Original paper
Abstract
Contrary to what traditional asset pricing would imply, a strategy that bets against beta, i.e. long in low beta stocks and short in high beta stocks, tends to out-perform the market. This puzzling empirical fact can be explained through the concept of relative arbitrage. Considering a market in which diversity is maintained, i.e. no single stock can dominate the entire market, we show that beta-arbitrage strategies out-perform the market portfolio with unit probability in finite time. We use the theoretical decomposition of beta-arbitrage excess return to provide empirical support to our explanation on equity country indices, equity sectors and individual stocks. Finally we show how to construct optimal beta-arbitrage strategies that maximize the expected return relative to a given benchmark.
Keywords:Â Relative arbitrage, Market diversity, Beta
Trading rules
- Investment universe: All available country ETFs.
- Beta assessment: Use a 1-year dynamic window, relative to MSCI US Equity Index.
- Arrange ETFs: From the lowest to the highest beta.
- Assign portfolios: Group into either high-beta or low-beta categories.
- Determine weights of assets: Rely on the beta rankings.
- Monthly adjustments: Make changes to portfolios at the start of each month.
- Normalize portfolios: Ensure both high and low-beta portfolios begin with a beta value of one.
- Implement the Betting-Against-Beta approach: Take long positions in low-beta and short positions in high-beta (ensuring zero cost and zero beta).
- Potential modifications: Adopt long/short positions in the most extreme beta deciles to potentially boost outcomes.
Python code
Backtrader
import backtrader as bt
import pandas as pd
import numpy as np
class BettingAgainstBeta(bt.Strategy):
params = (
('beta_window', 252), # 1-year rolling window for beta calculation
('rebalance_freq', 20) # Monthly rebalancing
)
def __init__(self):
self.etfs = self.datas[1:] # All country ETFs
self.msci_index = self.datas[0] # MSCI US Equity Index
self.counter = 0
def next(self):
self.counter += 1
if self.counter % self.params.rebalance_freq == 0:
etf_betas = []
for etf in self.etfs:
returns_etf = pd.Series(etf.get_array('close')[-self.params.beta_window:]).pct_change().dropna()
returns_index = pd.Series(self.msci_index.get_array('close')[-self.params.beta_window:]).pct_change().dropna()
beta = np.cov(returns_etf, returns_index)[0][1] / np.var(returns_index)
etf_betas.append((etf, beta))
etf_betas.sort(key=lambda x: x[1]) # Sort ETFs in ascending order by beta
low_beta_etfs = etf_betas[:len(etf_betas) // 2]
high_beta_etfs = etf_betas[len(etf_betas) // 2:]
# Adjust positions - long on low-beta, short on high-beta
for etf, beta in low_beta_etfs:
self.order_target_percent(etf, target=1 / len(low_beta_etfs))
for etf, beta in high_beta_etfs:
self.order_target_percent(etf, target=-1 / len(high_beta_etfs))
if __name__ == '__main__':
cerebro = bt.Cerebro()
# Load data for MSCI US Equity Index and Country ETFs
# Assuming historical data is available as csv files
msci_index_data = bt.feeds.YahooFinanceCSVData(dataname='path_to_msci_index_csv')
etf_list = ['path_to_etf1_csv', 'path_to_etf2_csv', 'path_to_etf3_csv', ...]
cerebro.adddata(msci_index_data)
for etf_data_path in etf_list:
etf_data = bt.feeds.YahooFinanceCSVData(dataname=etf_data_path)
cerebro.adddata(etf_data)
cerebro.addstrategy(BettingAgainstBeta)
cerebro.broker.setcash(100000.0)
cerebro.broker.setcommission(commission=0.001) # Set broker commission
print('Starting Portfolio Value: %.2f' % cerebro.broker.getvalue())
cerebro.run()
print('Final Portfolio Value: %.2f' % cerebro.broker.getvalue())Note: Make sure to replace the ‘path_to_msci_index_csv’, ‘path_to_etf1_csv’, ‘path_to_etf2_csv’, ‘path_to_etf3_csv’, … with the actual file paths for the MSCI US Equity Index and country ETFs historical data.
Keywords:Â Relative arbitrage, Market diversity, Beta
Contrary to what traditional asset pricing would imply, a strategy that bets against beta, i.e. long in low beta stocks and short in high beta stocks, tends to out-perform the market. This puzzling empirical fact can be explained through the concept of relative arbitrage. Considering a market in which diversity is maintained, i.e. no single stock can dominate the entire market, we show that beta-arbitrage strategies out-perform the market portfolio with unit probability in finite time. We use the theoretical decomposition of beta-arbitrage excess return to provide empirical support to our explanation on equity country indices, equity sectors and individual stocks. Finally we show how to construct optimal beta-arbitrage strategies that maximize the expected return relative to a given benchmark.