# 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.

**January 26, 2017** - Animal Spirits, Calendar Effects, Equity Premium

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:

- January (highest average excess return of all months) represents good mood, while October (lowest average excess return of all months) represents bad mood.
- Friday (highest average excess return of all days) represents good mood, while Monday (lowest average excess return of all days) represents bad mood.
- 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

**January 25, 2017** - Bonds, Equity Premium, Momentum Investing, Strategic Allocation

“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:

- Buy and hold.
- TSMOM Long Only – Each month, hold the asset (cash) if its own 12-month past return is positive (negative).
- TSMOM Long or Short – Each month, hold (short) the asset if its own 12-month past return is positive (negative).
- 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.
- 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

**January 24, 2017** - Bonds, Equity Premium, Momentum Investing, Strategic Allocation

Are stock and bond markets mutually reinforcing with respect to time series (intrinsic or absolute return) momentum? In their December 2016 paper entitled “Cross-Asset Signals and Time Series Momentum”, Aleksi Pitkajarvi, Matti Suominen and Lauri Vaittinen examine a strategy that times each of country stock and government bond (constant 5-year maturity) indexes based on past returns for both. Specifically:

- For stocks, they each month take a long (short) position in a country stock index if past returns for both the country stock and government bond indexes are positive (negative). If past stock and bond index returns have different signs, they take no position.
- For bonds, they each month take a long (short) position in a country government bond index if past return for bonds is positive (negative) and past return for stocks is negative (positive). If past stock and bond index returns have the same sign, they take no position.

They call this strategy cross-asset time series momentum (XTSMOM). For initial strategy tests, they consider past return measurement (lookback) and holding intervals of 1, 3, 6, 9, 12, 24, 36 or 48 months. For holding intervals longer than one month, they average monthly returns for overlapping positions. For most analyses, they focus on lookback interval 12 months and holding interval 1 month. For a given lookback and holding interval combination, they form a diversified XTSMOM portfolio by averaging all positions for all countries. They measure excess returns relative to one-month U.S. Treasury bills. They employ the MSCI World Index and the Barclays Capital Aggregate Bond Index as benchmarks. Using monthly stock and government bond total return indexes for 20 developed countries as available during January 1980 through December 2015, *they find that:* Keep Reading

**January 12, 2017** - Big Ideas, Equity Premium

How many factors are optimal for modeling future returns of individual stocks? How do these factors relate to conventionally used factors (market, size, value, momentum, investment, profitability…)? In the June 2016 version of their paper entitled “Multifactor Models and the APT: Evidence from a Broad Cross-Section of Stock Returns”, Ilan Cooper, Paulo Maio and Dennis Philip derive mathematically an optimal set of factors for predicting returns of 278 stock portfolios created by sorting U.S. stocks into tenths (deciles) according to 28 market anomalies encompassing aspects of value, momentum, investment, profitability and intangibles. They apply asymptotic principal components analysis to these portfolios to identify the factors. They quantify the premium of each of these factors as the average return spread between extreme deciles of monthly sorts of the 278 source portfolios on the factor. They then examine interactions between this mathematical factor set and several widely used empirical multi-factor models: the Fama-French 3-factor model (market, size, book-to-market); a 4-factor model (adding momentum to the 3-factor model); a second 4-factor model (adding liquidity to the 3-factor-model); a third 4-factor model (market, size, investment, profitability); and, a 5-factor model (adding investment and profitability to the 3-factor model). Using monthly returns for the 278 source stock portfolios during January 1972 through December 2013, *they find that:* Keep Reading

**January 10, 2017** - Equity Premium, Strategic Allocation

Do any equity asset allocation strategies convincingly outperform equal weighting (1/N) after accounting for data snooping bias and portfolio maintenance frictions? In their December 2016 paper entitled “Asset Allocation Strategies, the 1/N Rule, and Data Snooping”, Po-Hsuan Hsu, Qiheng Han, Wensheng Wu and Zhiguang Cao apply tests based on White’s Reality Check to compare out-of-sample performances of 23 basic allocation strategies and 5,490 combinations of these strategies to that of equal weighting (1/N) after accounting for snooping bias and portfolio frictions. The 23 basic strategies encompass: conventional mean-variance optimization; mean-optimization with parameter shrinkage (to avoid extreme allocations); the capital asset pricing (1-factor) model (CAPM); the Fama-french 3-factor model (market, size, book-to-market); the related 4-factor model (adding momentum); CAPM augmented with a cross-sectional volatility factor; a missing factor extension of CAPM; minimum variance; maximum diversification; equal risk contribution; volatility timing; and, reward-to-risk timing. Strategy combinations use two or three of the basic strategies with weights varied in increments of 10%. They apply these strategies to each of seven sets of equity assets: (1) 25 size and book-to-market sorted U.S. stock portfolios; (2) 49 industry U.S. stock portfolios; (3) the stocks in the Dow Jones Industrial Average; (4) 22 developed country stock indexes; (5) the combination of (1) and (2); (6) 93 long-lived stocks from the S&P 500 Index; and, (7) 100 size and book-to-market sorted U.S. stock portfolios. Specifically, they each month estimate model parameters and asset weights in each dataset based on the most recent 60 months, and then calculate respective strategy performances the next month. They set one-way trading frictions for all assets at either 0.05% or 0.50% to estimate net returns. They focus on associated Sharpe ratios and certainty equivalent returns (CEQ) as strategy performance metrics. Using the specified monthly data mostly since July 1969 (but since July 1990 for developed country markets and since July 1996 for S&P 500 Index stocks) through December 2014, *they find that:* Keep Reading

**December 29, 2016** - Big Ideas, Equity Premium, Volatility Effects

How should investors balance expected return and expected risk in allocating between risky and risk-free assets? In their short December 2016 paper entitled “Optimal Trade Sizing in a Game with Favourable Odds: The Stock Market”, Victor Haghani and Andrew Morton apply a simple rule of thumb related to mean-variance optimization to estimate the optimal allocation to risky assets. They also note several implications of this rule. Based on assumptions about investor motivation and straightforward mathematics, *they conclude that:* Keep Reading

**December 8, 2016** - Equity Premium

Does trading in exchange-traded funds (ETF) by authorized participants (who may create and redeem ETF shares by exchanging underlying assets) predict associated ETF returns? In their November 2016 draft paper entitled “ETF Arbitrage and Return Predictability”, David Brown, Shaun Davies and Matthew Ringgenberg examine the relationship between ETF share creation/redemption and ETF returns. For their principal analysis, they each week or each month rank ETFs into fifths (quintiles) based on change in shares outstanding and then calculate future returns by value-weighted or equal-weighted quintile. Using daily prices, share creation/redemption data, net asset values, volumes, bid-ask spreads, underlying asset characteristics and fund characteristics for approximately 1,200 ETFs, along with contemporaneous equity factor model returns, during January 2007 through December 2015, *they find that:* Keep Reading

**December 6, 2016** - Equity Premium

Is there a unique stock risk factor associated with expectations of a bear market? In the November 2016 version of their paper entitled “Bear Beta”, Zhongjin Lu and Scott Murray relate a put option-based indicator of the risk that the U.S. equity market will enter a bear state to individual stock returns. This indicator is based on two near-term out-of-the-money S&P 500 Index put options: a short position in a put option with strike price 1.5 standard deviations (based on S&P 500 implied volatility, VIX) below a zero excess (relative to the risk-free rate) index return; and, a long position in a put option 1.0 standard deviation below a zero excess index return. Using S&P 500 Index option prices, S&P 500 Index levels, VIX levels, risk-free rates, returns for a broad sample of U.S. stocks and various factor returns during January 1996 through August 2015, *they find that:* Keep Reading

**November 25, 2016** - Equity Premium

Is the recent Fama-French augmentation of their classic three-factor (market, size, book-to-market) model of stock returns with profitability and investment factors a major advance? In their November 2016 paper entitled “Five Concerns with the Five-Factor Model”, David Blitz, Matthias Hanauer, Milan Vidojevic and Pim van Vliet identify five concerns regarding the five-factor model. Based on empirical and theoretical (rationale) grounds, *they note that:* Keep Reading

**November 16, 2016** - Commodity Futures, Equity Premium

Does the term structure of crude oil futures predict stock market returns? In their October 2016 paper entitled “Do Oil Futures Prices Predict Stock Returns?”, I-Hsuan Chiang and Keener Hughen examine the ability of crude oil futures prices to predict U.S. stock market returns. They identify the first three principal components of the nearest six oil futures prices. After finding that one of these components (related to the term structure) predicts stock market returns, they define a simple oil futures term structure curvature factor as:

- Short-term slope (natural logarithm of the second nearest price minus natural logarithm of the nearest price), minus
- Long-term slope (natural logarithm of the sixth nearest price minus natural logarithm of the third nearest price).

They test the ability of this curvature factor to predict U.S. stock market performance and industry performance in-sample (based on returns) and out-of-sample (based on R-squared explanatory power) at a one-month horizon. They compare its out-of-sample predictive power with those of nine other widely used predictors: dividend-price ratio, dividend yield, earnings-price ratio, book-to-market ratio, long-term U.S. Treasuries yield, long-term U.S. Treasuries return, U.S. Treasuries yield spread, U.S. Treasury bills yield and default yield spread. Using daily prices for the six nearest WTI light crude oil futures contracts and monthly returns for the broad U.S. stock market, 49 value-weighted industries and stocks in four crude oil subsectors during March 1983 through December 2014, *they find that:* Keep Reading