# 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.

**September 7, 2017** - Equity Premium, Momentum Investing, Size Effect, Value Premium, Volatility Effects

How many, and which, factors should investors include when constructing multi-factor smart beta portfolios? In their August 2017 paper entitled “How Many Factors? Does Adding Momentum and Volatility Improve Performance”, Mohammed Elgammal, Fatma Ahmed, David McMillan and Ali Al-Amari examine whether adding momentum and low-volatility factors enhances the Fama-French 5-factor (market, size, book-to-market, profitability, investment) model of stock returns. They consider statistical significance, economic sense and investment import. Specifically, they:

- Determine whether factor regression coefficient signs and values distinguish between several pairs of high-risk and low-risk style portfolios (assuming stock style portfolio performance differences derive from differences in firm economic risk).
- Relate time-varying factor betas across style portfolios to variation in economic and market risks as proxied by changes in U.S. industrial production and S&P 500 Index implied volatility (VIX), respectively.
- Test an out-of-sample trading rule based on extrapolation of factor betas from 5-year historical rolling windows to predict next-month return for five sets (book-to-market, profitability, investment, momentum, quality) of four style portfolios (by double-sorting with size) and picking the portfolio within a set with the highest predicted returns.

Using monthly factor return data during January 1990 through October 2016, *they find that:* Keep Reading

**April 6, 2017** - Equity Premium, Size Effect, Volatility Effects

Does disentangling measures of stock illiquidity and market capitalization (size) support belief in an illiquidity premium (a reward for holding illiquid assets)? In the December 2016 version of their paper entitled “The Value of True Liquidity”, Robin Borcherding and Michael Stein investigate this question by controlling the most widely used stock illiquidity metric for size. Specifically they define and calculate true stock liquidities by:

- Calculating for each stock the conventional Amihud monthly measure of illiquidity (average absolute price impact of dollar trading volume during a month).
- Capture unexplained residuals from a regression that controls for the linear relationship (negative correlation) between this conventional illiquidity metric and size.
- Sorting stocks by size and capturing more detail regression residuals within size ranges to control for the non-linear relationship between conventional illiquidity and size.

They then form double-sorted portfolios to compare interactions of conventional and true liquidity with stock volatility and size. Using daily returns, trading data and characteristics for 4,739 U.S. common stocks during January 1990 through September 2015, *they find that:* Keep Reading

**March 22, 2017** - Equity Premium, Momentum Investing, Size Effect, Value Premium, Volatility Effects

How diversifying are different equity factors within and across country stock markets? In his January 2016 paper entitled “The Power of Equity Factor Diversification”, Ulrich Carl analyzes diversification properties of six equity factors (market excess return, size, value, momentum, low-beta and quality) across 20 developed stock markets. He defines each factor conventionally as returns to a portfolio that is each month long (short) stocks with the highest (lowest) expected returns based on that factor. He considers: (1) cross-country correlations for each factor; (2) cross-factor correlations for each country; (3) cross-country, cross-factor correlations; (4) dynamics of cross-country correlations for each factor based on rolling 36-month windows of returns; and, (5) cross-country correlations for each factor for the 30% lowest and 30% highest market excess returns (tail events). He also applies principal component analysis as another way to evaluate how diverse the 120 country-factor return streams are. Finally, he constructs cross-factor and cross-country portfolios to assess economic value of diversification properties. Using monthly returns in U.S. dollars for the six factors in each of the 20 countries during January 1991 through April 2015, *he finds that:* Keep Reading

**March 15, 2017** - Equity Premium, Momentum Investing, Size Effect, Value Premium, Volatility Effects

Are the returns of factors widely used to predict the cross-section of stock returns themselves predictable? In the January 2016 draft of his paper entitled “Equity Factor Predictability”, Ulrich Carl analyzes predictability of market, size (market capitalization), value (book-to-market ratio), momentum (returns from 12 months ago to one month ago), low beta (betting against beta) and quality factor returns. All factor returns derive from hedge portfolios that are long (short) stocks with high (low) expected returns based on their factor values. He employs a broad range of economic and financial variables in four sets and multiple ways of testing predictability to ensure robustness of findings and limit model/data snooping bias. Predictability tests he applies include: combinations of simple forecasts (mean or median of single-variable regression forecasts); principal component analysis to distill forecasting variables into a few independent predictive factors; and, methods that adjust variable emphasis according to their respective past performances. He considers several predictability evaluation metrics, including: mean squared error compared to that of the historical average return; utility gain of timing based on predictability; and, information ratio (difference in return divided by difference in risk) relative to the historical average return. He mostly examines next-month forecasts with a one-month gap between predictive variable measurement and forecasted return over two test periods: 1975-2013 and 1950-2013. Using monthly returns for the six factors (start dates ranging from 1928 to 1958), a large set of financial variables since 1928 and a large set of economic variables since 1962, all through November 2013, *he finds that:* Keep Reading

**March 10, 2017** - Equity Premium, Momentum Investing, Size Effect, Value Premium, Volatility Effects

How attractive are purified factor portfolios, constructed to focus on one factor by avoiding exposures to other factors? In their January 2017 paper entitled “Pure Factor Portfolios and Multivariate Regression Analysis”, Roger Clarke, Harindra de Silva and Steven Thorley explore a multivariate regression approach to generating pure factor portfolios. They consider five widely studied factors: value (earnings yield); momentum (cumulative return from 12 months ago to one month ago); size (market capitalization); equity market beta; and, profitability (gross profit margin). They also consider bond beta (regression of stock returns on 10-year U.S. Treasury note returns) to examine interest rate risk. They each month reform two types of factor portfolios:

- Primary – a factor portfolio with weights that deviate simply from market weights based on analysis of just one factor, with differences from market portfolio weights scaled by market capitalization.
- Pure – a factor portfolio derived from a multiple regression that isolates each factor, ensuring that it has zero exposures to all other factors.

They measure factor portfolio performance based on: average difference in monthly returns between each factor portfolio and the market portfolio; annualized standard deviation of the underlying monthly return differences; 1-factor (market) alpha; and, information ratio (alpha divided by incremental risk to the market portfolio). Using return and factor data for the 1,000 largest U.S. stocks during 1967 through 2016, *they find that:* Keep Reading

**February 27, 2017** - Momentum Investing, Size Effect, Value Premium, Volatility Effects

Are there plenty of exchange-traded funds (ETF) offering positive or negative exposures to widely accepted factor premiums? In his February 2017 paper entitled “Are Exchange-Traded Funds Harvesting Factor Premiums?”, David Blitz analyzes the exposures of U.S. equity ETFs to market, size, value, momentum and volatility factors. Specifically, he calculates factor betas (exposures) from a multi-factor regression of monthly excess (relative to the risk-free rate) total returns for each ETF versus market, small-minus-big size (SMB), high-minus-low value (HML), winners-minus-losers momentum (WML) and low-minus-high volatility (LV-HV) factor returns during 2011 through 2015. His overall sample consists of 415 U.S. equity ETFs with least 36 months of return history as of the end of 2015. He also considers subsamples consisting of: (1) 103 smart beta ETFs that explicitly target factor premiums, including fundamentally weighted and high-dividend funds; and, (2) the remaining 312 conventional ETFs, including sector funds and funds with conflicting factor exposures. He includes lists of the 10 ETFs with the most positive and the 10 ETFs with the most negative exposures to each factor from among the 100 largest ETFs. Using monthly Assets under Management (AuM) and total returns for the specified 415 ETFs, along with the monthly risk-free rate and the selected factor premiums during January 2011 through December 2015, *he finds that:* Keep Reading

**February 3, 2017** - Equity Premium, Momentum Investing, Size Effect, Value Premium, Volatility Effects

Does suppressing unrelated risks from stock factor portfolios improve performance? In their January 2017 paper entitled “Diversify and Purify Factor Premiums in Equity Markets”, Raul Leote de Carvalho, Lu Xiao, François Soupé and Patrick Dugnolle investigate how to improve the capture of four types of stock factor premiums: value (12 measures); quality (16 measures); low-risk (two measures); and, momentum (10 measures). They standardize the different factor measurement scales based on respective medians and standard deviations, and they discard outliers. Their baseline factors portfolios employ constant leverage (CL) by each month taking 100% long (100% short) positions in stocks with factor values associated with the highest (lowest) expected returns. They strip unrelated risks from baseline portfolios by:

- SN – imposing sector neutrality by segregating stocks into 10 sectors before ranking them for assignment to long and short sides of the factor portfolio.
- CV – replacing constant leverage by each month weighting each stock in the portfolio to target a specified volatility based on its actual volatility over the past three years.
- HB – hedging the market beta of the portfolio each month based on market betas of individual stocks calculated over the past three years by taking positions in the capitalization-weighted market portfolio and cash.
- HS – hedging the size beta of the portfolio each month based on size betas of individual stocks calculated over the past three years by taking positions in the equal-weighted market portfolio and the capitalization-weighted market portfolio.

They examine effects of combining measures within factor types, combining types of factors and excluding short sides of factor portfolios. They also look at U.S., Europe and Japan separately. Their portfolio performance metric is the information ratio, annualized average return divided by annualized standard deviation of returns. Using data for stocks in the MSCI World Index since January 1997, in the S&P 500 Index since January 1990, in the STOXX Europe 600 Index since January 1992 and in the Japan Topix 500 Index since August 1993, all through November 2016, *they find that:* Keep Reading

**October 19, 2016** - Size Effect, Volatility Effects

Is the finding in “Expected Stock Market Volatility and the Size Effect” that the size effect concentrates in intervals after months of very high stock market volatility robustly evident from liquid exchange-traded funds (ETF)? To investigate, we define the size effect as the difference in returns between iShares Russell 2000 (IWM) and iShares Russell 1000 (IWB) at a monthly frequency and use the CBOE Volatility Index (VIX) as expected market volatility. To check robustness of cited research, we consider:

- Thresholds for high VIX ranging from above average to two standard deviations above average.
- Out-of-sample identification of high monthly VIX values using either inception-to-date (ITD) or rolling 120-month (Rolling120) historical windows of monthly VIX closes.
- Lags between VIX measurements and size effect returns ranging from zero to two months.

We focus on differences in average monthly IWM-IWB returns, standard deviations of IWM-IWB monthly returns and IWM-IWB monthly reward-to-risk ratio (average return divided by standard deviation of returns) for months after high versus not-high values of VIX. Using monthly levels of VIX during January 1990 (inception) through September 2016 and monthly total returns for IWM and IWB during May 2000 (inception) through September 2016, *we find that:* Keep Reading

**October 18, 2016** - Size Effect, Volatility Effects

Is the size effect (small stocks tend to outperform large stocks) related to level of market risk as indicated by expected stock market volatility? In their September 2016 paper entitled “High Risk Episodes and the Equity Size Premium”, Naresh Bansal, Robert Connolly and Chris Stivers investigate the relationship between the size effect and two measures of expected stock market volatility: (1) during 1960 through 1989, realized volatility (RV) calculated from daily stock market returns over the prior 66 trading days; and, (2) during 1990 through 2014, the CBOE Volatility Index (VIX). To measure the size effect, they focus on Fama-French SMB factor portfolio monthly returns (return of the tenth, or decile, of stocks with the smallest market capitalizations minus the return of the decile of stocks with the largest market capitalizations). They also study return differences between each of the next three smallest deciles and the return of the largest decile. They consider both value weighting and equal weighting of stock deciles. They insert a skip-month between the volatility measurement interval and size effect return measurement intervals of 1, 3, 6 or 12 months. Using the specified monthly and daily data, *they find that:* Keep Reading

**September 26, 2016** - Size Effect, Value Premium, Volatility Effects

Do investors exploiting common stock return anomalies risk extraordinarily large drawdowns during market crashes? In their May 2016 paper entitled “Can Exposure to Aggregate Tail Risk Explain Size, Book-to-Market, and Idiosyncratic Volatility Anomalies?”, Sofiane Aboura and Eser Arisoy investigate whether portfolios based on the size, book-to-market ratio and idiosyncratic volatility effects bear elevated stock market tail risk. They measure market tail risk as change in VIX Tail Hedge Index (VXTH), which hedges extreme drops in the S&P 500 Index by holding the index and one-month far out-of-the-money (30-delta) call options on the CBOE Volatility Index (VIX). They test sensitivity of size and book-to-market factors to overall risk and tail risk by adding change in VIX (market volatility risk factor) and change in VXTH (market tail risk factor) to the Fama-French three-factor (market, size, book-to-market) model of stock returns. They consider two equal subperiods, one containing the 2008 financial crisis, to check robustness of findings. Using monthly values of VIX and VXTH, factor model returns and U.S. Treasury bill yields during January 2007 through February 2016 (110 months), *they find that:* Keep Reading