Original paper
Abstract
We investigate asset returns using the concept of beta herding, which measures cross-sectional variations in betas induced by investors whose beliefs about the market are biased due to changes in confidence or sentiment. Overconfidence or optimistic sentiment causes beta herding (compression of individual assets’ betas towards the market beta), while under-confidence or pessimistic sentiment leads to adverse beta herding (dispersion of betas away from the market beta). We find that beta herding is related to the low-beta anomaly, as high beta stocks underperform low beta stocks on a risk-adjusted basis exclusively following periods of adverse beta herding. As an explanation of the low-beta anomaly, we propose the persistence of bias in betas (i.e., a large difference in betas) that lasts for more than one year as market uncertainty continues.
Keywords:Â Beta, Herding, Overconfidence, Sentiment, Low-beta Anomaly
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.