Original paper

SSRN-id2593519.pdf1536.9KB

Abstract

This paper shows that low risk anomalies in the CAPM and in traditional factor models arise when investors require compensation for coskewness risk. Empirically, we find that option-implied ex-ante skewness is strongly related to ex-post residual coskewness, which allows us to construct coskewness factor mimicking portfolios. Controlling for skewness renders the alphas of betting-against-beta and -volatility insignificant. We also show that the returns of beta- and volatility-sorted portfolios are largely driven by a single principal component, which is in turn largely explained by skewness.

Keywords: low risk anomaly, coskewness, skewness, risk premia, equity options

Trading rules

Use the CRSP database for picking stocks.
Calculate a 1-year moving beta for each stock compared to the MSCI US Equity Index.
Order stocks from lowest to highest based on their computed beta.
Segment stocks into either high or low beta groupings.
Assign stock weights based on their beta rankings.
Adjust portfolios on a monthly basis.
Normalize both portfolios to achieve a beta value of one at the inception.
Construct a zero-cost, zero-beta Betting-Against-Beta strategy: take a long position in low-beta stocks and a short position in high-beta stocks.
Explore basic adjustments (e.g., long position in the lowest beta decile, short position in the highest beta decile) for potential performance improvement.

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.

Low Risk Anomalies?

Original paper

Abstract

Trading rules

Python code

Backtrader