Original paper

SSRN-id2080404.pdf179.3KB

Abstract

We start this paper by providing a detailed study of how the mean monthly return on the Small-Minus-Big (SMB) Fama-French factor is affected by the January effect and the stock market return during the preceding month and preceding calendar year. We then proceed to building a predictive model for the monthly SMB factor return that incorporates the January effect and the dependence on both the market return during the preceding month and preceding calendar year. Our findings suggest that a positive small stock premium appears mainly during the years following the years with a negative return on the market as the result of a delayed and stronger reaction of small stocks to good news and a stronger January effect. We also argue that the January effect constitutes a much lesser part of the size effect than it was previously supposed.

Keywords: size effect, size premium, January effect, stock return predictability

Trading rules

Construct a portfolio focused on the Small-minus-Big factor using the 6 Fama-French portfolios, which are value-weighted based on size and the book-to-market ratio.
Form the portfolios monthly as intersections of 2 portfolios formed on size and 3 portfolios formed on the ratio of book equity to market equity.
Define SMB as the mean return of the trio of smaller portfolios minus the mean return of the trio of larger portfolios.
In practical scenarios, the portfolio can be built by simultaneously investing in a set of small-cap stocks and betting against a set of large-cap stocks. Alternatively, use a blend of two ETFs (representing small and large caps) or futures contracts, like the S&P 500 and Russell 2000.
Use the past 12-month return as a trading indicator.
Hold an SMB factor portfolio only if the past 12-month return were negative.
Rebalance the portfolio monthly.

Python code

Backtrader

import backtrader as bt

# Create a Small-minus-Big factor portfolio using 6 Fama-French value-weighted portfolios formed on size and book-to-market
class FamaFrench(bt.Strategy):
    params = (
        ("size_dict", {"S": ["S1", "S2", "S3"], "B": ["B1", "B2", "B3"]}),
        ("beme_dict", {"L": ["L1", "L2", "L3"], "H": ["H1", "H2", "H3"]}),
    )

    def __init__(self):
        self.size_dict = self.p.size_dict
        self.beme_dict = self.p.beme_dict
        self.portfolio = {}
        self.weights = {}
        for size, size_portfolios in self.size_dict.items():
            for beme, beme_portfolios in self.beme_dict.items():
                for portfolio in size_portfolios:
                    if portfolio in beme_portfolios:
                        self.portfolio[portfolio] = bt.Security(self.data[portfolio])
                        self.weights[portfolio] = 0    # Form the portfolios monthly as intersections of 2 portfolios formed on size and 3 portfolios formed on the ratio of book equity to market equity

    def next(self):
        if self.datas[0].datetime.date().month == self.datas[0].datetime.date().replace(day=1).month:
            for size, size_portfolios in self.size_dict.items():
                for beme, beme_portfolios in self.beme_dict.items():
                    portfolios_intersection = list(set(size_portfolios) & set(beme_portfolios))
                    if len(portfolios_intersection) == 2:
                        big_portfolio, small_portfolio = sorted(portfolios_intersection)[::-1]
                        long = self.portfolio[small_portfolio]                        short = self.portfolio[big_portfolio]
                        long_exposure = self.weights[small_portfolio]
                        short_exposure = self.weights[big_portfolio]
                        # SMB is defined as the average return on the three small portfolios minus the average return on the three big portfolios
                        long_return = long[-1] / long[-2] - 1  # Daily return
                        short_return = short[-1] / short[-2] - 1  # Daily return
                        long_weight = long_exposure + long_return / 3
                        short_weight = short_exposure + short_return / 3
                        self.weights[small_portfolio] = long_weight
                        self.weights[big_portfolio] = short_weight
            total_exposure = sum([abs(weight) for weight in self.weights.values()])
            for portfolio in self.portfolio.values():
                portfolio_weight = self.weights[portfolio._name]
                portfolio_exposure = portfolio_weight / total_exposure if total_exposure != 0 else 0
                portfolio.set_exposure(portfolio_exposure)

    # Use the past 12-month return as a trading indicator
    def compute_return(self, portfolio):
        annual_return = portfolio[-1] / portfolio.buysell[0][0] ** (365.25 / len(portfolio)) - 1  # Annualized return
        return annual_return

    # Hold an SMB factor portfolio only if the past 12-month return were negative
    def should_hold(self):
        for size, size_portfolios in self.size_dict.items():
            for beme, beme_portfolios in self.beme_dict.items():
                portfolios_intersection = list(set(size_portfolios) & set(beme_portfolios))
                if len(portfolios_intersection) == 2:
                    big_portfolio, small_portfolio = sorted(portfolios_intersection)[::-1]
                    long = self.portfolio[small_portfolio]
                    short = self.portfolio[big_portfolio]
                    long_return = self.compute_return(long)
                    # Hold an SMB factor portfolio only if the past 12-month return were negative
                    if long_return >= 0:
                        self.weights[small_portfolio] = 0
                        self.weights[big_portfolio] = 0

    # Rebalance the portfolio monthly
    def rebalance_portfolio(self):
        for portfolio in self.portfolio.values():
            if portfolio.buysell:
                if (
                    self.datas[0].datetime.date().month
                    == self.datas[0].datetime.date().replace(day=1).month
                ):
                    if self.should_hold():
                        portfolio.sell()
                    else:
                        portfolio.buy()

cerebro = bt.Cerebro()
for ticker in TICKERS:
    data = bt.feeds.YahooFinanceData(
        dataname=ticker,
        fromdate=FROM_DATE,
        todate=TO_DATE,
        reverse=False
    )
    cerebro.adddata(data)cerebro.addstrategy(FamaFrench,
                    size_dict={"S": ["S1", "S2", "S3"], "B": ["B1", "B2", "B3"]},
                    beme_dict={"L": ["L1", "L2", "L3"], "H": ["H1", "H2", "H3"]})
cerebro.run()

Predictable Dynamics in the Small Stock Premium

Original paper

Abstract

Trading rules

Python code

Backtrader