How to Implement a CAPM-Style Alpha Calculation Using Linear Regression in Python

Here is a complete, runnable algorithm for QuantConnect that calculates the Alpha ($\alpha$) of a specific stock (e.g., TSLA) against the SPY benchmark using Linear Regression.

This example uses numpy to perform the Ordinary Least Squares (OLS) regression.

Python Code

# region imports
from AlgorithmImports import *
import numpy as np
# endregion

class LinearRegressionAlphaAlgorithm(QCAlgorithm):

    def initialize(self):
        self.set_start_date(2023, 1, 1)  # Set Start Date
        self.set_end_date(2024, 1, 1)    # Set End Date
        self.set_cash(100000)            # Set Strategy Cash

        # 1. Add the Benchmark (SPY) and the Target Stock
        self.benchmark = self.add_equity("SPY", Resolution.DAILY).symbol
        self.target = self.add_equity("TSLA", Resolution.DAILY).symbol
        
        # 2. Define the lookback period for the regression (e.g., 60 trading days)
        self.lookback = 60

        # 3. Schedule the calculation to run every day after market open
        self.schedule.on(
            self.date_rules.every_day(self.benchmark),
            self.time_rules.after_market_open(self.benchmark, 10),
            self.calculate_alpha
        )

    def calculate_alpha(self):
        # 4. Fetch historical data for both symbols
        history = self.history(
            [self.benchmark, self.target], 
            self.lookback, 
            Resolution.DAILY
        )
        
        # Guard clause: ensure we have data
        if history.empty:
            return

        # 5. Process Data: Unstack to separate symbols into columns and calculate returns
        # .unstack(level=0) pivots the data so symbols are columns
        closes = history['close'].unstack(level=0)
        
        # Calculate percentage change (returns) and drop NaN values created by the shift
        returns = closes.pct_change().dropna()

        # Ensure we have enough data points after dropping NaNs
        if len(returns) < 30:
            return

        # 6. Prepare Regression Variables
        # Y = Target Stock Returns
        # X = Benchmark Returns
        y = returns[self.target].values
        x = returns[self.benchmark].values

        # 7. Linear Regression Logic
        # We stack the X values with a column of 1s to allow the calculation of the intercept.
        # Equation: Y = Beta * X + Alpha * 1
        A = np.vstack([x, np.ones(len(x))]).T
        
        # np.linalg.lstsq returns the solution (slope and intercept), residuals, rank, and singular values
        # We only need the solution (index 0).
        # result[0] is the slope (Beta), result[1] is the intercept (Alpha)
        result = np.linalg.lstsq(A, y, rcond=None)[0]
        
        beta = result[0]
        alpha = result[1]

        # 8. Log the results
        self.log(f"Date: {self.time.date()} | {self.target} vs SPY | Beta: {beta:.4f} | Alpha: {alpha:.6f}")
        
        # Optional: Trade based on Alpha
        # Example: If Alpha is positive, go long
        if alpha > 0:
            self.set_holdings(self.target, 1.0)
        else:
            self.liquidate(self.target)

Key Implementation Details

1. Data Preparation (unstack and pct_change):
   The self.history call returns a MultiIndex DataFrame. Using .unstack(level=0) converts the Symbols from index levels to columns. This makes it easy to calculate returns for the entire DataFrame at once using .pct_change().

2. The Math ($Y = \beta X + \alpha$):

   • Y (Dependent Variable): The returns of the stock you are analyzing.
   • X (Independent Variable): The returns of the benchmark (SPY).
   • Alpha ($\alpha$): The intercept of the regression line. It represents the return of the asset that is not explained by the market movement.

3. Numpy vstack:
   Standard linear algebra libraries usually solve for $Ax = B$. To find the intercept (Alpha), we must augment the $X$ matrix with a column of ones.

   • Without the column of ones, the regression is forced through the origin (0,0), assuming Alpha is 0.
   • np.vstack([x, np.ones(len(x))]).T creates a 2D array where the first column is the SPY returns and the second column is all 1s.

4. rcond=None:
   This parameter is passed to lstsq to silence a specific Numpy FutureWarning regarding the handling of singular matrices. It tells Numpy to use the machine precision default for determining the cutoff for small singular values.

Q&A: Quantitative Analysis on QuantConnect

Q: Why do we use pct_change() instead of raw prices for regression?
A: Linear regression for Alpha/Beta requires stationary data. Stock prices are non-stationary (they trend), whereas returns (percentage changes) are generally stationary. Using raw prices would result in a spurious correlation based on the trend rather than the actual relationship between the asset's performance and the market.

Q: How can I calculate the Alpha for a list of stocks (Universe)?
A: You can iterate through a list of symbols. In the calculate_alpha method, instead of selecting a single self.target, you would loop through self.securities.keys(), extract the returns for each, and run the np.linalg.lstsq function inside the loop.

Q: What is the difference between Resolution.DAILY and Resolution.MINUTE for Alpha calculation?
A: Resolution.DAILY captures the relationship based on end-of-day moves, which is standard for calculating "Jensen's Alpha" for portfolio management. Resolution.MINUTE would calculate intraday Alpha, which is noisier and typically used for High-Frequency Trading (HFT) execution algorithms rather than fundamental strategy assessment.

Q: Can I use scipy.stats.linregress instead of numpy?
A: Yes. You can import from scipy import stats and use beta, alpha, r_value, p_value, std_err = stats.linregress(x, y). This is often more readable as it handles the intercept automatically, but numpy is generally slightly faster and requires no additional imports beyond standard scientific stack usage.