This week's analysis of selected financial research paper contains more text and no picture, but we still think it's worth reading …

Authors: Siddiqi

Title: CAPM: A Tale of Two Versions

Link: https://papers.ssrn.com/sol3/papers.cfm?abstract_id=3350280

Abstract:

Given that categorization is the core of cognition, we argue that investors do not view firms in isolation. Rather, they view them within a framework of categories that represent prior knowledge. This involves sorting a given firm into a category and using categorization-induced inferences to form earnings and discount-rate expectations. If earnings-aspect is categorization-relevant, then earnings estimates are refined, whereas discount-rates are confounded with the category-exemplar. The opposite happens when discount-rates are categorization relevant. Earnings-focused approach such as DCF, generally used by institutional investors, leads to a version of CAPM in which the relationship between average excess return and stock beta is flat (possibly negative). Value effect and size premium (controlling for quality) arise in this version. Discount-rate focused approach such as multiples or comparables valuation, typically used by individual investors, leads to a second version in which the relationship is strongly positive with growth stocks doing better. The two-version CAPM accounts for several recent empirical findings including fundamentally different intraday vs overnight behavior, as well as behavior on macroeconomic announcement days. Momentum is expected to be an overnight phenomenon, which is consistent with empirical findings. We argue that, perhaps, our best shot at observing classical CAPM in its full glory is a laboratory experiment with subjects who have difficulty categorizing (such as in autism spectrum disorders).

Notable quotations from the academic research paper:

"Consider the following two empirical observations:

Firstly, stock prices behave very differently with respect to their sensitivity to market risk (beta) at specific times. Typically, average excess return and beta relationship is flatter than expected. It could even be negative. However, during specific times, this relationship is strongly positive, such as on days when macroeconomic announcements are made or during the night.

Secondly, a hue, which is halfway between yellow and orange, is seen as yellow on a banana and orange on a carrot. In this article, we argue that the two observations are driven by the same underlying mechanism.

The second observation is an example of the implications of categorization for color calibration. In this article, we argue that the first observation is also due to categorization, which gives rise to two versions of CAPM. In one version, the relationship between expected return and stock beta is flatter than expected or could even be negative, whereas in the second version, this relationship is strongly positive.

Categorization is the mental operation by which brain classifies objects and events. We do not experience the world as a series of unique events. Rather, we make sense of our experiences within a framework of categories that represent prior knowledge. That is, new information is only understood in the context of prior knowledge.

Here, in accord with cognitive science literature, we present a view of categorization that has both an upside as well as a downside, and apply this nuanced perspective to the capital asset pricing model (CAPM). If categorization is fundamental to how our brains make sense of information, then investor behavior, like any other domain of human behaviour, should also be viewed through this lens. This means that the traditional view that each firm is viewed in isolation needs to be altered. When an investor considers a firm, she views it within a framework of categories that represent prior knowledge. This involves sorting a given firm into a category based on attributes that are deemed categorization-relevant. Categorization-induced inferences help refine such attributes while confounding categorization-irrelevant attributes with the category-exemplar.

Valuation requires estimating earnings (cash-flows) potential and estimating discount-rates. Even among firms that sell similar products (same sector) some may have more similar earnings potential, whereas other may have more similar discount-rates. The former type may include firms with similar earnings-related fundamentals but very different levels of debt ratio and equity betas. Also, their multiples (generally related to inverse of the discount-rate) such as P/E, EV/Sales or EV/EBITDA could be very different. The latter type may include firms with similar debt ratios and equity betas or similar P/E and EV/EBITDA but quite different earnings or cash-flows fundamentals.

We argue that, an earnings-focused approach, such as discounted cash-flows (DCF), tends to categorize the former type of firms together, whereas, the relative valuation approach (RV) based on multiples such as P/E or EV/EBITDA tends to categorize the latter types of firms together. In other words, the choice of a valuation approach introduces a bias in how firms are categorized.

In this paper, we take discounted cash-flows (DCF) as the prototype of an earnings-potential focused approach, and valuation by multiples or relative valuation (RV) as the prototype discount-rate focused approach.

We show that when earnings aspect is categorization-relevant (as in DCF analysis), a version of CAPM is obtained, which displays a flatter or even negative relationship between stock beta and expected excess returns. Betting-against-beta anomaly is observed along with the value effect, as well as the size premium after controlling for quality (consistent with the findings in Asness et al 2018). We argue that this is the default version which typically prevails. While categorizing firms, if investors are focused on the discount rate aspect (as in RV analysis), then the discount-rates are refined whereas earnings estimates are confounded with the category-exemplar. A second version of CAPM arises. In this version, there is a strong positive relationship between beta and expected excess return.

One way to make sense of the co-existence of two versions is to classify investors as either earnings-focused or discount rate-focused. If earnings-focused investors dominate, then the first version is observed. If the discount-rate-focused investors dominate, then the second version is observed. Note, that earnings-focused approach (such as DCF) is typically employed by large institutional investors, whereas RV approach is associated with individual investors (and with sell-side equity analysts who publish research reports for individual investors).

If institutional investors are earnings-focused and individual investors are discount rate-focused, then the trading behavior of each type can be observed to make specific predictions:

1) Institutional investors typically avoid trading at the open and prefer to trade in the afternoon near the market close. The objective is to time the trade when the market is most liquid to avoid any adverse price impact. This means that trade at open is dominated by individual investors. So, one expects to see the relationship between stock beta and average return to be strongly positive (second version) overnight and flat or even negative (first version) intraday.

2) Institutional traders typically trade in the right direction prior to macroeconomic announcement days (suggesting superior information) with institutional trading volume falling sharply on macro-announcement days. As trade on such days is dominated by individual investors, one expects to see a strongly positive relationship (second version) on macro-announcement days.

3) The first version generally dominates intraday due to institutional investors being dominant. As the corresponding CAPM version comes with size and value effects, the prediction is that size and value are primarily intraday phenomena.

4) We show that, all else equal, discount rate-focused investors have higher willingness-to-pay than earnings-focused investors. If discount rate-focused investors dominate trade at open, whereas earnings-focused investors are active intraday, then one expects prices to typically rise overnight from close-to-open and fall intraday between open-to-close.

5) If momentum traders, who buy past winners and short past losers, are primarily individual investors, then one expects momentum to be an overnight phenomenon observed between close-to-open. This is because individual traders dominate trade at or near open.

"

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A new research paper related to all equity factor strategies …

Authors: Zaremba, Maydybura, Czapkiewicz, Arnaut

Title: Explaining Equity Anomalies In Frontier Markets: A Horserace of Factor Pricing Models

Link: https://papers.ssrn.com/sol3/papers.cfm?abstract_id=3378785

Abstract:

We are the first to compare the explanatory power of the major empirical asset pricing models over equity anomalies in the frontier markets. We replicate over 160 stock market anomalies in 23 frontier countries for years 1996–2017, and evaluate their performance with the factor models. The Carhart’s (1997) four-factor model outperforms both the recent Fama and French (2015) five-factor model and the q-model by Hou, Xue, and Zhang (2015). Its superiority is driven by the ability to explain the momentum-related anomalies. Inclusion of additional profitability and investment factors lead to no further major improvement in the performance. Nonetheless, none of the models is able to fully explain the abnormal returns on all of the anomaly portfolios.

Notable quotations from the academic research paper:

"In times of soaring correlations among global stocks and increasing controversies on anomaly performance in emerging stock markets, one specific asset class may offer a remedy: frontier equities. Deemed the least developed emerging markets, the frontier countries are scattered around the globe, with presence in Africa, Asia, Europe, and Latin America. Being very diverse both economically and geographically, they range from the wealthy oil-producing kingdoms in the Gulf to some of the poorest countries in Africa. While the current size of the frontier stock markets is still fairly small – the total capitalization of the MSCI Frontier Market Index constituents equaled $134 billion in May 2018 (MSCI, 2018), accounting for less than 0.4% of developed markets – yet, the interest in them is growing quickly.

Considering the future potential, along with the soaring interest of the international community, and the investment opportunities, it is surprising how underresearched – if not ignored – the frontier equities are. The number of academic studies on this stock market class seems astonishingly modest. This leaves numerous important questions, which may be of huge importance for global investors, unanswered. Which equity anomalies – discovered originally in developed countries – work also in the frontier stock markets? Could they be translated into profitable strategies using easily-to-implement quantitative methods? Finally, which asset pricing models and factors best summarize the cross-sectional return patterns and the equity anomalies in frontier countries? Could the recent five-factor framework by Fama and French (2015) or the q-model by Hou, Xue, and Zhang (2015) be also applied in this growing asset class? The principal target of this study is to close this gap in the existing body of literature at least partially.

The elevated liquidity constraints, higher trading costs, short sale unavailability accompanied by less sophisticated investors may potentially result in larger mispricing and more pronounced stock market anomalies.

Our research aims to contribute in three primary ways. Our first goal is to conduct the most comprehensive test on which equity anomalies, discovered originally in the developed countries, are also present in the frontier equities. Thus, we examine the performance of 167 anomalies from the finance literature, encompassing different classes of patterns related to value, trend following, investment, profitability, risk, and many others. The large-scale analysis available in broadly-accessible journals was either limited to the few most prominent strategies, such as size, value, and momentum (Blackburn and Cakici 2017, De Groot, Pang, and Swinkels 2012). Our study aims to take a substantial leap forward in understanding the multidimensionality of equity returns in the frontier markets.

Second, we research which of the broadly-acknowledged asset pricing models serve best in explaining the cross-section of anomaly returns in the frontier markets. In particular, we consider seven factor pricing models: the capital asset pricing model (Sharpe 1964), abbreviated CAPM, the three-factor model (Fama and French 1993), abbreviated FF3, the four-factor model (Carhart 1997), abbreviated C4, the five-factor model (Fama and French 2015), abbreviated FF5, the q-model by Hou, Xue, and Zhang (2015), the six-factor model by Fama and French (2018), abbreviated FF6, and the six-factor model by Barillas and Shanken (2018), abbreviated BS6.

Last but not least, our research may be regarded as a large out-of-sample test of equity anomalies.

To answer our research questions, we replicate the 167 equity anomalies from Zaremba et al. (2018) in an extensive sample of over 3,600 companies from frontier markets from all over the world for years 1996 – 2017. We form the long-short anomaly portfolios and evaluate their returns using the seven considered factor pricing models: CAPM, FF3, C4, FF5, Q4, FF6, and BS6. We compare the models’ performance by employing a range of tools and statistics that assess their ability to explain the risk and mean returns jointly.

The principal findings of this study could be summarized as follows. First, out of the 167 anomaly portfolios, only 38% (19%) of the equal-weighted (value-weighted) long-short strategies produce profits significantly departing from zero at the 5% level. The successful return patterns are usually linked to the “value vs. growth” or trend following effects, verifying positively the arguments of Asness, Moskowitz, and Pedersen (2013) that value and momentum are everywhere.

Second, we demonstrate that Carhart’s (1997) four-factor model best explains the anomaly returns in frontier markets, outperforming other models in many ways. It displays lower average absolute intercepts and largest number of explained anomalies. Its cross-sectional and time-series R2 is higher CAPM, FF3, FF5, or Q4, and only marginally lower than in the case of FF6 and BS6.

"

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A new research paper related to:

#33 – Post-Earnings Announcement Effect

Authors: Sojka

Title: 50 Years in PEAD Research

Link: https://papers.ssrn.com/sol3/papers.cfm?abstract_id=3281679

Abstract:

Analysing earning’s predictive power on stock returns was in the heart of academic research since late 60’s. First introduced to academic world in 1967 during seminar “Analysis of Security Prices” by Chicago University Professors Ray Ball and Philip Brown. In the next four decades was extensively analysed by many academics and is now a well-documented anomaly and is referred to as Post Earnings Announcement Drift (PEAD). This phenomenon is still at the centre of academic research because it stands at odds with efficient market hypothesis which assumes that all information is instantaneously reflected in stock prices. Professional investors are also closely looking at PEAD as it implies that it is easy to beat the market average by simply ranking stocks based on their earnings surprise and investing in the top decile, quintile or quartile and shorting the bottom part. Academic evidence shows that this strategy produces an abnormal return of somewhere between 2.6% and 9.37% per quarter, according to various authors. In this paper I will present existing evidence supporting and contradicting “PEAD”, the history of academic research in that field and various techniques used to verify the phenomenon. The paper is organised as follows: first the history of the PEAD academic research is presented, in the second more recent evidence and research techniques used by authors are presented and finally conclusions and various critics of PEAD are shown.

Notable quotations from the academic research paper:

"Post Earnings Announcement Drift is a measure of markets inability to price correctly information contained in earnings report. Since it was first spotted by Ball and Brown (1968), it went through rigorous academic scrutiny, first to test if it really exists (Ball (1978), Latane and Jones (1977)), then to measure its magnitude in various time frames, to offer explanations for its existence and find more PEAD variations. On average academics found that the postponed response to earnings information produces about 6% abnormal 60 days return (Dechow et al (2013)). The whole market reaction attributed to earnings report, measured from 60 days prior to earnings release to 60 days after is estimated at 18%, which means that about a third of the whole market response is delayed – Dechow et al (2013).

Figure 18 presents cumulative PEAD strategy abnormal returns for a 40-years period from 1971 to 2011. The total abnormal return of the strategy is an astonishing 350%, which is beat only by BTM (Book-to-Market) strategy. PEAD profits are very consistent up to late 90’s, then we can observe dips in the abnormal returns during internet bubble (1991-2001) and then during market recovery after 2008 crash. Since the middle of the 90’s PEAD returns became riskier and much lower than in the previous 25 years, it may be attributed to wider academic research in the field and wider recognition of the phenomenon among investors.

The PEAD strategy is not easy to implement in practice as it requires large scale data collection and data processing, more recent advancements in information processing technologies may also affect the magnitude of PEAD exploitation. A dominant part of research on PEAD was conducted in the US and based on US stock market data. The magnitude of PEAD computed by academics across time, since 1968 when first academic paper mentioning PEAD was published, up to the most recent evidence, are shown in Table 26.

PEAD premium computed based on US market data by academics is not easily comparable. There are differences in period studied, subset of stocks used, definitions of expected earnings or unexpected earnings signal altogether. Among the results presented in Table 26, the highest return 14.03% in 120 days presented by Balakrishnan et al (2009) and the lowest is Chordia and Shivakumar (2005) 0.9% in 1 month. Both of those research papers confirm PEAD premium existence, but Chordia and Shivakumar (2005) focus their attention on explaining joint anomalies of momentum and PEAD, and form portfolios each month regardless of profit announcement date, taking last announced earnings in their SUE ranking, which obviously weakens the earnings signal."

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We at Quantpedia are not the only ones who are interested in finding strategies that can be used to mitigate the impacts of the large equity corrections. We have already written a short article about a lottery/skewness strategy in commodities, which offers some protection in a time of crisis. Our users can also screen a list of strategies that can be used as a hedge/diversification for equity markets during downturns. A new research paper written by Harvey, Hoyle, Rattray, Sargaison, Taylor and Van Hemert explores the same question and analyzes the performance of different tools that investors could deploy during equity bear markets. We sincerely recommend it …

Authors: Harvey, Hoyle, Rattray, Sargaison, Taylor, Van Hemert

Title: The Best of Strategies for the Worst of Times: Can Portfolios be Crisis Proofed?

Link: https://papers.ssrn.com/sol3/papers.cfm?abstract_id=3383173

Abstract:

In the late stages of long bull markets, a popular question arises: What steps can an investor take to mitigate the impact of the inevitable large equity correction? However, hedging equity portfolios is notoriously difficult and expensive. We analyze the performance of different tools that investors could deploy. For example, continuously holding short-dated S&P 500 put options is the most reliable defensive method but also the most costly strategy. Holding ‘safe-haven’ US Treasury bonds produces a positive carry, but may be an unreliable crisis-hedge strategy, as the post-2000 negative bond-equity correlation is a historical rarity. Long gold and long credit protection portfolios sit in between puts and bonds in terms of both cost and reliability. Dynamic strategies that performed well during past drawdowns include: futures time-series momentum (which benefits from extended equity sell-offs) and a quality strategy that takes long/short positions in the highest/lowest quality company stocks (which benefits from a ‘flight-to-quality’ effect during crises). We examine both large equity drawdowns and recessions. We also provide some out-of-sample evidence of the defensive performance of these strategies relative to an earlier, related paper.

Notable quotations from the academic research paper:

"The typical investment portfolio is highly concentrated in equities leaving investors vulnerable to large drawdowns. We examine the performance of a number of candidate defensive strategies, both active and passive, between 1985 and 2018, with a particular emphasis on the eight worst drawdowns (the instances where the S&P 500 fell by more than 15%) and three US recessions. To guard against overfitting, we provide out-of-sample evidence of the performance of these strategies in the 2018Q4 drawdown that occurred after we wrote an earlier, related paper.

We begin with two passive strategies, both of which benefit directly from a falling equity market. A strategy that buys, and then rolls, one-month S&P 500 put options performs well in each of the eight equity drawdown periods. However, it is very costly during the ‘normal’ times, which constitute 86% of our sample and expansionary (non-recession) times, which constitute 93% of our observations. As such, passive option protection seems too expensive to be a viable crisis hedge. A strategy that is long credit protection (short credit risk) also benefits during each of the eight equity drawdown periods, but in a more uneven manner, doing particularly well during the 2007-2009 Financial Crisis, which was a credit crisis. Nevertheless, the credit protection strategy is less costly during normal times and non-recessions than the put buying strategy.

Next, we consider so-called ‘safe-haven’ investments. A strategy that holds long positions in 10-year US Treasuries performed well in the post-2000 equity drawdowns, but was less effective during previous equity sell-offs. This is consistent with the negative bond-equity correlation witnessed post-2000, which is atypical from the longer historical perspective. As we move beyond the extreme monetary easing that has characterized the post-Financial Crisis period, it is possible that the bond-equity correlation may revert to the previous norm, rendering a long bond strategy a potentially unreliable crisis hedge. A long gold strategy generally performs better during crisis periods than at normal times, consistent with its reputation as a safe-haven security. However, its appeal as a crisis hedge is diminished by the fact that its long-run return, measured over the 1985-2018 period, is close to zero and that it carries substantial idiosyncratic risk unrelated to equity markets.

We then turn our attention to dynamic strategies.

Time-series momentum strategies add to winning positions (ride winners) and reduce losing positions (cut losers), much like a dynamic replication of an option straddle strategy. We show that such strategies performed well over the eight equity drawdowns and three recessions. We also explore limiting the equity exposure (no long positions allowed), which we find enhances the crisis performance.

Next, we consider long-short US equity strategies. A review of the factors proposed in the academic literature suggests that those that take long positions in high-quality and short positions in low-quality companies are most promising as crisis hedges, since they benefit from flights to quality when panic hits markets. The definition of a quality business is, of course, open to debate. However, broadly speaking, such companies will be profitable, growing, have safer balance sheets, and run investor-friendly policies in areas such as payout ratios. We examine a host of quality metrics, and illustrate the importance of a beta-neutral (common in practice) rather than a dollar-neutral (common in academic studies) portfolio construction.

"

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A new research paper related to:

#118 – Time Series Momentum

Authors: Lim, Zohren, Roberts

Title: Enhancing Time Series Momentum Strategies Using Deep Neural Networks

Link: https://papers.ssrn.com/sol3/papers.cfm?abstract_id=3369195

Abstract:

While time series momentum is a well-studied phenomenon in finance, common strategies require the explicit definition of both a trend estimator and a position sizing rule. In this paper, we introduce Deep Momentum Networks — a hybrid approach which injects deep learning based trading rules into the volatility scaling framework of time series momentum. The model also simultaneously learns both trend estimation and position sizing in a data-driven manner, with networks directly trained by optimising the Sharpe ratio of the signal. Backtesting on a portfolio of 88 continuous futures contracts, we demonstrate that the Sharpe-optimised LSTM improved traditional methods by more than two times in the absence of transactions costs, and continue outperforming when considering transaction costs up to 2-3 basis points. To account for more illiquid assets, we also propose a turnover regularisation term which trains the network to factor in costs at run-time.

Notable quotations from the academic research paper:

"While numerous papers have investigated the use of machine learning for financial time series prediction, they typically focus on casting the underlying prediction problem as a standard regression or classification task – with regression models forecasting expected returns, and classification models predicting the direction of future price movements. This approach, however, could lead to suboptimal performance in the context time-series momentum for several reasons.

Firstly, sizing positions based on expected returns alone does not take risk characteristics into account – such as the volatility or skew of the predictive returns distribution — which could inadvertently expose signals to large downside moves. This is particularly relevant as raw momentum strategies without adequate risk adjustments, such as volatility scaling, are susceptible to large crashes during periods of market panic. Furthermore, even with volatility scaling – which leads to positively skewed returns distributions and long-option-like behaviour – trend following strategies can place more losing trades than winning ones and still be profitable on the whole – as they size up only into large but infrequent directional moves. The fraction of winning trades is a meaningless metric of performance, given that it cannot be evaluated independently from the trading style of the strategy. Similarly, high classification accuracies may not necessarily translate into positive strategy performance, as profitability also depends on the magnitude of returns in each class. In light of the deficiencies of standard supervised learning techniques, new loss functions and training methods would need to be explored for position sizing – accounting for tradeoffs between risk and reward.

In this paper, we introduce a novel class of hybrid models that combines deep learning-based trading signals with the volatility scaling framework used in time series momentum strategies – which we refer to as the Deep Momentum Networks (DMNs). This improves existing methods from several angles.

Firstly, by using deep neural networks to directly generate trading signals, we remove the need to manually specify both the trend estimator and position sizing methodology – allowing them to be learnt directly using modern time series prediction architectures.

Secondly, by utilising automatic differentiation in existing backpropagation frameworks, we explicitly optimise networks for risk-adjusted performance metrics, i.e. the Sharpe ratio, improving the risk profile of the signal on the whole.

Lastly, retaining a consistent framework with other momentum strategies also allows us to retain desirable attributes from previous works – specifically volatility scaling, which plays a critical role in the positive performance of time series momentum strategies. This consistency also helps when making comparisons to existing methods, and facilitates the interpretation of different components of the overall signal by practitioners.

Referring to the cumulative returns plots for the rescaled portfolios in Exhibit 4, the benefits of direct outputs with Sharpe ratio optimisation can also be observed – with larger cumulative returns observed for linear, MLP and LSTM models compared to the reference benchmarks. Furthermore, we note the general underperformance of models which use standard regression and classification methods for trend estimation – hinting at the difficulties faced in selecting an appropriate position sizing function, and in optimising models to generate positions without accounting for risk."

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It looks that time series momentum is one of the most prevalent effects in finance. Once again, an academic paper shows that it works in every corner of financial markets – in traditional assets, alternative assets and even in long short equity factors …

Authors: Babu, Levine, Ooi, Pedersen, Stamelos

Title: Trends Everywhere

Link: https://papers.ssrn.com/sol3/papers.cfm?abstract_id=3386035

Abstract:

We provide new out-of-sample evidence on trend-following investing by studying its performance for 82 securities not previously examined and 16 long-short equity factors. Specifically, we study the performance of time series momentum for emerging market equity index futures, fixed income swaps, emerging market currencies, exotic commodity futures, credit default swap indices, volatility futures, and long-short equity factors. We find that time series momentum has worked across these asset classes and across several trend horizons. We examine the co-movement of trends across asset classes and factors, the performance during different market environments, and discuss the implications for investors.

Notable quotations from the academic research paper:

"Our full data contains 156 assets, of which 58 are the “traditional assets” studied in the literature cited above, 82 are “alternative assets,” meaning futures, forwards, and swaps not previously studied, and 16 are “factors” constructed as long-short equity portfolios. In other words, we collect so much new data that the number of new assets outnumbers the “traditional assets” studied in the literature. While we broaden the universe, we only consider investable liquid assets or strategies.

We find strong evidence for time series momentum across the assets and factors that we study. Over our sample period, the gross Sharpe ratio of 12-month time series momentum for traditional assets is 1.17, and the strategy delivers an even higher Sharpe ratio of 1.34 for the alternative assets. The Sharpe ratio for long-short equity factors is 0.95, and, when we diversify across all three asset groups, the combined trend-following strategy yields a gross Sharpe ratio of 1.60.

Figure 1 reports the t-statistics from the regression, using lags ranging from 1 month to 60 months. Panel A reports the results for traditional assets. The positive t-statistics for the first 12 months indicate return continuation – that is, trends – and t-statistics larger than 2 in magnitude are statistically significant, consistent with earlier findings. For lags above 12 months, we see some negative coefficients, indicating trend reversals, although these tend to be statistically insignificant. Panel B extends the analysis to alternative assets, which also display strong return continuation for the first 12 months, and more mixed returns beyond 12 months. Panel C extends the analysis to equity factor portfolios, showing that time series predictability is feature of more than just traditional and alternative assets, but also of equity factors, with positive t-statistics across the most recent 12 months. These results demonstrate the remarkable pervasiveness of return continuation for the most recent 12 months, but not for returns beyond 12 months, across a range of assets and equity factors.

Traditional assets. Our data for traditional assets are the prices of 58 liquid futures and forwards, consisting of 9 developed equity index futures, 13 developed bond futures, 12 cross-currency forward pairs (from nine underlying currencies), and 24 commodity futures.

Alternative assets. Our data for alternative assets consist of prices for 7 emerging market equity index futures, 17 fixed income swaps, 24 emerging market cross currency pairs, 21 commodity futures, 5 credit default swap indices, and 8 volatility futures.

Equity factors. For equity factors, our data consist of 16 of the most well-cited and robust single-name stock selection factors

"

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