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In the real world, some investors are prohibited from using leverage, and other investors’ leverage is limited by margin requirements. Therefore, they over-weigh risky stocks instead of using leverage, which makes these stocks more expensive. This behavior suggests that high-beta (risky) stocks should deliver lower risk-adjusted returns than low-beta stocks. Investors not limited in leverage (arbitrageurs) could exploit this inefficiency by “betting against beta”, i.e., by going long on a portfolio of low-beta stocks, leveraged to a beta of 1, and short on a portfolio of high-beta stocks, de-leveraged to a beta of 1.

This long-short portfolio delivers substantial risk-adjusted returns and works well for the US and Global equities, but nevertheless, caution is needed in implementing this strategy (costs, slippage, etc.) as research also suggests that this effect is the strongest in small-cap stocks. Research also shows that the effect isn‘t limited to stocks but also works well in other asset classes (even between asset classes).

Fundamental reason

The reason for the anomaly functionality was already stated in the short description – a lot of the investors are prohibited from using leverage and their only way to achieve higher returns is to buy more risky stocks, which is the main cause for their overvaluation. Investors not facing these constraints could earn above-average returns by exploiting this phenomenon.

Other papers

**Berrada, Messikh, Oderda, Pictet: Beta-Arbitrage Strategies: When Do They Work, and Why?**

**- Abstract**

Contrary to what traditional asset pricing would imply, a strategy that bets against beta, i.e. long in low beta stocks and short in high beta stocks, tends to out-perform the market. This puzzling empirical fact can be explained through the concept of relative arbitrage. Considering a market in which diversity is maintained, i.e. no single stock can dominate the entire market, we show that beta-arbitrage strategies out-perform the market portfolio with unit probability in finite time. We use the theoretical decomposition of beta-arbitrage excess return to provide empirical support to our explanation on equity country indices, equity sectors and individual stocks. Finally we show how to construct optimal beta-arbitrage strategies that maximize the expected return relative to a given benchmark.

**Frazzini, Kabiller, Pedersen: Buffett’s Alpha**

**- Abstract**

http://www.econ.yale.edu/~af227/pdf/Buffett%27s%20Alpha%20-%20Frazzini,%20Kabiller%20and%20Pedersen.pdf

Abstract:

Berkshire Hathaway has a higher Sharpe ratio than any stock or mutual fund with a history of more than 30 years and Berkshire has a significant alpha to traditional risk factors. However, we find that the alpha become statistically insignificant when controlling for exposures to Betting-Against-Beta and quality factors. We estimate that Berkshire’s average leverage is about 1.6-to-1 and that it relies on unusually low-cost and stable sources of financing. Berkshire’s returns can thus largely be explained by the use of leverage combined with a focus on cheap, safe, quality stocks. We find that Berkshire’s portfolio of publicly-traded stocks outperform private companies, suggesting that Buffett’s returns are more due to stock selection than to a direct effect on management.

**Berrada, Messikh, Oderda, Pictet: Beta-Arbitrage Strategies: When Do They Work, and Why?**

**- Abstract**

Contrary to what traditional asset pricing would imply, a strategy that bets against beta, i.e. long in low beta stocks and short in high beta stocks, tends to out-perform the market. This puzzling empirical fact can be explained through the concept of relative arbitrage. Considering a market in which diversity is maintained, i.e. no single stock can dominate the entire market, we show that beta-arbitrage strategies out-perform the market portfolio with unit probability in finite time. We use the theoretical decomposition of beta-arbitrage excess return to provide empirical support to our explanation on equity country indices, equity sectors and individual stocks. Finally we show how to construct optimal beta-arbitrage strategies that maximize the expected return relative to a given benchmark.

**Frazzini, Pedersen: Embedded Leverage**

**- Abstract**

Many financial instruments are designed with embedded leverage such as options and leveraged exchange traded funds (ETFs). Embedded leverage alleviates investors’ leverage constraints and, therefore, we hypothesize that embedded leverage lowers required returns. Consistent with this hypothesis, we find that asset classes with embedded leverage offer low risk-adjusted returns and, in the cross-section, higher embedded leverage is associated with lower returns. A portfolio which is long low-embedded-leverage securities and short high-embedded-leverage securities earns large abnormal returns, with t-statistics of 8.6 for equity options, 6.3 for index options, and 2.5 for ETFs. We provide extensive robustness tests and discuss the broader implications of embedded leverage for financial economics.

**Asness, Frazzini, Pedersen: Low-Risk Investing Without Industry Bets**

**- Abstract**

The strategy of buying safe low-beta stocks while shorting (or underweighting) riskier high-beta stocks has been shown to deliver significant risk-adjusted returns. However, it has been suggested that such “low-risk investing” delivers high returns primarily due to its industry bet, favoring a slowly changing set of stodgy, stable industries and disliking their opposites. We refute this. We show that a betting against beta (BAB) strategy has delivered positive returns both as an industry-neutral bet within each industry and as a pure bet across industries. In fact, the industry-neutral BAB strategy has performed stronger than the BAB strategy that only bets across industries and it has delivered positive returns in each of 49 U.S. industries and in 61 of 70 global industries. Our findings are consistent with the leverage aversion theory for why low beta investing is effective.

**Chow, Hsu, Kuo, Li: A Survey of Low Volatility Strategies**

**- Abstract**

This paper replicates various low volatility strategies and examines their historical performance using U.S., global developed markets, and emerging markets data. In our sample, low volatility strategies outperformed their corresponding cap-weighted market indexes due to exposure to the value, betting against beta (BAB), and duration factors. The reduction in volatility is driven by a substantial reduction in the portfolios’ market beta. Different approaches to constructing low volatility portfolios, whether optimization or heuristic based, result in similar factor exposures and therefore similar long-term risk-return performance. For long-term investors, low volatility strategies can contribute to a considerably more diversified equity portfolio which earns equity returns from multiple premium sources instead of market beta alone. While the lower risk and higher return seem persistent and robust across geographies and over time, we identify flaws with naïve constructions of low volatility portfolios. First, naïve low volatility strategies tend to have very high turnover and low liquidity, which can erode returns significantly. They also have very concentrated country/industry allocations, which neither provide sensible economic exposures nor find theoretical support in the more recent literature on the within-country/industry low volatility effect. Additionally, there is concern that low volatility stocks could become expensive, a development which would eliminate their performance advantage. This highlights the potential danger of a portfolio construction methodology that is unaware of the fundamentals of the constituent stocks — after all, low volatility investing is useful only if it comes with superior risk-adjusted performance. That many naïve low volatility portfolios are no longer value portfolios today bodes poorly for their prospective returns. More thoughtful portfolio construction research is necessary to produce low volatility portfolios that are more likely to repeat the historical outperformance with reasonable economic exposure and adequate investability.

**Bali, Brown, Murray, Tang: Betting Against Beta or Demand for Lottery**

**- Abstract**

Frazzini and Pedersen (2014) document that a betting against beta strategy that takes long (short) positions in low (high) beta stocks generates large abnormal returns of 6.6% per year, and attribute this phenomenon to funding liquidity risk. We investigate alternative explanations for this effect, and find that it is caused by demand for lottery-like assets, a behavioral phenomenon. Requiring betting against beta portfolios to be neutral to lottery demand eliminates the abnormal returns. Controlling for lottery-demand, multivariate analyses detect a theoretically consistent positive relation between beta and returns. Factor models that include our lottery-demand factor explain the abnormal returns of betting against beta portfolios. We conclude that the betting against beta phenomenon is driven by demand for lottery-like stocks.

**Bianchi: Looking Under the Hood: What Does Quantile Regression Tell Us About the Low-Beta Anomaly?**

**- Abstract**

In a CAPM world, the expected return of every portfolio is linearly related to its market beta. Further, the market portfolio attains the maximum Sharpe ratio among all portfolios of risky assets. Consequently, low-beta portfolios are predicted to earn a lower rate of return and to have Sharpe ratios no greater than the market portfolio. A low-beta portfolio of risky assets with beta B is predicted to earn the same rate of return as a portfolio that invests B in the market portfolio and 1 – B in the risk-free asset. Empirically, neither of these predictions has been realized. Low-beta (B < 1) portfolios have earned higher returns than their market portfolio plus risk-free asset counterparts, and they have achieved higher Sharpe ratios than the market portfolio. In the literature, this is referred to as the low-beta anomaly. This paper uses quantile regression to examine other dimensions of risk beyond beta and volatility, and finds that low-beta stocks and portfolios bear additional compensated risk in the form of excess kurtosis.
Wang: Institutional Holding, Low Beta and Idiosyncratic Volatility Anomalies
http://papers.ssrn.com/sol3/papers.cfm?abstract_id=2512411
Abstract:
Institutional investors subject to benchmarking, short-selling and leverage constraints have asymmetric effects on both low beta and low volatility anomalies documented by previous studies. Specifically, institutional investors prefer high-beta stocks to low-beta stocks to minimize the tracking error and utilize the embedded leverage of high beta stocks, leading to low-beta anomaly. They can act as the supply source of security lending to the short-sellers, mitigating the overpricing induced negative effect on expected returns from idiosyncratic volatility. Using size effect adjusted institutional ownership as a proxy for institutional limits to arbitrage, I confirm that mandated and financial constrained institutional investors contribute positively to the low beta anomaly but mitigate the low IVOL anomaly using sorting and Fama-MacBeth regressions. I distinguish the highly correlated low beta and low volatility anomalies and find a significantly positive risk premium for institutional holding. A strong January reversal effect of idiosyncratic volatility on expected return is also documented.
Baker, Wugler: The Risk Anomaly Tradeoff of Leverage
http://papers.ssrn.com/sol3/papers.cfm?abstract_id=2577948
Abstract:
The “low risk anomaly” refers to the empirical pattern that apparently high-risk equities do not earn commensurately high returns. In this paper, we consider the possibility that the risk anomaly represents mispricing, not a misspecification of risk, and develop the implications for corporate capital structure. The risk anomaly generates a simple tradeoff model: Starting at zero leverage, the overall cost of capital initially falls as leverage increases equity risk. As debt becomes risky, however, the marginal benefit of increasing equity risk declines. The optimum is reached at lower leverage for firms with high asset risk. Consistent with a risk anomaly tradeoff, firms with low-risk assets choose higher leverage. In addition, leverage is inversely related to systematic risk, holding constant total risk; a large number of firms maintain small or zero leverage despite high marginal tax rates; and many others maintain high leverage despite little tax benefit.
Baker, Wurgler: Do Strict Capital Requirements Raise the Cost of Capital? Bank Regulation, Capital Structure, and the Low Risk Anomaly
http://papers.ssrn.com/sol3/papers.cfm?abstract_id=2577963
Abstract:
Traditional capital structure theory predicts that reducing banks’ leverage reduces the risk and cost of equity but does not change the weighted average cost of capital, and thus the rates for borrowers. We confirm that the equity of better-capitalized banks has lower beta and idiosyncratic risk. However, over the last 40 years, lower risk banks have not had lower costs of equity (lower stock returns), consistent with a stock market anomaly previously documented in other samples. A calibration suggests that a binding ten percentage-point increase in Tier 1 capital to risk-weighted assets could double banks’ risk premia over Treasury bills.
Cannon: The Idiosyncratic Volatility Puzzle: A Behavioral Explanation
http://digitalcommons.usu.edu/cgi/viewcontent.cgi?article=1475&context=gradreports
Abstract:
In this study, I propose an alternative explanation for the idiosyncratic volatility puzzle. I postulate that the negative coefficient observed between idiosyncratic volatility and future returns is driven by investor sentiment. The results obtained from these analyses support the idea that the idiosyncratic volatility puzzle can be explained by investor sentiment. In periods of high investor sentiment, investors are optimistic in choosing stocks. Such effects lead investors to flock to assets with high idiosyncratic volatility, creating the negative relationship with return. Furthermore, in periods of lowest investor sentiment, results indicate a natural, positive relationship between idiosyncratic volatility and future returns, supporting standard risk-return theory.
Schneider, Wagner, Zechner: Low Risk Anomalies?
http://papers.ssrn.com/sol3/papers.cfm?abstract_id=2593519
Abstract:
This paper shows theoretically and empirically that beta- and volatility-based low risk anomalies are driven by return skewness. The empirical patterns concisely match the predictions of our model that endogenizes the role of skewness for stock returns through default risk. With increasing downside risk, the standard capital asset pricing model (CAPM) increasingly overestimates expected equity returns relative to firms' true (skew-adjusted) market risk. Empirically, the profitability of betting against beta/volatility increases with firms' downside risk, and the risk-adjusted return differential of betting against beta/volatility among low skew firms compared to high skew firms is economically large. Our results suggest that the returns to betting against beta or volatility do not necessarily pose asset pricing puzzles but rather that such strategies collect premia that compensate for skew risk. Since skewness is directly connected to default risk, our results also provide insights for the distress puzzle.
Buchner, Wagner: The Betting Against Beta Anomaly: Fact or Fiction?
http://papers.ssrn.com/sol3/papers.cfm?abstract_id=2703752
Abstract:
This paper suggests an alternative explanation for the recently documented betting against beta anomaly. Given that the equity of a levered firm is equivalent to a call option on firm assets and option returns are non-linearly related to underlying stock returns, linear CAPM-type regressions are generally misspecified. We derive theoretical expressions for the pricing error and analyze its magnitude using numerical examples. Consistent with the empirical findings of Frazzini and Pedersen (2014), our pricing errors are negative, increase with leverage, and become economically significant for higher levels of firm leverage.
Andricopoulos: Leverage As A Weapon of Mass Shareholder-Value Destruction; Another Look at the Low-Beta Anomaly
https://papers.ssrn.com/sol3/papers.cfm?abstract_id=2859363
Abstract:
The 'low-beta' or 'low-volatility anomaly' is one of the most researched in the field of 'alternative beta'. Despite strong published evidence going back to the 1970s that high beta/volatility stocks underperform relative to expectations generated by the Capital Asset Pricing Model (CAPM), the anomaly still persists. The explanations given for this are all behavioural; that investor biases lead to overpricing of high volatility stocks. This paper shows that investor biases cannot be the explanation for the anomaly. Instead, it is proposed that the anomaly stems from a destruction of shareholder value. The strong implication is that the more market leverage a firm has, the more shareholder value is destroyed. Although the prevailing view for a long time has been that adding debt is good for shareholders, making balance sheets more 'efficient', there is in fact a considerable volume of evidence that the opposite is true; evidence which has been incorrectly interpreted for many years. Some possible mechanisms for this shareholder-value destruction are proposed.
Hedegaard: Time-Varying Leverage Demand and Predictability of Betting-Against-Beta
https://papers.ssrn.com/sol3/papers.cfm?abstract_id=3194626
Abstract:
The leverage aversion theory implies that returns to the betting-against-beta (BAB) strategy are predictable by past market returns: An outward shift in investors' aggregate demand function simultaneously increases market prices and increases the expected future BAB return. I confirm the prediction empirically and find that the BAB strategy performs better in times when and in countries where past market returns have been high. I construct timing-strategies that are long BAB half the time and short BAB half the time, based on past market returns, and show that these timing strategies have realized strong historical performance.
Hwang, Rubesam, Salmon: Overconfidence, Sentiment and Beta Herding: A Behavioral Explanation of the Low-Beta Anomaly
https://papers.ssrn.com/sol3/papers.cfm?abstract_id=3224321
Abstract:
We investigate asset returns using the concept of beta herding, which measures cross-sectional variations in betas induced by investors whose beliefs about the market are biased due to changes in confidence or sentiment. Overconfidence or optimistic sentiment causes beta herding (compression of individual assets’ betas towards the market beta), while under-confidence or pessimistic sentiment leads to adverse beta herding (dispersion of betas away from the market beta). We find that beta herding is related to the low-beta anomaly, as high beta stocks underperform low beta stocks on a risk-adjusted basis exclusively following periods of adverse beta herding. As an explanation of the low-beta anomaly, we propose the persistence of bias in betas (i.e., a large difference in betas) that lasts for more than one year as market uncertainty continues.
Liu: Asset Pricing Anomalies and the Low-Risk Puzzle
https://papers.ssrn.com/sol3/papers.cfm?abstract_id=3258015
Abstract:
The original observation in Black, Jensen and Scholes (1972) that the security market line is too flat - the beta anomaly - is a driving force behind a number of well-documented cross-sectional asset pricing puzzles. I document that returns to a broad set of anomaly portfolios are negatively correlated with the contemporaneous market excess return. I show that this negative covariance implicitly embeds the beta anomaly in these cross-sectional return puzzles. Taking into account the exposure to the beta anomaly either attenuates or eliminates the economic and statistical significance of the risk-adjusted returns to a large set of asset pricing anomalies.
Novy-Marx, Velikov: Betting Against Betting Against Beta
https://papers.ssrn.com/sol3/papers.cfm?abstract_id=3300965
Abstract:
Frazzini and Pedersen’s (2014) Betting Against Beta (BAB) factor is based on the same basic idea as Black’s (1972) beta-arbitrage, but its astonishing performance has generated academic interest and made it highly influential with practitioners. This performance is driven by non-standard procedures used in its construction that effectively, but non-transparently, equal weight stock returns. For each dollar invested in BAB, the strategy commits on average $1.05 to stocks in the bottom 1% of total market capitalization. BAB earns positive returns after accounting for transaction costs, but earns these by tilting toward profitability and investment, exposures for which it is fairly compensated. Predictable biases resulting from the paper’s non-standard beta estimation procedure drive results presented as evidence supporting its underlying theory.