# Size Effect

Do the stocks of small firms consistently outperform those of larger companies? If so, why, and can investors/traders exploit this tendency? These blog entries relate to the size effect.

**March 11, 2019** - Big Ideas, Equity Premium, Momentum Investing, Size Effect, Strategic Allocation, Value Premium, Volatility Effects

How should investors feel about factor/multi-factor investing? In their February 2019 paper entitled “Alice’s Adventures in Factorland: Three Blunders That Plague Factor Investing”, Robert Arnott, Campbell Harvey, Vitali Kalesnik and Juhani Linnainmaa explore three critical failures of U.S. equity factor investing:

- Returns are far short of expectations due to overfitting and/or trade crowding.
- Drawdowns far exceed expectations.
- Diversification of factors occasionally disappears when correlations soar.

They focus on 15 factors most closely followed by investors: the market factor; a set of six factors from widely used academic multi-factor models (size, value, operating profitability, investment, momentum and low beta); and, a set of eight other popular factors (idiosyncratic volatility, short-term reversal, illiquidity, accruals, cash flow-to-price, earnings-to-price, long-term reversal and net share issuance). For some analyses they employ a broader set of 46 factors. They consider both long-term (July 1963-June 2018) and short-term (July 2003-June 2018) factor performances. Using returns for the specified factors during July 1963 through June 2018, *they conclude that:*

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**February 12, 2019** - Momentum Investing, Size Effect, Value Premium

How well do mutual funds exploit theoretical (academic) equity factor premiums, and how well do investors exploit such exploitation? In their January 2019 paper entitled “Factor Investing from Concept to Implementation”, Eduard Van Gelderen, Joop Huij and Georgi Kyosev examine: (1) how performances of mutual funds that target equity factor premiums (low beta, size, value, momentum, profitability, investment) compare to that of funds that do not; and, (2) flow-adjusted performances, indicating how much of any outperformance accrues to fund investors. They classify funds empirically based on factor exposures. Using monthly returns and total assets and quarterly turnover and expense ratios for 3,109 actively managed long-only U.S. equity mutual funds with assets over $5 million (1,334 dead and 1,775 live) since January 1990 and for 4,859 (2,000 dead and 2,859 live) similarly specified global mutual funds since January 1991, all through December 2015, along with contemporaneous monthly equity factor returns, *they find that:* Keep Reading

**January 22, 2019** - Size Effect, Volatility Effects

Do unconventional portfolio construction techniques obscure how, and how well, betting against beta (BAB) works? In their November 2018 paper entitled “Betting Against Betting Against Beta”, Robert Novy-Marx and Mihail Velikov revisit the BAB factor, focusing on interpretation of three unconventional BAB construction techniques:

- Rank weighting of stocks – BAB employs rank weighting rather than equal or value weighting, with each stock in high and low estimated beta portfolios weighted proportionally to the difference between its estimated beta rank and the median rank.
- Hedging by leveraging – BAB seeks market neutrality by deleveraging (leveraging) the high (low) beta portfolio based on estimated betas rather than borrowing to buy the market portfolio to offset BAB’s short market tilt.
- Novel beta estimation – BAB measures stock betas by combining market correlations based on five years of overlapping 3-day returns with volatilities based on one year of daily returns, rather than using slope coefficients of daily stock returns versus daily market returns.

Based on mathematical analysis and empirical results using returns for a broad sample of U.S. stocks during January 1968 through December 2017, *they find that:* Keep Reading

**January 8, 2019** - Fundamental Valuation, Momentum Investing, Size Effect, Strategic Allocation, Value Premium

Does active stock factor exposure management boost overall portfolio performance? In their November 2018 paper entitled “Optimal Timing and Tilting of Equity Factors”, Hubert Dichtl, Wolfgang Drobetz, Harald Lohre, Carsten Rother and Patrick Vosskamp explore benefits for global stock portfolios of two types of active factor allocation:

- Factor timing – exploit factor premium time series predictability based on economic indicators and factor-specific technical indicators.
- Factor tilting – exploit cross-sectional (relative) attractiveness of factor premiums.

They consider 20 factors spanning value, momentum, quality and size. For each factor each month, they reform a hedge portfolio that is long (short) the equal-weighted fifth, or quintile, of stocks with the highest (lowest) expected returns for that factor. For implementation of factor timing, they consider: 14 economic indicators standardized by subtracting respective past averages and dividing by standard deviations; and, 16 technical indicators related to time series momentum, moving averages and volatilities. They suppress redundancy and noise in these indicators via principal component analysis separately for economic and technical groups, focusing on the first principal component of each group. They translate any predictive power embedded in principal components into optimal factor portfolio weights using augmented mean-variance optimization. For implementation of factor tilting, they overweight (underweight) factors that are relatively attractive (unattractive) based on valuations of factor top and bottom quintile stocks, top-bottom quintile factor variable spreads, prior-month factor returns (momentum) and volatilities of past monthly factor returns. Their benchmark portfolio is the equal-weighted combination of all factor hedge portfolios. For all portfolios, they assume: monthly portfolio reformation costs of 0.75% (1.15%) of turnover value for the long (short) side; and, annual 0.96% cost for an equity swap to ensure a balanced portfolio of factor portfolios. For monthly factor timing and tilting portfolios only, they assume an additional cost of 0.20% of associated turnover. Using monthly data for a broad sample of global stocks from major equity indexes and for specified economic indicators during January 1997 through December 2016 (4,500 stocks at the beginning and 5,000 stocks at the end), *they find that:* Keep Reading

**November 26, 2018** - Bonds, Calendar Effects, Equity Premium, Momentum Investing, Size Effect, Strategic Allocation, Value Premium

Is the U.S. equity turn-of-the-month (TOTM) effect exploitable as a diversifier of other assets? In their October 2018 paper entitled “A Seasonality Factor in Asset Allocation”, Frank McGroarty, Emmanouil Platanakis, Athanasios Sakkas and Andrew Urquhart test U.S. asset allocation strategies that include a TOTM portfolio as an asset. The TOTM portfolio buys each stock at the open on the last trading day of each month and sells at the close on the third trading day of the following month, earning zero return the rest of the time. They consider four asset universes with and without the TOTM portfolio:

- A conventional stocks-bonds mix.
- The equity market portfolio.
- The equity market portfolio, a small size portfolio and a value portfolio.
- The equity market portfolio, a small size portfolio, a value portfolio and a momentum winners portfolio.

They consider six sophisticated asset allocation methods:

- Mean-variance optimization.
- Optimization with higher moments and Constant Relative Risk Aversion.
- Bayes-Stein shrinkage of estimated returns.
- Bayesian diffuse-prior.
- Black-Litterman.
- A combination of allocation methods.

They consider three risk aversion settings and either a 60-month or a 120-month lookback interval for input parameter measurement. To assess exploitability, they set trading frictions at 0.50% of traded value for equities and 0.17% for bonds. Using monthly data as specified above during July 1961 through December 2015, *they find that:*

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**November 20, 2018** - Economic Indicators, Fundamental Valuation, Momentum Investing, Size Effect, Technical Trading, Value Premium

Which economic and market variables are most effective in predicting U.S. stock market returns? In his October 2018 paper entitled “Forecasting US Stock Returns”, David McMillan tests 10-year rolling and recursive (inception-to-date) one-quarter-ahead forecasts of S&P 500 Index capital gains and total returns using 18 economic and market variables, as follows: dividend-price ratio; price-earnings ratio; cyclically adjusted price-earnings ratio; payout ratio; Fed model; size premium; value premium; momentum premium; quarterly change in GDP, consumption, investment and CPI; 10-year Treasury note yield minus 3-month Treasury bill yield (term structure); Tobin’s q-ratio; purchasing managers index (PMI); equity allocation; federal government consumption and investment; and, a short moving average. He tests individual variables, four multivariate combinations and and six equal-weighted combinations of individual variable forecasts. He employs both conventional linear statistics and non-linear economic measures of accuracy based on sign and magnitude of forecast errors. He uses the historical mean return as a forecast benchmark. Using quarterly S&P 500 Index returns and data for the above-listed variables during January 1960 through February 2017, *he finds that:* Keep Reading

**November 2, 2018** - Calendar Effects, Size Effect

Is there a reliable and material market capitalization (size) effect among U.S. stocks around the turn-of-the-year (TOTY)? To check, we track cumulative returns from 20 trading days before through 20 trading days after the end of the calendar year for the Russell 2000 Index, the S&P 500 Index and the Dow Jones Industrial Average (DJIA) since the inception of the Russell 2000 Index. We also look at full-month December and January returns for these indexes. Using daily and monthly levels of all three indexes from December 1987 through January 2018 (31 December and 31 January observations), *we find that:* Keep Reading

**October 18, 2018** - Size Effect

Why do simple stock portfolios such as equal weighting and random weighting beat market capitalization weighting over the long run? In their June 2018 paper entitled “Diversification, Volatility, and Surprising Alpha”, Adrian Banner, Robert Fernholz, Vassilios Papathanakos, Johannes Ruf and David Schofield tackle this question by decomposing expected stock portfolio log-return into average growth rate and excess growth rate (EGR). They focus on average log-return because, unlike arithmetic and geometric averages, it is an unbiased estimator of long-term performance. They apply two formulas derived in prior work to estimate portfolio log-returns:

- Expected portfolio log-return = weighted average stock log-return + EGR
- EGR = (weighted average stock return variance – portfolio return variance)/2

They apply these formulas to the following five portfolios, each consisting of monthly overlapping sub-portfolios formed from the 1,000 U.S. stocks with the (each day) largest market capitalizations and rebalanced annually with stock weights normalized to a sum of one:

- Capitalization-weighted (CW) – stock weights are proportional to their respective market capitalizations.
- Equal-weighted (EW) – weight of each stock is 1/1000.
- Large-overweighted (LO) – stock weights are proportional to the square of their respective market capitalizations.
- Random-weighted (RW) – stock weights are proportional to random values between zero and one (median of 1,000 trials).
- Inverse random-weighted (IRW) – stock weights are proportional to the reciprocals of random values between zero and one (median of 1,000 trials).

EGR quantifies the extent to which portfolio volatility is less than constituent stock volatilities and is always positive for long-only portfolios. Higher constituent stock volatilities generate higher portfolio EGRs. Using daily prices for the 1,000 U.S. stocks with the largest market capitalizations each day during 1964 through 2012 (5,384 distinct stocks over 49 years), *they find that:*

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**October 15, 2018** - Equity Premium, Momentum Investing, Sentiment Indicators, Size Effect, Value Premium, Volatility Effects

Quantitative investing involves disciplined rule-based approaches to help investors structure optimal portfolios that balance return and risk. How has such investing evolved? In their June 2018 paper entitled “The Current State of Quantitative Equity Investing”, Ying Becker and Marc Reinganum summarize key developments in the history of quantitative equity investing. Based on the body of research, *they conclude that:* Keep Reading

**July 20, 2018** - Equity Premium, Momentum Investing, Size Effect, Value Premium

The many factor-based indexes and exchange-traded funds (ETFs) that track them now available enable investors to construct multi-factor portfolios piecemeal. Is such piecemeal construction suboptimal? In their July 2018 paper entitled “The Characteristics of Factor Investing”, David Blitz and Milan Vidojevic apply a multi-factor expected return linear regression model to explore behaviors of long-only factor portfolios. They consider six factors: value-weighted market, size, book-to-market ratio, momentum, operating profitability and investment(change in assets). Their model generates expected returns for each stock each month, and further aggregates individual stock expectations into factor-portfolio expectations holding all other factors constant. They use the model to assess performance differences between a group of long-only single-factor portfolios and an integrated multi-factor portfolio of stocks based on combined rankings across factors. The focus on gross monthly excess (relative to the 10-year U.S. Treasury note yield) returns as a performance metric. Using data for a broad sample of U.S. common stocks among the top 80% of NYSE market capitalizations and priced at least $1 during June 1963 through December 2017, *they find that:* Keep Reading