly distributed. This raises questions about persistence.

13.12.3. Buffett: Multi-Factor Analysis (Think and Code)#

💡 Click to see answer

Adding factors typically reduces Buffett’s alpha relative to CAPM, because some of his apparent “skill” is actually systematic factor exposure.

Buffett’s factor loadings:

  • Positive HML — value investor (buys cheap stocks)

  • Positive RMW — quality preference (profitable firms)

  • Slightly negative CMA — likes firms that invest

  • Low/negative MOM — contrarian, patient

He is the anti–Cathie Wood: conservative, value-oriented, high-quality. After controlling for factors, his remaining alpha represents genuine stock-picking skill.

Buffett = value + quality + patience; Wood = growth + innovation + momentum.

13.12.4. Abnormal Returns Shortcut (Think and Code)#

💡 Click to see answer

The residuals from the regression ARE the abnormal returns:

abnormal_returns = model.resid  # exactly R_t - Σβ_j f_t^j

The alpha (intercept) is simply the average of these residuals.

If abnormal returns are clustered in one period, the “skill” may be period-specific rather than persistent — a red flag for forward-looking investment decisions.

13.12.5. Exercise 1: Factor Attribution#

💡 Click to see answer 
# Example with QQQ
ticker = "QQQ"
df_etf = get_daily_wrds_multiple_ticker([ticker], conn)
df_fac = get_factors("FF6", "daily")
df_etf, df_fac = df_etf.align(df_fac, join="inner", axis=0)
df_etf = df_etf.subtract(df_fac["RF"], axis=0)

y = df_etf[ticker].dropna()
X = sm.add_constant(df_fac.drop(columns=["RF"]).loc[y.index])
model = sm.OLS(y, X).fit()
print(model.summary())
print(f"Alpha (annualized): {model.params['const']*252:.4f}")
print(f"t-stat: {model.tvalues['const']:.2f}")
print(f"R²: {model.rsquared:.3f}")

13.12.6. Exercise 2: Bottom-Up vs Top-Down#

💡 Click to see answer

The top-down regression averages over the entire sample, so it mixes the tech and retail periods — the estimated betas are a blend that doesn’t accurately represent either regime.

The bottom-up approach correctly shows the sharp shift in exposures at the rebalancing date (end-2015). The biggest discrepancies will be in:

  • HML — tech stocks are growth (negative HML), retail stocks are closer to value

  • SMB — tech mega-caps vs. mid-cap retailers

For a high-turnover hedge fund, bottom-up is strictly better because top-down estimates lag behind actual exposure changes. The regression needs months of data to detect a shift that happened overnight.

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