Michael Brenndoerfer | Data & AI, Private Equity, Technology

Michael Brenndoerfer

Arbitrage Pricing Theory and Multi-Factor Models › Estimating Factor Loadings for a Stock

Michael BrenndoerferDecember 15, 20254 min read

Quantitative FinanceAll chapters

Part 9 of 15Portfolio Theory and Investment Analysis

Chapter 48Arbitrage Pricing Theory and Multi-Factor Models › Estimating Factor Loadings for a Stock

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Now let's estimate factor loadings for a specific stock. We'll create synthetic return data that mimics a real stock's behavior, then estimate its factor exposures using regression.

In[14]:

Code

# Generate synthetic stock returns with known factor exposures
np.random.seed(42)

# True factor loadings for our synthetic stock
true_betas = {
    "MKT": 1.2,  # Higher market exposure than average
    "SMB": 0.3,  # Slight small-cap tilt
    "HML": -0.4,  # Growth stock characteristics
}
true_alpha = 0.001  # Small positive alpha (0.1% monthly)

# Generate returns using the factor model
n_obs = len(ff_factors)
idiosyncratic_vol = 0.03  # 3% monthly idiosyncratic volatility
epsilon = np.random.normal(0, idiosyncratic_vol, n_obs)

stock_excess_returns = (
    true_alpha
    + true_betas["MKT"] * ff_factors["MKT"].values
    + true_betas["SMB"] * ff_factors["SMB"].values
    + true_betas["HML"] * ff_factors["HML"].values
    + epsilon
)

# Create a DataFrame
stock_data = pd.DataFrame(
    {"excess_return": stock_excess_returns}, index=ff_factors.index
)

In[15]:

Code

import statsmodels.api as sm

# Estimate factor loadings via OLS regression
X = ff_factors[["MKT", "SMB", "HML"]]
X = sm.add_constant(X)
y = stock_data["excess_return"]

model = sm.OLS(y, X).fit()

# Extract results for display
r_squared = model.rsquared
adj_r_squared = model.rsquared_adj
params = model.params
std_errs = model.bse
t_stats = model.tvalues
p_values = model.pvalues

Out[16]:

Console

Factor Model Regression Results
==================================================

R-squared: 0.6820
Adjusted R-squared: 0.6762

Estimated Factor Loadings:
--------------------------------------------------
Parameter      Estimate  Std Error     t-stat    p-value
--------------------------------------------------
const           -0.0005     0.0023      -0.20     0.8432
MKT              1.3249     0.0795      16.66     0.0000
SMB              0.3070     0.0768       4.00     0.0001
HML             -0.4512     0.0672      -6.71     0.0000

The regression successfully recovers the true factor loadings. The R-squared indicates what fraction of return variance is explained by the factors. The remaining variance represents idiosyncratic risk that could be diversified away in a portfolio.

Interpreting the ResultsLink Copied

Let's compare our estimates to the true values and interpret the stock's factor profile.

In[17]:

Code

# Compare estimated vs true parameters
comparison_data = []

# Alpha comparison
alpha_est = model.params["const"]
comparison_data.append(
    {
        "Factor": "Alpha",
        "True": true_alpha,
        "Estimated": alpha_est,
        "Difference": alpha_est - true_alpha,
    }
)

# Factor betas comparison
for factor in ["MKT", "SMB", "HML"]:
    beta_est = model.params[factor]
    beta_true = true_betas[factor]
    comparison_data.append(
        {
            "Factor": factor,
            "True": beta_true,
            "Estimated": beta_est,
            "Difference": beta_est - beta_true,
        }
    )

comparison_df = pd.DataFrame(comparison_data).set_index("Factor")

Out[18]:

Console

Comparison: Estimated vs True Factor Loadings
==================================================
         True  Estimated  Difference
Factor                              
Alpha   0.001    -0.0005     -0.0015
MKT     1.200     1.3249      0.1249
SMB     0.300     0.3070      0.0070
HML    -0.400    -0.4512     -0.0512

Out[19]:

Visualization

Two-bar chart comparing true and OLS-estimated monthly alpha in percent. — True and OLS-estimated monthly alpha shown separately from the dimensionless factor betas so the small intercept remains visible.

The estimated loadings closely match the true values. The small differences arise from estimation error due to finite sample size and idiosyncratic noise. This stock has:

High market beta (1.2): More volatile than the market, amplifying gains in bull markets and losses in bear markets
Positive SMB loading (0.3): Behaves somewhat like a small-cap stock, gaining when small stocks outperform
Negative HML loading (-0.4): Behaves like a growth stock, gaining when growth outperforms value

This profile is typical of a technology growth stock: high market sensitivity, modest small-cap characteristics, and strong growth orientation.