Scott Murray Presentation
Betting against Beta or Demand for Lottery
Turan G. Bali1 Stephen J. Brown2
Scott Murray3 Yi Tang4
1McDonough School of Business, Georgetown University
2Stern School of Business, New York University
3College of Business Administration, University of Nebraska – Lincoln
4School of Business, Fordham University
March 30, 2015
Most Persistent Anomaly
Security Market Line is Too Flat
- High β stocks generate negative abnormal returns
- Low β stocks generate positive abnormal returns
- Anomaly has persisted for more than 40 years
- Black, Jensen, and Scholes (1972)
- Blume and Friend (1973)
- Fama and MacBeth (1973)
Betting Against Beta: Frazzini and Pedersen
- Long low-β, short high-β portfolio generates abnormal returns
- Explanation: Leverage constrained investors buy high β
- Only way to increase expected return (can’t use leverage)
- Pension funds, mutual funds
Alternative Explanation – Lottery Demand
We propose that lottery demand causes betting against beta phenomenon
- Lottery investors want high probability of large up move
- Up moves partially driven by market sensitivity
- Lottery demanders likely to invest in high-β stocks
- Upward (downward) price pressure on high-β (low-β) stocks
- Future returns of high-β (low-β) stocks depressed (increased)
Lottery demand strong in equity markets
- Bali, Cakici, and Whitelaw (2011)
- Kumar (2009)
Capital Market Line

Results
Lottery Demand Explains Phenomenon
Lottery demand proxied by MAX
- Average of top 5 daily returns in month
Bivariate portfolio analysis
- Controlling for MAX, betting against beta disappears
- No other variable explains betting against beta
Fama and MacBeth (1973) Regressions
- β positively related to returns when MAX included
Orthogonal Component of β to MAX
- Does not generate betting against beta phenomenon
Results
Lottery Demand is the Channel
Lottery demand falls predominantly on high-β stocks
- β and MAX positively correlated in cross-section
Lottery demand generates betting against beta
- Strong in high-β,MAX correlation months
- Non-existent in low-β,MAX correlation months
Concentrated in low institutional holdings stocks
- Lottery demand driven by retail investors – Kumar (2009)
- Leverage constraints by mutual and pension funds
Aggregate lottery demand
- High correlation when aggregate lottery demand high
Results
Lottery Demand Factor (FMAX)
Long High-MAX Stocks, Short Low-MAX Stocks
- Proxies for returns associated with lottery investing
FMAX explains betting against beta phenomenon
- Alpha of high-low β portfolio is zero when FMAX included
FMAX explains alpha of FP’s BAB factor
- Alpha of BAB is zero when FMAX included in model
BAB factor cannot explain FMAX
- Alpha of FMAX large and significant when BAB in model
Data Sources
CRSP
- Daily and monthly stock data
Compustat
Kenneth French’s Data Library
- Daily and monthly factor returns
Global Insight
- LIBOR and U.S. Treasury bill yields
Pastor and Stambaugh (2003) Liquidity Factor
Institutional Holdings Data
- Thomson-Reuters Institutional Holdings (13F) database
Variables – Beta, Lottery Demand, Returns
Beta, Lottery Demand, and Returns
Beta (β)
- One-factor market model regression
- 12-month’s of daily return data
- Require minimum of 200 daily return observations
Lottery demand (MAX)
- Average of 5 highest daily returns in past month
Monthly stock excess returns
- Adjusted for delisting following Shumway (1997)
Variables – Firm Characteristics
Firm Characteristics
Market Capitalization (MKTCAP)
- Size is log of MktCap (in millions)
Book-to-market ratio (BM):Fama and French (1992, 1993)
Momentum (MOM):Jegadeesh and Titman (1993)
- Return in months t – 11 through t – 1
Illiquidity (ILLIQ):Amihud (2002)
Idiosyncratic Volatility (IVOL):Ang et al. (2006)
Variables – Risk Measures
Risk Measures
Co-skewness (COSKEW): Following Harvey and Siddique (2000)
Total skewness (TSKEW): Skewness of daily returns in past year
Downside beta (DRISK): Ang, Chen, Xing (2006)
- Stock beta on days when market return is below average
Tail beta (TRISK): Kelly, Jiang (2013), Ruenzi, Weigert (2013)
- Stock beta on days in bottom 10% of market returns
We require minimum of 200 daily return observations in past year for each of the risk variables
Variables – Funding Liquidity Measures
Funding Liquidity Measures
TED spread sensitivity (βTED
- TED spread is three-month LIBOR rate – 3-month T-bill rate
Sensitivity to TED spread volatility (βVOLTED, 1979-2012)
- VOLTED is standard deviation of daily TED spreads in month
T-bill rate sensitivity (βTBILL)
- TBILL is 3-month T-bill rate
Financial sector leverage sensitivity (βFLEV )
- FLEV is financial sector total assets / market value of equity
Calculated using 5 years of monthly data (minimum 24 months)
Sample
Monthly Sample, Aug. 1963 – Dec. 2012
- 593 months
- U.S. based common stocks
- Traded on NYSE/AMEX/Nasdaq
- Price at end of previous month ≥ $5
Univariate Portfolios Sorted on β
Excess Returns and 4-Factor Alphas

High-Low β portfolio generates negative alpha
- -0.51% per month
- Similar to FP (0.55% per month)
- Both high and low β portfolios generate significant alpha
Univariate Portfolio Firm Characteristics
Average Firm Characteristics

- MAX, MKTCAP, MOM, IVOL positively related to β
- BM, ILLIQ negatively related to β
Univariate Portfolio Risk Measures
Average Risk Measures

- COSKEW, DRISK, TRISK positively related to β
- TSKEW negatively related to β
Univariate Portfolio Funding Liquidity Measures
Average Funding Liquidity Measures

- βTED and βVOLTED positively related to β
- βTBILL and βFLEV negatively related to β
Univariate Portfolios Sorted on MAX
Excess Returns and 4-Factor Alphas

High-Low MAX generates negative returns and alpha
- Average return is -1.15% per month
- FFC4 alpha -1.40% per month
- Both high and low MAX portfolios generate significant alpha
Bivariate Portfolios Procedure
Bivariate Dependent Sort Portfolio Analysis
Sort first on control variable
- Firm characteristic, risk measure, or funding liquidity measure Then sort on β
- Generates dispersion in β, holds first sort variable constant
Table reports excess return for β decile portfolios
- Average across all deciles of control variable
- Results show conditional relation between β and future returns
Bivariate Portfolios – Control for Firm Characteristics

- Controlling for MAX explains the betting against beta effect
- Other firm charactersistics fail to explain phenomenon
Bivariate Portfolios – Control for Risk

- Risk fails to explain betting against beta phenomenon
Bivariate Portfolios – Control for Funding Liquidity

- Funding liquidity sensitivity fails to explain betting against beta phenomenon
Fama-MacBeth (1973) Regressions
Regressions with and without MAX
- Specification indicated at bottom
- Full results on next slide

- MAX included ! β positively related to future stock returns
Full Fama-MacBeth (1973) Regression Results

Bivariate Independent Sort Portfolios
Sort Independently on β and MAX
- High-Low β portfolio gives returns driven by β
- High-Low MAX portfolio gives returns driven by MAX
- Results on next slide
Results
MAX explains betting against beta effect
- High-Low β portfolios have insignificant alphas
Lottery demand effect persists after controlling for β
- High-Low MAX portfolios have large and significant alphas
Bivariate Independent Sort Portfolio Returns

Univariate β⊥MAX Portfolio Excess Returns
β⊥MAX is portion of β that is orthogonal to MAX
- Run cross-sectional regression of β on MAX
- β⊥MAX is intercept plus residual

β⊥MAX unrelated to returns
- High-Low alpha of 0.05% small and insignificant
- MAX explains betting against beta phenomenon
Univariate MAX⊥β Portfolio Excess Returns
MAX⊥β is portion of MAX that is orthogonal to β
- Run cross-sectional regression of MAX on β
- MAX⊥β is intercept plus residual

MAX⊥β negatively related to returns
- High-Low alpha of -1.44% large and significant
- Similar to unconditional result (FFC4 α = -1.40%)
- β fails to explain lottery demand phenomenon
High and Low β, MAX Correlation Months
Univariate Portfolios for Months with High and Low Correlation Between β and MAX:ρβ;MAX
- Median cross-sectional correlation is 0.29
- Low correlation months: correlation < median
- High correlation months: correlation > median
- Correlation measured during portfolio formation month
- Returns from month after measured correlation
High and Low β, MAX Correlation – β Portfolios
Univariate Portfolios Sorted on β

- Betting against beta effect driven by high correlation months
- Phenomenon does not exist in low correlation months
High and Low β, MAX Correlation – MAX Portfolios

- Lottery demand effect present in both correlation regimes
- Effect not driven by relation between MAX and β
Institutional Holdings and Betting against Beta
Bivariate Portfolios Sorted on INST then β

- Betting against beta only works in low INST stocks
- Not held by mutual funds, pension funds, etc.
Institutional Holdings and Lottery Demand
Bivariate Portfolios Sorted on INST then MAX

- Lottery demand stronger in low INST stocks
- Consistent with retail phenomenon
Lottery Demand Factor
Lottery Demand Factor (FMAX)
- Sort stocks into 2 market capitalization groups
- Breakpoint is median NYSE market capitalization
- Independently sort stocks into 3 MAX groups
- Breakpoints are 30th and 70th percentiles of MAX
- Calculated using all NYSE/AMEX/Nasdaq stocks
- FMAX factor is average return of 2 high MAX portfolios
- minus average return of 2 low MAX portfolios
FMAX Factor Returns
- -0.54% average monthly returns
- 4.83% monthly return standard deviation
- Newey and West (1987) t-statistic = -2.55
Factor Analysis of High-Low β Portfolio
Factor Sensitivities Using 4 Different Factor Models
- PS is Pastor and Stambaugh (2003) liquidity factor
- Only available 1968 – 2011

Lottery demand factor explains alpha of High-Low β portfolio
β Decile Portfolio Alphas
Alphas of β Sorted Decile Portfolios

FMAX explains alpha of high-β and low-β portfolios
BAB Factor
BAB Factor
- Return of long-short beta portfolio
- Long stocks with low beta
- Short stocks with high beta
- Breakpoint is median beta
- Weights determined by distance from median
- More extreme betas have higher weight
- Positive abnormal returns using standard factor models
- Data from Lasse Pedersen’s website
- Covers August 1963 – March 2012
BAB Factor Sensitivities
Factor Analysis of BAB Factor Returns

FMAX factor explains returns of BAB factor
FMAX Factor Sensitivities
Factor Analysis of FMAX Factor Returns

FMAX factor returns not explained by BAB factor
Proxy for Risk-Factor Sensitivity?
Does MAX capture a factor sensitivity?
βFMAX
- Sensitivity to FMAX factor
- Calculated using five years of monthly data
Proxy for Risk-Factor Sensitivity?
Does MAX capture a factor sensitivity?
βFMAX
- Sensitivity to FMAX factor
- Calculated using five years of monthly data
Univariate Portfolio Analysis

Fama-MacBeth (1973) Regressions
Regressions with and without MAX
- Full results on next slide

- βFMAX has no relation with future stock returns
- β remains positively related to future stock returns
- MAX remains negatively related to future stock returns
Full Fama-MacBeth (1973) Regression Results

Characteristics of high-MAX and low-MAX stocks
Lottery stocks characterizations – Kumar (2009)
- Low prices, high idiosyncratic vol, high idiosyncratic skew
Contemporaneous Characteristics

Future Characteristics

MAX captures lottery qualities of stocks