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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.

Equity Factor Risk-return Predictability

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

Which Equity Factors Are Predictable?

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

When Potential Upside Is Bigger Than Potential Downside

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

Purified Factor Portfolios

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:

  1. 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.
  2. 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

Integrated Approach to Factor Investing

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:

  1. Which characteristics are jointly significant from a portfolio perspective on an in-sample, gross basis?
  2. How does accounting for trading costs (an in-sample, net basis) affect the answer?
  3. 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

Illiquidity as a Stock Return Factor

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

Equity Market and Treasuries Variance Risk Premiums as Return Predictors

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

Suppressing Unrelated Risks from Stock Factor Portfolios

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

Mood Beta as Stock Return Predictor

Do individual stocks react differently and persistently to aggregate investor mood changes? In their December 2016 paper entitled “Mood Beta and Seasonalities in Stock Returns”, David Hirshleifer, Danling Jiang and Yuting Meng investigate whether some stocks have higher sensitivities to investor mood changes (higher mood betas) than others, thereby inducing calendar effects in the cross-section of returns. They specify mood based on three calendar-based U.S. stock market return anomalies:

  1. January (highest average excess return of all months) represents good mood, while October (lowest average excess return of all months) represents bad mood.
  2. Friday (highest average excess return of all days) represents good mood, while Monday (lowest average excess return of all days) represents bad mood.
  3. The two days before holidays (abnormally high average excess return) represent good mood, while the two days after holidays (abnormally low average excess return) represent bad mood.

They structure their investigation via a factor model of stock returns, with mood as a factor. They measure a stock’s mood beta by regressing its returns during high and low mood intervals versus contemporaneous equal-weighted market returns over a rolling historical window. Each year, they regress a stock’s monthly January and October returns versus monthly equal-weighted market returns for those months over the last 10 years. Each week, they regress a stock’s daily Friday and Monday returns versus contemporaneous equal-weighted market returns for those days over the last ten weeks. Each holiday, they regress a stocks pre-holiday and post-holiday daily returns versus versus equal-weighted market returns for those days over the last year (including the same holiday the previous year. They then use the stock’s mood betas to predict its returns during subsequent times of good and bad mood. Using daily and monthly stock returns for a broad sample of U.S. common stocks during January 1963 through December 2015, they find that: Keep Reading

Simple, Practical Test of Cross-asset Class Intrinsic Momentum

“Cross-asset Class Intrinsic Momentum” summarizes research finding that past country stock index (government bond index) returns relate positively (positively) to future country stock market index returns and negatively (positively) to future country government bond index returns. Is this finding useful for specifying a simple strategy using exchange-traded fund (ETF) proxies for the U.S. stock market and U.S. government bonds? To investigate we test the following five strategies:

  1. Buy and hold.
  2. TSMOM Long Only – Each month, hold the asset (cash) if its own 12-month past return is positive (negative).
  3. TSMOM Long or Short – Each month, hold (short) the asset if its own 12-month past return is positive (negative).
  4. XTSMOM Long Only – Each month hold stocks if 12-month past returns for stocks and government bonds are both positive, and otherwise hold cash. Each month hold bonds if 12-month past returns are negative for stocks and positive for government bonds, and otherwise hold cash. 
  5. XTSMOM L-S-N (Long, Short or Neutral) – Each month hold (short) stocks if 12-month past returns for both are positive (negative), and otherwise hold cash. Each month hold (short) bonds if 12-month past returns are negative (positive) for stocks and positive (negative) for bonds, and otherwise hold cash.

We use SPDR S&P 500 (SPY) and iShares 7-10 Year Treasury Bond (IEF) as proxies for the U.S. stock market and U.S. government bonds. We use the 3-month U.S. Treasury bill (T-bill) yield as the return on cash. We apply the five strategies separately to SPY and IEF, and to an equally weighted, monthly rebalanced combination of the two for a total of 15 scenarios. Using monthly total returns for SPY and IEF and monthly T-bill yield during July 2002 (inception of IEF) through December 2016, we find that: Keep Reading

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