# 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.

**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

**September 6, 2017** - Big Ideas, Equity Premium

Are more data, higher levels of signal statistical significance and more sophisticated prediction models better for financial forecasting? In their August 2017 paper entitled “Practical Significance of Statistical Significance”, Ben Jacobsen, Alexander Molchanov and Cherry Zhang perform sensitivity testing of forecasting practices along three dimensions: (1) length of lookback interval (1 to 300 years); (2) required level of statistical significance for signals (1%, 5%, 10%…); and, (3) different signal detection methods that rely on difference from an historical average. They focus on predicting whether returns for specific calendar months will be higher or lower than the market, either excluding or including January. Using monthly UK stock market returns since 1693 and U.S. stock market returns since 1792, both through 2013, *they find that:*

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**August 31, 2017** - Big Ideas, Equity Premium

How serious is the snooping bias (p-hacking) derived from brute force mining of stock trading strategy variations? In their August 2017 paper entitled “p-Hacking: Evidence from Two Million Trading Strategies”, Tarun Chordia, Amit Goyal and Alessio Saretto test a large number of hypothetical trading strategies to estimate an upper bound on the seriousness of p-hacking and to estimate the likelihood that a researcher can discover a truly abnormal trading strategy. Specifically, they:

- Collect historical data for 156 firm accounting and stock price/return variables as available for U.S. common stocks in the top 80% of NYSE market capitalizations with price over $3.
- Exhaustively construct about 2.1 million trading signals from these variables based on their levels, changes and certain combination ratios.
- Calculate three measures of trading signal effectiveness:
- Gross 6-factor alphas (controlling for market, size, book-to-market, profitability, investment and momentum) of value-weighted, annually reformed hedge portfolios that are long the value-weighted tenth, or decile, of stocks with the highest signal values and short the decile with the lowest.
- Linear regressions that test ability of the entire distribution of trading signals to explain future gross returns based on linear relationships.
- Gross Sharpe ratios of the hedge portfolios used for alpha calculations.

- Apply three multiple hypothesis testing methods that account for cross-correlations in signals and returns (family-wise error rate, false discovery rate and false discovery proportion.

They deem a signal effective if it survives both statistical hurdles (alpha t-statistic 3.79 and regression t-statistic 3.12) and has a monthly Sharpe ratio higher than that of the market (0.12). Using monthly values of the 156 specified input variables during 1972 through 2015, *they find that:*

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**August 10, 2017** - Equity Options, Equity Premium

Do aggregate positions in put and call options on individual stocks, as indicators of sentiment of informed traders, predict future market returns? In their July 2017 paper entitled “Stock Return Predictability: Consider Your Open Options”, Farhang Farazmand and Andre de Souza examine the power of average value-weighted put option open interest divided by average value-weighted call option open interest in individual U.S. stocks (PC-OI) to predict U.S. stock market returns. Specifically, they:

- Compute for each stock each day total put option open interest and total call option open interest.
- Average daily values for each stock by month and weight by market capitalization.
- Calculate PC-OI by dividing the sum of monthly capitalization-weighted average put option open interest by the sum of monthly capitalization-weighted call option open interest.
- Each month, relate via regression monthly PC-OI to stock market return the next three months to determine the sign of the future return coefficient.
- Each month, create a net signal from the sum of the signs of these coefficients from the last three monthly regressions. A positive (negative) sum indicates a long (short) position in the stock market and an offsetting short (long) position in the risk-free asset.

They further test whether PC-OI predictive power concentrates in stocks with unique informativeness as represented by high idiosyncratic volatility (individual stock return volatility unexplained via regression versus market returns). For comparison, they also test their model with S&P 500 index options. Using daily open interest for options on AMEX, NYSE and NASDAQ common stocks and on the S&P 500 Index with moneyness 0.8-1.2 and maturities 30-90 days, associated stock characteristics, and contemporaneous U.S. stock market returns during January 1996 through August 2014, *they find that:*

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**August 2, 2017** - Bonds, Equity Premium, Strategic Allocation

How differently does the “Simple Asset Class ETF Value Strategy” (SACEVS) perform when the U.S. stock market rises and falls? This strategy seeks to exploit relative valuation of the term risk premium, the credit (default) risk premium and the equity risk premium via exchange-traded funds (ETF). To investigate, because the sample period available for mutual funds is much longer than that available for ETFs, we use instead data from “SACEVS Applied to Mutual Funds”. Specifically, each month we reform a Best Value portfolio (picking the asset associated with the most undervalued premium, or cash if no premiums are undervalued) and a Weighted portfolio (weighting assets associated with all undervalued premiums according to degree of undervaluation, or cash if no premiums are undervalued) using the following four assets:

The benchmark is a monthly rebalanced portfolio of 60% stocks and 40% U.S. Treasuries (60-40 VWUSX-VFIIX). We say that stocks rise (fall) during a month when the return for VWUSX is positive (negative) during the SACEVS holding month. Using monthly risk premium estimates, SR and LR, and Best Value and Weighted returns during June 1980 through June 2017 (444 months), *we find that:*

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**August 1, 2017** - Bonds, Equity Premium, Strategic Allocation

A subscriber asked how the “Simple Asset Class ETF Value Strategy” (SACEVS) performs when interest rates rise. This strategy seeks to exploit relative valuation of the term risk premium, the credit (default) risk premium and the equity risk premium via exchange-traded funds (ETF). To investigate, because the sample period available for mutual funds is much longer than that available for ETFs, we use instead data from “SACEVS Applied to Mutual Funds”. Specifically, each month we reform a Best Value portfolio (picking the asset associated with the most undervalued premium, or cash if no premiums are undervalued) and a Weighted portfolio (weighting assets associated with all undervalued premiums according to degree of undervaluation, or cash if no premiums are undervalued) using the following four assets:

The benchmark is a monthly rebalanced portfolio of 60% stocks and 40% U.S. Treasuries (60-40 VWUSX-VFIIX). We use the T-bill yield as the short-term interest rate (SR) and the 10-year Constant Maturity U.S. Treasury note (T-note) yield as the long-term interest rate (LR). We say that each rate rises or falls when the associated average monthly yield increases or decreases during the SACEVS holding month. Using monthly risk premium estimates, SR and LR, and Best Value and Weighted returns during June 1980 through June 2017 (444 months), *we find that:*

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**July 10, 2017** - Equity Premium, Volatility Effects

Is the Capital Asset Pricing Model (CAPM), which relates the return of an asset to its non-diversifiable risk, called beta, worth learning? In his June 2017 paper (provocatively) entitled “Is It Ethical to Teach That Beta and CAPM Explain Something?”, Pablo Fernandez tackles this question. Based on the body of relevant research, *he concludes that:*

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**June 12, 2017** - Equity Premium, Strategic Allocation, Volatility Effects

Can investors predict the return of a stock from its relationship with the dispersion of returns across all stocks? In their May 2017 paper entitled “Building Efficient Portfolios Sensitive to Market Volatility”, Wei Liu, James Kolari and Jianhua Huang examine a 2-factor model which predicts the return on a stock based on its sensitivity to (1) the value-weighted stock market return (beta risk) and (2) the standard deviation of value-weighted returns for all stocks (zeta risk). They first each month estimate zeta for each stock via regressions of daily data over the past year. They then rank stocks by zeta into quantile portfolios and calculate next-month equal-weighted returns across these portfolios and various long-short combinations of these portfolios (hedge portfolios) to measure dependence of future returns on zeta. Finally, they generate performance data for aggregate zeta risk portfolios by adding value-weighted market index returns to returns for each of the long-short zeta-sorted portfolios. Using daily and monthly returns for a broad sample of U.S. stocks in the top 90% of market capitalizations for that year, monthly equity market returns and monthly U.S. Treasury bill yields as the risk-free rate during January 1965 through December 2015, *they find that:* Keep Reading

**June 7, 2017** - Calendar Effects, Equity Premium

Are there opportunities to trade S&P 500 Index additions in the current market environment? In her May 2017 paper entitled “The Diminished Effect of Index Rebalances”, Konstantina Kappou examines returns for S&P 500 Index additions before and after the 2008 financial crisis. She focuses on additions because deletions generally involve confounding information such as restructuring, bankruptcy or merger. Current index management practices are to announce changes after market hours about five days in advance (announcement date – AD) and to implement changes at the specified close (event date – ED). She investigates returns during an event window from 15 trading days before AD through 252 trading days after ED. She calculates abnormal returns as differences between returns for added stocks and contemporaneous market returns. She considers 276 index additions during January 2002 through November 2013, with October 2008 separately pre-crisis from post-crisis. She excludes 48 of the additions due to lack of data or confounding information. Using daily returns for the remaining 228 S&P 500 Index additions during the specified sample period,* she finds that:* Keep Reading

**May 30, 2017** - Equity Premium

Are equity sector and factor investing complementary? In their May 2017 paper entitled “Factors vs. Sectors in Asset Allocation: Stronger Together?”, Marie Briere and Ariane Szafarz compare efficient sector investing (diversifying economic risks) and efficient factor investing (diversifying across risk factors) for U.S. stocks, and then assess advantages of combining the two approaches. They first construct two efficient frontiers (sets of portfolios with the highest expected returns across the range of volatilities), one from 10 sectors and the other from 10 factors. Their sector set consists of long-only portfolios covering (1) non-durable consumer goods, (2) durable consumer goods, (3) manufacturing, (4) energy, (5) technology, (6) telecommunications, (7) shops, (8) health care, (9) utilities and (10) other. Their factor set consists of the long and short portfolios separately for (1) size, (2) book-to-market, (3) momentum, (4) profitability and (5) investment. They consider six scenarios consisting of three samples (full period, crisis subperiods and non-crisis subperiods) for long-only and long-short efficient portfolios. They define crises by combining NBER recession dates and Forbes Magazine bear market dates. Using monthly returns for sectors and factors as specified from Kenneth French’s data library and the broad market, along with yields for 1-month U.S. Treasury bills as the risk-free rate, during July 1963 through December 2016, *they find that:* Keep Reading