Original paper
Abstract
In a CAPM world, the expected return of every portfolio is linearly related to its market beta. Further, the market portfolio attains the maximum Sharpe ratio among all portfolios of risky assets. Consequently, low-beta portfolios are predicted to earn a lower rate of return and to have Sharpe ratios no greater than the market portfolio. A low-beta portfolio of risky assets with beta B is predicted to earn the same rate of return as a portfolio that invests B in the market portfolio and 1 - B in the risk-free asset. Empirically, neither of these predictions has been realized. Low-beta (B < 1) portfolios have earned higher returns than their market portfolio plus risk-free asset counterparts, and they have achieved higher Sharpe ratios than the market portfolio. In the literature, this is referred to as the low-beta anomaly. This paper uses quantile regression to examine other dimensions of risk beyond beta and volatility, and finds that low-beta stocks and portfolios bear additional compensated risk in the form of excess kurtosis.
Trading rules
- Utilize CRSP database for stock selection
- Compute 1-year rolling beta for each stock relative to MSCI US Equity Index
- Rank stocks in ascending order based on estimated beta
- Allocate stocks to low beta or high beta portfolios
- Weight securities by their ranked betas
- Rebalance portfolios monthly
- Rescale both portfolios to have a beta of one at formation
- Establish zero-cost, zero-beta Betting-Against-Beta portfolio: long low-beta, short high-beta
- Consider simple modifications (e.g., long bottom beta decile, short top beta decile) to enhance performance
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.