Original paper
Abstract
The “low risk anomaly” refers to the empirical pattern that apparently high-risk equities do not earn commensurately high returns. In this paper, we consider the possibility that the risk anomaly represents mispricing, not a misspecification of risk, and develop the implications for corporate capital structure. The risk anomaly generates a simple tradeoff model: Starting at zero leverage, the overall cost of capital initially falls as leverage increases equity risk. As debt becomes risky, however, the marginal benefit of increasing equity risk declines. The optimum is reached at lower leverage for firms with high asset risk. Consistent with a risk anomaly tradeoff, firms with low-risk assets choose higher leverage. In addition, leverage is inversely related to systematic risk, holding constant total risk; a large number of firms maintain small or zero leverage despite high marginal tax rates; and many others maintain high leverage despite little tax benefit.
Keywords: leverage, capital structure, anomaly, risk, tradeoff, corporate finance
Trading rules
- Use the CRSP database for choosing stocks.
- Calculate a rolling 1-year beta for every stock relative to the MSCI US Equity Index.
- Organize stocks from lowest to highest based on their calculated beta values.
- Distribute stocks into either low beta or high beta portfolios.
- Assign weights to stocks according to their beta rankings.
- Adjust the portfolios on a monthly basis.
- Rescale both portfolios to have a beta of one at formation.
- Create a cost-neutral, beta-neutral Betting-Against-Beta strategy: take a long position in low-beta stocks and a short position in high-beta stocks.
- Explore basic tweaks (for instance, go long on the stocks in the lowest beta decile and short on those 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.