Original paper

SSRN-id2703752.pdf136.1KB

Abstract

This paper suggests an alternative explanation for the recently documented betting against beta anomaly. Given that the equity of a levered firm is equivalent to a call option on firm assets and option returns are non-linearly related to underlying stock returns, linear CAPM-type regressions are generally misspecified. We derive theoretical expressions for the pricing error and analyze its magnitude using numerical examples. Consistent with the empirical findings of Frazzini and Pedersen (2014), our pricing errors are negative, increase with leverage, and become economically significant for higher levels of firm leverage.

Keywords: Asset Pricing, Performance Measurement, Abnormal Return, Systematic Risk

Trading rules

Use the CRSP database for choosing stocks.
Calculate a rolling 1-year beta for every stock relative to the MSCI US Equity Index.
Organize stocks from lowest to highest based on their calculated beta values.
Distribute stocks into either low beta or high beta portfolios.
Assign weights to stocks according to their beta rankings.
Adjust the portfolios on a monthly basis.
Rescale both portfolios to have a beta of one at formation.
Create a cost-neutral, beta-neutral Betting-Against-Beta strategy: take a long position in low-beta stocks and a short position in high-beta stocks.
Explore basic tweaks (for instance, go long on the stocks in the lowest beta decile and short on those in the highest beta decile) to boost returns.

Python code

Backtrader

Here’s a Backtrader Python code implementing the trading rules you described:

import backtrader as bt
import backtrader.feeds as btfeeds
import pandas as pd

class BetaRanking(bt.Strategy):
    params = (
        ('lookback', 252),
        ('rebalance_freq', 21),
        ('num_long', 10),
        ('num_short', 10)
    )

    def __init__(self):
        self.betas = {}
        self.rankings = []
        self.rebalance_day = 0
        
        # Compute beta for each stock
        for d in self.datas:
            self.betas[d._name] = bt.indicators.Beta(d, self.data0, period=self.params.lookback)

    def next(self):
        self.rebalance_day += 1
        if self.rebalance_day % self.params.rebalance_freq != 0:
            return

        self.rankings = sorted(
            self.datas,
            key=lambda d: self.betas[d._name][0]
        )
        
        longs = self.rankings[:self.params.num_long]
        shorts = self.rankings[-self.params.num_short:]
        
        for d in longs:
            self.order_target_percent(d, target=1.0 / self.params.num_long)

        for d in shorts:
            self.order_target_percent(d, target=-1.0 / self.params.num_short)

        for d in self.datas:
            if d not in longs and d not in shorts:
                self.order_target_percent(d, target=0.0)


if __name__ == '__main__':
    cerebro = bt.Cerebro()

    # Add strategy
    cerebro.addstrategy(BetaRanking)

    # Load CRSP database
    # Replace this part with your actual CRSP data
    for stock_data in CRSP_data:
        data = btfeeds.PandasData(dataname=stock_data)
        cerebro.adddata(data)

    # Load MSCI US Equity Index data
    msci_data = btfeeds.PandasData(dataname=pd.read_csv('msci_us_equity_index.csv', parse_dates=['date'], index_col='date'))
    cerebro.adddata(msci_data, name='msci_us_equity_index')

    # Set initial capital and run
    cerebro.broker.setcash(100000.0)
    cerebro.run()

This code implements a Backtrader strategy that calculates rolling betas for each stock relative to the MSCI US Equity Index, ranks stocks based on their betas, and rebalances the portfolio monthly. Remember to replace the data loading part with your actual CRSP data and the MSCI US Equity Index data in a suitable format.

The Betting Against Beta Anomaly: Fact or Fiction?

Original paper

Abstract

Trading rules

Python code

Backtrader