1. Adrien Verdelhan Presentation

    The International CAPM Redux

    Francesca Brusa (Oxford)
    Tarun Ramadorai (Oxford and CEPR)
    Adrien Verdelhan (MIT Sloan and NBER)

    March 2015

    Aggregate Foreign Equity Holdings (as % of global GDP)

    adrienp1

    Motivation

    • Large increase in financial integration over the past 30 years.
      • Aggregate foreign equity holdings, as a % of global GDP: from 3% in the 1980s to approximately 30% in 2011.
      • Foreign equity holdings of the U.S.: $6 trillion.
    • What are the expected returns on those foreign equity holdings?
      • Finance logic:
        • If an asset’s return tends to be low in bad times, the asset is risky. Its expected return should be above the risk-free rate.
    • How to measure bad times?
      • I Capital Asset Pricing Model (CAPM): Aggregate market return
    • How to compare returns?
      • Foreign equity returns are in foreign currencies

    Currency Risk

    • Exchange rate risk should matter. . .
      • When investors invest abroad, but consume at home, and domestic and foreign purchasing powers differ.
    • Yet, empirical evidence is diffcult to obtain.
      • International CAPM of Dumas and Solnik (1995): world stock market return + three bilateral exchange rates (Yen, Mark, and Pound).
    • Recent findings: Two risk factors account for a large share of systematic changes in bilateral exchange rates
      • Idea: Instead of using raw bilateral exchange rates, use currency risk factors that summarize their systematic variation.

    This Paper

    • 1. Three factors describe the time-series of foreign equity benchmark returns
      • Smaller difference between expected and realized returns than in the competitors’ models (World CAPM, International CAPM, and Fama-French factors)
    • 2. Those three factors account for a large share of foreign equity mutual funds’ and macro/emerging hedge funds’ returns
    • 3. A simple reduced-form model to study the interaction between currency and equity risk

    Realized vs Predicted Excess Returns

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    Literature

    • World CAPM:
      • Sharpe (1964) and Lintner (1965) to Stehle (1977), Solnik (1974), . . .
      • Korajczyk and Viallet (1989), Harvey (1991), Chan, Karolyi and Stulz (1992), Bekaert and Harvey (1995), Karolyi and Stulz (1996).
    • International CAPM:
      • Solnik (1974), Sercu (1980), Stulz (1981), Adler and Dumas (1983), . . .
      • Bekaert and Hodrick (1992), Ferson and Harvey (1993), Ferson and Harvey (1994), Bekaert (1995, 1996), Dumas and Solnik (1995), De Santis and Gerard (1998), Harvey, Solnik, and Zhou (2002)
    • New equity factors and frictions:
      • Chaieb and Errunza (2007), Bekaert, Harvey and Lundblad (2007); Hou, Karolyi, and Kho (2011), Karolyi, Lee and van Dijk (2012), Karolyi and Wu (2014), Malkhozov, Mueller, Vedolin, and Venter (2014)
    • Currency drivers:
      • order
        ows (e.g., Evans and Lyons, 2002, and Froot and Ramadorai, 2005) and risk factors (e.g., Lustig, Roussanov and Verdelhan, 2011, 2014, Menkhoff, Sarno, Scheming, and Schrimpf, 2012, Maggiori, 2012, Lettau, Maggiori, and Weber, 2013, and Verdelhan, 2014)

    Road map

    • Empirical evidence:
      • Equity benchmark indices
      • Hedge funds and mutual funds
    • Theoretical framework:
      • Intuition
      • Replication in a reduced-form model
    • Conclusion

    Empirical Asset Pricing

    Asset Pricing

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    Equity Literature Summary

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    Currency Literature Summary

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    Intuition

    adrienp6

    This Paper: International CAPM Redux

    adrienp7

    Data and Estimation

    • Test assets: from 46 developed and emerging countries, spanning value, growth, and country index returns from
      1/1976 to 12/2013.
    • Risk factors: global equity factor (built from local currency equity returns) and two currency factors: Dollar and Carry
    • Estimation: time-varying betas obtained on rolling windows

    adrienp8

    adrienp9

    Expected vs Realized Equity Excess Returns

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    Time-varying Factor Betas: Developed Markets

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    Time-varying Factor Betas: Emerging Markets

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    Realized vs Predicted Excess Returns

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    Robustness Checks

    • Other comparisons of the pricing errors:
      • Histograms and kernel density estimates of rolling alphas
      • Rolling mean absolute alphas
    • Even if the market prices of risk are estimated (Fama-McBeth procedure), the CAPM Redux still compares favorably:
      • the market prices of risk are positive and significant for our three factors
      • equity risk is priced for the CAPM, International CAPM, and FF models, but not the other factors (except GBP); the prices of risk appear further removed from the mean of the risk factors.
    • Subsamples:
      • time-windows (increasing sizes starting from 1976 or backwards from 2013)
      • test assets (country aggregates, developed vs emerging markets)
      • length of the rolling windows

    Kernel Density of Rolling Alphas

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    Rolling Mean Absolute Alphas

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    Correlation Between Equity Returns & Currency Factors

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    Mutual and Hedge Fund Returns

    Hedge Funds and Mutual Funds

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    Mutual Funds’ Significant Exposure to Currency Factors

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    \Macro/Emerging” Hedge Funds’ Significant Exposure to Currency Factors

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    Statistical Significance of Positive HF and MF’s Alphas

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    Theoretical Framework

    Intuition

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    Framework

    • Assume that financial markets are complete (thus log changes in exchange rates are differences in log SDF)
    • Start from the law of motion of each SDF, which depends on country-specific and global shocks
    • Assume that each country-aggregate dividend growth ratedepends on the same shocks as the SDF
    • Two key assumptions:
      • Prices of risk are time-varying (time-varying currency risk premia)
      • One global shock affects all SDF in the same way (role for a pure equity risk factor)

    Reduced-Form Model with Endogenous Exchange Rates

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    Model Solution

    • Simple and tractable reduced-form model:
      • Interest rates, exchange rates, and price-dividend ratios in closed-forms, as well as realized and expected equity and currency excess returns, and equity and currency risk factors.
    • Implications:
      • The world stock market return in U.S. dollars depends on all global shocks.
      • No role for bilateral exchange rates in theory
      • But in the model the market price of global equity risk depends on several state variables.
        • Those state variables are unknown to the econometrician.
        • Quantities and market prices of risk evolve at different frequencies.
    • Role for currency factors in practice, even in simulated data

    Realized vs Predicted Excess Returns: Simulated Data

    adrienp20

    Conclusion

    • Three factors (Global Equity, Dollar, and Carry) describe the time-series of foreign equity benchmark returns.
      • Predicted and realized equity excess returns are closer than in the competitors’ models.
    • Currency risk matters for foreign equity mutual fund and \macro/emerging” hedge fund returns.
    • Interaction between currency and equity risk in a simple reduced-form model:
      • Time-variation in quantities and prices of risk is key.


  2. Alberto Cavallo Presentation

     

    PPPs and Exchange Rates
    Evidence from Online Data in Seven Countries

     

    Alberto Cavallo
    MIT and NBER

    March 2015

     

    Agenda

    • motivation
    • Data
    • RERs
    • Shock Adjustments(half-lives, persistence comparisons)
    • Relative prices vs E shocks

    Conclusions

    • RERs shock adjustment is much faster than previously recorded âžœ months, not years
      • micro data necessary to compare relative price levels
      • Tradable, perfectly matched goods
    • Deviations of RER from normal levels increase pressure to adujust either Relative Prices or E
    • Adjustment margin and speeds vary by country
      • Argentina, Chaina, Australia âžœ mostly prices
      • Brazil, South Africa, UK âžœ mostly E

    Motivation

    albertop32

    Motivation

    albertop33

    Motivation

    albertop34

    Online Data and the Billion Prices Project

    • Academic Project at Mit to collect data from retailers that post prices online.
    • Started in 2008, joint with Roberto Rigobon (MIT)
    • Objective: inflation measurement and macro/international research
    • We collect daily data from hundreds of large retailers, for all goods sold, in 50 counteries.

    Albertop1

    How does Data Scraping work

    • Every day, our software downloada a public webpage, analyses its HTML code, extract price data, and stores it in a database

    Albertop2

    Alternatinve Data Sources for PPPs

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    Are Online Price Indices match CPIs

    • Our Online Price Indices match CPIs âžœ implies price change are similar

    albertop4

    But for PPP comparisons, price levels are key.

    Are Price Levels Different?

    • Cavallo (2015) âžœ simultaneous random sampling of online and offline prices using barcode-scanning app and crowdsourced workers

    albertop5

    Are Price Levels Different?

    • Cavallo (2015) âžœ simultaneous random sampling of online and offline prices using barcode-scanning app and crowdsourced workers

    Albertop6

    PPP Series with Online Data

    albertop7

    Methodology

    albertop8

    A Country-level RER For tradable goods

    albertop9

    Methodology

    albertop10

    Methodology

    albertop11

    Methodology

    albertop12

    Methodology

    albertop13

    Methodology

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    Methodology

    albertop15

    Aggregate Results

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    Aggregate Results

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    Aggregate Results

    • Most RERs appear to fluctuate around certain levels (relative PPP)
      • Are these levels reasonable
      • Comparison to ICP

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    Rates of Convergence and Half-lives of RERs Shocks

    • There is much faster mean reversion in these RERs than in the literature

    albertop19

    • Why faster adjustment?
      • Micro date more volatile âžœ control for sale events, stockouts
      • Only tradable goods
      • Identical Goods across country âžœ perfect matching

    Co-movement in Relative prices and E

    • Deviation of RER from normal level increase pressure to adjust

    albertop19

    • Adjustment may come through relative prices or the exchange rate
      • Magnitude persistence of each shock could matter
      • Implies that E co-moves with Pus/PLc the PPP exchange rate

    Argentina

    albertop19

    Brazil

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    UK

    albertop19

    Australia

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    China

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    South Africa

    albertop19

    A Vector Error Correction Model

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    Error Correction Model

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    Extension

    • We can further:
      •  Restrict the analysis to periods when the deviation is big (RERs over a given threshold)
        • Speeds of adjustment increase
      • Distinguish between devation in one sector/product vs deviation multiple sectors/products
        • Single-sector Deviation (eg gas prices) tend to be corrected via prices, while multi-sector deviations leads to mecro adjustments via the nominal exchange rate


    Conclusions

    • RERs shock adjustment is much faster than previously recorded months, not years
      • Micro-date necessary to compare relative price levels
      • Tradable perfectly matched goods
    • Deviation of RER from normal levels increase pressure to adjust either relative price or E
    • Adjustment margin and speeds vary by country
    • Argentina, China, Australia mostly Prices
    • Brazil, South Africa, UK mostly E

  3. Using CPIs:Relative Prices and E

    albertop19

    Devation in Many Products

    albertop19


  • lan cooper Paper

    The behaviour of sentiment-induced share returns:
    Measurement when fundamentals are observable

     

    Richard A Brealey
    London Business School

    Ian A Cooper*
    London Business School
    Sussex Place
    Regent’s Park

    London NW1 4SA
    United Kingdom
    +44-2070008259
    icooper@london.edu
    Evi Kaplanis
    London Business School
    February 2015

     

     

    Abstract

    We test the effect of sentiment on returns using a sample of upstream oil stocks where we have a good proxy for fundamental value. For this sample, the influence of sentiment is highly time-varying, appearing only after the post-2000 increased interest in oil-related assets. Sentiment affects returns on these stocks principally through their
    fundamentals rather than through deviations from fundamentals. Retail investor sentiment predicts short-term momentum of fundamentals and Baker-Wurgler sentiment predicts mean reversion of fundamental factors. These effects appear in a
    portfolio that is long hard-to-arbitrage stocks and short easy-to-arbitrage stocks, but only because this portfolio has net exposure to fundamentals.

    JEL classification: G11, G12, G14
    Key words: Investor sentiment, return predictability, arbitrage

    *Corresponding author

    In this article we investigate the relationships between investor sentiment and deviations of share prices from fundamental values. To do this we use a sample of shares for which a large part of the fundamental value is observable: upstream oil stocks. We measure their fundamental values using oil and gas prices and the forward oil price contango.

    We focus on upstream oil stocks because there is a direct relationship between the present value of these stocks and the oil price. In a world where output prices minus extraction costs obey the Hotelling Principle, the
    value of natural resource companies depends only on current prices less extraction costs. Miller and Upton (1985a, 1985b) test this proposition and find that it provides a good explanation of the variation in value of a sample of oil producers. Hence a large part of the fundamental value of upstream oil stocks is observable. We make use of this present value condition to split the return on our sample into the part that represents fundamentals and the part that is a deviation from fundamentals.

    Following Baker-Wurgler (2006), we test the impact of sentiment using a portfolio that is long high-variance stocks and short low-variance stocks (the “Hi-Lo” portfolio). We find that two types of sentiment predict returns: retail sentiment, which predicts momentum, and the Baker-Wurgler (2006) measure of sentiment, which predicts reversion to fundamentals. We find that the influence of sentiment in each case is time-varying. In particular, the ability of sentiment to predict returns appears only after 2000. Contrary to the theory that sentiment mainly affects the deviations from fundamental value of hard-to-arbitrage stocks, we find that both measures of sentiment influence prices through the fundamentals themselves rather than through deviations from fundamentals.

    If the Hi and Lo portfolios had similar loadings on fundamental variables, the net Hi-Lo portfolio would be hedged against fundamental effects and we should not observe fundamentals affecting the returns on this portfolio. However, in our data the loadings on fundamental factors of the Hi and Lo portfolios are different, so the combination of a long and short position in this portfolio does not eliminate its exposure to fundamentals. Methodologically, this raises the issue that tests based on such portfolios do not avoid the need to control for fundamentals.

    The results are more consistent with the idea that sentiment predicts market-wide returns rather than deviations of individual stocks from their fundamental value (Arif and Lee (2014)).

    The remainder of the article is organized as follows. In Section 1 we give a brief review of related literature. In section 2 we develop our tests and in section 3 we describe our data. Section 4 presents our main tests of the influence of sentiment on returns with and without controls for fundamentals.
    Section 5 provides some robustness tests. Section 6 concludes.

    1. Related literature

    Our work is broadly related to a number of studies that have found evidence of serial dependence in returns. Evidence of momentum over periods of six to twelve months is provided by amongst others Lehmann (1990), Jegadeesh (1990), Jegadeesh and Titman (1993, 2001), Asness et al (2013), and Moskowitz et al (2011). Evidence that this medium-term momentum is followed by longer-term mean reversion comes from variance-ratio tests (Poterba and Summers 1988, Lo and MacKinlay 1988, Cutler, Poterba, and Summers 1991) and autocorrelation tests (Fama and French 1988). Evidence that high short-term variance is related to deviations from fundamentals
    comes from excess variance tests (Shiller 1981, LeRoy and Porter 1981).

    Sentiment-based explanations of momentum, mean-reversion, and deviations from fundamentals envisage these effects as arising from behavioural biases by naïve investors combined with costs of arbitrage.For example, Daniel, Hirshleifer, and Subrahmanyam (1998) present a model in which a combination of overconfidence and biased self-attribution create both under- and over-reaction. Similarly, Barberis, Shleifer, and Vishny (1998) appeal to the behaviour al biases of representativeness and conservatism to show how these can result in under- and over-reaction.In both papers asset prices can be decomposed into one part that reflects fundamentals and another consisting of deviations from fundamentals. The effect of sentiment on asset prices operates through the deviations from fundamentals.

    Empirical evidence on the link between sentiment and returns requires a measure of sentiment. A number of suggestions have been proposed. Many of these reflect the view that sentiment changes are driven by retail investors. Possible proxies include flows into mutual funds (Brown et al 2003), buy/sell imbalances by retail investors (Kumar and Lee 2006), IPO volume and initial returns (Baker and Wurgler 2006), market turnover (Baker and Wurgler 2006), closed-end fund discounts, (Lee, Shleifer, Thaler 1991, Chen et al 1993, Swaminathan 1996, and Neal and Wheatley 1998), the growth stock premium (Baker and Wurgler 2006), and survey data (Qiu and Welch 2004, Brown and Cliff 2004, 2005). These data have been used either singly or in combination as sentiment measures to test hypotheses about the relationship between sentiment and subsequent stock returns.

    Our tests of the effect of sentiment are most closely related to Baker and Wurgler (2006, 2007, 2012) and Baker, Wurgler, and Yuan (2012). Baker and Wurgler divide their sample of stocks into ten portfolios on the basis of their prior volatility, which serves as a proxy for difficulty of arbitrage. They find that returns on the more volatile stocks are lower following a time of optimism, and that returns are higher following a time of pessimism. For the less volatile stocks that are easier to arbitrage the reverse is true. They develop a measure of investor sentiment which they find predicts returns for portfolios that are long the more volatile stocks and short less volatile stocks. This finding is consistent with a combination of behavioural biases and limits to arbitrage.

    Barberis, Shleifer and Wurgler (2005) stress the importance of controlling for fundamentals when measuring the effect of sentimen on security prices. For example, Derrien and Kecskés (2009) show that the effect of sentiment on equity issuance disappears once controls for fundamentals are included. Baker and Wurgler’s use of a
    Hi-Lo portfolio will be effective in controlling for fundamentals only if the long and short positions have equal loadings on fundamental factors. The alternative is to attempt to control directly for fundamentals. Brown and Cliff (2005) use as their dependent variable estimates of deviations from fun
    damental value based on the dividend discount model. They find that these deviations are positively related to a sentiment measure derived from survey data. However, the dividend discount model gives a very noisy observation of fundamental value. For our sample of stocks, we have a more direct measure of fundamental value than Brown and Cliff and so are able to perform a more powerful test of the way that sentiment is transmitted to share prices and returns.

    2. Hypotheses and tests

    We test the implications of the hard-to-arbitrage hypothesis using a simple empirical procedure that relates sentiment measures to deviations of share prices from fundamentals and also to the fundamentals themselves. We split the log share price, Pt, into a component reflecting fundamental value, Ft, and a separate component, NFt, which is the deviation from fundamental value:

    NFt=Pt-Ft        (1)

    We assume that prices are affected by the actions of two types of traders. One type is an arbitrageur, whose behaviour is captured by the Baker-Wurgler sentiment measure St. The other type is a naïve trend-follower, whose behaviour is captured by a bullish retail sentiment measure, Bt. Fundamentals and non-fundamentals may respond to both sentiment variables:

    oil-paper-Q-GROUP-500

    oil-paper-Q-GROUP-5001

    We model the response to Baker-Wurgler sentiment by assuming that arbitrageurs push prices down when this sentiment variable is high, making θS.FS,NFB,F>0 and θ,NF>0.

    The Baker-Wurgler sentiment variable is a measure of mispricing and so should rise when the deviation from fundamentals increases:

    oil-paper-Q-GROUP-5002

    oil-paper-Q-GROUP-5002

    where 0S,NF>0. We expect St to be highly persistent, reflecting the long cycle of swings in mispricing We hypothesize that the bullish sentiment indicator reflects trend-following behavior:

    oil-paper-Q-GROUP-6001

    θB,P>0. We expect Bt to be less persistent than St, reflecting shorter cycles in momentum sentiment.

    We stack Equations (2)-(5) to form the system shown in Table 1, which we estimate using VAR. Our null hypothesis is that the sentiment variables affect deviations from fundamentals but not the fundamentals themselves. This would be consistent with the limits-to-arbitrage hypothesis, whereby sentiment causes deviations from fundamentals when such deviations are hard to arbitrage. The indicated signs of the key parameters under the null hypothesis are shown in Table 1. In particular, the standard hypothesis is that sentiment affects deviations from fundamentals, implying θS,F=0, θB,F=0, θS,NFB,NF>0. As an alternative, we test the hypothesis that the influence of sentiment occurs via its effect on fundamentals, which implies that θS,F<0, θB,F>0, θS,NF=0, θB,NF=0.

     

    TABLE 1 HERE

     

     

    3. Data

    Our sample consists of the stocks of the 121 U.S. oil exploration and production companies quoted on the NYSE during the period March 1983 to January 2011. We define an exploration or production company as one with a North American Industrial Classification (NAIC) of 211111 or a Standard Industrial Classification (SIC) of 1311. We assume that fundamental value is a function of the month-end spot price of West Texas Intermediate oil and the spot wellhead price of West Texas natural gas. We also use the change in the contango in oil prices, where contango is measured as the log of the price of the sixth most distant futures contract less that of the price of the closest futures contract.1The Appendix gives our data sources.

    To investigate the behaviour of returns we form portfolios of oil stocks. We proxy the behaviour of all upstream oil stocks by the equally weighted portfolio of all stocks with return data for a given month (the “All-stocks” portfolio). The hard-to-arbitrage hypothesis suggests that sentiment-induced deviations from fundamental value shouldbe greater in the more volatile stocks. We therefore form sub-portfolios of our sample of stocksconsisting of the tercile of stocks with the highest variance of returns over the preceding 60 months and the tercile with the lowest variance. Following Baker-Wurgler, we form a portfolio that is long the tercile of high-variance stocks and short the tercile of low-variance stocks (the “Hi-Lo” portfolio). These portfolios are formed out of sample, and so represent a viable trading strategy.

    We employ two measures of sentiment: the Baker-Wurgler index (“BW sentiment”) and the proportion of individual investors who report that they are bullish in the regular survey conducted by the American Association of Individual Investors (“Bullish sentiment”). Both measures are available for the period July 1987 to January
    2011. To facilitate comparison between the two sentiment measures, we recalibrate the index values in terms of the number of standard deviations from the mean for the total period. Figure 1 provides a plot of these two rescaled measures. The Baker-Wurgler index is characterised by long swings in sentiment with a marked peak in value in February 2001. By contrast, Bullish sentiment is more noisy and less persistent. The first-order autocorrelation coefficient in the Baker-Wurgler index is .96, and the serial correlation in the index persists at least through lag 6. By contrast, the first-order autocorrelation coefficient in the AAII measure is .45 and the lower order serial correlations fall away rapidly.

     

    FIGURE 1 HERE

     

     


    1Our results are robust to varying the definition of these variables. For example, using spot Brent prices or a closer futures contract does not affect our conclusions. Equally, we obtain qualitatively

    The monthly levels of the two sentiment indexes are only weakly related with a correlation of .09. The Baker-Wurgler index more closely resembles a cumulative sum of past values of the Bullish measure,2 which is consistent with the Baker-Wurgler index capturing cumulative deviation from fundamentals rather than short-term swings in sentiment.

    Table 2 shows descriptive statistics for our variables. Panel A shows the means and standard deviations of the oil portfolio returns and of the changes in the fundamental values. Although the portfolios are formed out-of-sample, the standard deviation of the high-volatility portfolio is 50% higher than that of the low-volatility portfolio. The volatility of the Hi-Lo portfolio which should, in principle, be hedged against changes in fundamentals is almost as high as that of the Hi volatility portfolio, suggesting that the long-short strategy may have only limited effect in controlling risk.

     

    TABLE 2 HERE

     

    Panel B of Table 2 shows the correlation matrix for the entire period. Two features of the matrix are of interest and point to issues that are explored in more depth later.

    1. The correlations between the returns on oil stocks and the three fundamental variables are quite high. Taken together, the three fundamental variables
      explain 41% of the variance in the returns on the portfolio of all oil stocks.
    2. The long-short portfolio of oil stocks (Hi-Lo) is not well hedged against fundamentals, and its returns remain quite highly correlated with all three
      fundamental variables.In other words, the high-volatility stocks are not
      only more difficult to arbitrage, but they also have different loadings on the fundamental factors.

    Panel C of Table 2 shows the correlations between lagged sentiment and portfolio
    returns, and lagged sentiment and fundamentals. For our entire data period lagged Baker-Wurgler sentiment negatively predicts returns across all portfolios and the gas and oil fundamentals. Lagged Bullish sentiment positively predicts returns across all


    similar, but somewhat less strong, results using the Datastream index of U.S. oil stocks rather than our equally weighted portfolio of upstream stocks only

    portfolios and the oil and gas fundamentals. Consistent with the cost of arbitrage hypothesis, the correlation with Baker-Wurgler sentiment is higher for the high variance portfolio than for the low variance portfolio. However, the correlation with fundamental variables is even higher. For the Bullish sentiment correlations, the high and low variance portfolios have equal correlations with sentiment, but the Hi-Lo portfolio has an even higher correlation.

    Thus overall the data are consistent with sentiment predicting returns in the hypothesized ways. However, they do not suggest that the mechanism is necessarily via deviations from fundamentals rather than through the fundamentals themselves.

    4. Sentiment and returns

    In this section we examine the influence of sentiment on returns. In Section 4.1 we provide evidence that the returns on the portfolios of oil stocks are characterised by momentum and longer-term mean reversion. We then examine the returns to see whether these patterns in returns are a function of our two measures of sentiment. In Sections 4.2 and 4.3 we go on to decompose the returns into a fundamental and residual component and we analyse the relationship between these two components and our sentiment measures. In Section 4.4 we then examine the relationship between stock returns and “deep” fundamentals based on demand and supply in the oil market.

    4.1 The influence of sentiment on the Hi-Lo portfolio

    Before testing for the effect of sentiment on returns, we first examine the serial properties of returns on the All-stock portfolio and the relationship of these returns to fundamentals. Table 3 shows for our portfolio of oil stocks the variance rates at differing intervals expressed as a proportion of the 1-month variance rate using the variance ratio test with overlapping data proposed by Lo and Mackinlay (1988). Consistent with standard results, the variance ratio rises for 6-9 months reflecting medium-term momentum and then falls back over the following year reflecting longer-term mean reversion.


    A regression of the Baker-Wurgler index on the concurrent and 9 lagged values of the AAII measure
    gives a positive loading on each of the independent variables with a multiple correlation of .34.

     

    TABLE 3 HERE

     

    We examine the influence of sentiment by regressing total returns on the two lagged sentiment measures:

    oil-paper-Q-GROUP-1001

    To investigate the role of fundamentals we augment this regression with controls for the fundamental variables:

    oil-paper-Q-GROUP-1002

    when ΔFt is the vector of fundamental variables.3

    We estimate the VAR system (2)-(5) using GMM with the Newey-West correction for standard errors. Table 5 shows the results for the entire period 1988-2011.6 Contrary to the “deviations from fundamentals” hypothesis all the coefficients of the regression of deviations on lagged variables in column (2) are insignificant, and the Rbar2is negative. In contrast, the regression of fundamental returns on lagged sentiment (column (1)) has an Rbar2 of 5%. There is a negative coefficient on lagged B-W sentiment and a positive coefficient on the lagged bullish variable. Both coefficients are strongly significant. This result is consistent with sentiment-based trading operating largely through the fundamentals themselv
    es, rather than through the deviations in the share price from fundamentals. The negative coefficient on lagged B-W sentiment is consistent with high sentiment signalling that the oil market is above its equilibrium and likely to fall. The positive coefficient on the Bullish variable is consistent with short-term momentum pushing the oil market upwards when retail investors are bullish.

     

    TABLE 5 HERE

     

    Column (3) shows that the Baker-Wurgler sentiment variable is persistent, with a partial serial correlation of 0.97 consistent with
    a half-life of 20 months. The variable also responds to lagged changes in fundamentals, but not to lagged changes in deviations from fundamentals. Again, this is consistent with sentiment operating through the fundamentals themselves and not through deviations of share prices from fundamentals. Column (4) shows that the Bullish
    variable is much less persistent, with a half-life of less than a month. It has positive serial correlation and positively responds to past fundamental returns.

    We also estimate the system for two sub-periods divided at 2000. The motivation for looking separately at the sub-periods is the sharp increase in institutional investment in oil futures after 2000 (Buyuksahin and Robe (2012) and Singleton (2011)). In the


    6The period is reduced by 5 years because the first 60 months are used to estimate equation 8.

    ten years to 2010 open interest in crude oil futures by non-commercial traders was 5.6 times its level over the previous 15 years (Figure 2A). Figure 2B shows that there was also a sharp rise in the cumulative cash flows into managed futures funds, which increased from $9.3 billion in September 2002 to $137.0 billion in March 2008 before losing most of these gains in 2009.

     

    FIGURE 2A and B HERE

     

    Table 6 decomposes the Hi-Lo portfolio data into two sub-periods, divided at the end of 2000. Column (1) shows the results for the entire period, and columns (2) and (3) for the two sub-periods. The two sentiment variables have no significant effect on deviations from fundamentals in any period, though the coefficients on the Bullish variable consistently have the correct sign. The coefficients from the regression of the fundamental component of returns on the two lagged sentiment variables have the correct sign but are insignificant in the first sub-period. By contrast, in the second period the corresponding coefficients are strongly
    significant. Although the time-series behavior of the sentiment variables appears to be the similar in the two sub-periods, the effect of sentiment on stock returns changes completely in the second period. Consistent with the result that sentiment affects prices largely via fundamentals, the effect occurs only once there is significant investment interest in the fundamental markets post-2000.7

     

    TABLE 6 HERE

     

     

    4.3 The effect of the differencing interval

    Table 7 shows the effect of increasing the differencing interval to 3 months and 12 months. The results are shown for the entire period (Panel A) and the two sub-periods (Panels B and C). The VAR is estimated using GMM with overlapping observations


    7The sharp changes in cumulative flows into commodity hedge funds prompted us to examine (more in hope than expectation) the effect within the VAR of interacting the cumulative flows with the sentiment variables. There was no evidence that the impact of sentiment on returns was related to the cumulative flows into managed futures funds.

    and Newey-West corrected standard errors. In the regression of fundamental returns the effect of moving from a 1-month to 3-month differencing interval is to increase the magnitude and significance on both of the lagged sentiment measures in all three periods. The Rbar2 of the regression of fundamentals on sentiment increases dramatically, rising to .29 for a 12-month differencing interval in the second sub-period. Thus the sentiment measures appear to have a prolonged effect on the fundamental returns. In contrast, the longer differencing intervals have almost no effect on the coefficients for the deviations from fundamentals, which remain insignificant at all intervals and in all periods.

     

    TABLE 7 HERE

     

     

    4.4 “Deep Fundamentals”

    Our measure of the fundamental return on the portfolio of oil stocks is equivalent to a weighted average of the contemporaneous change in the price of oil and gas and the change in the contango. The evidence that this weighted average is a function of the prior level of sentiment implies that oil and gas prices are themselves influenced by sentiment (Pindyck (1993)). Thus it appears that sentiment drives oil prices away from equilibrium values in a way that leads to predictable returns on oil stocks. This effect increases after 2000 when interest in commodities as an asset class increased significantly. One-month returns are slightly predictable using sentiment, but returns over a 1-year horizon are highly predictable. Overall, the results appear to reflect a slowly changing but predictable component of oil and gas prices that is related to sentiment and generates a predictable return on oil stocks. Sentiment appears to have no effect on the price of oil stocks other than through the prices of the commodities themselves.

    We can gain some further insight by examining the relationship between the change in oil prices and prior sentiment while controlling for the deeper fundamentals that determine oil prices.

    oil-paper-Q-GROUP-14002

    where ΔDFt is the vector of the underlying determinants of the change in oil prices.

    The main problem in estimating (9) is the lack of good proxies for deeper fundamentals that are available at sufficiently high frequency. We proxy the fundamental determinants oil prices by changes in world oil production and consumption, changes in world proven reserves (annual data only), changes in oil inventories (monthly data only), and a measure of economic growth. We estimate (9) using annual data, overlapping 12-month data and overlapping 3-month data. In the case of the annual data estimates are for the period 1988 to 2011 and in the case of the regressions using monthly data estimates are for the period 1994 to 2011. The results are summarized in Table 8 Panel A. For comparison, Panel B shows the same regressions with only the sentiment variables included.

     

    TABLE 8 HERE

     

    With relatively few independent observations, the tests lack power, but they give an indication that the sentiment variables predict not only the change in oil prices but also the change in deep fundamentals. In particular, if the deep fundamentals are excluded the Baker-Wurgler sentiment predicts the oil price, but when the deep fundamentals are included it does not. Thus the table provides mild support that sentiment predicts not only energy prices but also the fundamental variables that drive oil prices. Through these it predicts the return on oil stocks.

    5. Robustness tests

    We have already noted that our findings are robust to (a) pre-whitening the fundamental variables, (b) using different definitions of the crude oil price and the oil contango, (c) using the Datastream index of oil stocks.

    5.1 Long-only portfolios

    To the extent that the Hi-Lo portfolio is better hedged against fundamental factors than long-only portfolios, the fundamental component of returns will be relatively small. We therefore repeated the VAR estimates with long-only portfolios. The results for the tercile of stocks with the highest variance were very similar to those for the Hi-Lo portfolio. In particular, the effects of sentiment on returns were significant only for the second period, and sentiment impacted
    returns largely through the fundamental component.

    5.2 Nasadaq stocks

    To test further the robustness of these results to a different sample of oil stocks, we extended the sample to include 274 stocks of U.S. oil and exploration companies that were traded on Nasdaq. Although this produced a larger sample, the quality of the Nasdaq data appears to be inferior with shorter time series for many stocks, leptokurtic returns and more zero returns. Unsurprisingly, the high variance portfolio tended to have a high concentration in Nasdaq stocks.

    The expanded portfolio is better hedged against fundamental factors and the addition of these factors has therefore less effect on estimates of the sentiment effect. Otherwise, the results are similar to those reported in Tables 4 and 5. The coefficients on B-W are consistently negative and those on Bullish consistently positive. However, there continues to be a big difference between the two periods with the coefficients being significant only in the later period.

    5.3 The effect of lagged market returns

    To evaluate the role of market returns in generating sentiment, we augmented the VAR system by including the lagged market return in each of the regressions. This did not significantly change the relationships between either of the sentiment variables and either of the returns. It did not
    increase the R2‘s for the prediction of returns. The Bullish sentiment variable is not significantly related to the lagged market return in the second sub-period, where the main sentiment effect is apparent. This suggests that the momentum generated by the positive relationship between fundamental returns and lagged Bullish sentiment is not simply a proxy for the effect of lagged market returns.

    5.4 Lagged fundamentals

    We also added to the VAR system more lags of the fundamental returns. These were generally insignificant and did not change the basic results. The sentiment variables remained significant in the second sub-period and the effect of sentiment showed up only in the fundamental regression and not the deviations from fundamentals.

    6. Conclusions

    Using a sample of upstream oil stocks where we have a good proxy for fundamental value, we show that sentiment predicts returns. However, the effect is highly time-varying, appearing only after the post-2000 increased interest in oil-related assets.

    Sentiment effects come it two forms: retail investor sentiment predicts short-term momentum, and Baker-Wurgler sentiment predicts medium-term mean reversion of fundamental factors. Whilst the sentiment variables explain only a negligible proportion of the variance of returns, the additional return due to a change in sentiment is not unimportant. For example, in the second period for the portfolio of all oil stocks a one standard deviation rise in the level of investor sentiment added about .3 % to the following month’s return; during the same period a similar one-standard-deviation rise in the Baker-Wurgler index reduced return by about .3%

    Contrary to the hard-to-arbitrage hypothesis, sentiment affects returns on these stocks through fundamentals rather than through deviations from fundamentals. Overall, it appears that retail sentiment drives the prices of oil and gas futures away from their deeper fundamental values until the deviation is sufficiently large that arbitrageurs drive the prices back towards their equilibrium values. This process for the fundamentals is then reflected in the prices of upstream oil stocks

    These effects appear even in a portfolio that is long hard-to-arbitrage stocks and short easy-to-arbitrage stocks, because this portfolio has a net exposure to fundamentals. This has implications for tests of the hard-to-arbitrage hypothesis, showing that it is important to have effective controls for fundamentals even when the long-short portfolio is used.

    Our finding that sentiment affects upstream oil stocks through the fundamentals raises the issue as to whether this is also the case with other industries that invest inassets that are traded in speculative markets. Obvious examples would be stocks in other extractive industries but a similar effect could characterize financial institutions. It also prompts the question whether the magnitude of any sentiment effects depends on the extent to which the fundamentals are tradeable. If this is the case, sentiment effects might vary not just with ease of arbitrage but with the nature of the company’s fundamentals. The sharp increase in the significance of sentiment effects in the post-2000 period was accompanied by an increase in speculative activity in energy futures. If these effects are truly linked, then it raises the question as to the effect of trading activity on the influence of sentiment. These would appear to be fruitful, if difficult, areas for future research.

    REFERENCES

    Arif, Salman, and Charles M.C. Lee, 2014, “Aggregate Investment and Investor Sentiment,” Review of Financial Studies, forthcoming.

    Asness, C.S., T.J. Moskowitz, and L.H. Pedersen, 2013, “Value and Momentum Everywhere,” Journal of Finance 68, 929-981

    Baker, M., and J. Wurgler, 2006. “Investor Sentiment and the Cross-section of Stock Returns,” Journal of Finance 61, 1645-1680.

    Baker, M., and J. Wurgler, 2007. “Investor Sentiment in the Stock Market.” Journal of Economic Perspectives 21, 129-151.

    Baker, M., and J. Wurgler, 2012. “Comovement and Predictability Relationships between Bonds and the Cross-section of Stocks,” Review of Asset Pricing Studies, 57-87

    Baker, M., J. Wurgler, and Y. Yuan 2012. “Global, Local, and Contagious Investor Sentiment,” Journal of Financial Economics, 104, 272-287.

    Barberis, N., A. Shleifer, and J. Wurgler, 2005, “Comovement,” Journal of Financial Economics, 75, 283-317

    Barberis, N., A. Shleifer, and R. Vishny, 1998, “A Model of Investor Sentiment,” Journal of Financial Economics 49, 307-343.

    Brown, G.W., and M.T. Cliff, 2004, “Investor Sentiment and the Near-Term Stock Market,” Journal of Empirical Finance, 11, 1-27.

    Brown, G.W., and M.T. Cliff, 2005, “Investor Sentiment and Asset Valuation,” Journal of Business, 78, 405-440.

    Brown, S.J., W.N. Goetzmann, T.Hiraki, N.Shirishi, and M.Watanabe, 2003, “Investor Sentiment in U.S. and Japanese Daily Mutual Fund Flows,” NBER working paper No. W9470.

    Buyuksahin, S. and M.A. Robe, 2012, “Speculators, Commodities and Cross-Market Linkages,” Available at SSRN: http://ssrn.com/abstract=1707103 or http://dx.doi.org/10.2139/ssrn.1707103.

    Chen, N-f, R. Kan, and M. Miller, 1993, “Are the Discounts on Closed End Funds a Sentiment Index?” Journal of Finance 48, 795-800.

    Cutler, D.M., J.M.Poterba, and L.H. Summers, 1991, “Speculative Dynamics,” Review of Economic Studies 58, 529–546.

    Daniel, K.D., D.A. Hirshleifer, and A.Subrahmanyam, A., 1998, “A Theory of Overconfidence, Self-attribution, and Security Market Under- and Over-reactions,” Journal of Finance 53, 1839-1885.

    Derrien, F. and A. Kecskés, 2009, “How Much Does Investor Sentiment Really Matter for Equity Issuance Activity?” European Financial Management 15, 787-813.

    Fama, E.F. and K.R. French, 1988, “Permanent and Temporary Components of Stock Prices,” Journal ofPolitical Economy, 96, April, 246–273.

    Haushalter, G.D., 2000, “Financing Policy, Basis Risk, and Corporate Hedging,” Journal of Finance, 55, 107-152

    Jegadeesh, N, 1990, “Evidence of Predictable Behavior of Stock Returns,” Journal of Finance, 45, 881-898.

    Jegadeesh, N., and S. Titman., 1993. “Returns to Buying Winners and Selling Losers: Implications for Stock Market Efficiency,” Journal of Finance, 48, 65–91.

    Jegadeesh, N., and S. Titman., 2001. Profitability of Momentum Strategies: An Evaluation of Alternative Explanations. Journal of Finance, 56, 699–720.

    Kumar, A., and C.M.C. Lee, 2006, “Retail Investor Sentiment and Return Comovements,” Journal of Finance, 61, 2451-2486.

    Lee, C.M.C., A. Shleifer, and R.H. Thaler, 1991, “Investor Sentiment and the Closed-End Fund Puzzle,” Journal of Finance 46, 75–109.

    Lehmann, B., 1990 “Fads, Martingales, and Market Efficiency Quarterly Journal of Economics, 105 February, 1-28.
    LeRoy, S.F. and R.D. Porter, 1981 “The Present Value Relation: Tests Based on Implied Variance Bounds,” Econometrica, 49, 97-113.

    Lo, A.W. and C. MacKinlay, 1988 “Stock Market Prices Do Not Follow Random Walks: Evidence from a Simple Specification Test,” Review of Financial Studies, 1, 41-66

    Miller, M.H. and C.W. Upton, 1985a “A Test of the Hotelling Valuation Principle,” Journal of Political Economy, 93, February, 1-25.

    Miller, M.H. and C.W. Upton, 1985b, “The Pricing of Oil and Gas: Some Further Results,” Journal of Finance 40, 1009-1018.

    Moskowitz, T.J.,Y.H. Ooi and L.H. Pedersen, 2011, “Time Series Momentum,” Journal of Financial Economics, 104, 228-250.

    Neal R. and S.M.Wheatley, 1998 “Do Measures of Sentiment Predict Returns?” Journal of Financial and Quantitative Analysis, 33, 523-535.

    Pindyck, R S, 1993, “The Present Value Model of Rational Commodity Pricing,” The Economic Journal 103, 511-530.

    Poterba, J.M. and L.H. Summers, 1988, “Mean Reversion in Stock Prices: Evidence and Implications,” Journal of Financial Economics 22 October, 27-60.

    Qiu, L. and I. Welch, 2004, “Investor Sentiment Measures” NBER Working Paper No. W10794.

    Shiller, R.J., 1981, “Do Stock Prices Move Too Much to be Justified by Subsequent Changes in Dividends?” American Economic Review, June, 71, 421–436

    Singleton, K.J., 2011, “Investor Flows and the 2008 Boom/Bust in Oil Prices,” Available at SSRN: http://ssrn.com/abstract=1793449 or http://dx.doi.org/10.2139/ssrn.1793449.

    B, Swaminathan, 1996, “Time Varying Expected Small Firm Returns and Closed-End Fund Discounts,” The Review of Financial Studies, 845-887.

    APPENDIX: PRINCIPAL DATA SOURCES

    Stock samples All common stocks with NAIC code of 211111 or SIC code of 1311 (oil production or exploration) that were listed on the NYSE or Nasdaq and whose issuers were incorporated in the USA. Returns data were taken from the CRSP monthly database. Portfolio returns were constructed from equally weighted holdings in all stocks with valid returns data for that month. Portfolio returns were then converted to continuously compounded returns.

    Oil prices Month-end spot prices for West Texas Intermediate taken from the Energy Administration website at www.eia.gov

    Natural gas prices Monthly spot prices for Natural Gas Wellhead Price taken from Globalfindata. Prices are month averages from March 1983-December 1995 and end-of-month from January 1996- January 2011.

    Contango NYMEX futures prices are taken from Quandl at www.quandl.com.. The contango is defined as the price of the contract that is sixth nearest to delivery divided by the price of the contract that is closest to delivery. The change in contango is defined as ln (contangot)– ln (contangot).

    Baker-Wurgler Sentiment Index SENT1constructed from IPO volume, IPO first-day returns, market turnover, and the market-book ratio of high-volatility stocks relative to that of low-volatility stocks. See http://people.stern.nyu.edu/jwurgler/. The index is rescaled to have mean zero and unit standard deviation

    American Association of Individual Investors (AAII) Investor Sentiment Survey Proportion of investors reporting they are bullish divided by the total proportion reporting that they are either bullish or bearish (i.e. not neutral). Taken from final week’s survey in each month as reported on www.aaii..com/sentimentsurvey. The index is rescaled to have mean zero and unit standard deviation. Data are available from July 1987.

    Figure 1: Measures of sentiment

    This figure shows plots of the Baker-Wurgler and Bullish sentiment indexes from July 1987 to January 2011. In both cases the index has been standardized to have a mean of zero and standard deviation of unity.

    Cooper-oil-presentation-Q-group-2015-10

    Cooper-oil-presentation-Q-group-2015-101

    Figure 2A: Open interest in oil futures

    This figure shows on a log scale the level of open interest in crude oil futures by non-
    commercial traders from January 1986 to December 2012.

    oil-paper-Q-GROUP-2425

    Source: CFTC

    Figure 2B: Cumulative flows into commodity hedge funds

    This figure shows on a log scale the cumulative quarterly cash flows into Managed Futures Funds in millions of dollars from first quarter 1994 to the third quarter 2010.

    oil-paper-Q-GROUP-2525

    Source: TASS Research

    Table 1: Structure of the VAR and hypotheses about coefficients

    oil-paper-Q-GROUP-2626

    Table 2: Summary statistics

    The table shows summary statistics for portfolio returns and log changes in the fundamental variables, monthly data 1988.01-2011.01. The underlying stock prices are for a balanced sample of 121 upstream oil firms. Lo is the return on the portfolio of stocks in the lowest volatility tercile. Hi is the return on the portfolio in the highest volatility tercile. All is the return on the portfolio of the entire sample. Hi-Lo is the return on the portfolio that is long the Hi portfolio and short the Lo portfolio. Δ WTI is the first difference of the log spot price of WTI.Δ Gas is the first difference of the log spot natural gas price. ΔCont is the first difference in the log of the WTI contango, where contango is defined as the ratio of the price of the sixth most distant futures contract to that of the nearest contract.

    Panel A: Means and standard deviations of portfolio returns and changes in fundamental variables.

    oil-paper-Q-GROUP-2727

    Panel B: Correlation coefficients. Below the diagonal is the Pearson coefficient and above the diagonal in italics is the Spearman’s rank coefficient. Bold face indicates coefficient is significantly different from zero at the 90% confidence level.

    oil-paper-Q-GROUP-2728

    Panel C: Correlation coefficients. Correlations with lagged sentiment indices. Entries show the correlations between the return in a month and the level of the two sentiment indices at the beginning of that month. St-1 is the Baker Wurgler sentiment index lagged 1 month, and Bt-1 is the percentage of bullish respondents to AAII survey lagged 1 month.

    oil-paper-Q-GROUP-2828

    Table 3: Variance ratios of the All-stocks portfolio

    The table shows the variance of the All-stocks portfolio over different intervals as a proportion of the 1-month variance rate, measured over 1983.01-2012.12. The portfolio invests equal amounts each month in all U.S. oil production and exploration stocks listed on the New York Stock Exchange.

    oil-paper-Q-GROUP-2929

    Table 4: Regression of portfolio returns on lagged sentiment with and without controls for fundamentals

    The table summarizes the results of a regression of returns on each of two oil portfolios against two lagged sentiment measures, St-1 and Bt-1, for the period 1988.01-2011.01. The coefficients on the sentiment measures are shown without and with controls for the effect of fundamentals. The All-stocks portfolio invests equal amounts each month in all U.S. oil production and exploration stocks listed on the New York Stock Exchange. The Hi-Lo portfolio has long positions in the tercile of stocks with the highest variance over the prior 60-month period and short positions in the tercile of stocks with the lowest variance over the prior 60-month period. The regressions are estimated from monthly data using OLS with a constant (not reported). t-statistics in parentheses. * denotes 10% significance, ** 5% significance, and *** 1% significance.

    oil-paper-Q-GROUP-3030parentheses. * denotes 10% significance, ** 5% significance, and *** 1% significance.

    Table 5: VAR of fundamental values for the Hi-Lo portfolio, deviations from fundamentals, Baker-Wurgler sentiment, and Bullish sentiment

    The table shows a VAR of the return to fundamentals (Ft – Ft-1) for the Hi-Lo portfolio,
    return to deviations from fundamentals (NFt – NFt-1) for the Hi-Lo portfolio, Baker-Wurgler sentiment (St-1), and Bullish sentiment (Bt-1). Monthly data 1988.01 to 2011.01. The Hi-Lo portfolio has long positions in the tercile of stocks with the highest variance over the prior 60-month period and short positions in the tercile of stocks with the lowest variance over the prior 60-month period. Returns on the portfolio each month are split between fundamentalreturns and deviations from fundamentals. Fundamental returns are based on a regression of the portfolio return on the oil price return, gas price return, and the change in contango in the oil market, using a rolling window of 60 months prior to the month for which the stock return is split. The system is estimated using GMM with Newey-West correction for the standard errors. t-statistics in parentheses. * denotes 10% significance, ** 5% significance, and *** 1% significance.

    oil-paper-Q-GROUP-3131

    Table 6: Sub-period VAR results

    The table shows a VAR of the return to fundamentals (Ft – Ft-1) for the Hi-Lo portfolio, return to deviations from fundamentals (NFt – NFt-1) for the Hi-Lo portfolio, Baker-Wurgler sentiment, S, and bullish sentiment, B. Monthly data 1988.01 to 2011.01, divided at 2000.12. Stock returns each month are split between fundamental returns and deviations from fundamentals based on a regression of the stock return on the oil price return, gas price return, and the change in contango in the oil market, using a rolling window of 60 months prior to the month for which the stock return is split. Coefficient θi,j measures the autoregression coefficient of variable j on the lagged value of variable i.The system is estimated using GMM with Newey-West correction for the standard errors. t-statistics in parentheses. * denotes 10% significance, ** 5% significance, and *** 1% significance.

    oil-paper-Q-GROUP-3232

    Table 7:  Effect of sentiment on returns using different measurement intervals

    The table shows results from a VAR of the return to fundamentals (Ft – Ft-1) for the Hi-Lo portfolio, return to deviations from fundamentals (NFt – NFt-1) for the Hi-Lo portfolio, Baker-Wurgler sentiment (S), and Bullish sentiment (B). The differencing interval in months is shown in the left column. Monthly data 1988.-01 to 2011.01, divided at 2000.12. Stock returns each month are split between fundamental returns and deviations from fundamentals based on a regression of the stock return on the oil price return, gas price return, and the change in contango in the oil market, using a rolling window of 60 months prior to the month for which the stock return is split. The system is estimated using GMM with Newey-West correction for the standard errors.t-statistics in parentheses. * denotes 10% significance, ** 5% significance, and *** 1% significance.

    oil-paper-Q-GROUP-3333

    Table 8: Regression of changes in the oil price on “deep” fundamental factors and lagged sentiment (second period)

    Panel A summarizes the results of a regression of changes in the price of WTI against two lagged sentiment measures, St-1 and Bt-1, and controls for fundamental factors. Oil consumption is OECD consumption. Economic activity is US Industrial Production. The regressions consist of 121 rolling 12-month and 3-month changes. The regressions are estimated using OLS with a constant (not reported). Standard errors in the overlapping regressions are calculated using a Newey-West correction. t-statistics in parentheses. * denotes 10% significance, ** 5% significance, and *** 1% significance. Panel B shows the results of the same regressions without the deep fundamental controls and only the sentiment variables.

    oil-paper-Q-GROUP-3434


  • Ian Cooper Presentation

    The behaviour of
    sentiment-induced share returns:
    Measurement when
    fundamentals are observable

    Richard Brealey
    Ian Cooper
    Evi Kaplanis
    London Business School

     

     

    Share prices and sentiment

    • Many theories about the effect of sentiment on share prices are expressed in terms of deviations from fundamentals
      — Daniel, Hirshleifer, and Subrahmanyam (1998),
      Barberis, Shleifer, and Vishny (1998)
    • Barberis et al (2012):
      — Sentiment pushes prices away from fundamentals in the
      short term
      — In the medium-term they mean-revert
    • The effect should be concentrated in “hard-to-arbitrage” stocks
    • On the other hand, there is evidence of market-wide sentiment effects (Arif and Lee (2014)) which may affect all stocks

     

     

    The importance of controlling for fundamentals

    • Results about sentiment and returns can be different if controls for fundamentals are not included (Barberis, Shleifer, Wurgler (2005))
    • Various methods have been tried:
      — Use a sample where fundamentals are presumed to stay the same (Barberis et al)
      — Use control variables: Derrien and Kecskés (2009) show the effect of sentiment on equity issuance disappears when controls are included
      — Use a DCF estimate of fundamental value: Brown and Cliff (2005) show that DDM values are positively related to sentiment. But DDM values are hard to estimate and depend on a discount rate model.
      — Use long-short portfolios: BW use long hard-to-arbitrage, short easy-to-arbitrage portfolios, depends on long-short portfolio being neutral with respect to fundamental factors
    • Our study: Use a sample where we believe we have a good proxy for fundamental value

    Our study

    • Use a sample of firms where we should have an excellent proxy for fundamental value: Upstream oil stocks
      — Empirically, oil price explains cross-section of values (Miller and Upton (1985a), (1985b))
      – Theory says discount rate uncertainty is not so important for these stocks once the oil price is controlled for (Hotelling)
    • Split the stock return into the part caused by changes in fundamentals and the part that represents the deviation from fundamentals
    • Test whether sentiment predicts the behavior of the deviation from fundamentals in the way that theory says it should
    • Two measures of sentiment:
      — Baker-Wurgler index should predict reversion to fundamentals
      — Retail investor sentiment index should predict momentum trading

    Main results

    • Sentiment predicts returns in the expected way
      a. Baker-Wurgler (“BW”) sentiment predicts mean-reversion
      b. Retail “Bullish” sentiment predicts momentum
    • The effect appears only after the major increase in interest in commodities, particularly oil, in 2000
    • Only then do fundamentals explain a large part of returns
    • The effect appears to work via the fundamentals themselves rather than through deviations from fundamentals
    • Economic size is about 0.3% per month from bullish sentiment and about 0.3% per month from BW sentiment
    • The effect appears in a Hi-Lo portfolio, but because it has a net exposure to fundamentals

    Data

    • 121 US oil exploration and production stocks (SIC 1311)
    • Monthly data 1983-2011 (CRSP)
    • Fundamentals measured by spot oil price, spot gas price, oil contango
    • Equally weighted portfolio of all stocks
    • Sub-portfolios sorted by variance over prior 60 months
    • Highest tercile variance portfolio minus lowest tercile variance portfolio (“Hi-Lo”)
    • BW sentiment
    • American Association of Individual Investors “bullishness” survey

    Serial properties of our data

    • Our data show standard properties of returns data:
      — Short-term momentum
    • Lehman (1990), Jegadeesh (1990), Jagadeesh and Titman (1993)
      — Longer term mean-reversion
    • Poterba and Summers (1988), Lo and MacKinlay (1998), Cutler, Poterba, and Summers (1991)
      — Consistent with excess volatility relative to fundamentals
    • Shiller (1981), LeRoy and Porter (1981)

    Variance ratios of our sample

    Variance ratios of the All-stocks portfolio
    relative to the 1-month variance rate
    Equally-weighted all US oil production and exploration
    stocks listed on NYSE (121 stocks)

    sssss12

    Sentiment measures

    • Baker-Wurgler composite index of sentiment (index of IPO volume and returns, market volume, relative pricing of high and low volatility stocks)
    • Survey data (Qiu and Welch (2004), Brown and Cliff (2004, 20025))
      — We use American Association of Individual Investors survey: Proportion who report that they are bullish (“Bullish” sentiment)
    • Hypothesis is that BW sentiment is an index of the level of mispricing, so returns following high BW sentiment will be low, reflecting reversion to fundamental value, and that this will be more pronounced for hard-to-arbitrage stocks
    • Hypothesis is that Bullish sentiment picks up momentum trading and high bullish sentiment will be followed by high returns

    wurgler

    Baker Wurgler

    • BW sentiment
      monthly serial
      correlation
      0.96
    • Bullish
      sentiment
      monthly serial
      correlation
      0.45
    • Monthly
      correlation
      between them
      0.09

     

    Sub-period analysis: Motivation

    • For our sub-period analysis, we split the period at the end of 2000
    • Open interest in oil futures (CFTC) and flows into commodity hedge
      funds (Managed Futures Funds, TASS data) increased significantly
      post-2000

    Cooper-oil-presentation-Q-group-2015-11

    Preliminary results

    • Across the sample period the fundamentals explain 41% of the
      variance in the All-stocks portfolio returns
    • Hi-Lo portfolio returns are not well hedged against fundamentals, oil
      and gas price are highly significant factors for Hi-Lo returns
    • Correlation between returns and lagged sentiment are larger for high-
      variance portfolio and the Hi-Lo portfolio: appears at first sight to be
      consistent with hard-to arbitrage hypothesis
    • Including controls reduces considerably the significance of sentiment
      variables are reduced considerably for both portfolios for entire
      period and for both sub-periods

    Splitting returns

    • To avoid overfitting the coefficients, each month we split
      the returns into fundamentals and non-fundamentals using
      lagged parameter values
    • Estimate fundamentals regression for prior 60 months:
      Rit = ai + bi Δ WTIt + ci Δ GASt + di Δ Conti + uit
    • Fundamentals: Returns to WTI spot price, Gas spot price,
      change in Contango of 6th-1st futures price
    • Use estimated coefficients and actual realizations of{ΔWTIt, ΔGASt, ΔContt}to estimate fundamental return for that month
    • Remainder of return is non-fundamental
    • Roll the windows

    Regression structure

    • Fundamental return from t-1 to t is: Ft – Ft-1
    • Non-fundamental return from t-1 to t is: NFt – NFt-1
    • Regress Ft – Ft-1 and NFt – NFt-1 on lagged returns and lagged sentiment measures
    • Include BW sentiment, St, and Bullish sentiment, Bt, in a VAR

    VAR Structure and Hypotheses

    Cooper-oil-presentation-Q-group-2015-152

     

    VAR for entire period: Hi-Lo portfolio

    Cooper-oil-presentation-Q-group-2015-162

    VAR Results: Summary

    • BW predicts reversion to mean
    • Bullish predicts momentum
    • Both effects operate through fundamentals (i.e. oil and gas prices)
    • Hi-Lo portfolio is exposed to fundamentals
    • No effect of sentiment through deviations from fundamentals
    • BW sentiment responds to fundamental return, not non-fundamental return

    Sub-period results: Hi-Lo portfolio

    Cooper-oil-presentation-Q-group-2015-181

    Different measurement intervals

    Cooper-oil-presentation-Q-group-2015-194

    Sentiment transmission in the second period

    Cooper-oil-presentation-Q-group-2015-20

    Robustness

     

    • Including lagged market return does not change results
    • Looking only at high variance portfolio alone does not change results
    • Including 274 NASDAQ exploration and production stocks:- High vol portfolio is almost exclusively NASDAQ
      – Hi-Lo is better hedged against fundamentals
      – Effect of sentiment still appears only in second period
      – Predictability of Hi-Lo return is lower
      – Effect of sentiment appears only through fundamentals
      when return is split (not in current paper)

    What if the oil price is not “fundamental”?

    • Evidence that oil and gas prices are affected by sentiment (Pindyck (1993))
    • Regress oil price change on sentiment and “deep” fundamentals
    • Deep fundamentals are changes in:
      – Oil production, oil consumption, gdp, proven oil
      reserves (annual data), oil inventories (quarterly)
    • Low power test, but coefficient on BW is mildly significant and negative, coefficient on bullish is mildly significant and positive
    • Consistent with channel of influence of sentiment being
      through oil price

    Regression of changes in the oil price on deep
    fundamentals and sentiment

    Cooper-oil-presentation-Q-group-2015-23

    Results: Summary

    • Sentiment predicts returns in the expected way
      a. Baker-Wurgler sentiment predicts mean-reversion
      b. Retail bullish sentiment predicts momentum
    • Economic size is about 0.3% per month from bullish sentiment and about 0.3% per month from BW sentiment
    • The sentiment effect for these stocks operates through fundamentals(i.e. oil and gas prices)
    • Significant effects appear only after 2000 when:
      – interest in oil as an asset increased considerably
      – fundamentals became more important in determining upstream oil stock returns
    • Inconsistent with theories where sentiment causes deviations from
      fundamental value
    • Inconsistent with with hard-to-arbitrage theories about sentiment
      effects

    Caveats and issues

    1. Maybe our results are sector-specific because, in this
      instance, what we use as fundamentals are themselves
      tradeable (Basak and Pavlova (2014))
    2. The highly time-varying effect of sentiment here has a
      clear cause that may not be present for other stocks
    3. Our result that that the Hi-Lo portfolio loads on fundamental factors may not apply in other cases
    • Raises the question of what “fundamental value” means in
      this instance
    • Raises question of why sentiment should predict returns
      in “easy-to-arbitrage” assets like oil


  • Robert Stambaugh Paper

    Do Funds Make More When They Trade More?

    ˇLuboˇs P´astor
    Robert F. Stambaugh
    Lucian A. Taylor *

    February 9, 2015

    Abstract

    We find that active mutual funds perform better after trading more. This time-series relation between a fund’s turnover and its subsequent benchmarkadjusted return is especially strong for small, high-fee funds. These results are consistent with high-fee funds having greater skill to identify time-varying profit opportunities and with small funds being more able to exploit those opportunities. In addition to this novel evidence of managerial skill and fund-level decreasing returns to scale, we find evidence of industry-level decreasing returns: The positive turnover-performance relation weakens when funds act more in concert. We also identify a common component of fund trading that is correlated with mispricing proxies and helps predict fund returns.

    *P´astor is at the University of Chicago Booth School of Business. Stambaugh and Taylor are at the Wharton School of the University of Pennsylvania. Email:lubos.pastor@chicagobooth.edu, stambaugh@ wharton.upenn.edu, luket@wharton.upenn.edu. We are grateful for comments from Jonathan Berk, Gene Fama, Vincent Glode, Todd Gormley, Christian Hansen, Marcin Kacperczyk, David Musto, Jonathan Reuter, Sergei Sarkissian, Clemens Sialm, and the audiences at the 2014 German Finance Association conference and the following universities and other institutions: Aalto, BI Oslo, Cass, Cheung Kong, Chicago, Copenhagen, Houston, Mannheim, McGill, NBIM, NHH Bergen, SAIF, Tsinghua PBCSF, Tsinghua SEM, and Wharton. We are also grateful to Yeguang Chi and Gerardo Manzo for superb research assistance.

    1. Introduction

    Mutual funds invest trillions of dollars on behalf of retail investors. The lion’s share of this money is actively managed, despite the growing popularity of passive investing.1 Whether skill guides the trades of actively managed funds has long been an important question, given active funds’ higher fees and trading costs. We take a fresh look at skill by analyzing time variation in active funds’ trading activity. We explore a simple idea: A fund trades more when it perceives greater profit opportunities. If the fund has the ability to identify and exploit those opportunities, then it should earn greater profit after trading more heavily.
    We find that funds do earn more after trading more heavily. Specifically, a fund’s turnover positively predicts the fund’s subsequent benchmark-adjusted return. This new evidence of skill comes from our sample of 3,126 active U.S. equity mutual funds from 1979 through 2011. The result is significant not only statistically but also economically: a one-standarddeviation increase in turnover is associated with a 0.65% per year increase in performance for the typical fund. Funds seem to know when it’s a good time to trade.

    The turnover-performance relation is stronger for funds that charge higher fees as well as funds that are smaller in size. These results further support the presence of skill and provide novel evidence of decreasing returns to scale at the individual-fund level. Because smaller funds trade smaller amounts, decreasing returns to scale associated with liquidity costs allow smaller funds to better exploit time-varying profit opportunities. Identifying those opportunities requires skill, and managers with greater skill should earn higher fees. Our results suggest that higher-fee funds are more skilled at identifying time-varying profit opportunities, and that smaller funds can better exploit those opportunities. Small, high-fee funds also have especially volatile turnover, consistent with their having greater abilities to identify and exploit time-varying opportunities.

    Fund trading has a common component that appears to be related to mispricing in the stock market. Average turnover across funds-essentially the first principal component of turnover-is significantly related to three proxies for potential mispricing: investor sentiment, cross-sectional dispersion in individual stock returns, and aggregate stock market liquidity. Funds trade more when sentiment or dispersion is high or liquidity is low, suggesting that stocks are more mispriced when funds collectively perceive greater profit opportunities.

    These perceptions seem justified, because average turnover positively predicts a fund’s

    1As of 2013, mutual funds worldwide have about $30 trillion of assets under management, half of which is managed by U.S. funds. About 52% of U.S. mutual fund assets are held in equity funds, and 81.6% of the equity funds’ total net assets are managed actively (Investment Company Institute, 2014).

    future return, even when we control for the fund’s own turnover. This predictive relation is highly significant: a one-standard-deviation increase in average turnover is associated with a 0.80% per year increase in fund performance. A fund’s performance can thus be predicted not only by its own turnover but also by other funds’ turnover. More trading by other funds appears to indicate greater profit opportunities in general. Any opportunity identified by a given fund is likely to be more profitable if there is generally more mispricing at that time, as indicated by other funds’ heavy trading.

    The positive relation between average turnover and future fund returns is weaker in periods when funds act more in concert, as measured by the average correlation among the funds’ benchmark-adjusted returns. This evidence is consistent with decreasing returns to scale at the level of the active-management industry (see P´astor and Stambaugh, 2012, and P´astor, Stambaugh, and Taylor, 2014). When funds act in concert, pursuing the same profit opportunities, prices move and those opportunities become harder to find.

    The relation between a fund’s return and lagged average turnover becomes even stronger when the average is calculated only across funds in the same size-fee category as the given fund. The turnover of other same-category funds thus seems to be a better signal of the fund’s profit opportunities than the turnover of different-category funds. This result suggests that funds in different size-fee categories pursue somewhat different opportunities.

    We provide an investment perspective on the turnover-performance relation by applying a novel mapping between time-series regressions and investment strategies. We show that the estimated slope coefficient from our regression of a fund’s return on its lagged turnover is closely related to the average return of a strategy that dynamically allocates between the fund and its benchmark. The strategy invests more in the fund following higher turnover by the fund. When implemented for all funds and combined with a short position in the strategy’s static counterpart, this “timing” strategy produces an impressive annual Sharpe ratio of 0.79, which exceeds the Sharpe ratios of the market, size, value, and momentum factors over the same 1979–2011 period. This finding provides an additional perspective on the economic significance of the turnover-performance relation identified by our study.

    Finally, we consider a cross-sectional investment strategy that captures an element of the time-series relation between turnover and performance. Every month, we sort funds into portfolios based on the ratio of a fund’s recent turnover to the trailing historical average of that fund’s turnover. We find that funds whose turnover is high based on this ratio tend to outperform funds whose turnover is low. Funds that have recently traded more than usual perform especially well when there are better profit opportunities in the market, as judged by high sentiment. This strategy’s performance is statistically significant, but it is weaker than the performance of the timing strategy, which is more directly motivated by our turnover-performance relation.

    The literature investigating the skill of active mutual funds is extensive. Average past performance delivers a seemingly negative verdict, since many studies show that active funds have underperformed passive benchmarks, net of fees.2 Yet active funds can have skill. Skilled funds might charge higher fees, and some funds might be more skilled than others. Moreover, with fund-level or industry-level decreasing returns to scale, skill does not equate to average performance, either gross or net of fees.3

    Our investigation of skill adds a new dimension to the literature on the relation between mutual fund turnover and fund performance. The empirical evidence on this relation is mixed. For example, Elton, Gruber, Das, and Hlavka (1993) and Carhart (1997) find a negative turnover-performance relation, Wermers (2000) and Edelen, Evans, and Kadlec (2007) find no significant relation, and Dahlquist, Engstr¨om and S¨oderlind (2000) and Chen, Jagadeesh and Wermers (2001) find a positive relation. The main difference between these studies and ours is that all of these studies examine the cross-sectional relation between turnover and performance, whereas we focus on the time-series relation.4

    We obtain our time-series results from panel regressions of fund returns on lagged fund turnover and various controls, including fund fixed effects. Fund fixed effects are crucial for finding a positive turnover-performance relation. With fund fixed effects, identification comes from within-fund time variation in turnover and performance, not from the crosssectional variation exploited in the prior studies mentioned above. If we drop fund fixed effects, the positive turnover-performance relation weakens to marginally significant. The relation wanes further if we replace fund fixed effects with month fixed effects, thereby isolating pure cross-sectional variation. These results underline the time-series nature of the turnover-performance relation. We do not find that higher-turnover funds perform better; we find that a given fund performs better when it trades more.

    To help interpret our results from panel regressions with fixed effects, we present a formula for the slope coefficient from such regressions estimated in an unbalanced panel. The slope

    2See, for example, Jensen (1968), Elton, Gruber, Das, and Hlavka (1993), Malkiel (1995), Gruber (1996), Carhart (1997),Wermers (2000), P´astor and Stambaugh (2002), and Fama and French (2010), among others. 3See Berk and Green (2004), P´astor and Stambaugh (2012), Berk and van Binsbergen (2014), P´astor, Stambaugh, and Taylor (2014), and Stambaugh (2014). 4The two studies that find a positive relation differ from our study in other ways as well. Dahlquist, Engstr¨om and S¨oderlind find this positive relation in a small sample of Swedish mutual funds (80 funds in 1993–1997); our sample of U.S. funds is much larger. Chen, Jagadeesh and Wermers find this relation based on the returns of the funds’ disclosed stock holdings, whereas we analyze the returns of the funds themselves.

    from a panel regression with fund fixed effects is a weighted average of the slopes estimated fund by fund in pure time-series regressions of return on turnover. Greater weight is given to funds with longer samples and more volatile turnover. Analogously, with month fixed effects, the panel regression slope is a weighted average of the month-by-month estimates in pure cross-sectional regressions of return on turnover. More generally, our formula also clarifies the relation between a panel regression slope and the well-known estimator of Fama and MacBeth (1973). The Fama-MacBeth estimator emerges as a special case from a panel regression with month fixed effects if the panel is balanced and the cross-sectional variance of the independent variable is constant over time.

    While we find that funds perform better after increasing their trading activity, others have related fund activity to performance in different ways. Kacperczyk, Sialm, and Zheng (2005) find that funds that are more active in the sense of having more concentrated portfolios perform better. Kacperczyk, Sialm, and Zheng (2008) find that a fund’s actions between portfolio disclosure dates, as summarized by the “return gap,” positively predict fund performance. Cremers and Petajisto (2009) find that funds that deviate more from their benchmarks, as measured by “active share,” perform better. Cremers, Ferreira, Matos, and Starks (2014) find similar results. In the same spirit, Amihud and Goyenko (2013) find better performance among funds having lower R-squareds from benchmark regressions. These studies are similar to ours in that they also find that more active funds perform better, but there are two important differences. First, all of these studies measure fund activity in ways different from ours. Second, all of them identify cross-sectional relations between activity and performance, whereas we establish a time-series relation.

    Given this time-series perspective, our study is also related to the literature on time variation in mutual fund performance. Some authors, inspired by Ferson and Schadt (1996), model performance as a linear function of conditioning variables (e.g., Avramov and Wermers, 2006). Others relate fund performance to the business cycle (e.g., Moskowitz, 2000, Glode, 2011, Kosowski, 2011, and Kacperczyk, van Nieuwerburgh, and Veldkamp, 2013, 2014), to aggregate market returns (Glode, Hollifield, Kacperczyk, and Kogan, 2012), and to time variation in fund risk (e.g., Brown, Harlow, and Starks, 1996, and Huang, Sialm, and Zhang, 2011). None of these studies relate fund performance to fund turnover.

    While we analyze funds’ ability to time their turnover, others have investigated the value of active fund management by examining different fund actions. Chen, Jegadeesh, and Wermers (2000) find that stocks recently bought by funds in aggregate outperform stocks recently sold, suggesting that funds have stock-picking skill. Baker, Litov, Wachter, and Wurgler (2010) find that much of this outperformance takes place around corporate earnings announcements, indicating one likely source of the funds’ skill. Cohen, Coval, and P´astor (2005) find that funds whose portfolio decisions are similar to those of other funds with strong track records perform better. Cohen, Frazzini, and Malloy (2008) find that fund managers perform better when they trade shares of firms they are connected to through their educational networks. Like us, all of these studies report that active management adds value, but they examine different dimensions of fund skill. Our finding that funds are able to successfully time their trading activity seems new in the literature.

    Lastly, our analysis of the common variation in fund turnover is related to the literature on correlated trading behavior of mutual funds, or “herding.” Early studies include Nofsinger and Sias (1999) and Wermers (1999). More recently, Koch, Ruenzi, and Starks (2010) and Karolyi, Lee, and van Dijk (2012) argue that such correlated trading gives rise to commonality in liquidity among stocks. Commonality in individual stock turnover is analyzed by Lo and Wang (2000), Cremers and Mei (2007), and others. None of these studies examine fund turnover. Our analysis of the common component of fund turnover seems novel.

    The rest of the paper is organized as follows. Section 2 documents the basic turnoverperformance relation. Section 3 examines this relation across categories of funds based on size and fees. Section 4 analyzes the common component of turnover and its predictive power for fund returns. Section 5 provides two investment perspectives on the turnover-performance relation. Section 6 concludes.

    2. The Turnover-Performance Relation

    Active mutual funds pursue alpha-returns in excess of their benchmarks. The funds’ managers perceive opportunities for producing alpha and trade to exploit them. A manager trades more when he identifies more alpha-producing opportunities, so a skilled manager should perform better after he trades more. We look for such evidence of skill by estimating the relation between a fund’s turnover and its subsequent return. We specify this turnover-performance relation for a given fund i as the linear regression

    stambaugh1

    The skill we investigate is an ability to exploit opportunities in period t − 1 for which a nontrivial fraction of the payoff occurs in period t. A prime example is a purchase of an underpriced security in period t − 1 followed by the correction of the mispricing in period t. One can imagine other forms of skill that we would not detect. For example, a fund could have skill to identify short-horizon opportunities, such as liquidity provision, that deliver all of their profits in the same period in which the fund trades to exploit those opportunities. Such skill would impart no time-series relation between turnover in period t−1 and performance in period t. Similarly, the turnover-performance relation would be very weak, possibly undetectable, for a fund skilled only in identifying long-horizon opportunities that deliver most of their payoffs after the next period. Moreover, detecting skill using the turnover-performance relation requires variation over time in the extent to which profitable opportunities arise. In principle, a fund could be skilled at identifying opportunities that arise to the same extent every period. Such skill would impart no variation over time in trading and expected payoffs. Although the turnover-performance relation cannot detect all forms of skill, it nevertheless provides novel insights into the ability of funds to identify and exploit time-varying profit opportunities.

    We explore the turnover-performance relation using the dataset constructed by P´astor, Stambaugh, and Taylor (2014), who combine CRSP and Morningstar data to obtain a sample of 3,126 actively managed U.S. domestic equity mutual funds covering the 1979–2011 period. To measure the dependent variable Ri,t, we follow the above study in using GrossRi,t, the fund’s net return minus the return on the benchmark index designated by Morningstar, plus the fund’s monthly expense ratio taken from CRSP. We use gross rather than net returns because our goal is to measure a fund’s ability to outperform a benchmark, not the value delivered to clients after fees. We estimate all regressions at a monthly frequency, but a fund’s turnover is reported only as the total for its fiscal year. Thus, we measure turnover, Xi,t−1, by the variable FundTurni,t−1, which is the fund’s turnover for the most recent 12- month period that ends before month t. This measure, reported by CRSP, is defined by the SEC as the lesser of the fund’s total purchases and sales, divided by the fund’s 12-month average total net asset value. By largely excluding turnover arising from flows to and from the fund, this measure reflects portfolio decisions to replace some holdings with others. We winsorize FundT urni,t−1 at the 1st and 99th percentiles.

    To increase the power of our inferences in equation (1), we estimate a pooled time-series and cross-sectional regression that imposes the restriction

    b1 = b2 = · · · = b ,               (2)

    either across all funds or across funds within size-fee categories discussed later. We include fund fixed effects, so that b reflects only the contribution of within-fund time variation in turnover. The fund fixed effects correspond to the ai’s in equation (1) when the restriction in (2) is imposed across all funds. When later allowing b to differ across size-fee categories, we also include fixed effects for those categories, in which case a fund’s ai equals the sum of its fund and category fixed effects. The regression specification combining equations (1) and (2), which isolates the time-series contribution of turnover to performance, is our main specification. For comparison, we also consider other specifications, as we explain next.

    stambaugh2

    stambaugh3

    stambaugh4

    2.2. Robustness

    The positive turnover-performance relation documented above, which is our main result, is robust to a variety of specification changes. We summarize the robustness results here and report them in detail in the online appendix, which is available on our websites. We have already shown that the turnover-performance relation obtains whether or not month fixed effects are included in the panel regression, which rules out all aggregate variables as the source of this relation. Furthermore, the relation obtains when we include benchmarkmonth fixed effects, ruling out any variables measured at the benchmark-month level.7 An example of such a variable is benchmark turnover, which can be reflected in a fund’s turnover to the extent that some of the fund’s trading passively responds to reconstitutions of the fund’s benchmark index. Adding benchmark-month fixed effects has a tiny effect on the estimated turnover-performance relation, strengthening our interpretation of this relation as being driven by skilled active trading. The relation also obtains, and is equally strong, when gross fund returns are replaced by net returns.

    Importantly, the positive turnover-performance relation does not obtain in a placebo test in which we replace active funds by passive index funds, as identified by Morningstar. When we produce the counterpart of Table 1 for the universe of passive funds, we find no slope coefficient significantly different from zero. In fact, the estimated slope coefficients in the specifications with fund fixed effects are not even positive (the corresponding t-statistics in the bottom row of Table 1 are -0.51 and -1.07). This result is comforting because passive funds should not exhibit any skill in identifying time-varying profit opportunities. The fact that the turnover-performance relation emerges for active funds but not passive funds supports our skill-based interpretation of this relation.

    Additional support for our interpretation comes from another placebo test, in which we replace our turnover measure, FundT urn, by flow-driven turnover. Funds often trade in response to inflows and outflows of capital. Such flow-driven trading is fairly mechanical in that its timing is determined mostly by the fund’s investors rather than the fund’s manager. Therefore, we expect flow-driven turnover to exhibit a weaker relation to fund performance compared to FundT urn, which largely excludes flow-driven trading, as noted earlier. To test this hypothesis, we construct two measures of flow-driven fund turnover. Both measures rely on monthly dollar flows, which we back out from the monthly series of fund size and fund returns, and both cover the same 12-month period as FundT urn. The first measure is

    7Gormley and Matsa (2014), among others, advocate the use of a fixed-effects estimator as a way of controlling for unobserved group heterogeneity in finance applications.

    the sum of the absolute values of the 12 monthly dollar flows, divided by the average fund size during the 12-month period. The second measure is the smaller of two sums, one of all positive dollar flows and one of all negative flows during the 12-month period, divided by average fund size. Consistent with our hypothesis, we find that neither measure of flowdriven turnover has any predictive power for fund returns, whether or not we include various controls such as FundT urn. Moreover, the inclusion of flow-driven turnover does not affect the significant predictive power of FundT urn for fund returns.

    We estimate the turnover-performance relation at the monthly frequency. Even though funds report their turnover only annually, most of the variables used in our subsequent analysis, such as fund returns, fund size, industry size, sentiment, volatility, liquidity, correlation, and business-cycle indicators, are available on a monthly basis. Therefore, we choose the monthly frequency in an effort to utilize all available information. Nonetheless, when we reestimate the turnover-performance relation by using annual fund returns, we find a positive and highly significant time-series relation, just like in Table 1. In addition, we consider a specification that allows the slope coefficient from the monthly turnover-performance regression to depend on the number of months between the end of the 12-month period over which FundT urn is measured and the month in which the fund return is computed. Specifically, we add a term to the right-hand side of the regression that interacts the above number of months with FundT urn. We find that the interaction term does not enter significantly, suggesting that our constant-slope specification is appropriate.

    Our turnover-performance relation captures the predictive power of FundT urn in a given fiscal year for fund performance in the following fiscal year (e.g., turnover in 2014 predicts returns in 2015). In principle, some fund trades could take longer to play out (e.g., a trade in 2014 could lead to profits in 2016).8 To test for such long-horizon effects, we add two more lags of FundT urn to the right-hand side of regression (7). We find that neither of those additional lags has any predictive power for returns after controlling for the most recent value of FundT urn, which retains its positive and significant coefficient. Therefore, we use only the most recent FundT urn in the rest of our analysis.
    The positive turnover-performance relation emerges not only from the panel regression in Table 1, which imposes the restriction (2), but also from fund-by-fund regressions. For each fund i, we estimate the slope coefficient bi from the time-series regression in equation (1) in the full sample. We find that 61% of the OLS slope estimates bi are positive. Moreover, 9% (4%) of the bi‘s are significantly positive at the 5% (1%) confidence level. A weighted

    8The relations between fund performance and funds’ investment horizons are analyzed by Yan and Zhang (2009), Cremers and Pareek (2014), and Lan and Wermers (2014), among others. average of these bi‘s appears in the bottom left cell of Table 1, as shown in equation (8).

    Mutual funds sometimes benefit from receiving allocations of shares in initial public offerings (IPOs) at below-market prices. Lead underwriters tend to allocate more IPO shares to fund families from which they receive larger brokerage commissions (e.g., Reuter, 2006). To the extent that higher commissions are associated with higher turnover, this practice could potentially contribute to a positive turnover-performance relation. This contribution is unlikely to be substantial, though. Fund families tend to distribute IPO shares across funds based on criteria such as past returns and fees rather than turnover (Gaspar, Massa, and Matos, 2006). In addition, the high commissions that help families earn IPO allocations often reflect an elevated commission rate rather than high family turnover, and they are often paid around the time of the IPO rather than over the previous fiscal year.9 Moreover, the contribution of IPO allocations to fund performance seems modest. For each year between 1980 and 2013, we calculate the ratio of total money left on the table across all IPOs, obtained from Jay Ritter’s website, to total assets of active domestic equity mutual funds, obtained from the Investment Company Institute. This ratio, whose mean is 0.30%, exceeds the contribution of IPO allocations to fund performance because mutual funds receive only about 25% to 41% of IPO allocations, on average.10 IPOs thus boost average fund performance by only about 7.5 to 12 basis points per year. Furthermore, the IPO market has cooled significantly since year 2000. Money left on the table has decreased to only 0.10% of fund assets on average, so that IPOs have boosted average fund performance by only 2.5 to 4 basis points per year since January 2001. Yet the turnover-performance relation remains strong during this cold-IPO-market subperiod: the slope estimates in the bottom row of Table 1 remain positive and significant, with t-statistics in excess of 3.2.

    We benchmark each fund’s performance against the index selected for each fund category by Morningstar. For example, for small-cap value funds, the benchmark is the Russell 2000 Value Index. Such an index-based adjustment is likely to adjust for fund style and risk more precisely than the commonly used loadings on the three Fama-French factors. The Fama-French factors are popular in mutual fund studies because their returns are freely available, unlike the Morningstar benchmark index data. Yet the Fama-French factors are not obvious benchmark choices because they are long-short portfolios whose returns cannot be costlessly achieved by mutual fund managers. Cremers, Petajisto, and Zitzewitz (2013) argue that the Fama-French model produces biased assessments of fund performance, and they recommend using index-based benchmarks instead. We follow this advice. But we find very similar results when we adjust fund returns by using the three Fama-French factors or

    9See, for example, Nimalendran, Ritter, and Zhang (2007) and Goldstein, Irvine, and Puckett (2011). 10These estimates are from Reuter (2006), Ritter and Zhang (2007), and Field and Lowry (2009).

    the four factors that also include momentum. In both cases, the slope coefficients in the top row of Table 1 remain insignificant while the slopes in the bottom row continue to be highly significant, with t-statistics ranging from 7.14 to 8.76.

    We assess fund performance by subtracting the benchmark return from the fund’s return, effectively assuming that the fund’s benchmark beta is equal to one. This simple approach is popular in investment practice, and it circumvents the need to estimate the funds’ betas. When we estimate those betas using OLS, we find very similar results. To avoid using imprecise beta estimates for short-lived funds, we replace OLS betas of funds having track records shorter than 24 months by the average beta of funds in the same Morningstar category. Just as in Table 1, we find that the slope estimates in the top row are insignificant while the slopes in the bottom row are highly significant, with t-statistics of about 7.6.

    The test described in the previous paragraph assumes that each fund’s beta is timeinvariant. In a separate test, we allow fund betas to vary over time. This test helps us assess whether the turnover-performance relation could be driven by time variation in systematic risk. If high turnover were associated with more risk, then the higher returns following high turnover could simply represent compensation for risk. However, it is not clear a priori why higher turnover should be followed by more as opposed to less systematic risk. Moreover, we do not fund any such relation in the data. When we model fund betas as a linear function of FundT urn, we find results very similar to those in Table 1.

    We report all of our results based on the full sample period of 1979–2011. In addition, we verify the robustness of our results in the 2000–2011 subperiod, motivated by two potential structural changes in the data. The first change relates to the way CRSP reports turnover. Prior to September 1998, all funds’ fiscal years are reported as January–December, raising the possibility of inaccuracy, since after 1998 the timing of funds’ fiscal years varies across funds.11 The second change, identified by P´astor, Stambaugh, and Taylor (2014), relates to the reporting of fund size and expense ratios before 1993. Using the 2000–2011 subperiod provides a robustness check that is conservative in avoiding both potential structural changes. We find that all of our main conclusions are robust to using the 2000–2011 subperiod. For example, the time-series turnover-performance relation in Table 1 remains positive and significant, with t-statistics of 4.37 and 3.74 in the bottom row. In the online appendix, we report the counterparts of all of our tables estimated in the 2000–2011 subperiod.

    11In private communication, CRSP explained that this change in reporting is related to the change in its fund data provider from S&P to Lipper on August 31, 1998. CRSP has also explained the timing convention for turnover, which is the variable turn ratio in CRSP’s fund fees file. If the variable fiscal yearend is present in the file, turnover is measured over the 12-month period ending on the fiscal yearend date; otherwise turnover is measured over the 12-month period ending on the date marked by the variable begdt.

    3. Fund Size and Fees Matter

    Our evidence so far reveals that the typical fund performs better after it trades more. Next, we ask whether this turnover-performance relation differs across funds. We focus on two readily observed fund characteristics, a fund’s size and its expense ratio (or “fee,” for short). Both characteristics are related to fund performance. Fund size matters because larger funds tend to trade larger amounts. In the presence of decreasing returns to scale associated with liquidity costs, larger funds are less able to exploit alpha-producing opportunities (e.g., Perold and Salomon, 1991). Identifying those opportunities requires skill, and managers with greater skill should receive greater fee revenue in equilibrium (e.g., Berk and Green, 2004). Among funds of similar size, a manager with greater skill should thus have a higher expense ratio.

    We explore the roles of fund size and fees in the turnover-performance relation. For each month t, we compute the terciles of FundSizei,t−1 and ExpenseRatioi,t−1, the most recent values of size and fees available from CRSP prior to month t. Each fund in the sample in month t is assigned to one of the resulting size and fee categories. We then estimate the turnover-performance relation, first separately within each of the size and fee categories and then within each of the nine size-fee categories for the 3 Ö 3 classification. To do so, we add fixed effects for the categories to the previous specification containing fund fixed effects. We estimate separate slopes on FundT urni,t−1 for each category, thereby imposing the restriction in equation (2) only within a category.

    Table 2 reports the estimated slope coefficients on turnover. We see that both fund size and fees matter in the turnover-performance relation: the turnover coefficient is decreasing in fund size and increasing in expense ratio. The role of fund size is dramatic. In the one-way sort, small funds have a turnover coefficient of 0.00186 (t-statistic: 7.56), whereas mediumsized funds have a coefficient not even half as big, equal to 0.00086 (t = 3.74). The coefficient for large funds is lower by another half and insignificant-only 0.00043 (t = 1.46). Fees also play a strong role, even though the turnover-performance relation is significantly positive in all fee categories. In the one-way sort, the turnover coefficients increase monotonically in fees, producing a significant high-low difference (t = 4.06) and a high-fee coefficient three times higher than the low-fee value (0.00170 versus 0.00058).

    The results in Table 2 for the 3 Ö 3 two-way sort are consistent with the effects of size and fees discussed above. For a given level of one characteristic, the other characteristic matters in the same direction as in the one-way sort results, judging by the signs of the small-large (size) and high-low (fee) differences. The joint roles of size and fees also imply a larger turnover coefficient for small, high-fee funds than for large, low-fee funds. That difference is indeed positive, with a t-statistic of 3.55. Small, high-fee funds have the largest t-statistic and the second largest slope coefficient among the nine fund categories.

    The strong turnover-performance relation for small, high-fee funds has especially large economic significance because turnover is most volatile for those funds. Table 3 reports summary statistics for turnover within the size and fee categories. Panel B shows that FundT urni,t−1 for small, high-fee funds has a standard deviation of 0.547, as compared to 0.438 for all funds and only 0.379 for large, low-fee funds. In general, turnover volatility is increasing in fees and decreasing in fund size. Suppose we translate a one-standard-deviation difference in turnover to a difference in subsequent return. This measure of economic significance is especially large for small, high-fee funds, because their turnover has not only a large slope but also high volatility. Combining these values from Tables 2 and 3 implies that a one-standard-deviation increase in turnover for small, high-fee funds translates to an increase in expected return of 1.25% per year (= 0.00191 Ö 0.547 Ö 1200). This is a large effect, both relative to other funds and in absolute terms. Relative to other funds, the corresponding value for large, low-fee funds is only 0.21% (= 0.00046 Ö 0.379 Ö 1200), and the value for all funds, reported earlier, is 0.65%. In absolute terms, the 1.25% expected return increase is comparable to the average GrossRi,t of small, high-fee funds, which is 0.0938% per month, or 1.13% per year, as shown in Table 4. Interestingly, Table 4 also shows that smaller funds outperform larger funds, and high-fee funds outperform low-fee funds, in gross returns (though not in net returns). These patterns are similar to those in Table 2, suggesting that the turnover-performance relation might play a role in overall fund performance.

    The importance of fund size in the turnover-performance relation in Table 2 presents novel evidence of decreasing returns to scale at the fund level. Prior studies of fund-level decreasing returns generally look for a direct negative relation between a fund’s return and its size.12 While point estimates from such approaches are often consistent with decreasing returns, statistical significance is elusive when applying methods that avoid econometric biases (see P´astor, Stambaugh, and Taylor, 2014). The disadvantage of a fund’s being large instead emerges here as a weaker relation, or even no relation, between the fund’s trading and its subsequent performance, in sharp contrast to the strong positive relation for small funds. Unlike the directly estimated relation between fund size and return, the role of fund size in the turnover-performance relation is highly significant, both economically and statistically.

    More trading should produce higher returns the greater is the manager’s skill in identifying profitable opportunities. Managers with more skill should receive more fee revenue, as

    12See, for example, Chen, Hong, Huang, and Kubik (2004), Yan (2008), and Reuter and Zitzewitz (2013).

    noted earlier. Fee revenue is proportional to the expense ratio for a given fund size, implying a positive partial correlation between skill and expense ratio, conditional on size. Recall from the results of the two-way sort that within each size category, the turnover-performance relation is stronger for high-fee funds, consistent with such funds having greater skill. A one-way sort similarly reveals a stronger relation for high-fee funds, consistent with a positive simple correlation between skill and expense ratio. The latter correlation does not necessarily follow, as it depends on how size covaries with fees and skill in the cross-section, but it seems reasonable for managers with greater skill to charge higher fee rates.

    Besides fees, we consider two additional proxies for fund skill. First, we calculate gross alpha adjusted for both fund-level and industry-level returns to scale, following P´astor, Stambaugh, and Taylor (2014). Second, we take the unadjusted gross alpha over the fund’s lifetime. For both proxies, we find that high-skill funds exhibit a significantly stronger turnover-performance relation than low-skill funds. These results, which are consistent with those in Table 2 based on fees, are in the online appendix. The appendix also shows additional robustness results. For example, all the conclusions from Table 2 continue to hold if we replace benchmark-adjusted fund returns by returns adjusted for the three Fama-French factors or the four factors that also include momentum. The same is true if we allow the factor model betas to vary over time as a linear function of fund turnover. In addition, while the regressions in Table 2 exclude month fixed effects, including such fixed effects produces very similar results, and so does including benchmark-month fixed effects. Overall, our results suggest that high-fee funds have greater skill in identifying time-varying profit opportunities, and small funds are more able to exploit those opportunities.

    Small funds also have higher average turnover than large funds. This result, shown in Panel A of Table 3, is consistent with a natural skill-based sorting of managers. Some managers are more skilled at identifying short-lived opportunities yielding profits over short horizons, while others are more adept at identifying opportunities with longer holding periods that allow patient trading. In a competitive market for managerial talent, one would expect the short-horizon managers to manage small funds: the liquidity constraints that bite when trades must be done quickly render their talents less useful in trading large amounts. In contrast, one would expect the long-horizon managers to manage large funds: their skills can be more profitably exploited by trading larger amounts, since their trades can be executed more patiently. Therefore, this sorting mechanism implies that smaller funds should hold their positions over shorter periods. This implication is supported by Panel A of Table 3, because the higher average turnover of smaller funds suggests those funds have shorter holding periods. The sorting mechanism also implies that larger funds should have more persistent turnover due to more patient trading. Indeed, Panel C of Table 3 shows that turnover of larger funds exhibits higher autocorrelation.

    Our analysis focuses on two salient fund characteristics, size and fees. In addition, we ask whether the strength of the turnover-performance relation varies with fund style. Following the 3 Ö 3 Morningstar style box, we split funds into small-cap, mid-cap, and large-cap categories, and also separately into value, blend, and growth categories. For each style, we calculate a turnover-performance regression slope coefficient, producing a table analogous to Table 2. We report this table in the online appendix. The table shows that the turnoverperformance relation is positive and significant across all fund styles with the sole exception of mid-cap growth, for which the t-statistic is 1.60. The relation is about equally strong for value and growth funds, but it is significantly stronger for small-cap funds than for large-cap funds. This result is consistent with the common argument that mispricing is more likely to be found among small-cap stocks, which tend to exhibit lower institutional ownership and less analyst coverage compared to large-cap stocks. It makes sense for funds’ ability to spot and exploit trading opportunities to be stronger in stocks that are more mispriced.

    4. What Other Funds Do Matters

    We have shown that if Fund ABC trades more than usual this period, the fund typically performs better than usual next period. Suppose now that many other funds trade more than usual this period. Are there implications for the performance of Fund ABC? On the one hand, this heavier trading by other funds could be good news for Fund ABC. If there is more mispricing this period, as indicated by many funds trading more, then any opportunities identified by Fund ABC could be more profitable. On the other hand, if the other funds are identifying the same opportunities and thus acting in concert, their heavier trading could produce especially large price impacts, reducing mispricing that would otherwise benefit Fund ABC. This section considers both of these potential effects in exploring whether the turnover-performance relation depends on the trading activity of other funds. We begin by considering a mispricing-based explanation for common variation in fund turnover. We then investigate how that common variation impacts the turnover-performance relation.

    4.1. Mispricing and Trading

    When do funds, viewed collectively, trade more than usual? If alpha-producing opportunities arise from mispricing, then periods with more mispricing should be those when funds trade more. A simple measure of the common component in fund trading is the cross-sectional average of individual fund turnover. We let AvgTurnt denote average turnover contemporaneous with month t, that is, the average turnover across funds’ 12-month fiscal periods that contain month t. AvgTurnt, plotted in Panel A of Figure 1, fluctuates between 59% and 102% per year from 1979 to 2011.13 This series has a 95% correlation with the first principal component of individual fund turnover. We ask whether AvgTurnt is higher when mispricing is more likely. We use three proxies for the likelihood of mispricing: Sentimentt, V olatilityt, and Liquidityt. The three series are plotted in Panel B of Figure 1.

    The first mispricing proxy, Sentimentt, is the value in month t of Baker and Wurgler’s (2006, 2007) investor-sentiment index. If sentiment-driven investors participate more heavily in the stock market during high-sentiment periods, the mispricing such investors create is more likely to occur during those periods (e.g., Stambaugh, Yu, and Yuan, 2012). We thus expect funds exploiting such mispricing to trade more when sentiment is high. Consistent with this prediction, a regression of AvgTurnt on Sentimentt produces a significantly positive coefficient (t = 3.17), as shown in the first column of Table 5. We include a time trend in the regression, given the positive trend in AvgTurnt evident in Figure 1. As reported in the last row, the R2 in the regression including Sentimentt exceeds the R2 when regressing on just the time trend by 0.171.

    The second mispricing proxy, V olatilityt, is the cross-sectional standard deviation in month t of the returns on individual U.S. stocks.14
    The rationale for this variable is that higher volatility corresponds to greater uncertainty about future values and thus greater potential for investors to err in assessing those values. As a result, periods of high volatility admit greater potential mispricing, and we expect funds exploiting such mispricing to trade more when volatility is high. Consistent with this prediction, a regression of AvgTurnt on V olatilityt produces a significantly positive coefficient (t = 7.23), as shown in column 2 of Table 5. The R2 in that regression, which again includes a time trend, exceeds the R2 in the trend-only regression by 0.189.

    The third proxy, Liquidityt, is the value in month t of the stock-market liquidity measure of P´astor and Stambaugh (2003). Empirical evidence suggests that higher liquidity is accompanied by greater market efficiency (e.g., Chordia, Roll, and Subrahmanyam, 2008, 2011). In other words, periods of lower liquidity are more susceptible to mispricing. Therefore, we might expect funds to trade more when liquidity is lower. On the other hand, lower liquidity

    13CRSP turnover data are missing in 1991 for unknown reasons. We therefore treat AvgTurn as missing in 1991 in our regressions. In Figure 1, though, we fill in average turnover in 1991 by using Morningstar data, for aesthetic purposes. We rely on CRSP turnover data in our analysis because Morningstar is ambiguous about the timing of funds’ fiscal years. 14We thank Bryan Kelly for providing this series.
    also implies higher transaction costs, which could discourage funds from trading. Our evidence suggests that the former effect is stronger: Regressing AvgTurnt on Liquidityt yields a significantly negative coefficient (t = −4.14), reported in column 3 of Table 5. Including Liquidityt increases the R2 versus the trend-only regression by 0.024, a more modest increase than produced by the other two proxies.

    When all three mispricing proxies are included simultaneously as regressors, each enters with a coefficient and t-statistic very similar to when included just by itself. This allinclusive regression, reported in column 4 of Table 5, also adds two additional variables that control for potential effects of the business cycle and recent stock-market returns, but neither variable enters significantly. (The two variables are the Chicago Fed National Activity Index and the return on the CRSP value-weighted market index over the previous 12 months.) The combined ability of the three mispricing proxies to explain variance in AvgTurnt is substantial: the R2 exceeds that of the trend-only regression by 0.324.15 Overall, the results make sense: funds trade more when there is more mispricing.

    What mispricing are funds exploiting? To see whether funds trade based on well-known market anomalies, we regress the returns of eleven such anomalies, as well as their composite return, on lagged average fund turnover. The eleven anomalies, whose returns we obtain from Stambaugh, Yu, and Yuan (2012), involve sorting stocks based on two measures of financial distress, two measures of stock issuance, accruals, net operating assets, momentum, gross profitability, asset growth, return on assets, and the investment-to-assets ratio. We find no significant slopes on average turnover. To the extent that funds trade more when there is more mispricing, they must be exploiting mispricing beyond these eleven anomalies.

    Finally, we consider the role of stock market turnover in explaining AvgTurnt. We measure market turnover as total dollar volume over the previous 12 months divided by total market capitalization of ordinary common shares in CRSP. Market turnover reflects trading by all entities, includingmutual funds, so it could potentially be related to AvgTurnt. It could also be related to Sentimentt, which is constructed as the first principal component of six variables that include NYSE turnover. However, when we add market turnover to the all-inclusive specification in Table 5, it does not enter significantly, whereas the slope on Sentimentt remains positive and significant. The other two mispricing proxies also retain their signs and significance, and the remaining variables remain insignificant. In short,

    15If we exclude the time trend from the regressions, we find results similar to those reported in Table 5. V olatility and Liquidity continue to enter significantly with the same signs as in Table 5, and the business cycle and market return remain insignificant. The only difference relates to Sentiment, whose coefficient retains the positive sign but loses statistical significance. This evidence suggests that Sentiment is better at capturing deviations of AvgTurn from its trend than in capturing the raw variation in AvgTurn.

    adding market turnover does not affect any of our inferences in Table 5.

    4.2. Other Funds: Evidence That They Matter

    Next, we investigate how the trading of other funds enters the turnover-performance relation. We first ask whether average lagged fund turnover, which reflects commonality in fund trading, helps predict a given fund’s subsequent performance. We denote average lagged fund turnover, or the average of FundTurni,t−1 across i, by AvgTurnt−1.16 The first column of Table 6 reports the result of replacing FundTurni,t−1 by AvgTurnt−1 and then repeating the regression from the bottom left cell of Table 1, which includes fund fixed effects.17 We see a significantly positive coefficient on AvgTurnt−1 (t = 2.13), indicating that the common component of fund trading helps predict individual fund performance. The estimated slope coefficient, 0.00741, implies substantial economic significance. Given the time-series standard deviation of AvgTurnt−1, 0.090, a one-standard-deviation increase in the variable translates to an increase in expected return of 0.80% per year (= 0.00741 Ö 0.090 Ö 1200).

    The information in AvgTurnt−1 about a fund’s subsequent performance is undiminished by conditioning on the fund’s own turnover. The results in column 2 of Table 6 reveal that the coefficient and t-statistic for AvgTurnt−1 are little changed by controlling for FundTurni,t−1. The importance of either variable is insensitive to whether the other is included, because the average correlation between FundTurni,t−1 and AvgTurnt−1 is a modest 0.131 (the “all-all” value in Panel A of Table 7). Both turnover variables remain significant also after controlling for additional variables described below (columns 3 through 7 of Table 6).

    A simple story emerges from the joint abilities of FundTurni,t−1 and AvgTurnt−1 to predict fund performance. A given fund’s turnover, FundTurni,t−1, is higher-and its subsequent performance is better-when its own manager identifies more alpha-producing opportunities. When many managers identify such opportunities, AvgTurnt−1 is higher, and there is more mispricing in general. Even when a fund’s own manager does not identify

    16Note that AvgTurnt−1 uses only information available before month t because it is the average of turnovers computed over 12-month periods that end before month t. It is thus reasonable to use AvgTurnt−1 to predict performance in month t. Also note that the notation for time subscripts is complicated by the fact that funds report turnover only annually. In Section 4.1, we use the notation AvgTurnt to denote average turnover across funds’ 12-month fiscal periods that contain month t. That notation is slightly inconsistent with the notation in this section because given our definition of FundTurni,t, the contemporaneous average turnover in Section 4.1 is the average of FundTurni,t+11 across i. We prefer to use the notation AvgTurnt (instead of AvgTurnt+11) in Section 4.1 to emphasize the contemporaneous nature of the analysis in that section. We hope the reader will pardon this slight abuse of notation. 17Month fixed effects must be omitted because a common time series is used as a regressor for each fund. Also, the regressions in Table 6 exclude a time trend, but the results are very similar if we include one.

    unusually many opportunities in a given period, the opportunities he does identify are likely to be more profitable if there is generally more mispricing in that period.

    Heavier trading by other funds is not necessarily all good news, however. Other funds are competitors whose trades can move prices. A stronger presence of active managers in the stock market produces greater price corrections and thus lowers the active managers’ alphas. That is the idea behind the concept of industry-level decreasing returns to scale, introduced by P´astor and Stambaugh (2012). In line with that concept, P´astor, Stambaugh, and Taylor (2014) find empirically that fund performance is negatively related to the size of the active management industry. Following that study, we define IndustrySizet−1 as the value in month t − 1 of the total assets managed by all funds in our sample, divided by the total market value of U.S. stocks. Our evidence, shown in column 3 of Table 6, confirms a significantly negative relation between fund performance and IndustrySizet−1.

    Industry size is one way of measuring the degree of competition among funds. Another way, which we introduce here, is to gauge the extent to which funds act in concert. We use a simple return-based measure, AvgCorrt−1, which is the average pairwise correlation between all individual funds’ benchmark-adjusted gross returns in the 12 months ending in month t− 1. AvgCorrt−1 fluctuates between 0.01 and 0.26 from 1979 to 2011. We interpret higher values of AvgCorrt−1 as indicating more concerted active trading by funds. We find that AvgCorrt−1 is negatively related to fund performance (t = −2.42; see column 4 of Table 6). This result is consistent with the interpretation that when funds trade more in concert, prices are impacted and profit opportunities are reduced.

    The price impact of funds’ concerted trading should be stronger when those funds trade more heavily. Therefore, when AvgTurnt−1 is higher, the effect of AvgCorrt−1 on performance should be less favorable. Similarly, the more funds act in concert (i.e., the higher AvgCorrt−1), the less favorable should be their heavier trading (AvgTurnt−1). Either way, we expect fund performance to be negatively related to the interaction term AvgTurnt−1 Ö AvgCorrt−1. This is indeed the case, as shown in column 5 of Table 6 (t = −2.69). At the same time, the slope on AvgTurnt−1 remains positive and significant. We thus see simultaneous support for both the positive and negative aspects of heavier trading by other funds. This evidence suggests that the benefit of the greater mispricing reflected in other funds’ heavier trading is countered by greater price correction when those funds act more in concert.

    This opposing effect of funds acting in concert is consistent with the previously discussed concept of industry-level decreasing returns to scale. The underlying mechanism behind that concept is that a larger industry implies more money chasing the same alpha-producing opportunities, thereby moving prices more and reducing each active fund’s alpha. Our measure of AvgTurnt−1 Ö AvgCorrt−1 directly addresses this effect of more money acting in concert. The significantly negative relation between this term and performance provides additional and novel evidence of industry-level decreasing returns to scale.

    Interestingly, adding the interaction term in column 5 of Table 6 changes the sign of the slope on AvgCorrt−1 from negative to positive (t = 2.55). This result highlights a positive aspect of concerted fund trading, as measured by AvgCorrt−1. If funds choose to make similar trades at the same time, they must perceive those trades as attractive. If those funds are skilled, their perceptions are correct and their concerted trading thus indicates more mispricing, with favorable implications for performance. This positive effect of concerted trading is opposed by the negative effect discussed earlier. Which effect prevails depends on the amount of other funds’ trading. The estimates in column 5 indicate that when fund trading is light (i.e., AvgTurnt−1 is low), the positive effect prevails and AvgCorrt−1 is positively related to performance. When trading is heavy, though, the negative effect prevails. The negative effect also prevails on average, as shown in column 4. We thus see that the benefit of the greater mispricing signaled by other funds’ concerted trading is more than offset by the cost of greater price correction when those funds trade more heavily.

    To explore the robustness of our inferences about how performance relates to what other funds do, we add the three mispricing proxies to the regression in Table 6 (column 6). We also allow FundTurni,t−1 to enter differently across the nine size-fee categories, as in Section 3 (column 7). In both specifications, the statistical significance of AvgTurnt−1, AvgCorrt−1, and their interaction is unaffected by the inclusion of the mispricing proxies, and the corresponding slope coefficients are close to those reported earlier. The three mispricing proxies enter with the same signs in predicting performance (Table 6) as they do in explaining the variation in AvgTurnt−1 (Table 5), though in Table 6 only Sentimentt−1 is significant (t = 3.42). This result suggests that turnover, with both its common and fund-specific dimensions, largely subsumes performance-relevant mispricing information in the other two mispricing proxies, V olatilityt−1 and Liquidityt−1. In contrast, Sentimentt−1 contains additional information about future fund performance.

    So far we have treated other funds as being all other funds, for simplicity. For a given fund, “other” funds can also be defined more narrowly as those sharing the fund’s characteristics. Motivation for this alternative definition arises from Table 7. Panel A reports the correlation between FundTurni,t−1 and AvgTurnt−1, with the correlation averaged across all funds as well as across just the funds within a size and fee category. Panel B repeats the same calculations while replacing AvgTurnt−1 with OwnCellAvgTurnt−1, the average lagged turnover calculated across only those funds belonging to the same size and fee tercile as fund i in month t. When comparing Panels A and B, we see that FundTurni,t−1 has a lower average correlation with AvgTurnt−1 than it has with OwnCellAvgTurnt−1. This inequality holds for every one of the six one-way and nine two-way size and fee categories. These results reveal greater common variation in turnover among funds with similar sizes and fees than among all funds taken together.

    If trading by other funds signals the presence of greater mispricing, then heavier trading by funds similar to one’s own could signal greater mispricing that is especially relevant. In other words, heavier trading by less similar funds could be less relevant to one’s own fund. This possibility is consistent with the above evidence that the turnover of one’s own fund typically comoves more with the turnover of funds that have similar size and fees. It is also consistent with the natural sorting mechanism discussed at the end of Section 3.

    Since there is more commonality in turnover among similar funds, it is natural to ask whether similar funds’ trading helps explain performance. We run regressions similar to those in Table 6 but replace AvgTurnt−1 with OwnCellAvgTurnt−1. The results are in Table 8. We also add AvgTurnt−1 as an independent variable, allowing a horse race between it and its own-cell counterpart. The latter wins, consistent with the greater relevance of what other similar funds do. The t-statistics for OwnCellAvgTurnt−1, which range from 3.11 to 7.02, are uniformly higher than the corresponding t-statistics for AvgTurnt−1 in Table 6. Moreover, AvgTurnt−1 becomes insignificant, driven out by OwnCellAvgTurnt−1. Finally, AvgCorrt−1 plays a similar role as before: the interaction term OwnCellAvgTurnt−1 Ö AvgCorrt−1 enters significantly negatively (t = −2.90).

    Overall, our results show that a fund’s performance is related not only to its own turnover but also to that of other funds, especially other similar funds. Heavier trading by other funds signals greater mispricing and is positively related to performance, but there is also an opposing negative relation to the extent that funds act in concert.

    5. Investment Perspectives

    In this section, we take an investment perspective to assess the economic significance of our regression results. We examine the performance of two investment strategies designed to exploit the turnover-performance relation.

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    Moskowitz, Tobias J., 2000, Discussion, Journal of Finance 55, 1695–1703. Nimalendran, Mahendrarajah, Jay R. Ritter, and Donghang Zhang, 2007, Do today’s trades affect tomorrow’s IPO allocations? Journal of Financial Economics 84, 87–109.

    Nofsinger, John R., and RichardW. Sias, 1999, Herding and feedback trading by institutional and individual investors, Journal of Finance 54, 2263–2295.

    P´astor, ˇ Luboˇs, and Robert F. Stambaugh, 2002, Mutual fund performance and seemingly unrelated assets, Journal of Financial Economics 63, 315–349.

    P´astor, ˇLuboˇs, and Robert F. Stambaugh, 2003, Liquidity risk and expected stock returns, Journal of Political Economy 111, 642–685.

    P´astor, ˇLuboˇs, and Robert F. Stambaugh, 2012, On the size of the active management industry, Journal of Political Economy 120, 740–781.

    P´astor, ˇLuboˇs, Robert F. Stambaugh, and Lucian A. Taylor, 2014, Scale and skill in active management, Journal of Financial Economics, forthcoming.

    Perold, Andr´e F., and Robert S. Salomon, 1991, The right amount of assets under management, Financial Analysts Journal 47, 31–39.

    Reuter, Jonathan, 2006, Are IPO allocations for sale? Evidence from mutual funds, Journal of Finance 61, 2289–2324.

    Reuter, Jonathan, and Eric Zitzewitz, 2013, How much does size erode mutual fund performance? A regression discontinuity approach, Working paper, Boston College.

    Ritter, Jay R., and Donghang Zhang, 2007, Affiliated mutual funds and the allocation of initial public offerings, Journal of Financial Economics 86,
    337–368.

    Stambaugh, Robert F., 2014, Investment noise and trends, Journal of Finance 69, 1415–1453. Stambaugh, Robert F., Jianfeng Yu, and Yu Yuan, 2012, The short of it: Investor sentiment and anomalies, Journal of Financial Economics 104, 288–302.

    Wermers, Russ, 1999, Mutual fund herding and the impact on stock prices, Journal of Finance 54, 581–622.

    Wermers, Russ, 2000, Mutual fund performance: An empirical decomposition into stockpicking talent, style, transactions costs, and expenses, Journal of Finance 55, 1655–1695.

    Yan, Xuemin, 2008, Liquidity, investment style, and the relation between fund size and fund performance, Journal of Financial and Quantitative Analysis 43, 741-768.

    Yan, Xuemin, and Zhe Zhang, 2009, Institutional investors and equity returns: Are shortterm institutions better informed?, Review of Financial Studies 22, 893–924.


  • Robert Stambaugh Presentation

    Do Funds Make More When They Trade More?

    ˇLuboˇs P´astor (Chicago Booth)
    Rob Stambaugh (Wharton)
    Luke Taylor (Wharton)

    Motivation

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    Main Result

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    Other Results

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    Literature

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    Sample

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    Sample

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    Turnover-Performance Relation

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    Turnover-Performance Relation

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    Role of Fund Size and Fees

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    Turnover-Performance Relation in Size and Fee Categories

    pstambaugh10

    Volatility of Fund Turnover

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    Average Fund Turnover

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    Autocorrelation of Fund Turnover

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    Average Benchmark-Adjusted Gross Fund Returns

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    Average Benchmark-Adjusted Net Fund Returns

    pstambaugh16

    Role of Other Funds

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    Is Average Turnover Related to Mispricing?

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    What Helps Explain Fund Performance?

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    Commonality in Turnover

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    What Helps Explain Fund Performance?

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    Investment Perspectives

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    Timing Strategy

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    Equivalence Between Timing Strategy and Regression

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    Average Returns of Timing Strategy

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    Economic Significance

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    Cross-Sectional Strategy

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    Average Gross Returns of Cross-Sectional Strateg

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    Average Net Returns of Cross-Sectional Strategy

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    Conclusions

    pstambaugh31


  • Anna Scherbina paper

    Cross-Firm Information Flows and the Predictability of
    Stock Returns

    Anna Scherbina*       Bernd Schluschey
    UC Davis       Federal Reserve Board

    First draft: April 10, 2013
    This draft: January 7, 2015

    ABSTRACT


  • Anna Scherbina Presentation

    Cross-Firm Information Flows

    Anna Scherbina (joint with Bernd Schlusche)

    The Q Group Conference

    April 1, 2015

    Motivation

    Anna-p

    Preview of the results

    Anna-p1

    Relevance to pracititoners

    Anna-p2

    Motivation

    Anna-p3

    Motivation

    Anna-p4

    Related literature

    Anna-p5

    Identifying information leaders for each stock

    Anna-p6

    Leadership summary

    Anna-p7

    Aggregating leader signals

    Anna-p8

    Example

    Anna-p9

    Timeline: portfolio formation

    Anna-p10

    Portfolio formation

    Anna-p11

    Equal-weighted portfolio returns

    Anna-p12

    Value-weighted portfolio returns

    Anna-p13

    Cumulative return: monthly portfolios

    Anna-p14

    Alternative specifications

    Anna-p15

    Other results

    Anna-p16

    Higher frequencies

    Anna-p17

    Weekly-frequency leaders, weekly portfolios (1980-2011)

    Anna-p18

    Cumulative return: weekly portfolios

    Anna-p19

    Conditioning on prior-week return, weekly portfolios

    Anna-p20

    Break-even trading costs for the weekly strategy

    Anna-p21

    Annual news counts, TRNA dataset, average over 1996-2011

    Anna-p22

    Explaining the number of followers (1997-2011)

    Anna-p23

    Do sophisticated investors trade on this strategy?

    Anna-p24

    Utilisation

    Anna-p25

    Do sophisticated investors trade on this strategy?

    Anna-p26

    Summary

    Anna-p27

    Possible extensions

    Anna-p28


  • Kent Smetters Paper

    A Sharper Ratio: A General Measure for Correctly
    Ranking Non-Normal Investment Risks

    Kent Smetters * Xingtan Zhang †
    This Version: February 3, 2014

    Abstract

    While the Sharpe ratio is still the dominant measure for ranking risky investments, much effort has been made over the past three decades to find more robust measures that accommodate non- Normal risks (e.g., “fat tails”). But these measures have failed to map to the actual investor problem except under strong restrictions; numerous ad-hoc measures have arisen to fill the void. We derive a generalized ranking measure that correctly ranks risks relative to the original investor problem for a broad utility-and-probability space. Like the Sharpe ratio, the generalized measure maintains wealth separation for the broad HARA utility class. The generalized measure can also correctly rank risks following different probability distributions, making it a foundation for multi-asset class optization. This paper also explores the theoretical foundations of risk ranking, including proving a key impossibility theorem: any ranking measure that is valid for non-Normal distributions cannot generically be free from investor preferences. Finally, we show that approximation measures, which have sometimes been used in the past, fail to closely approximate the generalized ratio, even if those approximations are extended to an infinite number of higher moments.
    Keywords: Sharpe Ratio, portfolio ranking, infinitely divisible distributions, generalized ranking measure, Maclaurin expansions
    JEL Code: G11

    *Kent Smetters: Professor, The Wharton School at The University of Pennsylvania, Faculty Research Associate at the NBER, and affiliated faculty member of the Penn Graduate Group in Applied Mathematics and Computational Science. By Email: smetters@wharton.upenn.edu.
    †Xingtan Zhang, PhD (Penn Mathematics), and first-year PhD student in Applied Economics at The Wharton School at The University of Pennsylvania. By email: xingtan@wharton.upenn.edu.

    1 Introduction

    Bill Sharpe’s seminal 1966 paper demonstrated that picking a portfolio with the largest expected risk premium relative to its standard deviation is equivalent to picking the portfolio that maximizes the original investor’s expected utility problem, assuming that portfolio returns are Normally distributed.1 This simple mean-variance investment ranking measure – the “Sharpe ratio” – is, therefore, a sufficient statistic for the investor’s problem that does not rely on the investor’s preferences or wealth level.

    The immense power of the Sharpe ratio ranking measure stems from the fact that it allows the investment management process to be decoupled from the specific attributes of the heterogeneous investor base. The multi-trillion dollar money management industry relies heavily on this separation. Investors in a mutual fund or hedge fund might differ in their levels of risk aversion and wealth (including assets held outside the fund). Nonetheless, an investment manager only needs to correctly estimate the first two moments of the fund’s return in order to pick the single risky portfolio that is best for each underlying investor.2 It is not surprising, therefore, that the Sharpe ratio is tightly integrated into the modern investment management practice and embedded into virtually all institutional investment analytic and trading platforms. Even consumer-facing investment websites like Google Finance reports the Sharpe ratio for most mutual funds along with just a few other basic statistics, including the fund’s alpha, beta, expected return, R2 tracking (if an indexed fund), and standard deviation.

    Of course, it is well known that investment returns often exhibit “higher order” moments that might differ from Normality (Fama 1965; Brooks and Kat 2002; Agarwal and Naik 2004, and Malkiel and Saha 2005).3 In practice, investment professionals, therefore, often look for investment opportunities that would have historically-that is, in a “back test”-produced unusually large Sharpe ratios under the belief that large values provide some “buffer room” in case the underlying distribution is not Normal. This convention, though, is misguided. Outside of the admissible utility-probability space (“admissible space” for short) where the Sharpe ratio is valid, it is easy to create portfolios with large Sharpe ratios that are actually first-order stochastically dominated by portfolios with smaller Sharpe ratios (Leland 1999; Spurgin 2001; and Ingersoll et al. 2007).4 Besides the presence of the usual “fat tails,” it is now well known that non-Normally distributed risks easily emerge at the investment fund level with modern trading strategies and the use of financial derivatives (Section 6), which have exploded in use over time.

    The historical debate over the ability of the Sharpe ratio to correctly rank risky investments following non-Normal probability distributions (e.g., Tobin 1958, 1969; Borch 1969; Feldstein 1969) has led to a long-standing interest in producing more robust ranking measures. The first line of work dates back to at least the work by Paul Samuelson (1970), who, at the time, also expressed some skepticism that such extensions were actually needed in practice. (The subsequent explosion of modern trading strategies, novel asset classes and financial derivatives might have changed his mind.) A short list of other contributors include the well-cited paper by Kraus and Litzenberger (1976), Scott and Horvath (1980), Owen and Rabinovitch (1983), Ingersoll’s (1987) classic textbook, Brandt et al (2005), Jurczendo and

    1Of course, the Sharpe ratio builds on the pioneering mean-variance work by Markowitz (1952, 1959) and Tobin (1958).
    2The Sharpe ratio, however, only ranks risky portfolios in order to determine the best one. The ratio itself does not determine the optimal division of an investor’s wealth between this best risky portfolio and the risk-free instrument. That division must be determined in a second stage using consumer-specific information. The Sharpe ratio, therefore, supports the standard division between the “investment manager,” who determines the best risky portfolio, and the “financial planner” who, knowing each client in more detail, helps decide the share of the client’s wealth that should be invested into this best risky asset based on the client’s specific circumstances.
    3A related literature has examined how disaster risk can explain equilibrium pricing within the neoclassical growth model (Barro 2009; Gabaix 2012; Gourio 2012; Wachter 2012)
    4In other words, the portfolio with the smaller Sharpe ratio would be preferred by all expected utility maximizers with positive marginal utility in wealth.

    Maillet (2006), Zakamouline and Koekebakker (2008), Dávila (2010) and Pierro and Mosevich (2011). This line of work, however, imposed fairly strong restrictions on investor utility preferences and/or the risk distribution. The current paper contributes to this line of research by deriving a ranking measure that is valid over a broad admissible space.5

    A second line of research bypasses the investor’s expected utility problem altogether and produces risk measures that satisfy certain mathematical properties such as “coherence.”6 Examples of coherent risk measures include “average VaR,” “entropic VaR,” and the “superhedging price.” While these measures satisfy certain axioms, a portfolio that maximizes one or more of these measures does not necessarily maximize the standard investor expected utility problem, as considered by Sharpe and many others. The application of these measures for the actual investor is, therefore, unclear, which might help explain the continued popularity of the Sharpe ratio.

    A third line of work, which is actually the largest line in scope, has evolved more from practitioners. It has produced heuristic measures that have a more “intuitive” interpretation in nature than the axiomatic-based measures. Common heuristic measures include “value at risk (VaR),”7 Omega, the Sortino ratio, the Treynor ratio, Jensen’s alpha, Calmar ratio, Kappa, Roy’s safety-first criterion, numerous tail risk measures, various upside-downside capture metrics, and many more.8 These metrics, however, tend to be especially problematic. Not only do they fail to satisfy any sort of reasonably mathematical properties, there is no apparent relationship between a reasonable description of the investor problem and these measures. In practice, therefore, investment managers often combine the Sharpe ratio with one or more of these measures when attempting to account for non-Normal risk (e.g., maximize the Sharpe ratio subject to the investment’s “value at risk” being less than some threshold). Despite its limitations, the Sharpe ratio, therefore, remains the gold standard of the investment industry.

    This paper makes three contributions. First, as summarized in our Lemma 2, we demonstrate how to solve an infinite-order Maclaurin expansion for its correct asymptotic root when no closed form solution exists. We can then derive a generalized ranking measure (the “generalized ratio”) that correctly ranks risky returns under a much broader admissible utility-probability space consistent with the Sharpe ratio or previous extensions. By “correctly ranks,” we mean it in the tradition of Sharpe: the generalized ratio picks the portfolio preferred by the original investor expected utility problem.

    It is easy to motivate the importance of allowing for a broad admissible utility-probability space. A broad utility space captures realistic investor attitudes toward risk. For example, while the common assumption of Constant Absolute Risk Aversion is useful for obtaining various theoretical insights, it is also fairly implausible for modeling risky investment decisions. Similarly, allowing for a broad set of risk distributions is, of course, important for accommodating more extreme risks with “fat tails.”

    But our generalized ratio can also rank between risks that follow different probability distributions. The generalized ratio, therefore, can be used as the foundation for multi-asset class optimization. For example, it can pairwise rank a risky portfolio without financial derivatives that follows one distribution

    5Throughout this paper, we will write expressions like “ranking measure ABC is valid over admissible space XYZ” even though such terminology is a bit redundant since admissibility implies validity. However, we believe that such terminology is generally understood and more readable than various alternatives.
    6A “coherent” risk measure satisfies monotonicity, sub-additivity, homogeneity, and translational invariance (Artzner et al 1999). More recent work has emphasized risk measures that avoid “worst case” scenarios and are monotonic in first-order stochastic dominance. See, for example, Aumann and Serrano (2008); Foster and Hart (2009); and Hart (2011).
    7Standard VaR is not coherent, whereas the variants on VaR noted in the previous paragraph are coherent.
    8Modigliani (1997) proposed a transformation of the Sharpe ratio, which became known as the “risk-adjusted performance measure.” This measure attempts to characterize how well a risk rewards the investor for the amount of risk taken relative to a benchmark portfolio and the risk-free rate. This measure is not included in the list in the text because it mainly provides a way of interpreting the unit-free Sharpe ratio rather than offering an alternative measure in the presence of non- Normally distributed risk.

    against another risky portfolio with option overlays following a different distribution. This flexibility is much more powerful than simply assuming that all potential portfolio combinations follow the same probability distribution form, even if that distribution is more extensible than Normality.

    Like the original Sharpe ratio, our generalized measure preserves wealth separation under the broad functional form of HARA utility, which includes many standard utility functions as special cases.9 Unlike the original Sharpe ratio, however, our generalized ratio does not preserve separation from investor preferences. But we show that this limitation is not a function of the generalized risk measure. Rather, we prove a key impossibility theorem: preference separation is generically impossible in the presence of non-Normal risk. Fortunately, as a practical matter, the generalized ratio still supports the decoupled investment management process noted above: instead of reporting a single ratio, each fund can report a small tuple of ratios corresponding to different standardized levels of risk aversion (see Section 6). By law, financial advisors must already actively test for the level of risk aversion of each client.

    Second, using some of the machinery that we developed, we then “backtrack” to explore the theoretical foundations of the classic Sharpe ratio in more detail. Despite its extensive usage in academics and industry, very little is actually known about the Sharpe ratio beyond the few cases where it is well known to correctly rank risks (e.g., Normally distributed risk or quadratic utility). We show that the Sharpe ratio is actually valid under a larger admissible space than currently understood. We also explore why it is challenging to actually write down a necessary condition for the Sharpe ratio to be a valid ranking measure. In the process, we are also able to generalize the Kraus-Litzenberger (1976) “preference for skewness” result to an unlimited number of higher moments. This generalization is useful because plausible utility functions produce an infinite number of non-zero higher-order derivatives, and there does not exist any probability distribution that can be fully described by any finite number of cumulants greater than two.

    Third, we derive a linear approximation of the investor problem in the presence of non-Normal higher-order moments. This formulation accommodates a simple closed-form solution, and it nests some previous attempts to generalize the Sharpe ratio. Our computations, however, show that approximations can be very inaccurate. Accurate ranking, therefore, requires using the generalized ratio.

    The paper is organized as follows. Section 2 provides an overview of the standard investor problem. Section 3 derives the generalized ratio described earlier. Section 4 explores the theoretical foundations of the Sharpe ratio in more detail. Section 5 derives the linear approximation. Section 6 provides numerical examples comparing the Sharpe ratio, the generalized ratio, and the linear approximation for a range of potential investment applications. Section 7 concludes. Proofs of lemmas and theorems are provided in the Appendices.

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    7 Conclusions

    The Sharpe ratio correctly ranks risky investments, consistent with the original investor problem, if risks are Normally distributed. Considerable past effort has been made to develop new measures that are robust to non-Normally distributed risks, which emerge with “fat tails,” modern trading strategies and the modern extensive use of financial derivatives in hedging portfolio risk. Some of this effort has started with the original investor problem and added some higher moments under fairly strong restrictions on the risk distribution and/or utility function. Even more ranking measures have been developed that satisfy certain mathematical properties or are purely ad hoc. Those measures, however, do not map back to an original investor problem, making their interpretation unclear. Not surprisingly, the Sharpe ratio, therefore, remains the gold standard in the industry, despite its lack of robustness to more general risk distributional assumptions.

    This paper derives a generalized ranking measure that is valid under a broad admissible utilityprobability space and yet preserves wealth separation for the broad HARA utility class. Our ranking measure can be used with non-Normal distributions. Because it can also pairwise compare composite risks following different distributions, it can also serve as the foundation for multi-asset class portfolio optimization, thereby replacing the mixture of other measures that are currently being used in industry. We demonstrate that the generalized ratio can produce very different optimal allocations than the Sharpe ratio, especially in the context of financial derivatives and other securities that produce non-Normal distributions. Along the way, we prove a key impossibility theorem: any ranking measure that is valid at non-Normal “higher moments” cannot generically be free from investor preferences. But, as a matter of practice, we demonstrate how the generalized ratio can be easily presented at the fund level for different risk tolerances, which already must be legally assessed by financial advisors for each investor.

    References

    [1] Agarwal, V. and Naik, N. Y. (2004). “Risk and Portfolio Decisions Involving Hedge Funds,” Review of Financial Studies, 17 (1): 63-98.

    [2] Artzner, P., Delbaen, F., Eber, J.M. and Heath, D. (1999). “Coherent Measures of Risk”. Mathematical Finance 9 (3): 203-228.

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    [39] Owen, J., and Rabinovitch, R. (1983). “On the class of elliptical distributions and their applications to the theory of portfolio choice.” Journal of Finance, 38(3): 745–752.

    [40] Pierro, M. D. and Mosevich, J. (2011). “Effects of skewness and kurtosis on portfolio rankings,” Quantitative Finance, 11(10): 1449 – 1453.

    [41] Roy, A.D. (1952). “Safety First and the Holding of Assets.” Econometrica 20: 431–450.

    [42] Samuelson, P. (1970). “The Fundamental Approximation Theorem of Portfolio Analysis in Terms of Means, Variances and Higher Moments.” The Review of Economic Studies, 37(4): 537–542.

    [43] Scott, R., and Horvath, P. (1980). “On the Direction of Preference for Moments of Higher Order than Variance.” Journal of Finance, 35: 915–919.

    [44] Shadwick, W. F. and Keating, C. (2002). “A Universal Performance Measure”, Journal of Performance Measurement, 6 (3): 59 – 84.

    [45] Sharpe, W.F. (1964). “Capital asset prices: A theory of market equilibrium under conditions of risk.” Journal of Finance 19: 425 – 442.

    [46] Sharpe, W.F. (1966). “Mutual Fund Performance,” Journal of Business 39: 119–138.

    [47] Sortino, F.A. and Price, L.N. (1994). “Performance Measurement in a Downside Risk Framework,” Journal of Investing 3(3): 59 – 65.

    [48] Spurgin, R.B. (2001). “How to Game Your Sharpe Ratio,” Journal of Alternative Investments 4(3): 38 – 46.

    [49] Tobin, J. (1958). “Liquidity Preference as Behaviour Towards Risk,” Review of Economic Studies 25: 65 – 86.

    [50] Tobin, J. (1969). “Comment on Borch and Feldstein,” Review of Econonmic Studies, 36: 13 – 14.

    [51] Wachter, J. (2012). “Can Time-Varying Risk of Rare Disasters Explain Aggregate Stock Market Volatility?,” Journal of Finance 68(3): 987-1035.

    [52] Zakamouline, V. and Koekebakker, S. (2009). “Portfolio Performance Evaluation with Generalized Sharpe Ratios: Beyond the Mean and Variance”, Journal of Banking and Finance 33(7): 1242-1254.

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  • Kent Smetters Presentation

    A Sharper Ratio: A General Measure for Ranking
    Investment Risks

    Kent Smetters1 Xingtan Zhang2

    1The Wharton School and NBER

    2The Wharton School

    March 2015

    Disclaimer

    No conflicts to report.

    Size of active and passive indexing

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    Rebalancing occurs at a higher frequency than risk measurement

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    Mismatch of trading – risk intervals under-explored

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    Example Baseline: 50 / 50 portfolio [1]

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    Example Baseline: 50 / 50 portfolio [2]

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    Example Baseline: 50 / 50 portfolio [3]

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    Andy Lo’s “Capital Decimation Partners”

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    But, you don’t invest in those strategies, right?

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    Simple trading rule [1]

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    Simple trading rule [2]

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    Simple trading rule [3]

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    Simple trading rule [4]

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    Simple trading rule [5]

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    Simple trading rule [6]

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    Myth 1: “Max drawdown” is informative

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    Myth 2: Multidimension Risk Management [1]

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    Myth 2: Multidimension Risk Management [2]

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    Myth 2: Multidimension Risk Management [3]

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    Myth 3: Higher moments are hard to estimate [1]

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    Myth 3: Higher moments are hard to estimate [2]

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    Toward a Summary Statistic

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    Sharpe Ratio [1]

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    Sharpe Ratio [2]

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    Sharpe Ratio [3]

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    Sharpe Ratio [4]

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    Sharpe Ratio [5]

    But when Sharpe Ratio is not valid, it often “breaks, not bends”

    Sharpe Ratio [5]

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    Past Attempts to Extend SR [1]

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    Past Attempts to Extend SR [2]

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    Our approach: Solve the original truncated Taylor problem [1]

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    Our approach: Solve the original truncated Taylor problem [2]

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    New Measure [1]

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    New Measure [2]

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    New Measure [3]

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    New Measure [4]

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    “1. Independent of wealth” [1]

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    “5. Ranking depends on r” [1]

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    “5. Ranking depends on r” [2]

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    “5. Ranking depends on r” [3]

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    Summary and Intuition [1]

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    Summary and Intuition [2]

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    Concern 1: Does qH N (tY ;b) work with little data? [1]

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    Concern 2: Is qH N (tY ;b) score too complicated? [1]

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    Concern 2: Is qH N (tY ;b) score too complicated? [2]

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    Concern 2: Is qH N (tY ;b) score too complicated? [3]

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    Application 1: Ranking Smart Betas

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    Market Cap Weighting

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    Market Cap vs. Equal, Div, and Fund

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    Adding Commercial Indices [1]

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    Adding Commercial Indices [2]

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    Intuition and Punchline

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    Application 2: Active Manager Selection [1]

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    Application 2: Active Manager Selection [2]

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    Application 2: Manager Selection [3]

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    Application 3: Multi-Asset Portfolio Optimization [1]

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    Application 3: Multi-Asset Portfolio Optimization [2]

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    Application 4: Is Piketty’s Endowment Example Right?

    [GETTING DATA NOW]

    Works Cited in Talk I