# Equity Premium

Governments are largely insulated from market forces. Companies are not. Investments in stocks therefore carry substantial risk in comparison with holdings of government bonds, notes or bills. The marketplace presumably rewards risk with extra return. How much of a return premium should investors in equities expect? These blog entries examine the equity risk premium as a return benchmark for equity investors.

**March 23, 2017** - Equity Premium

Are the profitability (Robust Minus Weak, RMW) and investment (Conservative Minus Aggressive, CMA) stock factor premiums observed post-1963 robust in earlier data? In the February 2017 version of his paper entitled “The Profitability and Investment Premium: Pre-1963 Evidence”, Sunil Wahal measures the profitability (revenue minus cost of goods sold, scaled by total assets) and investment (change in total assets) factor premiums using data hand-collected from Moody’s Manual that predates the sample commencing July 1963 used to discover these factors. He defines the premiums conventionally as returns to a portfolio that is each month long (short) stocks with relatively high (low) expected returns. Because of missing older data, inconsistent accounting methods and accounting data availability lag (at least six months between fiscal year end and returns), he starts his accounting data in 1938 and return data in July 1940. Using available firm accounting and monthly return data as specified through June 1963, *he finds 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 20, 2017** - Equity Premium

Do very long samples clarify the role of aggregate dividend yield as a stock market return predictor? In their January 2017 paper entitled “Four Centuries of Return Predictability”, Benjamin Golez and Peter Koudijs examine whether aggregate dividend yield (dividend-to-price ratio) predicts stock market return in a pieced sample spanning four centuries. They test predictability in the overall sample and the pieces separately, mostly based on real (inflation-adjusted) data. Using annual stock market price and dividend data and contemporaneous estimates of inflation and the risk-free rate for the Netherlands/UK during 1629-1812, for the UK alone during 1813-1870 and for the U.S. during 1871-2015, *they find that:* Keep Reading

**March 16, 2017** - Equity Premium

Do factors widely used to model cross-sectional returns of U.S. stocks exhibit reward-for-risk behaviors? In other words, are expected factor returns higher (lower) when factor return volatility is high (low)? In their January 2017 paper entitled “The Risk-Return Tradeoff Among Equity Factors”, Pedro Barroso and Paulo Maio examine reward-for-risk behaviors of the size (small minus big market capitalizations), value (high minus low book-to-market ratios), momentum (winners minus losers from 12 months ago to one month ago), profitability (robust minus weak gross profit) and investment (conservative minus aggressive) risk factors. They compute risk as realized variance based on the last 21 daily factor returns and predict factor returns via regressions that relate monthly returns to respective factor variances. They test out-of-sample predictive power by comparing forecast errors from inception-to-date regressions (minimum 10 years of data) to those of the historical average. They test out-of-sample economic significance of findings via a strategy that each month holds a broad stock market index plus a 150% or 200% long (short) position in a factor portfolio when the regression-predicted factor risk premium is positive (negative). They compare the performance of this strategy to buying and holding the broad market index. They also consider a long-only factor exposure strategy. Finally, they perform an ancillary test of the ability of realized factor variances to predict the equity risk premium. Using daily and monthly factor returns for the U.S. equity market during January 1964 through December 2015, *they find 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 14, 2017** - Equity Premium

How does the fact that many investments can go up more than they can go down interact with their optimal allocations? In their February 2017 paper entitled “What Our Market Return Forecasts Really Mean: Convexity in Equity Returns and its Implications for Investment Sizing”, Victor Haghani and James White examine how return convexity, return forecast and investment sizing tie together. They measure equity return convexity as arithmetic mean of returns over multiple intervals during a sample period minus geometric mean (compound annual growth rate) over the same sample period. They approximate its value generally as one-half the variance (square of standard deviation) of returns measured across the same multiple intervals. In the context of results from an early 2017 survey of 118 experienced finance professionals about expected U.S. stock market performance, they then consider implications of convexity return when applying the Merton rule for optimal allocation sizing (Sharpe ratio divided by the product of investor risk aversion and standard deviation of returns). Based on mathematical formulas and approximations, *they conclude 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

**March 8, 2017** - Equity Premium

Which stock market factors and stock characteristics contribute significantly to portfolio performance when considered jointly (accounting for interactions) on a net basis (accounting for offsetting trades)? In their February 2017 paper entitled “A Portfolio Perspective on the Multitude of Firm Characteristics”, Victor DeMiguel, Alberto Martin-Utrera, Francisco Nogales and Raman Uppal investigate which of 51 stock factors/characteristics matter on a net basis when considered jointly rather than individually. They focus on three research questions:

- Which characteristics are jointly significant from a portfolio perspective on an in-sample, gross basis?
- How does accounting for trading costs (an in-sample, net basis) affect the answer?
- Can investors exploit net findings out-of-sample?

They first construct single-characteristic hedge portfolios that are long (short) stocks with expected returns above (below) respective cross-sectional averages. They then construct a parametric multi-characteristic portfolio by adding to a benchmark portfolio the linear combination of single-characteristic hedge portfolios that maximizes mean-variance utility. They next determine which single-characteristic portfolio linear coefficients (the parameters) are significantly different from zero and decompose the contribution of each into gross return, risk and trading friction components. They measure the in-sample performance of a portfolio that exploits those characteristics with significant coefficients on a net basis. Finally, they perform an out-of-sample “big-data” strategy test that each month employs a rolling window of the last 100 months to specify the coefficients of the 51 long-short characteristic portfolios and holds the specified multi-characteristic portfolio the next month. They estimate proportional trading frictions for each stock as a function that decreases with each of two variables, market capitalization of the stock and time. Using monthly return and characteristics data for a broad sample of U.S. stocks (an average of about 3,000 per month) during January 1980 through December 2014, *they find that:* Keep Reading

**February 16, 2017** - Equity Premium

Does the original 1963-1997 study identifying (Amihud) illiquidity as a stock pricing factor hold in recent data? In their December 2016 paper entitled “Illiquidity and Stock Returns: Cross-Section and Time-Series Effects: A Replication”, Lawrence Harris and Andrea Amato replicate the original research and extend it to 1998-2015 data. As in the prior study, they: (1) each month measure Amihud illiquidity as the annual average ratio of a stock’s daily absolute return to its daily dollar volume; (2) use monthly regressions to relate stock illiquidity to next-month stock returns and other stock/firm characteristics; (3) quantify how next-month and next-year excess equally weighted stock market return varies with average expected (explained by autoregression) and unexpected (not explained by autoregression) stock illiquidity; and (4) compare the explanatory power of Amihud illiquidity to that of other simple illiquidity measures based on the same absolute returns and dollar volumes. Calculations exclude stocks with extreme (top and bottom 1%) illiquidities as unreliable. Using daily return and trading volume data and contemporaneous monthly characteristics for a broad sample of U.S. stocks during 1963 through 2015, *they find that:* Keep Reading

**February 8, 2017** - Bonds, Equity Premium, Volatility Effects

Do bonds, like equity markets, offer a variance risk premium (VRP)? If so, does the bond VRP predict bond returns? In their January 2017 paper entitled “Variance Risk Premia on Stocks and Bonds”, Philippe Mueller, Petar Sabtchevsky, Andrea Vedolin and Paul Whelan examine and compare the equity VRP (EVRP) via the S&P 500 Index and U.S. Treasuries VRP (TVRP) via 5-year, 10-year and 30-year U.S. Treasuries. They specify VRP generally as the difference between the variance indicated by past values of variance (realized) and that indicated by current option prices (implied). Their VRP calculation involves:

- To forecast daily realized variances at a one-month horizon, they first calculate high-frequency returns from intra-day price data of rolling futures series for each of 5-year, 10-year and 30-year Treasury notes and bonds and for the S&P 500 Index. They then apply a fairly complex regression model that manipulates squared inception-to-date returns (at least one year) and accounts for the effect of return jumps.
- To calculate daily implied variances for Treasuries at a one-month horizon, they employ end-of-day prices on cross-sections of options on Treasury futures. For the S&P 500 Index, they use the square of VIX.
- To calculate daily EVRP and TVRPs with one-month horizons, subtract respective implied variances from forecasted realized variances.

When relating VRPs to future returns for both Treasuries and the S&P 500 Index, they calculate returns from fully collateralized futures positions. Using the specified futures, index and options data during July 1990 through December 2014, *they find that:* Keep Reading