Original paper

SSRN-id3258015.pdf583.8KB

Abstract

The observation that low-risk assets on average have higher risk-adjusted returns relative to high-risk assets – the low-risk effect – is a driving force behind a broad set of well-documented cross-sectional asset pricing anomalies. I document that long-short strategies formed on a wide range of anomaly characteristics tend to buy low-risk and short high-risk assets. I show that this tendency implicitly embeds the low-risk effect in these cross-sectional long-short portfolios. Taking into account the exposure to the low-risk effect attenuates the economic and statistical significance of the risk-adjusted returns to a large set of asset pricing anomalies.

Keywords: Low-risk effect, cross-sectional stock returns, beta anomaly, volatility anomaly

Trading rules

Employ the CRSP database for selecting stocks.
Compute a 1-year rolling beta for each stock against the MSCI US Equity Index.
Sort stocks from lowest to highest based on their calculated betas.
Categorize stocks into either low or high beta portfolios.
Weight stocks in the portfolios based on their beta rankings.
Adjust portfolio compositions on a monthly basis.
Normalize both portfolios to achieve a beta value of one upon creation.
Formulate a zero-cost, zero-beta strategy: go long on low-beta stocks and short on high-beta stocks.
Test basic alterations (like long positions in the lowest beta decile and short positions 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.

Asset Pricing Anomalies and the Low-Risk Puzzle

Original paper

Abstract

Trading rules

Python code

Backtrader