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Essays on passive investing Grégoire, Vincent 2013

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Essays on Passive Investing by Vincent Grégoire B.Ing., Université Laval, 2003 M.Sc., Université Laval, 2006 M.Sc., Université Laval, 2007 a thesis submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy in the faculty of graduate studies (Business Administration) THE UNIVERSITY OF BRITISH COLUMBIA (Vancouver) June 2013 © Vincent Grégoire, 2013 Abstract This thesis contains two essays related to passive investing and passive investment vehicles. In the first essay, I introduce a general equilibrium model with active investors and index- ers. The presence of indexers causes market segmentation, and the degree of segmentation is linked to the relative wealth of indexers in the economy. Any shock to this relative wealth generates excess comovement by inducing correlated shocks to discount rates of index stocks. The wealthier the indexers are, the greater the resulting excess comovement is. In the data, I find that S&P 500 stocks tend to comove more with other index stocks and less with non-index stocks, but this was not the case until the 1970s when indexing gained in popularity. I use passive holdings of S&P 500 stocks as a proxy for the wealth of indexers and find that changes in passive holdings are positively related to changes of excess comovement in S&P 500 stocks. In the second essay, I use liquid exchange traded funds to study the issue of international mutual fund predictability. Mutual fund returns are predictable when the Net Asset Value is computed from prices that do not reflect all available information. This problem was brought to the public eye with the late trading and market timing scandal of 2003, which led to SEC intervention in 2004. Since these events, mutual fund managers have been more active in adjusting NAV, reducing predictability by about half. The simple trading strategy I present yields annual returns of 33% from 2001 to 2004 and 16% from 2005 to 2010. Even after accounting for trading restrictions in mutual funds, an arbitrager could earn annual returns of 2.73% from 2005 to 2010, suggesting the problem is not fully resolved. The main methodological contribution of this essay is to develop a filtering approach based on a state-space model that embeds the fund manager problem, thus accounting for unobserved actions of fund managers. I also show that predictability increases significantly when information sources suggested by prior literature, such as index and futures returns, are supplemented by premiums on related exchange traded funds. ii Preface This dissertation is original, unpublished, independent work by the author, Vincent Grégoire. iii Table of Contents Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ii Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii Table of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iv List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viii Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix Dedication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . x 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 2 Indexers and Comovement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2.2 Related literature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 2.3 An economy with indexers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 2.3.1 Information structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 2.3.2 Consumption space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 2.3.3 Securities market . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 2.3.4 Trading strategies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 2.3.5 Agent’s preferences and endowments . . . . . . . . . . . . . . . . . . . . . 11 2.3.6 Equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 2.3.7 Relation to international asset pricing models . . . . . . . . . . . . . . . . 11 2.3.8 Agents’ problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 2.3.9 Representative agent . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 2.3.10 Stock return dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 2.4 Indexing and stock return dynamics . . . . . . . . . . . . . . . . . . . . . . . . . 16 2.4.1 Parameter values . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 2.4.2 Stock prices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 iv 2.4.3 Comovement and the price of risk . . . . . . . . . . . . . . . . . . . . . . 18 2.5 Indexing and comovement in the S&P 500 . . . . . . . . . . . . . . . . . . . . . . 20 2.5.1 Data and variable construction . . . . . . . . . . . . . . . . . . . . . . . . 20 2.5.2 Passive ownership and comovement . . . . . . . . . . . . . . . . . . . . . . 23 2.5.3 Robustness checks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 2.6 Concluding remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 3 Do Mutual Fund Managers Adjust NAV for Stale Prices? . . . . . . . . . . 34 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 3.2 Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 3.2.1 Relationship between vMF , nMF , rI≠ and rI+ . . . . . . . . . . . . . . . 37 3.3 Filtering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 3.3.1 The mutual fund managers’ problem . . . . . . . . . . . . . . . . . . . . . 38 3.3.2 The arbitrager’s problem . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 3.4 Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 3.4.1 Hypotheses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 3.4.2 Data sources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 3.4.3 ETF premiums . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 3.4.4 Matching mutual funds with ETFs and predictors . . . . . . . . . . . . . 42 3.4.5 Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 3.5 Predictability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 3.6 Fees and trading restrictions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 3.6.1 Trading restrictions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 3.6.2 Fees . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 3.7 Concluding remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 4.1 Limitations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 4.2 Future work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 A Supporting Materials for Chapter 2 . . . . . . . . . . . . . . . . . . . . . . . . 66 A.1 Proofs and model derivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66 A.1.1 Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66 A.1.2 Agents’ problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 A.1.3 Optimal portfolios . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68 A.1.4 Proof of Proposition 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 A.1.5 Proof of Proposition 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71 A.1.6 Proof of Corollary 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 A.1.7 Proof of Proposition 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74 v A.1.8 Solving for fAi,t . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 A.1.9 Solving for fIi,t . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77 A.1.10 Matching moments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79 A.2 Vector notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80 A.2.1 Rotation matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81 A.2.2 Dividend basis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82 A.2.3 Market basis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82 B Supporting Materials for Chapter 3 . . . . . . . . . . . . . . . . . . . . . . . . 84 B.1 Kalman filter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84 B.2 The mutual fund managers’ problem . . . . . . . . . . . . . . . . . . . . . . . . . 85 B.2.1 Specification 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86 B.3 Arbitragers’ problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86 B.3.1 Specification 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86 B.3.2 Arbitragers’ problem: embedded filter . . . . . . . . . . . . . . . . . . . . 87 B.3.3 Specification 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88 B.3.4 Information set without ETF premiums . . . . . . . . . . . . . . . . . . . 89 vi List of Tables Table 2.1 Changes in comovement on lagged changes in passive ownership . . . . . . . . 26 Table 2.2 Changes in cross-sectional comovement on lagged changes in passive ownership 26 Table 2.3 Changes in comovement, excluding the 20 largest S&P 500 firms, on lagged changes in passive ownership . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 Table 2.4 Changes in comovement, computed from weekly returns, on lagged changes in passive ownership . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 Table 2.5 Changes in comovement on lagged changes in passive ownership for index and non-index stocks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 Table 3.1 Count and TNA of international ETFs and mutual funds . . . . . . . . . . . . 41 Table 3.2 Summary statistics of daily premiums on international ETFs . . . . . . . . . . 43 Table 3.3 Median correlation of returns for monthly mutual fund matches . . . . . . . . 44 Table 3.4 Median estimated staleness for each specification . . . . . . . . . . . . . . . . 47 Table 3.5 Median estimated staleness for the specification with no NAV adjustment . . . 48 Table 3.6 Simple trading strategy, for each specification . . . . . . . . . . . . . . . . . . 50 Table 3.7 Simple trading strategy for the best specification, by style . . . . . . . . . . . 51 Table 3.8 Simple trading strategy, comparison between specifications and time periods . 52 Table 3.9 Count and TNA of international mutual funds by fee structure . . . . . . . . . 52 Table 3.10 Complex trading strategy, for each specification . . . . . . . . . . . . . . . . . 53 Table 3.11 Complex trading strategy for the best specification, by style . . . . . . . . . . 54 Table 3.12 Complex trading strategy, comparison between specifications and time periods 55 Table 3.13 Median correlation of returns for monthly mutual fund matches with lagged predictors, by year and style . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 Table 3.14 Median estimated staleness for funds with and without rear-load fees, by year and style . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 Table 3.15 Simple trading strategy using funds with and without rear-load fees . . . . . . 58 vii List of Figures Figure 2.1 Share of passive AUM in US equity mutual funds and ETFs . . . . . . . . . . 6 Figure 2.2 Stock prices as a function of the indexer’s relative wealth . . . . . . . . . . . 18 Figure 2.3 Expected returns and Sharpe ratios as a function of the indexer’s relative wealth . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 Figure 2.4 Comovement as a function of the indexer’s relative wealth . . . . . . . . . . . 21 Figure 2.5 Price of risk as a function of the indexer’s relative wealth . . . . . . . . . . . 22 Figure 2.6 Annual passive ownership for the S&P 500 . . . . . . . . . . . . . . . . . . . 23 Figure 2.7 Average comovement of S&P 500 stocks from 1957 to 2010 . . . . . . . . . . 24 Figure 2.8 Average comovement of S&P 500 stocks from 1957 to 2010, for the 20 largest stocks and excluding the 20 largest stocks . . . . . . . . . . . . . . . . . . . . 27 Figure 2.9 Average comovement of S&P 500 stocks from 1957 to 2010 using weekly returns 28 Figure 2.10 Average comovement of smallest 250 S&P 500 stocks and largest 250 non- S&P 500 stocks from 1957 to 2010 . . . . . . . . . . . . . . . . . . . . . . . . 31 Figure 3.1 Median estimated staleness by month for the specification with no NAV ad- justment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 viii Acknowledgments The successful completion of my doctoral studies at the Sauder School of Business would not have been possible without the support of key faculty members and fellow students. First and foremost, I would like thank my two co-advisers, Murray Carlson and Adlai Fisher, for their continued support throughout the years I spent at UBC. Their mentoring and their support – both academic and financial – had been instrumental to the research presented in this thesis. I have also largely benefited from discussions, and the resulting feedback, with Jason Chen and Harjoat Bhamra. I would also like to thank Ali Lazrak, Ron Giammarino and Carolin Pflueger for their useful comments and support. Numbers of fellow students have also helped me develop the ideas presented in this thesis. More specifically, I would like to thank Alberto Romero, Thomas Ruf, Jake Wetzel, Charles Martineau and Oliver Boguth. Without them around, not only would the quality of my research not be the same, my stay in Vancouver would not have been so enjoyable. I was lucky enough to receive generous funding throughout my studies. I would like to acknowledge financial support from UBC, from the Fonds québécois de la recherche sur la société et la culture and from the Canadian Securities Institute Research Foundation. Finally, I would like to thank my family for their continued support and encouragement during my studies. I would like to thank my parents, my siblings and most importantly my three children and my wife Julie. They provide me an infinite source of motivation, and this is why I dedicate my thesis to them. ix Dedication Pour Julie, Clara, Louis et Caleb. x Chapter 1 Introduction Ever since the introduction of the CAPM by Sharpe (1964) and Lintner (1965), investors have looked to index-linked investing as a solution to the problem of implementing diversification and holding the market. While passive investing through indexing has been a common practice for decades, the growth of the share of index-linked investment has been strong in the past few years, in part due to the introduction of liquid, low-fee passive exchange traded funds (ETFs). The share of ETFs and index funds grew from 3.25% of all US equity mutual funds in 1993 to almost 25% in 2010.1 Passive investments in the S&P 500 led the trend; it is estimated that by 1990 around 10% of the market cap of the index was held by indexers. To some, the growth in indexing is not surprising. French (2008) makes an argument in favor of passive investment: it is cheaper than active investment and, since trading is a zero- sum game, it is not possible to beat the market on average. He finds that, compared to a world in which everyone holds the market, investors spend 0.67% of the aggregate value of the market searching for superior returns. The alternative to active investing proposed by French (2008) is to maximize diversification by tracking indices that cover the total market such as the Wilshire 5000 or the Russell 3000E. However, in practice the most popular US equity indices remain the S&P 500 and the Russell 2000, which are cap-based subsets of the total market. Since in equilibrium market must clear, if a group of investors chooses to diverge from optimal diversification, it imposes under-diversification to remaining investors who as a group must hold the remaining available shares. Savov (2009) argues that because of rebalancing, indexers fail to attain buy-and-hold index returns. He finds that high index fund flows forecast low index fund returns relative to active fund returns, and this can account for most of the dierential alphas between the two types of 1Source: 2011 Investment Company Institute Fact Book. http://www.icifactbook.org/. According to data from Standard and Poor’s and from Russell Investments, passive assets under management as a share of total market cap in 2010 were 13.5%, 8.6%, 6.5%, and 6.1% respectively for the S&P 500, the S&P MidCap 400, the S&P SmallCap 600 and the Russell 2000. For the full investment landscape including pension funds, public/private funds, endowments and hedge funds, data from French (2008) suggests that in 2006 at least 10% of all investments in US equities were passive. This is a lower bound since no data was available on passive investments by foreign investors and foreign holdings who held about 40% of US equities in 2006. 1 funds. Pastor and Stambaugh (2012) propose an explanation for the presence of active investing, despite historical evidence of negative alpha, based on the idea of decreasing returns to scale. In their model, active investing yields positive alpha if there are not many active investors. As the weight of active investors increases, alphas decrease and can even become negative.2 They show that after calibration, their model can explain the rise of indexing because investors learn about the parameters of the returns function. There is a growing literature presenting empirical evidence of eects related to indexing (see Wurgler (2011) for a summary). The eects of indexing on stock prices have been studied extensively following the early work of Harris and Gurel (1986), Shleifer (1986) and Jain (1987) who all found that stocks added to the S&P 500 encounter abnormal returns and trading volume following the announcement. Further research confirmed this fact and showed that these price increases do not fully revert back to the initial level as stated by Shleifer (1986), providing evidence that stocks are not perfect substitutes and thus have downward sloping demand curves.3 This has an impact on indexers as they are front-runned in their trading by arbitragers who anticipate the demand due to rebalancing. Petajisto (2011) labeled this eect the index premium and estimated the lower bound for this cost to be around 21-28bp annually for the S&P 500 and 38-77bp annually for the Russell 2000. In Chapter 2, I aim to explain another eect of indexing called comovemement. In a frictionless economy with rational investors, prices reflect fundamental value so any comovement in prices reflects common changes in fundamentals or discount rates. However, the literature on comovement presents evidence suggesting the presence of excess comovement among index stocks.4 I first present an asset-pricing model of indexing in which comovement arises endogenously. The continuous-time economy is populated by two rational agents, an active investor and an indexer, who must allocate their wealth between a riskless bond and three risky stocks. Two of the stocks form a value-weighted index, and the indexer is constrained to invest only in the bond and those two stocks, according to index weights. I obtain closed-form approximations for equilibrium prices and moments of stock returns. In the model, index stocks have lower Sharpe ratios and higher prices than non-index stocks and they comove more with other index stocks than with non-index stocks. These eects are stronger the larger the relative wealth of indexers is. 2The idea of decreasing returns to scale was made popular by Berk and Green (2004). The dierence is that Pastor and Stambaugh (2012) apply the idea to aggregate active investments, while Berk and Green (2004) propose a cross-sectional approach. 3See Pruitt and Wei (1989); Dhillon and Johnson (1991); Graham and Pirie (1994); Beneish and Whaley (1996); Lynch and Mendenhall (1997); Erwin and Miller (1998); Wurgler and Zhuravskaya (2002); Hegde and McDermott (2003); Denis, McConnell, Ovtchinnikov, and Yu (2003); Chen, Noronha, and Singal (2004); Becker- Blease and Paul (2006) for other S&P 500 related papers and Liu (2000); Kaul, Mehrotra, and Morck (2000); Biktimirov (2004); Chakrabarti, Huang, Jayaraman, and Lee (2005); Shankar and Miller (2006); Okada, Isagawa, and Fujiwara (2006); Mase (2007); Cai and Houge (2008); Becker-Blease and Paul (2010) for papers on other indices. 4See Section 2.2 for an extensive literature review. 2 There are three sources of comovement in the model. First, the fundamental dividend streams may be correlated. Second, when one dividend stream gets a positive shock it becomes larger relative to the other assets and investors demand more of the other assets to remain diversified, generating correlated price returns. These two sources of comovement are present in frameworks with unconstrained agents and aect index stocks and non-index stocks alike. The third source of comovement, which I call excess comovement, is due to the indexing constraint. Since the two agents hold dierent portfolios, a dividend shock to any asset changes their relative wealth. This induces them to change their demand for risky stocks, which aects the price of risk. The indexer has demand only for the index stocks, so the resulting discount rate shocks for the two index stocks are positively related. The discount rate shock for the non-index stock is negatively related to those for the index stocks. Furthermore, the sensitivity of the discount rate to changes in wealth is not constant; it is increasing in relative wealth of the indexer. This makes excess comovement more pronounced when the indexer is wealthier. I test the model predictions regarding excess comovement using S&P 500 stocks. I use the amount invested passively in the index divided by the total value of the market as a proxy for the relative wealth of indexers. I call this measure passive ownership (PO). Comovement of S&P 500 stocks with other index stocks increases as passive ownership increases. Inversely, comovement with non-S&P 500 stocks decreases as passive ownership increases. This relationship is not present in the largest 250 stocks that are not part of the S&P 500, even though their size is comparable to the smallest 250 stocks in the index. In Chapter 3, I use the information content of ETFs – which are transparent and very liquid index-linked securities – to study predictability in international mutual funds. Mutual fund returns are predictable when NAV is computed from stale prices.5 For example the last trade of some securities held by a small cap fund or a foreign equity fund can occur hours before markets close. Consider a mutual fund specialized in Japanese equities. Japanese stocks trade on the Tokyo Stock Exchange until it closes at 3 p.m. local time (2 a.m. ET). When NAV is computed at 4 p.m. ET the last transaction occurred more than 14 hours earlier. Any new information is not reflected in the NAV, unless the closing prices are correctly adjusted, so returns are predictable. This predictability imposes indirect costs to long term shareholders when exploited by ar- bitragers. It can lead to frequent trading of fund shares, imposing costs to the fund – such as transaction fees and liquidity issues – which are borne by all shareholders. Furthermore, net gains to arbitragers are net losses to long term shareholders.6 This dilution and fund outflows resulting from lower realized returns cause assets under management to decrease over time, which aects long-term compensation of fund managers.7 5See, e.g., Chalmers, Edelen, and Kadlec (2001); Bhargava and Dubofsky (2001); Goetzmann, IvkoviÊ, and Rouwenhorst (2001); Greene and Hodges (2002); Boudoukh, Richardson, Subrahmanyam, and Whitelaw (2002); Varela (2002); Zitzewitz (2003). 6Looking at flow of funds, Greene and Hodges (2002) find that dilution in international funds was 0.48% between 1998 and 2001 while Zitzewitz (2003) finds 1.14% in 2001. 7McCabe (2009) looks at fund managers incentives and concludes that after accounting for the short-term 3 A decade ago, Chalmers, Edelen, and Kadlec (2001) and Zitzewitz (2003) found that while fund managers where allowed to adjust NAV to reflect new information, most did not. Fair value adjustment solutions aim to fix the root cause of the problem: if there is no predictability, there will be no market timing arbitrage. Many such approaches have been suggested to adjust either individual securities or the fund’s overall NAV.8 I show that after 2004 mutual fund managers are more active in adjusting NAV. The simple trading strategy I present yields annual returns of 33% from 2001 to 2004 and 16% from 2005 to 2010. ETFs and mutual funds with similar investment styles suer correlated shocks. Observations of ETF premiums therefore provide valuable information on the fair value of funds. These premiums, defined as the log dierence between market prices and NAV, represent a direct proxy for the degree of staleness in the ETF. Prior studies of market timing and predictability use liquid US equity indices and market prices of futures as predictors of NAV returns.9 I show that predictability increases significantly when information sources suggested by prior literature are supplemented by premiums on related ETFs. gain due to increased investments, fund managers are still better o preventing market timing arbitrage because of the long term loss of revenue due to dilution and loss of clients. 8See Ciccotello, Edelen, Greene, and Hodges (2002) and Zitzewitz (2003) for an overview of the dierent methodologies. 9Zitzewitz (2003) discussed the possibility of using ETF returns as predictors, but dismissed it because of the relative illiquidity of international ETFs at the time. Jares and Lavin (2004) use the returns on country ETFs in the context of mutual fund predictability as an explanatory variable alongside returns on the S&P 500. 4 Chapter 2 Indexers and Comovement 2.1 Introduction The share of index-linked investments increased dramatically over the past 15 years, in part due to the introduction of liquid, low-fee passive exchange traded funds (ETFs). Figure 2.1 illustrates that the share of ETFs and index funds grew from 3.25% of all US equity mutual funds in 1993 to almost 25% in 2010.10 Passive investments in the S&P 500 led the trend; it is estimated that by 1990 around 10% of the market cap of the index was held by indexers. In their seminal study of stock return comovement, Barberis, Shleifer, and Wurgler (2005) show that the impact of joining the S&P 500 on comovement is stronger in the later half of their sample. It is natural to think the rise of indexing could have caused this increase in comovement of index stocks. I investigate this hypothesis and show, in a theoretical framework and empirically, that an increase in the level of indexing causes index stocks to comove more with other index stocks and less with non-index stocks. Understanding comovement is central to the way we think about risk in finance. The most common way to adjust returns for risk in the academic literature is to obtain excess returns after controlling for comovement with the Fama-French (or Fama-French-Cahart) risk factors. As Cremers, Petajisto, and Zitzewitz (2010) show, these factors are highly correlated with returns of popular indices. For example, correlation of the Market-Rf factor with the S&P 500 from 1986-2005 is 0.981. Correlation of the Small minus Big factor with the Russell 2000 minus the S&P 500 is 0.933. This alignment of common risk factors with popular indices, combined with an increase in index-linked comovement, causes index membership to emerge as a new risk factor. I aim to shed some light on how we should think about this new risk factor going 10Source: 2011 Investment Company Institute Fact Book. http://www.icifactbook.org/. For the full investment landscape including pension funds, public/private funds, endowments and hedge funds, data from French (2008) suggests that in 2006 at least 10% of all investments in US equities were passive. This is a lower bound since no data was available on passive investments by foreign investors and foreign holdings who held about 40% of US equities in 2006. Bhattacharya and Galpin (2011) use cross-sectional variance of turnover as a proxy for value-weighted investing and find that between 1995 and 2007 value weighting increased in popularity in 35 of the 46 countries under study. 5 forward by explaining how index-linked comovement arises. Many potential explanations for index-linked comovement have been proposed before. Some favor behavioral explanations, with noise traders or momemtum traders generating correlated trading and thus comovement (see, e.g., Barberis and Shleifer (2003) and Barberis, Shleifer, and Wurgler (2005)). Others proposed friction-based explanations, such as the dierent costs or eorts required to process information at the individual asset level versus the index level (see, e.g., Veldkamp (2006), Peng and Xiong (2006) and Mondria (2010)). Finally, some propose that price discovery is faster for index stocks, so comovement arise because of time-synchronicity in trading (see Barberis, Shleifer, and Wurgler (2005)). My explanation is that investors who focus on index investing create market segmentation, and this market segmentation generates comovement. I first present an asset-pricing model of indexing in which comovement arises endogenously. The continuous-time economy is populated by two rational agents, an active investor and an indexer, who must allocate their wealth between a riskless bond and three risky stocks. Two of the stocks form a value-weighted index, and the indexer is constrained to invest only in the bond and those two stocks, according to index weights. The third stock is only available to the active investor, so in equilibrium she holds the full supply. Having three stocks allows me to study the comovement of one index stock with the other index stock and with the non-index stock. I obtain closed-form approximations for equilibrium prices and moments of stock returns. Figure 2.1. Share of Passive AUM in US Equity Mutual Funds and ETFs 20101993 1996 1998 2000 2002 2004 2006 2008 25 % 0 2 4 6 8 10 12 14 16 18 20 22 Year Pa ss iv e In ve st m en ts  (% ) ETFs Mutual Funds Notes: This figure presents the share of assets under management in US equity mutual funds and exchange traded funds that are passively invested, broken down in the share of passive mutual funds (bottom) and ETFs (top). Data is from the 2011 Investment Company Institute Fact Book. http://www.icifactbook.org/ 6 In the model, index stocks have lower Sharpe ratios and higher prices than non-index stocks and they comove more with other index stocks than with non-index stocks. These eects are stronger the larger the relative wealth of the indexer is. There are three sources of comovement in the model. First, the fundamental dividend streams may be correlated. Second, as in Cochrane, Longsta, and Santa-Clara (2008) and Martin (2013), when one dividend stream gets a positive shock it becomes larger relative to the other assets (its dividend share increases). To remain diversified investors demand more of the other assets, generating correlated price returns. These two sources of comovement are present in frameworks with unconstrained agents and aect index stocks and non-index stocks alike. The third source of comovement, which I call excess comovement, is due to the indexing constraint. Since the two agents hold dierent portfolios, a dividend shock to any asset changes their relative wealth. This induces them to change their demand for risky stocks, which aects the price of risk. The indexer has demand only for the index stocks, so the resulting discount rate shocks for the two index stocks are positively related. The discount rate shock for the non-index stock is negatively related to those for the index stocks. Furthermore, the sensitivity of the discount rate to changes in wealth is not constant; it is increasing in relative wealth of the indexer. This makes excess comovement more pronounced when the indexer is wealthier. I test these model predictions regarding excess comovement using S&P 500 stocks. I use the amount invested passively in the index divided by the total value of the market as a proxy for the relative wealth of indexers. I call this measure passive ownership (PO). Comovement of S&P 500 stocks with other index stocks increases following passive ownership increases. Inversely, comovement with non-S&P 500 stocks decreases following passive ownership increases. This relationship is not present in the largest 250 stocks that are not part of the S&P 500, even though their size is comparable to the smallest 250 stocks in the index. 2.2 Related literature In a frictionless economy with rational investors, prices reflect fundamental value so any co- movement in prices reflects common changes in fundamentals or discount rates. Cochrane, Longsta, and Santa-Clara (2008) show that in an economy with two Lucas trees available in fixed supply, market clearing causes stock returns to be correlated even if the dividend streams are not. When one of the trees enjoys a positive shock, its share of aggregate dividend in- creases, lowering the other asset’s share. Lowering the share of dividend typically raises the price-dividend ratio – causing a positive return – because the asset provides more diversification benefit. However, the literature on comovement presents evidence suggesting the presence of excess comovement within index stocks.11 To explain this phenomenon, prior research developed friction-based and sentiment-based theories of comovement. Barberis and Shleifer (2003) analyze the idea of style investing (also 11See, for example, Vijh (1994), Barberis, Shleifer, and Wurgler (2005) and Boyer (2011) for evidence of comovement in S&P indices or Shiller (1989) for earlier evidence of excess comovement in equities. 7 called the category view) and its relationship to comovement. This theory states that investors group assets into dierent categories based on some characteristic, such as size or book-to- market ratio, in order to simplify portfolio decisions. They then allocate funds at the style level rather than at the individual asset level, generating correlated flows to assets in the same style. Peng and Xiong (2006) present a model in which investors have scarce cognitive resources. In- vestors prefer to learn category-level information, and the resulting endogenous style investing causes comovement. Another theory for comovement is the habitat view, which proposes that some investors restrict their investments to a subset of available assets due to frictions such as trading costs or information availability. When these investors change their portfolio because of liquidity needs, change in risk aversion, or other idiosyncratic events, they induce correlated trading for the subset of assets they invest in. Finally, the information diusion view of co- movement proposes that comovement arises because information is incorporated at dierent rate for dierent assets. Thus, assets prices for which information is incorporated at the same rate will comove more because of time-synchronicity in trading. A related view is that when faced with new information, investors for adjust their index exposure before evaluating stocks individually. Barberis, Shleifer, and Wurgler (2005) investigate these dierent views of comovement by looking at S&P 500 additions and deletions. They find evidence consistent with all three views of comovement. Focusing on stocks that should incorporate information at similar rate, Greenwood (2008) and Boyer (2011) provide strong evidence consistent with the style and habitat views of comovement. While my story is based on the category view, I do not reject the other explanations. I provide evidence suggesting that pure indexers play a role generating excess comovement in index stocks, but it is likely that other factors also contribute to this phenomenon. Another use of indices that could be related to comovement is benchmarking in delegated portfolio management.12 Basak and Pavlova (2012) find predictions that are similar to mine in the presence of benchmarking: discount rates are lower for index stocks, and index stocks comove more with other index stocks than with non-index stocks. However, in their model non- index stocks are not aected by the wealth of benchmarked investors. The wealth of indexers does not aect the correlation between index stocks and non-index stocks, which remains zero. Institutional investors have been associated with excess comovement early on. For example, Pindyck and Rotemberg (1993) show that excess comovement in equities is greater in stocks with high institutional ownership. Grouping institutional investors by their level of activity, Ye (2012) argues that active investors can eliminate excess comovement, but that passive investors play no role in comovement. This conclusion regarding the role of passive investors is contrary to my hypothesis, and I believe it is flawed by the way passive investors are identified. Using the Active Share measure of Cremers and Petajisto (2009), he ranks institutional investors based on the level of activity and labels as passive investors those in the lowest tercile. Under this 12 See, e.g., Cuoco and Kaniel (2011), Basak and Pavlova (2012) and Brennan, Cheng, and Li (2012). 8 definition, investors identified as passive could be true indexers, closet indexers or even active investors who are less active relative to other active investors. The underlying assumption is that investors in the bottom tercile – a proportion that is fixed through time – are passive, while my story is about the impact of changes in passive investing. 2.3 An economy with indexers The model consists of an infinite-horizon, continuous-time economy with three Lucas trees and two agent types, active investors and indexers. I extend Bhamra (2007), in which there are two agents and two risky assets, by adding an indexing constraint and an additional risky stock. This additional risky asset allows me to derive closed-form approximations for the comovement of an index stock with other index stocks and with non-index stocks as a function of state variables. One of these state variables is the relative wealth of indexers in the economy. The indexing constraint is exogenous. One might think of indexing as a self-imposed constraint motivated by the simplicity of index-linked investing. Veldkamp (2006) presents an information-based explanation; she shows that in a Grossman-Stiglitz framework with economies of scale in information production, it is rational for some agents to only acquire in- formation on an index rather than on individual stocks. These agents then endogenously choose to engage in index-linked investing, as they cannot distinguish between individual stocks.13 2.3.1 Information structure Uncertainty is represented by a filtered probability space (,F ,F,P) on which is defined a 3- dimensional vector of independent Brownian motions Z = [Z1 Z2 Z3]Õ. The filtration F= {Ft} is the augmentation under P of the filtration generated by Z. The sigma-field Ft represents the information available at time t and the probability measure P represents the agent’s common beliefs. Stochastic processes to follow are progressively measurable with respect to F and equalities involving random variables hold P-a.s. 2.3.2 Consumption space There is a single perishable good, the numeraire. The agents’ consumption set C is given by the set of non-negative progressively measurable consumption rate process ct with s T 0 |ct|dt < Œ, ’T œ [0,Œ). 2.3.3 Securities market The investment opportunities are represented by a locally riskless bond earning the instanta- neous interest rate r and three risky stocks, representing claims to exogenously given strictly 13 Peng and Xiong (2006) also find that categorizing assets is an endogenous outcome when investors have limited attention, as they choose to focus on market-wide and sector-wide information. Mondria (2010) presents another model where frictions on information lead to categorizing and generate price comovement. 9 positive dividend processes Di, i= 1,2,3, with dDi,t Di,t = µDdt+‡DdZDi,t , i œ {1,2,3}, (2.1) where ZD,i are standard Brownian motions14 with equal pairwise correlation coecients flD. The aggregate dividend if defined as DM,t = q3 i=1Di,t and the index dividend is defined as DI,t = q2 i=1Di,t.15 The initial bond value is normalized to unity so that the bond price process is given by Bt = exp 3⁄ t 0 rsds 4 . (2.2) The stock price processes can be defined as dSi,t = (Si,tµi,t≠Di,t)dt+Si,t‡i,tdZt, (2.3) where µt is the 3-dimensional column vector with µi,t as the ith element and ‡t is the 3◊ 3 matrix with ‡i,t as the ith column. The instantaneous covariance matrix is t = ‡t‡Õt. Both µt and ‡t are determined endogenously in equilibrium. Stock return processes can be defined from stock prices and dividends as dRi,t = dSi,t+Di,tdt Si,t = µi,tdt+‡i,tdZt. (2.4) The supply of each stock is normalized to one share, while the bond is in zero net supply. There exist a value-weighted index with stocks 1 and 2 as its constituents. The third stock is a non-index stock. 2.3.4 Trading strategies Trading takes place continuously. An admissible trading strategy is a 4-dimensional vector process (–,“), where “ is an 3-dimensional column vector with “i as its ith element and –t and “i,t denote the amounts invested at time t in the bond and in stock i, satisfying the required regularity conditions.16 A trading strategy (–,“) is said to finance the consumption plan c œ C if the corresponding wealth process W = –+1Õ“ satisfies the dynamic budget constraint dWt = [–trt+“Õtµt≠ ct]dt+[“Õt‡t]dZt, (2.5) where 1 is a 3-dimensional column vector of ones. 14ZD,i are linear combinations of the fundamental independent Brownian motions Zi defined in 2.3.1. For more details about the transformation, see Appendix A.2. 15See Appendix A.1.1 for details on the market and the index portfolios. 16See pp.234-235 of Back (2010) for a formal presentation of the required regularity conditions. 10 2.3.5 Agent’s preferences and endowments There are two representative agents, an active investor A and an indexer I, both with time- additive log-normal utility functions: Uj,t(c) = Et 5⁄ Œ 0 e≠”· log(cj,(t+·))d· 6 , j œ {A,I} (2.6) for some common rate of time preference ” > 0 and individual consumption cj . Agents dier by their endowment and by their investment opportunity set. The indexer is endowed with a fraction — of each index stocks while the active investor owns (1≠—) share of each index stock and one share of the non-index stock. The indexer faces an exogenous constraint that limits her investment opportunity set to the bond and the index portfolio, according to index weights, which are endogenous. The active investor is unconstrained and faces a complete market. 2.3.6 Equilibrium Let E = ((,F ,F,P),D1,D2,D3,U1,U2,—) denote the primitives for the economy. An equilib- rium for the economy E is an interest rate stock price process (r,S) and a set {cúj ,(–új ,“új )}, j œ {A,I} of consumption and admissible trading strategies for the two agents such that: (i) (–új ,“új ) finances cúj for j œ {A,I}; (ii) cúA maximizes UA over the set of consumption plans c œ C financed by an admissible trading strategy (–,“) œ “ with –0+“Õ01= (1≠—)[S1,0+S2,0]+S3,0; (iii) cúI maximizes UI over the set on consumption plans cœ C financed by an admissible trading strategy (–,“) œ “ with –0+“Õ01= —[S1,0+S2,0], “j,t = “I,t Si,tS1,t+S2,t for i= 1,2 where “I is the amount invested in the index and “3 © 0; (iv) all markets clear: cúA+ cúI =D, –úA+–úI = 0 and “úA+“úI = S. 2.3.7 Relation to international asset pricing models The setup is similar to models of international asset pricing with mild segmentation. It is conceptually related to a setup with two countries and with investors in one country being con- strained to invest at home while the investors in the other country are unconstrained. Bhamra (2007), which I extend, is one such model. In my model, the constrained investor faces the ad- ditional constraint of having to invest according to the index weights; i.e. her actions influence the price of the index but not the price of index stocks relative to one another. My model is also closely related to Pavlova and Rigobon (2008), in which there are three countries (with three agents and three goods). They find that portfolio restrictions and terms of trade generate excess comovement in asset prices. While their portfolio constraints are dierent, the economic mechanism through which excess comovement appears is the same as in my model. 11 2.3.8 Agents’ problem Agent j has instantaneous wealth Wj,t = –j,tBt+“Õj,tSt. Define fij,t a 3-dimensional vector with the i-th element equal to agent’s j proportion of wealth invested in stock i at time t (fij,t= “j,tWj,t ). For the constrained indexer I, this reduces to fiI,t = fiII,t[ÊI1,t ÊI2,t 0]Õ, where fiII,t is the indexer’s proportion of wealth invested in the index portfolio. It will be more convenient to deal with normalized wealth such that ‹j,t = Wj,tWA,t+WI,t is the share of wealth owned by agent j. Agent j’s optimization problem at time t is to maximizes her time additive utility: Uj,t = Et 5⁄ Œ t e≠”(s≠t) log(cj,s)ds 6 , (2.7) subject to her budget constraint, which gives maxUj,t subject to Et C⁄ Œ 0 ›j,s ›j,t cj,sds D ÆWj,t, (2.8) where ›j,t is the marginal utility of consumption of agent j at time t. This process can be written as d›j,t ›j,t =≠rj,tdt≠◊Õj,tdZt, (2.9) where rj,t is agent’s j implied risk-free rate and ◊j,t is her shadow price of risk. Both agents trade in the bond, so in equilibrium they will have the same risk-free rate (i.e. rI,t = rA,t = rt.) However their dierent investment opportunity sets means they will face dierent market prices of risk. Agent A is unconstrained, so her optimal portfolio proportions are given by fiA,t = ≠1t (µt≠ r1). (2.10) For agent I, using the convex duality approach of CvitaniÊ and Karatzas (1992) to derive optimal portfolio weights17, fiI,t coincides with the optimal portfolio in the incomplete market: fiI = SWWU fiIIÊI1 fiIIÊI2 0 TXXV , (2.11) where fiII,t = (µI,t≠ r)/‡2I,t. 17See Appendix A.1.2 for details. This methodology has been previously used in the finance literature for portfolio constraints, see for example Shapiro (2002) and Pavlova and Rigobon (2008) for similar applications or Chapters 5 and 6 of Karatzas and Shreve (1998) for a textbook treatment. 12 The market clearing condition imposes that Êt = fiA,t‹A,t+fiI,t‹I,t, (2.12) where Êt is a 3-dimensional vector with the i-th element equal to the value-weight of stock i in the economy (Êi = Si/ q3 k=1Sk). Substituting the optimal portfolio weights in the market clearing condition yields the following proposition: Proposition 1 In equilibrium, expected excess stock returns are as follows: µ1≠ rf = 1‡2IÊI [(µI ≠ rf )(‡1Ê1+fl1,2‡2Ê2) + Ê2Ê3 ‹AÊI 1 Ê1[cov(R1,R2)cov(R1,R3)≠‡21cov(R2,R3)] ≠Ê2[cov(R1,R2)cov(R2,R3)≠‡22cov(R1,R3)] 2È , (2.13) µ3≠ rf = Ê3‡23+(1≠Ê3)cov(RI ,R3)+Ê3‡23 5 ‹I ‹A (1≠fl2I,3) 6 , (2.14) µI ≠ rf = ÊI‡2I +(1≠ÊI)cov(RI ,R3), (2.15) where µI and ‡I denote the drift and variance of the index and ÊI = Ê1+Ê2. Result for stock 2 is omitted as it is symmetric to stock 1. Proof See Appendix A.1.4. Proposition 1 tells us that holding variances/covariances constant, the non-index stock ex- cess returns are increasing in the relative wealth of passive investors. This is due to the addi- tional risk active investors are taking as they become more under-diversified (compared with the case where they hold the market). It is highlighted by the term of correlation between the index and stock 3. This is consistent with the standard result from one period models of mild segmentation.18 I cannot however conclude from Proposition 1 on the actual equilibrium eect of an increase in ‹I as the variance and covariance terms are determined endogenously in equilibrium and thus also depend on ‹I . 2.3.9 Representative agent The equilibrium concept used is a multi-trees extension of Lucas (1978)19 in which I determine the equilibrium asset prices as the shadow prices from the pricing kernel determined by the investor’s marginal utility of consumption and market clearing. Market clearing implies that they must hold all the shares in the world, with the restriction that the indexer is constrained in her investment opportunity set, therefore forcing the unconstrained agent to hold all the shares of the non-index stock. 18See, e.g., Errunza and Losq (1985). 19See Cochrane, Longsta, and Santa-Clara (2008) and Martin (2013) for multi-trees extensions of Lucas (1978) with no investment constraints. 13 Following Cuoco and He (1994), I can still use a social planner to derive equilibrium prices, but the weight ⁄t will be stochastic: Ut = Et ⁄ Œ t e≠”(s≠t) (logcA,s+⁄s logcI,s)ds. (2.16) Agent A is unconstrained, so her state-price density must correspond to the equilibrium state-price density:20 ›t = ›A,t = ŸAe≠”t(‹A,tDM,t)≠1. (2.17) Solving for the equilibrium state-price density in terms of endogenous stock returns mo- ments, I obtain the following proposition: Proposition 2 The equilibrium risk-free rate and price of risk of the representative agent are rf = rıf + fl‹ADM‡‹A‡DM ‹A , (2.18) ◊ = ◊ı+ ‡‹A ‹A . (2.19) where rıf = ”+µDM ≠‡2DM and ◊ ı = ‡DM are the risk-free rate and price of risk in the uncon- strained economy (◊ and ‡ indicate 3-dimensional vectors) and fl‹ADM is the covariance between the aggregate dividend process and the consumption share process ‹A. Proof See Appendix A.1.5 for the derivation. From Corollary 2 it will be clear that when ‹A = 1 (there is no indexer) the term ‡‹A vanishes and we obtain rf = rıf and ◊ = ◊ ı. Proposition 2 states that the risk-free rate is increasing in ‹I (remember, ‹I = 1≠‹A if the covariance between the relative wealth process of the active investor with the aggregate dividend is positive. While I do not prove it formally, numerical results show that, with reasonable parameters, fl‹ADM is positive; the risk-free rate is increasing in ‹I as the indexer is a net borrower. The proposition also states that the price of each risk factor is increasing in the sensitivity of the relative wealth process to each risk factor and in ‹I . However, since the relative wealth process is endogenous to the model, it is not yet clear what the eect will be in equilibrium. The following corollary will shed a little more light on these results. Corollary 1 In equilibrium, the prices of risk of individual agents have the following relation- ship: 1 ◊A≠◊ı 2 =≠ ‹I ‹A 1 ◊I ≠◊ı 2 . (2.20) Proof See Appendix A.1.6. 20See Appendix A.1.2 for details of the derivation. 14 Corollary 1 illustrates where the excess comovement originates. Keeping in mind that ‹I and ‹A Ø 0, it states that individual agents’1 prices of risk will deviate from the unconstrained price of risk in opposite directions. In equilibrium, the components of ◊I corresponding to the index stocks are larger than the corresponding components of ◊ı, while the opposite is true for the component corresponding to the non-index stock. Any shock to the relative wealth of the two agents aects the equilibrium price of risk ◊A in opposite directions for index and non- index stocks. This generates excess comovement. The slope of the coecient ‹I‹A is (1≠‹I)≠2, so it is increasing in ‹I . This causes the excess comovement to increase with relative wealth of indexers. Corollary 2 In equilibrium, the share of aggregate wealth owned by the active investor follows the process: d‹A = µ‹Adt+‡‹AdZ⁄, (2.21) where µ‹A = ‹A‹2I‡2⁄, (2.22) ‡‹A = ‹A‹I‡⁄, (2.23) and ‡⁄ is the volatility of the stochastic weight in the representative agent’s problem. Proof Follows from the proof of Proposition 2. Corollary 2 illustrates that the equilibrium is not stationary. Since µ‹A is positive, over time the active investor will dominate, and ‹A = 1 (‹A = 0) is an absorbing state (in that case both µ‹A and ‡‹A are equal to 0). This is expected from a model with constrained investors; the unconstrained one will dominate over time since both agents have the same preferences but dier in their investment opportunity sets. Thus, in its current form the model cannot explain the rise of indexing of the past decades. A richer model could generate the observed level of indexing as an endogenous outcome in a general equilibrium setup. For example, this could be done by adding frictions such as the incremental cost of active investing and allowing one rational agent to invest both passively and actively at same time, as in Petajisto (2009). However, the additional complexity is not necessary for the current discussion. The current form of the model provides valuable insight on stock prices dynamics for given levels of indexing, which is the purpose of this paper. 2.3.10 Stock return dynamics Since both agents have time-additive log utility, it follows from the first order condition of the HJB equation that the aggregate stock market value SM,t = S1,t+S2,t+S3,t =DM,t/”, and thus aggregate stock market value is independent of the relative wealth of agents.21 21For infinite horizon log utility, J(t,w,x) = logw” + f(t,x). The FOC of the HJB equation is u Õ(ci) = Jwi , which yields ci = wi”. In the current setup, the representative agent must consume the aggregate dividend and 15 Proposition 3 The equilibrium stock prices have the form: Si,t = SAi,t+ ‹I,t ‹A,t XIi,t, (2.24) where SAi,t is the stock price in the unconstrained economy and XIi,t is an adjustment for the consumption of the constrained agent. Proof See Appendix A.1.7. Using a perturbation expansion22 I obtain a closed form approximation for the stock price in the unconstrained economy: SA1 = DM ” A s1+ s1 ! 1≠3s1+2s21≠2s2+2s1s2+2s22 " (1≠flD)‡2D 2” B +O(‘4), (2.25) where si =Di/DM is the share of dividends of asset i. I find SA2 by symmetry and SA3 by using the fact that SA3 = DM” ≠SA1 ≠SA2 . Solving for XIi,t is trickier as the relative wealth process ‹I depends on stock prices and stock prices in turn depend on ‹I , so I must solve for ‹I and XIi jointly. Using again a perturbation expansion, I obtain a closed form approximation for the stock price adjustment in the constrained economy: XI1 = DMs1 2(s1+s2)”2 1 2(s1+s2≠1) 1 2 1 s21≠s1+s1s2+s22 2 ≠s2 2 +(s1+s2) 1 1+2s21≠3s1+2s1s2+2s22≠2s2 2 ‹A 2 (1≠flD)‡2D+O(‘4). (2.26) As for the stock prices in the unconstrained economy, I find XI2 by symmetry and XI3 by XI1 +XI2 +XI3 = 0. 2.4 Indexing and stock return dynamics I present predictions of the model regarding dynamics of stock returns in the presence of index- ers, which I obtain from numerical results. Stock return dynamics are function of the relative level of each dividend stream and of the relative wealth of the two agents. Since I am mostly concerned with the eect of the relative weight of indexers in the economy, I fix the relative level of each dividend stream and look at conditional moments as a function of the relative weight of indexers ‹I . 2.4.1 Parameter values For the following discussion, I fix the parameter values. I set the rate of time preference ” = 0.01. I obtain from Datastream the historical dividend yield for the S&P 500 and the S&P own the aggregate stock market, thus cA+ cI =DM and wA+wI = SM . We thus have that SM =DM/”. 22See Hinch (1991) for a textbook treatment of perturbation methods. 16 600 Smallcap from 1999 to 2011. The correlation between the two series is 0.956 using annual data and 0.919 using daily data, so I set flD = 0.94. The average dividend yield for the S&P 500 from 1983 to 2011 is 2.60%, so I set µD = 0.026. The standard deviation of annual returns for the S&P 500 from 1983 to 2011 is 0.17, so I set ‡D =0.17. Cochrane, Longsta, and Santa-Clara (2008) show that in a model with fixed supply of assets, the dividend share influences stock return dynamics, so I look at three configurations for the relative level of dividend streams. In each of them, I set the current level of stock 1’s dividend to 1% of the current aggregate dividend (s1 = 0.01), so it represents a marginal index stock. In the first configuration, I set the level of stock 2 and 3’s dividends to equal shares of the rest (s2 = s3 = 0.495). In an unconstrained economy, stocks 2 and 3 would have the same dynamics since their dividend processes have the same parameters and their dividend levels are equal. However, this assumption of equal shares is far from the reality of the two most widely used US equity indices: the S&P 500 and the Russell 2000. The relative weight of the index in the economy is an important factor to consider when looking at implications of indexing. As of 2010, the S&P 500 represents about 78% of the total US market capitalization, while the Russell 2000 represents only 9%. The second and third specifications are set to mimic the S&P 500 and Russell 2000, with respective dividend levels set to s2 = 0.77, s3 = 0.22 and s2 = 0.08, s3 = 0.91. Stock prices also depend on the current level of aggregate dividends, so I normalize DM = 1 in order to get stock prices that reflect a relative weight of each stock’s value in the market (SM = DM/” = 100). The focus of this analysis should be qualitative rather than structural. I do not claim that the model quantitatively matches empirical evidence. As illustrated by Petajisto (2009), it is not possible to generate indexing eects of economically significant magnitude with this type of model. 2.4.2 Stock prices Figure 2.2 shows the stock price for the large index stock and non-index stock as a function of ‹I for the equal size specification. As the eect of segmentation becomes larger, the index stock increases in price while the non-index stock decreases in price. This illustrates that stocks are imperfect substitutes as one is not available to all investors. This is consistent with Shleifer (1986) who finds that stocks included in the S&P 500 experience positive abnormal returns on the event and that there is a permanent component to that eect. The results are qualitatively similar for the other specifications, but the price dierential due to the dividend share dierential makes it hard to see on a graph. Figure 2.3 shows that the expected return of the index stock is decreasing as a function of ‹I for all specifications while it is increasing for the non-index stock, which explains the relationship for prices. This decrease in expected return leads to a decrease in the Sharpe ratio for the index stock. Depending on the relative dividend shares, the Sharpe ratio of the non- index stock is increasing or decreasing with ‹I , but the slope is always negative and of greater magnitude for the index stock. 17 Figure 2.2. Stock Prices as a Function of the Indexer’s Relative Wealth 0.00 0.05 0.10 0.15 0.20 0.25 0.30n I 48.0 48.5 49.0 49.5 50.0 50.5 51.0 S non - in dex in dex Notes: This figure presents the stock price of the large index stock (stock 2, solid blue line) and the non-index stock (stock 3, dashed red line) as a function of ‹I , the share of wealth owned by the indexer. The parameters used are ” = 0.01, ‡D = 0.17, µD = .026, flD = 0.94 and DM = 1. Results are for the specification where the large index stock (stock 2) and the large non-index stock (stock 3) have the same dividend level (s2 = s3 = 0.495). 2.4.3 Comovement and the price of risk The idea of excess comovement is that index stocks comove more with other index stocks than they do with non-index stocks compared to the benchmark case with no indexers. Figure 2.4 shows the comovement — of index stock 1 with the other index stock and with the non-index stock, as measured with a bivariate regression in the style of Barberis, Shleifer, and Wurgler (2005). We can see that for each specification, the — with the other index-stock increases with ‹I while the — with the non-index stock decreases. The origin of the x axis (‹I = 0) shows the comovement in an unconstrained model. Deviations from those values as ‹I increases represent excess comovement. The model predicts that excess comovement is increasing in the relative wealth of the indexer. To understand what is driving this excess comovement, it is useful to look at the eect of indexers on the price of risk ◊. The components of ◊ represent the price of risk factors Zi, which are the shocks to the dividend streams. Z1 and Z2 are the shocks to the index stocks’ dividend streams while Z3 is the shock to the non-index stock’s dividend stream. Figure 2.5 presents the price of risk for a setup with equal dividend share between stock 2 and 3.23 Panel (a) presents the price of risk in the unconstrained economy (does not depend on ‹I) and Panel (b) presents the shadow price of risk for the indexer. As we can see, the indexer’s shadow price of risk for the index stocks is much higher than in the unconstrained case, while the opposite is true for the non-index stock. The resulting equilibrium price of risk, presented in Panel (c), is one where the price of risk of Z1 and Z2 are decreasing as ‹I increases, while the price of risk of Z3 increases. This means that for a given shock to the relative wealth of the indexer, the price of risks associated with index stocks reacts in the opposite direction than the one associated 23The values used are s1 = 0.1 and s2 = s3 = 0.45. s1 is set to a larger value than in the previous figures to scale the associated price of risk on the the graph, but results are qualitatively similar with s= 0.01. 18 Figure 2.3. Expected Returns and Sharpe Ratios as a Function of the Indexer’s Relative Wealth 0.05 0.10 0.15 0.20 0.25 0.30n I 0.0358 0.0360 0.0362 0.0364 m non - in dex in dex (a) 0.05 0.10 0.15 0.20 0.25 0.30n I 0.0357 0.0358 0.0359 0.0360 0.0361 m non - in dex in dex (b) 0.00 0.05 0.10 0.15 0.20 0.25 0.30n I 0.0345 0.0350 0.0355 0.0360 m non - in dex in dex (c) 0.05 0.10 0.15 0.20 0.25 0.30n I 0.167 0.168 0.169 0.170 Sharpe Ratio non - in dex in dex (d) 0.05 0.10 0.15 0.20 0.25 0.30n I 0.1680 0.1685 0.1690 0.1695 0.1700 0.1705 Sharpe Ratio non - in dex in dex (e) 0.05 0.10 0.15 0.20 0.25 0.30n I 0.155 0.160 0.165 0.170 Sharpe Ratio non - in dex in dex (f) Notes: This figure presents the expected returns and Sharpe ratios of the large index stock (stock 2, solid blue line) and the non-index stock (stock 3, dashed red line) as a function of ‹I , the share of wealth owned by the indexer. As the relative wealth of the indexer becomes larger, the Sharpe ratio of the index stock decreases. The parameters used are ” = 0.01, ‡D = 0.17, µD = .026 and flD = 0.94. Panels (a) and (d) present results for the setup where the large index stock (stock 2) and the large non-index stock (stock 3) have the same dividend level (s2 = s3 = 0.495). Panel (b) and (e) present results for the setup where the large index stock has a higher dividend level than the large non-index stock (s2 = 0.77, s3 = 0.22). Panel (c) and (f) present results for the setup where the large index stock has a lower dividend level than the large non-index stock (s2 = 0.08, s3 = 0.91). 19 with the non-index stock, generating excess comovement.24 Furthermore, the magnitudes of the slopes are increasing in ‹I , so the resulting excess comovement becomes larger as ‹I increases. 2.5 Indexing and comovement in the S&P 500 Using S&P 500 stocks, I test empirically the prediction that index stocks comove more with other index stocks and less with non-index stocks as the relative wealth of indexers increases. 2.5.1 Data and variable construction I obtain S&P 500 index constituents and individual stock returns and characteristics from CRSP. I consider all CRSP stocks listed on Amex, Nasdaq or NYSE with share codes 10 or 11. These restrictions exclude an average of 16 S&P 500 stocks per year after 1983. The annual estimates of passive assets under management in the S&P 500 for 1983-2010 are from Standard and Poor’s. I define the passive ownership (PO) of an index at year t as the total passive assets under management divided by the total CRSP market capitalization. I use this value as a proxy for the relative wealth of passive investors. Figure 2.6 presents the passive ownership of the S&P 500 for 1983 to 2010. Average comovement of index stocks I first estimate comovement from 1957, the year the S&P 500 was first published, and then restrict the study on the relationship with passive ownership to 1983-2011 (the years for which the data is available). Following Barberis, Shleifer, and Wurgler (2005), I estimate comovement of an index stock by regressing its returns on the returns of the index (net of the eect of the stock) and the market returns net of the index: Rj,t = –j+—j,SP500RSP500,t+—j,nonSP500RnonSP500,t+uj,t. (2.27) I compute value-weighted daily returns of the S&P 500 portfolio (excluding the stock under study) and of the non-S&P 500 portfolio using data from CRSP. This diers from the way Barberis, Shleifer, and Wurgler (2005) compute returns; they back out individual eect of the stock from S&P 500 returns. My approach avoids the need to estimate the daily weight of each index stock in order to remove its net contribution. It also keeps the weighting methodology constant throughout the sample, while the S&P 500 changed from value-weighted to float- weighted in 2005. For every year, I look at stocks that are in the index at the end of December. I estimate comovement —s using daily data for the months [+1,+12], with a minimum of 6 months availability for the estimate to be considered valid. The average S&P 500 stock comoves more with the index than with the rest of the stock market, but this was not always the case. Figure 2.7 illustrates how the average comovement 24From (2.20), we know that the slope is actually ≠(◊I ≠◊ı)/(1≠‹I). 20 Figure 2.4. Comovement as a Function of the Indexer’s Relative Wealth 0.05 0.10 0.15 0.20 0.25 0.30n 0.482 0.484 0.486 0.488 b1 j HBSWL non-index index (a) 0.05 0.10 0.15 0.20 0.25 0.30n 0.483 0.484 0.485 0.486 0.487 b1 j HBSWL non-index index (b) 0.05 0.10 0.15 0.20 0.25 0.30n 0.47 0.48 0.49 0.50 b1 j HBSWL non-index index (c) Notes: This figure presents the comovement of the marginal index stock (stock 1) with the large index stock (solid blue line) and the non-index stock (dashed red line) as a function of ‹I , the share of wealth owned by the indexer. As the relative wealth of the indexer becomes larger, the comovement of the index stock with the other index stock increases while the comovement of the index stock with the non-index stock decreases. The parameters used are ” = 0.01, ‡D = 0.17, µD = .026 and flD = 0.94. Panel (a) presents results for the setup where the large index stock (stock 2) and the large non-index stock (stock 3) have the same dividend level (s2 = s3 = 0.495). Panel (b) presents results for the setup where the large index stock has a higher dividend level than the large non-index stock (s2 = 0.77, s3 = 0.22). Panel (c) presents results for the setup where the large index stock has a lower dividend level than the large non-index stock (s2 = 0.08, s3 = 0.91). 21 Figure 2.5. Price of Risk as a Function of the Indexer’s Relative Wealth 0.00 0.05 0.10 0.15 0.20 0.25 0.30nI0.00 0.02 0.04 0.06 0.08 0.10 0.12 q¯ Z3 Z2 Z1 (a) 0.00 0.05 0.10 0.15 0.20 0.25 0.30nI0.00 0.05 0.10 0.15 0.20 0.25 qI Z3 Z2 Z1 (b) 0.00 0.05 0.10 0.15 0.20 0.25 0.30nI0.00 0.02 0.04 0.06 0.08 0.10 0.12 q Z3 Z2 Z1 (c) Notes: This figure presents the price of risk for each risk factor as a function of ‹I , the share of wealth owned by the indexer. The green and blue dashed lines correspond to the price of risks associated Z1 and Z2 respectively (shocks to index stocks dividends) while the solid red line corresponds to the price of risk associated with Z3 (shocks to the non-index stock dividends). As the relative wealth of the indexer becomes larger, the comovement of the index stock with the other index stock increases while the comovement of the index stock with the non-index stock decreases. The parameters used are ” = 0.01, ‡D = 0.17, µD = .026, flD = 0.94 s1 = .1 and s2 = s3 = .45. Panel (a) presents ◊ı, which corresponds to the price of risk in the unconstrained economy. Panel (b) presents the shadow price of risk for the indexer (◊I). Panel (c) presents the equilibrium price of risk. 22 Figure 2.6. Annual Passive Ownership for the S&P 500 20101983 1986 1988 1990 1992 1994 1996 1998 2000 2002 2004 2006 2008 9 0       1 2 3 4 5 6 7 8 Year Pa ss iv e O wn er sh ip  (% ) Notes: This figure presents the annual end of year passive ownership for the S&P 500. Passive ownership is defined as passive assets under management for the S&P 500 divided by total CRSP market capitalization. Passive assets under management is from Standard and Poor’s and market capitalization is from CRSP. —s of S&P 500 stocks changed over the years since the index was created in 1957. Prior to 1975, the average index stock comoved more with the rest of the market than with the index. This pattern reversed over the following 15 years. Consistent with predictions of the model, we can see that for almost all months since indexing has become non-negligible (early 1980s), the average comovement of index stocks with the index is greater than with the rest of the market. This is also consistent with Barberis, Shleifer, and Wurgler (2005) and Boyer (2011) who both find the a similar eect while focusing only on stocks that get added or removed from an index. Their focus is on the change in comovement surrounding index additions and deletion events, not on the time series of comovement. The model predicts that the — with the index should be increasing in passive investors’ wealth. It also predicts the opposite relationship for the — with the rest of the market. I use passive ownership (PO) as a proxy for passive investors’ wealth. 2.5.2 Passive ownership and comovement The goal of Barberis, Shleifer, and Wurgler (2005) is to investigate whether the excess co- movement of index stocks with other index stocks is caused by the index membership, so in their setup an event-study is appropriate. They find that after inclusion index stocks comove more with other index stocks and less with non-index stocks. These results are stronger later in their sample, supporting the idea that comovement is caused by indexing. My objective is to determine if this eect becomes stronger as the relative weight of indexers in the economy 23 Figure 2.7. Average Comovement of S&P 500 Stocks from 1957 to 2010 20101957 1965 1970 1975 1980 1985 1990 1995 2000 2005 1.2 -0.2 0 0.2 0.4 0.6 0.8 1 Year Co m ov em en t β nonSP500 SP500 β Notes: This figure presents the average annual comovement —s of S&P 500 stocks from 1957 to 2010 estimated using a [+1,+12] months window from the bivariate regression Rj,t = –j + —j,SP500RSP500,t + —j,nonSP500RnonSP500,t+ uj,t. Results are for stocks in the index on the last trading day of the year, using returns from the following year. increases. I test this hypothesis in a more direct manner, using regressions based on the average comovement of S&P 500 stocks. Looking at Figure 2.6, one may have reasonable concerns that the increase in comovement is due to a time trend that just happens to be correlated with the increase in passive ownership. In order to avoid this issue, the analysis focuses on absolute changes in comovement and in passive ownership. I regress changes in average —SP500,t and average —nonSP500,t (which I denote —SP500,t and —nonSP500,t) as dependent variables and the lagged changes in the index’s passive ownership (POt≠1) as the independent variable. Changes in — exhibit negative auto-correlation, so I also repeat each regression including the lagged value of the dependent variable, —t≠1 (which corresponds to —SP500,t≠1 or —nonSP500,t≠1.) Changes are computed as the absolute dierence between two annual observations. Table 2.1 presents regressions results. The null hypothesis is that all the coecients are 0. The model predicts a positive sign for the coecient on POt≠1 in the regression of —SP500 and a negative sign for the coecient on POt≠1 in the regression of —nonSP500. Coecients on POt≠1 are all of the expected signs and statistically significant. Furthermore, the adjusted R2 are quite high, suggesting that the change in passive ownership explain a good fraction of variation. This suggests that S&P 500 stocks do comove more with other S&P 500 stocks and less with non-S&P 500 stocks as indexer wealth increases. 24 2.5.3 Robustness checks Panel regression The previous regression results are strong, but one might wonder if the relationship is also present in the cross-section of comovement —s. I repeat the previous tests using a pooled regression with individual firms’ changes in comovement — as dependent variables. I also include firm-specific lagged changes in — as control variable. I include firm-fixed eects and cluster standard errors by year. Table 2.2 presents regressions results. Considering all observation within a year share the same change in passive ownership, it would be unlikely for POt≠1 to explain much of the variation. This is evident from the low R2. Nonetheless, the coecients POt≠1 are of the same sign and magnitude as in the regression based on mean —s. They are also statistically significant, with the exception of the last regression. Size eects Size has long been considered a driver of stock returns. If large firms are indeed fundamentally similar, we would expect them to comove more with other large firms even in the absence of indexers. Since the S&P 500 is a value-weighted index,25 comovement with the index means comovement with large S&P 500 stocks. If large S&P stocks indeed have comoving fundamen- tals, we should expect the eect of indexers on those stocks to be weakened or even negligible. On the other hand, if small S&P 500 stocks – which are still large stocks by Fama-French stan- dards – have fundamentals that comove more with stocks of similar size, including non-S&P 500 stocks like the large S&P 400 MidCap stocks, we should expect the eect of indexers to be more pronounced on them. The value-weighted RnonSP500 put more weight on the biggest non-S&P 500 stocks, which are more likely to be close in size to small S&P 500 stocks. To look at dierences due to size, I split the sample of S&P 500 stocks in two. Each year, the top portfolio includes the largest 20 stocks, which account on average for 32.0% of the index market cap, and the bottom portfolio includes the remaining stocks. Average comovement —s for both subsamples are presented in Figure 2.8. Clearly, it does appear that comovement with the index was higher for larger stocks than for smaller stocks at the beginning of the sample, when indexing was marginal. I repeat the previous tests on stocks in the bottom portfolio, thus excluding the largest 20 stocks from the analysis and present the results in Table 2.3. The coecients have the expected signs, are slightly more statistically significant and larger in magnitude than those for the full sample presented in Table 2.1 for the regressions on averages. Thus, results do not appear to be driven by the large S&P 500 stocks. Results for the top portfolio (not shown) are not statistically significant. 25The S&P 500 switched to float-weighting in 2005. Float weighting uses the market capitalization of the float – the shares available for trading – excluding shares held by large block-holders. 25 Table 2.1. Changes in Comovement on Lagged Changes in Passive Ownership —SP500,t —nonSP500,t Intercept -0.01 0.00 -0.01 0.02 -0.00 0.02 (-0.88) (0.18) (-0.90) (0.71) (-0.09) (0.70) POt≠1 7.45** 8.05** -10.53* -10.64* (2.09) (2.24) (-2.00) (-2.04) —t≠1 -0.33 -0.36* -0.02 -0.05 (-1.20) (-1.72) (-0.05) (-0.18) Adj. R2 0.12 0.07 0.22 0.11 -0.04 0.08 Obs. 26 26 26 26 26 26 Notes: This table presents results of regressions of changes in average comovement estimates for S&P 500 firms (—SP500,t and —nonSP500,t) on the lagged changes in passive ownership (PO), from 1983 to 2010 (annual), and on —t≠1 which is the lagged value of the dependent variables in the regression. The dependent variable are the changes in average —s. Comovement —s are estimated from the regression Rj,t = –j +—j,SP500,tRSP500,t+ —j,nonSP500,tRnonSP500,t + uj,t, where RSP500,t is the value-weighted return of the S&P 500 stocks portfolio (excluding stock j) and Rnon-SP500,t is the value-weighted return of the rest of the market. Comovement —s are estimated based on index membership at the end of December, using daily data for the following 12 months. PO is the total passive assets under management divided by the total CRSP market capitalization and acts as a proxy for the relative wealth of indexers. T-stats are presented in parenthesis. I use heteroscedasticity-consistent standard errors. *, **, *** indicate significance at 10%, 5% and 1% level respectively. Table 2.2. Changes in Cross-Sectional Comovement on Lagged Changes in Passive Ownership —SP500,i,t —nonSP500,i,t POt≠1 7.14** 6.69* -9.50* -8.94 (2.16) (1.84) (-1.92) (-1.51) —t≠1 -0.46*** -0.46*** -0.41*** -0.41*** (-9.80) (-9.81) (-13.38) (-13.59) R2 0.00 0.20 0.20 0.01 0.17 0.17 Obs. 11,190 11,190 11,190 11,190 11,190 11,190 Notes: This table presents results of regressions of changes in comovement estimates for S&P 500 firms (—SP500,i,t and —nonSP500,i,t) on the lagged changes in passive ownership (PO), from 1983 to 2010 (annual), and on —t≠1 which is the lagged value of the dependent variable in the regression. The pooled regressions use individual firm changes in —s as the dependent variables. Comovement —s are estimated from the regression Rj,t = –j +—j,SP500,tRSP500,t+—j,nonSP500,tRnonSP500,t+uj,t, where RSP500,t is the value-weighted return of the S&P 500 stocks portfolio (excluding stock j) and Rnon-SP500,t is the value-weighted return of the rest of the market. Comovement —s are estimated based on index membership at the end of December, using daily data for the following 12 months. PO is the total passive assets under management divided by the total CRSP market capitalization and acts as a proxy for the relative wealth of indexers. T-stats are presented in parenthesis. Standard errors are clustered by year and include firm fixed eects. *, **, *** indicate significance at 10%, 5% and 1% level respectively. 26 Figure 2.8. Average Comovement of S&P 500 Stocks from 1957 to 2010, for 20 Largest Stocks and Excluding 20 Largest Stocks 20101957 1965 1970 1975 1980 1985 1990 1995 2000 2005 1.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 1.2 1.4 Year Co m ov em en t β nonSP500 SP500 β (a) Average comovement for the largest 20 stocks in the S&P 500. 20101957 1965 1970 1975 1980 1985 1990 1995 2000 2005 1.2 -0.2 0 0.2 0.4 0.6 0.8 1 Year Co m ov em en t β nonSP500 SP500 β (b) Average comovement for S&P 500 stocks excluding the largest 20 stocks. Notes: The figures present the average annual comovement —s from 1957 to 2010 estimated using a [+1,+12] months window from the bivariate regression Rj,t = –j +—j,SP500RSP500,t+—j,nonSP500RnonSP500,t+uj,t. Re- sults are for stocks in the index on the last trading day of the year, using returns from the following year. 27 Correlated trading In the model, comovement is a product of changes in discount rate. An alternative explanation for index-linked comovement is correlated trading. When news is made public, it could be that index stocks are traded more quickly to reflect the new information, while trading in non-index stock may lag behind. Under this hypothesis, comovement should not be noticeable at lower frequencies. To test for this hypothesis, I repeat the tests using comovement based on weekly returns. The methodology for computing average comovement —s is the same as in Section 2.5.1, with the exception that I use weekly returns instead of daily returns. Figure 2.9 presents the time series of average comovement —s using weekly returns. The pattern is very similar to the one obtained with daily returns. Regression results are presented in Table 2.4. Even for comovement computed using weekly returns, the results are all of the expected sign and statistically significant. Note that this does not reject the correlated trading hypothesis. It merely illustrates that if correlated trading impacts comovement, the eect is likely orthogonal to the eect of passive ownership. Figure 2.9. Average Comovement of S&P 500 Stocks from 1957 to 2010 using Weekly Returns 20101957 1965 1970 1975 1980 1985 1990 1995 2000 2005 1.2 -0.2 0 0.2 0.4 0.6 0.8 1 Year Co m ov em en t β nonSP500 SP500 β Notes: This figure presents the estimates of comovement obtained from weekly returns. Average annual comove- ment —s of S&P 500 stocks from 1957 to 2010 estimated using a [+1,+12] months window from the bivariate regression Rj,t = –j +—j,SP500RSP500,t+—j,nonSP500RnonSP500,t+uj,t. Results are for stocks in the index on the last trading day of the year, using returns from the following year. Non-index stocks While the previous results are consistent with my hypothesis, it could also be that all stocks in the size range of small S&P 500 stocks began to comove more with large S&P 500 stocks, 28 Table 2.3. Changes in Comovement, Excluding the 20 Largest S&P 500 Firms, on Lagged Changes in Passive Ownership —SP500,t —nonSP500,t Intercept -0.02 0.00 -0.01 0.02 0.00 0.02 (-0.88) (0.19) (-0.89) (0.70) (-0.10) (0.69) POt≠1 7.86** 8.51** -11.11* -11.21* (2.10) (2.27) (-2.00) (-2.05) —t≠1 -0.32 -0.36 -0.01 -0.04 (-1.17) (-1.70) (-0.03) (-0.15) Adj. R2 0.12 0.07 0.22 0.12 -0.04 0.08 Obs. 26 26 26 26 26 26 Notes: This table presents results of regressions of changes in average comovement estimates for firms in the S&P 500 excluding the 20 largest (—SP500,t and —nonSP500,t) on the lagged changes in passive ownership (PO), from 1983 to 2010 (annual), and on —t≠1 which is the lagged value of the dependent variable in the regression, for S&P 500 firms. The dependent variables are the changes in average —s. Comovement —s are estimated from the regression Rj,t = –j+—j,SP500,tRSP500,t+—j,nonSP500,tRnonSP500,t+uj,t, where RSP500,t is the value-weighted return of the S&P 500 stocks portfolio (excluding stock j) and Rnon-SP500,t is the value- weighted return of the rest of the market. Comovement —s are estimated based on index membership at the end of December, using daily data for the following 12 months. PO is the total passive assets under management divided by the total CRSP market capitalization and acts as a proxy for the relative wealth of indexers. T-stats are presented in parenthesis. I use heteroscedasticity-consistent standard errors. *, **, *** indicate significance at 10%, 5% and 1% level respectively. Table 2.4. Changes in Comovement, Computed from Weekly Returns, on Lagged Changes in Passive Ownership —SP500,t —nonSP500,t Intercept -0.02 0.01 -0.02 0.03 -0.01 0.02 (-0.67) (0.31) (-0.66) (0.62) (-0.23) (0.53) POt≠1 12.36* 13.67** -14.21* -14.40* (1.80) (2.19) (-1.80) (-1.73) —t≠1 -0.47** -0.50*** -0.32 -0.33 (-2.49) (-3.09) (-1.13) (-1.38) Adj. R2 0.07 0.20 0.31 0.08 0.07 0.16 Obs. 26 26 26 26 26 26 Notes: This table presents results of regressions of changes in average comovement estimates for S&P 500 firms using weekly returns (—SP500,t and —nonSP500,t) on the lagged changes in passive ownership (PO), from 1983 to 2010 (annual), and on —t≠1 which is the lagged value of the dependent variable in the regression. The dependent variables are the changes in average —s. Comovement —s are estimated from the regression Rj,t = –j +—j,SP500,tRSP500,t+—j,nonSP500,tRnonSP500,t+uj,t, where RSP500,t is the value-weighted return of the S&P 500 stocks portfolio (excluding stock j) and Rnon-SP500,t is the value-weighted return of the rest of the market. Comovement —s are estimated based on index membership at the end of December, using daily data for the following 12 months. PO is the total passive assets under management divided by the total CRSP market capitalization and acts as a proxy for the relative wealth of indexers. T-stats are presented in parenthesis. I use heteroscedasticity-consistent standard errors. *, **, *** indicate significance at 10%, 5% and 1% level respectively. 29 contemporaneously with increases in passive ownership. In order to verify that this is an indexing eect, I form a sample consisting of the smallest 250 stocks in the S&P 500 and the largest 250 stocks in the non-S&P 500 portfolio. The size ratio of the bottom S&P 500 portfolio to the top non-S&P 500 portfolio varies between 0.51 and 1.18 with an average of 0.87.26 The average comovement —s for the two groups are presented in Figure 2.10. There is a clear dierence between the time series patterns of the two groups. I do a dierence-in-dierence analysis, in which the continuous treatment is the change in PO and the treated group (T ) is the portfolio of small S&P 500 stocks. The control group is the portfolio of large non-index stocks. The hypothesis is that only the POt≠1 for the treated group (POt≠1◊T ) should have statistically significant coecients with the expected signs. Results, shown in Table 2.5, support the hypothesis. In all regressions, the coecient on PO is small and not statistically significant. As expected, changes in passive ownership are are not related to changes in comovement for non-index stocks. For the regressions on —SP500 the coecients on PO◊T are of the expected sign and statistically significant. This suggests that passive ownership only aects comovement of index stocks. For the regressions on —nonSP500 the coecients on PO◊T are of the expected sign but not statistically significant. 2.6 Concluding remarks I present theoretical explanations and empirical evidence suggesting that comovement within index stocks increases with the relative wealth of indexers. This can explain the rise of co- movement in S&P 500 stocks that began in the 1970s. While the notion of excess comovement in index stocks is not new, this is the first paper to link it directly to the relative wealth of indexers and to its eect on discount rates. I first introduce an asset-pricing model where the economy is populated by two rational representative agents, an active investor and an indexer. Two of the stocks form an index, and the indexer is constrained to invest solely in the bond and the index. The model predicts that, as the relative wealth of the indexer increases, index stocks comove more with other index stocks and less with non-index stocks. While this model proved useful for my analysis, I am aware of its limitations. One such limitation of the model is its inability to explain the rise of indexing since the model is not stationary; over time the active investor dominates. Future research could address this issue, introducing frictions such as the incremental cost of active investing, for which French (2008) provide estimates, and allowing one rational agent to invest both passively and actively at same time through delegated portfolio management, as in Petajisto (2009). Second, I show that comovement of S&P 500 stocks with other S&P 500 stocks increases while comovement with non-S&P 500 stocks decreases as the share of passive holdings in the 26Market capitalization is not the only determinant for inclusion in the S&P 500. Other factors such as liquidity and market representation are also taken into consideration, which can exclude very large firms from the index. Probably the most notable example is Berkshire Hathaway Inc., which was only included in the S&P 500 (and S&P 100) in 2010 following the 50-1 split of its Class B shares, making it the 21st largest company in the index. 30 Figure 2.10. Average Comovement of Smallest 250 S&P 500 Stocks and Largest 250 Non-S&P 500 Stocks from 1957 to 2010 20101957 1965 1970 1975 1980 1985 1990 1995 2000 2005 1.6 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 1.2 1.4 Year Co m ov em en t β nonSP500 SP500 β (a) Average comovement of the smallest 250 stocks in the S&P 500. 20101957 1965 1970 1975 1980 1985 1990 1995 2000 2005 1.4 -0.2 0 0.2 0.4 0.6 0.8 1 1.2 Year Co m ov em en t β SP500 nonSP500β (b) Average comovement of the largest 250 stocks not in the S&P 500. Notes: This figure presents the average annual comovement —s from 1957 to 2010 estimated using a [+1,+12] months window from the bivariate regression Rj,t = –j +—j,SP500RSP500,t+—j,nonSP500RnonSP500,t+uj,t. Re- sults are based on index membership on the last trading day of the year, using returns from the following year. 31 index increases. I show that these results are robust to some popular alternative explanations of comovement, including the correlated trading view. 32 Table 2.5. Changes in Comovement on Lagged Changes in Passive Ownership for Index and Non-Index Stocks —SP500,t —nonSP500,t Intercept 0.00 0.00 0.00 0.00 (0.12) (0.06) (-0.01) (-0.01) T -0.03 -0.02 0.03 0.03 (-0.92) (-0.86) (0.55) (0.54) POt≠1 -2.12 -1.45 0.25 0.25 (-0.61) (-0.44) (0.04) (0.04) POt≠1◊T 13.81** 14.14** -14.72 -14.86 (2.13) (2.30) (-1.34) (-1.36) —t≠1 -0.35** -0.07 (-2.64) (-0.30) Adj. R2 0.07 0.18 0.01 -0.01 Obs. 52 52 52 52 Notes: This table presents results for dierence-in-dierence regressions with continuous treatment of changes in average comovement estimates (—SP500,t and —nonSP500,t) on the lagged changes in passive ownership (PO), from 1983 to 2010 (annual), and on —t≠1 which is the lagged value of the dependent variable in the regression. The sample includes the smallest 250 stocks in the S&P 500 (the treated group T ) and the largest 250 stocks not in the S&P 500 (the control group). Regressions use the changes in average —s (within-group averages) as the dependent variables. Comovement —s are estimated from the regression Rj,t = –j +—j,SP500,tRSP500,t+ —j,nonSP500,tRnonSP500,t + uj,t, where RSP500,t is the value-weighted return of the S&P 500 stocks portfolio (excluding stock j if j is in the index) and Rnon-SP500,t is the value-weighted return of the rest of the market (excluding stock j if j is not in the index). Comovement —s are estimated based on index membership at the end of December, using daily data for the following 12 months. PO is the total passive assets under management divided by the total CRSP market capitalization and acts as a proxy for the relative wealth of indexers. T-stats are presented in parenthesis. I use heteroscedasticity-consistent standard errors. *, **, *** indicate significance at 10%, 5% and 1% level respectively. 33 Chapter 3 Do Mutual Fund Managers Adjust NAV for Stale Prices? 3.1 Introduction In 2003 the U.S. Mutual Fund industry was shaken by the late trading and market timing scan- dal that led to a myriad of lawsuits and criminal convictions.27 This scandal was made possible by poor management of the price-setting mechanism. Fund managers set prices for fund shares, known as Net Asset Value (NAV), for all transactions of the day at 4 p.m. Eastern Time (ET). Some fund managers were found to allow execution of trades after this daily cuto, allowing investors to trade on new information. Other managers were complicit in market timing, waiv- ing restrictions on frequent trading that were intended to limit the ability of arbitragers to exploit mispricing. In May of 2004 the SEC introduced a new rule forcing the disclosure in prospectuses of risks associated with frequent trading and market timing. Funds must also disclose the measures (or lack of) taken to reduce improper trading and market timing. Market timing does not require late trading; mutual fund returns are predictable when NAV is computed from stale prices.28 For example the last trade of some securities held by a small cap fund or a foreign equity fund can occur hours before US markets close. Consider a mutual fund specialized in Japanese equities. Japanese stocks trade on the Tokyo Stock Exchange until it closes at 3 p.m. local time (2 a.m. ET). When NAV is computed at 4 p.m. ET the last transaction occurred more than 14 hours earlier. Any new information is not reflected in the NAV, unless the closing prices are correctly adjusted, so returns are predictable. This predictability imposes indirect costs to long term shareholders when exploited by ar- bitragers. It can lead to frequent trading of fund shares, imposing costs to the fund – such as transaction fees and liquidity issues – which are borne by all shareholders. Furthermore, net 27See, e.g., Houge and Wellman (2005) and Zitzewitz (2006) 28See, e.g., Chalmers, Edelen, and Kadlec (2001); Bhargava and Dubofsky (2001); Goetzmann, IvkoviÊ, and Rouwenhorst (2001); Greene and Hodges (2002); Boudoukh, Richardson, Subrahmanyam, and Whitelaw (2002); Varela (2002); Zitzewitz (2003). 34 gains to arbitragers are net losses to long term shareholders.29 This dilution and fund outflows resulting from lower realized returns cause assets under management to decrease over time, which aects long-term compensation of fund managers.30 Time has passed, but have things gotten better? A decade ago, Chalmers, Edelen, and Kadlec (2001) and Zitzewitz (2003) found that while fund managers where allowed to adjust NAV to reflect new information, most did not. Fair value adjustment solutions aim to fix the root cause of the problem: if there is no predictability, there will be no market timing arbitrage. Many such approaches have been suggested to adjust either individual securities or the fund’s overall NAV.31 I find that after 2004 mutual fund managers are more active in adjusting NAV. The simple trading strategy I present yields annual returns of 33% from 2001 to 2004 and 16% from 2005 to 2010. In order to prevent frequent trading, it is common practice to limit the number of round trips (buy and sell) by an investor in a given period and to impose high rear-load fees that disappear over time. While the implementation of the first solution is not straightforward (it is not always possible to track individual investors), the main issue with these restrictions is that they are imposed on all shareholders, reducing the liquidity of investments in mutual funds. Even after accounting for trading restrictions in mutual funds, my results indicate that an arbitrager could earn annual returns of 2.73% from 2005 to 2010. One challenge in testing for NAV adjustments by mutual fund managers is that these actions are not directly observable. To overcome this issue, I test for predictability in international mutual funds using trading strategies. If managers are adjusting NAV perfectly, there would be no predictability and the trading strategies would not provide abnormal returns. The results I obtain show that there remained predictability after 2004, thus I can reject this hypothesis of perfect adjustment. Alternatively, managers could be adjusting partially to the best of their abilities, but failing to use all available information such as exchange traded fund (ETF) premiums. In that case, predictability should go down after 2004 but not disappear completely. My results are consistent with this hypothesis. ETFs and mutual funds with similar investment styles suer correlated shocks. Observations of ETF premiums therefore provide valuable information on the fair value of funds. These premiums, defined as the log dierence between market prices and NAV, represent a direct proxy for the degree of staleness in the ETF. Prior studies of market timing and predictability use liquid US equity indices and market prices of futures as predictors of mutual fund NAV returns.32 I show that predictability increases significantly when information sources suggested 29Looking at flow of funds, Greene and Hodges (2002) find that dilution in international funds was 0.48% between 1998 and 2001 while Zitzewitz (2003) finds 1.14% in 2001. 30McCabe (2009) looks at fund managers incentives and concludes that after accounting for the short-term gain due to increased investments, fund managers are still better o preventing market timing arbitrage because of the long term loss of revenue due to dilution and loss of clients. 31See Ciccotello, Edelen, Greene, and Hodges (2002) and Zitzewitz (2003) for an overview of the dierent methodologies. 32Zitzewitz (2003) discussed the possibility of using ETF returns as predictors, but dismissed it because of the relative illiquidity of international ETFs at the time. Jares and Lavin (2004) use the returns on country ETFs 35 by prior literature are supplemented by premiums on related ETFs. The main methodological contribution of this paper is the filtering approach used to ac- count for fair-valuation adjustment of NAV. Recent literature dealing with funds prone to stale pricing has accepted the presence of predictability in NAV returns. Chen, Ferson, and Peters (2010) and Qian (2011) present methodology for performance evaluation of mutual funds in the presence of stale prices. These papers treat NAV predictability as a given that introduces bias in performance evaluation and then correct for that bias. I present a state-space model of mutual fund prices inspired by the one of Chen, Ferson, and Peters (2010). I extend the model by introducing an ETF and by allowing the fund manager to adjust NAV. Following Sargent (1989), I embed the fund manager’s problem of fair valuation in the model. I use two dierent specifications of the arbitrager’s filtering problem. In the first version, I assume that the fund manager does not adjust for stale prices. In the second version, I assume that the fund manager does adjust for stale prices, but does so using only a subset of publicly available information. The empirical investigation is based on the likelihood estimation of these two dierent dynamic linear models of mutual funds with stale pricing. Using likelihood estimation of the well-known Kalman filter, I show how ETF premiums can be used in conjunction with other know predictors such as S&P 500 returns to predict mutual fund returns. 3.2 Model This section presents a simple framework with stale pricing, which I use as the basis for the filtering problem presented in the next section. Let there be a mutual fund for which the true (fundamental) log value of assets vMF follows a random walk: vMFt = vMFt≠1 +ÁMFt , (3.1) where ÁMFt is i.i.d. normal with mean zero and variance ‡2MF . The return ÁMFt is linearly related to some publicly observable return rIt : ÁMFt = —rIt +ÁMF,It , (3.2) where ÁMF,It is i.i.d. normal with mean zero and variance ‡2MF,I . The publicly observable return is also observable at one point in time intra-period.33 Let rIt © rI≠t +rI+t , where rI≠t is the return during the first sub-period and rI+t is the return during the second sub-period. Let µ and (1≠µ) denote the approximate weights of rI+t and rI≠t on rIt , respectively. µ can be interpreted as the length of the second sub-period relative to one period. At each time t, the fund manager observes nMFt , the log net asset value of the mutual fund. This NAV represents the total log value of the fund assets based on the latest transaction price. If all underlying assets trade continuously, then nMFt = vMFt . However, if prices are stale, we in the context of mutual fund predictability as an explanatory variable alongside returns on the S&P 500. 33For example, daily returns can be divided in over-night and intra-day returns. 36 have nMFt = (1≠÷)vMFt +÷vMFt≠1 +Ánt , (3.3) where ÷ is the staleness parameter of the fund and Ánt is i.i.d. normal with mean zero and variance ‡2n. ÷ can take values between 0 (no staleness) and 1 (fully stale). This net asset value is private information available only to the fund manager. The manager uses this information to set the price nMFút that is eective for transaction orders received on day t before nMFút becomes public. Let there also exist an ETF with a similar objective as the mutual fund and for which both the true log value vETF and the log NAV nETF are publicly observable. The ETF premium is linearly related to the mutual fund premium, so we have vETFt ≠nETFt = “[vMFt ≠nMFt ]+ÁETFt , (3.4) where ÁETFt is i.i.d. normal with mean zero and variance ‡2ETF . 3.2.1 Relationship between vMF , nMF , rI≠ and rI+ By definition, full predictor returns rIt are linearly related to fundamental fund returns: rIt = 1 — [vMFt ≠vMFt≠1 ]+ÁIt , (3.5) where ÁIt = ≠1— Á MF,I t . This relationship can further be decomposed in two sub-period predictor returns: rI≠t + rI+t = 1 — [vMFt ≠nMFt ]+ 1 — [nMFt ≠vMFt≠1 ]+ÁIt . (3.6) The exact relationship between vMF , nMF , rI≠ and rI+ depends on the relative values of ÷ and µ: rI≠t = Y__]__[ 1 — [nMFt ≠vMFt≠1 ]+ÁI≠t if µ= ÷ 1 — 1≠µ 1≠÷ [nMFt ≠vMFt≠1 ]+ÁI≠t if µ > ÷ 1 —{[nMFt ≠vMFt≠1 ]+ ÷≠µ÷ [vMFt ≠nMFt ]}+ÁI≠t if µ < ÷ (3.7) rI+t = Y__]__[ 1 — [vMFt ≠nMFt ]+ÁI+t if µ= ÷ 1 —{µ≠÷1≠÷ [nMFt ≠vMFt≠1 ]+ [vMFt ≠nMFt ]}+ÁI+t if µ > ÷ 1 — µ ÷ [vMFt ≠nMFt ]+ÁI+t if µ < ÷ (3.8) where ÁI≠t +ÁI+t = ÁIt and ÁI≠t and ÁI+t are i.i.d. normal with mean zero, variance ‡2I≠= (1≠µ)‡2I and ‡2I+ = µ‡2I respectively, and mutually independent. Since µ and ÷ are fixed parameters of the model, only one of the three situations is valid and the model is linear. 37 3.3 Filtering The filtering problem of extracting valuable information from ETF premiums and other pre- dictors is further complicated by the adjustment that might be done by mutual fund managers. I test two specifications that represent dierent assumptions regarding the action of managers. I first start with a base specification that assumes fund managers make no adjustment before reporting NAV. Next, I introduce a richer specification in which they adjust NAV optimally using a predictor, but not ETF premiums. In the emprical section, I compare abnormal re- turns of trading strategies based on filtering results from both models. Finally, I test the base specification with the additional restriction that ETF premiums are not part of the arbitrager’s information set. 3.3.1 The mutual fund managers’ problem Each day, mutual fund managers need to set nMFút – the price mutual fund shares trade at on date t – as close as possible to vMFt so that investors get a fair price, thus keeping out market timers. Their problem is nMFút = E[vMFt |MFt ], where MFt is the mutual fund managers’ information set that includes nMFt and all the other information that fund managers look at. We are faced with a linear Gaussian model so fund managers can solve the problem optimally with Kalman filtering. If they do in fact solve the model optimally using all observables, econometricians would be unable to find evidence of predictability as managers have a strictly greater information set.34 This contradicts empirical evidence, but does oer an interesting starting point to investigate actions of mutual fund managers. Optimal filtering The fund manager’s information set includes all publicly available information at time t plus private information about nMFt , the NAV value of fund: MFt = {nMFt ,(vETFt ≠nETFt ), rI≠t , rI+,MFt≠1 }. The filtering problem of the fund manager is to estimate vMFt , whose process is defined in (3.1). The observations are related linearly to that state variable as defined in (3.3), (3.4), (3.7) and (3.8) respectively. Since ÁETFt , ÁI≠t , ÁI+t , Ánt , and ÁMFt are assumed to be Gaussian innovations, this problem can be solved optimally with a Kalman filter. Details of the implementation using the standard Kalman filter are presented in Appendix B.2. Suboptimal filtering There is no reason to assume the fund managers are solving the filtering problem optimally. Since we cannot empirically dierentiate optimal filtering from the absence of predictability, 34MFt includes nMFt , which is unobservable to econometricians. 38 there is no point in testing for it explicitly. However, empirical evidence of predictability in Section 3.5 suggests that they in fact do not solve optimally. Since manager have strictly better information than the econometricians, there would be no predictability if they were adjusting optimally. There is a multitude of ways fund managers can solve the problem suboptimaly. For example, the manager could ignore some of the information such as the ETF premium or the predictor return. Another possibility is that instead of relying on the Kalman filter, they could use other more user-friendly approximations or rule-of-thumbs. Alternatively, managers might not pay attention at all and set the price to the observed NAV. I test for two dierent specifications of suboptimal filtering, which are: 1. nMFút = nMFt , the fund manager does not adjust the NAV before setting the price. 2. The manager sets the price to the solution of the Kalman filter, without using the observa- tion of (vETFt ≠nETFt ), the ETF premium. This is a more ecient (in the context of this model) variation of the correction proposed by Goetzmann, IvkoviÊ, and Rouwenhorst (2001). Details of how a manager could implement the second specification using the standard Kalman filter are presented in Appendix B.2.1. 3.3.2 The arbitrager’s problem As external observers, arbitragers (and econometricians) cannot solve the manager’s problem since they only observe nMFút≠1 , not nMFt . Thus the problem is to find the best estimates v̂MFt and n̂MFút using publicly available information, including ETF premiums and nMFút≠1 , but not nMFt . If the resulting estimate v̂MFt is more accurate than nMFút , it means that mutual fund managers could get better estimates of the fair value of their funds by looking at extra information. Alternatively, it means that returns are predictable. Evaluating the quality of estimates directly with data is impossible since we do not observe vMFt . On the other hand, it is possible to test if there is economically significant predictability in adjusted returns (nMFút ≠nMFút≠1 ), by building trading strategies and looking at their returns. These strategies aim to mimic what mutual fund arbitragers are doing, so I call this problem the arbitrager’s problem. The arbitrager’s information set includes all publicly available information at time t: At = {nMFút≠1 ,(vETFt ≠nETFt ), rI≠t , rI+,At≠1}. The filtering problem of the fund manager is to estimate vMFt , whose process is defined in (3.1), and nMFút , which is the price that the manager will set for current day transactions. As for nMFút≠1 , it is the solution to the manager’s problem so the arbitrager has to make an assumption on how the manager solves the problem. An estimate of nMFt can also be found as a by-product of the filtering problem. The other observations are related linearly to the state variables vMFt and nMFt as defined in (3.3), (3.4), (3.7) and (3.8) respectively. Empirically, I test for the two specifications of the manager’s solution presented in Section 3.3.1, which are all linear functions of the state variables. The solution of the Kalman filter is linear, so the solution of one problem, such as in the second specification, can be 39 embedded as an observation in another Kalman filter as proposed by Sargent (1989). To get a sense of the benefits gained from using ETF premiums, I also test the first manager’s solution (no adjustment to NAV) without including ETF premiums in the arbitrager’s information set. The system is assumed to be linear with Gaussian innovations and is solved with a Kalman filter. Details of the implementation using the standard Kalman filter are presented in Appendix B.3. 3.4 Methodology 3.4.1 Hypotheses The main objective of this paper is to test the following three hypotheses regarding international equity mutual funds: H1: ETF premiums improve the predictability of mutual fund returns prone to stale pricing over the use of traditional predictors. H2: Over the period from 2000 to 2010, mutual fund managers have increased how they actively adjust NAV for stale pricing. H3: After 2004, there was still statistically and economically significant predictability in mu- tual fund returns. The main assumptions I rely on for testing are that the framework presented in Section 3.2 is a reasonable approximation of the real world and that the hypothesized adjustments to NAV are similar to the process presented in 3.3.1, which implicitly assumes that fund managers do not look at ETF premiums. 3.4.2 Data sources I have ETF data from January 1995 to December 2010, with data on international equity ETFs starting in 2000. Market prices for ETF shares are from the CRSP US Stock Database (security code 73). Over the sample period, there are 1116 ETFs, for which the last available listing was on AMEX (29), NYSE (2), NASDAQ (90) or NYSE ARCA (995)35. Net asset value information is from the CRSP Survivor-Bias-Free US Mutual Fund Database. In total, there are 1077 ETFs listed in the CRSP Mutual Fund Database in the sample period. ETFs from the two datasets are matched based on CUSIP for a total of 1075 matches. The 2011 Investment Company Fact Book36 states that the total number of ETFs in the US at 2010 year-end was 923, while 160 have been liquidated since 2000, for a total of 1083. It thus appears that my dataset is quite comprehensive. Mutual fund data is also from the CRSP Mutual Fund Database. The 35Over the sample period, most ETFs moved from AMEX and NYSE to NYSE Arca. As of December 2010, 2 where still actively trading on NYSE and 29 had been liquidated or delisted while still trading on AMEX. 36www.icifactbook.org 40 initial sample, which includes all open-ended funds with total net assets over $5 million and at least one year of observations (excluding the incubation period), consists of 27,003 mutual funds (unique share class). I adjust mutual fund NAVs for splits and dividends following the methodology from the “Survivor-Bias-Free US Mutual Fund Guide”. Daily index and futures data is from Datastream. The empirical part of this research focuses on international equity funds because those funds are known to be prone to stale prices due to non-synchronous trading. The style filtering is done using the Lipper Objective Code available in the CRSP Mutual Fund Database.37 After filtering, 192 ETFs and 2,482 mutual funds remain. Details about the size of funds as of December 2010 are presented in Table 3.1 (numbers of funds is lower than stated above because dead funds are not presented in the table). Among the 2,482 mutual funds, 1,344 are retail funds, 807 are institutional funds, 156 are index funds and 645 are alive during the full period (2000-2010).38 It is worth noting that TNA for Chinese and Japanese ETFs – the styles most aected by staleness – now surpass TNA in their mutual fund counterparts. Table 3.1. Count and TNA of International ETFs and Mutual Funds ETFs Mutual Funds Style Funds TNA Funds TNA China Region Funds 21 15,819 52 9,618 Emerging Markets Funds 50 111,047 318 179,293 European Region Funds 26 9,667 64 11,999 International Funds 37 63,623 1,145 784,379 International Small-Cap Funds 3 1,348 93 41,803 Japanese Funds 10 5,318 21 838 Pacific Region Funds 5 1,675 33 9,780 Pacific Ex Japan Funds 13 18,438 30 19,000 Total 165 226,936 1,756 1,056,710 Notes: This table presents the count and total net assets of international exchange traded funds and mutual funds, by Lipper Objective, according to the CRSP Survivor-Bias-Free US Mutual Fund Database as of December 31, 2010. Sample consists of ETFs and mutual funds with at least one year of returns prior to December 2010. Only mutual funds with at least $5 millions are included. Total net assets are in $ millions. 37 For international funds, all international funds are included except those trading mostly securities from Canada, Mexico and other Latin-American countries since trading hours in those countries overlap with US exchanges. Included are funds with the following Lipper Objective Codes: ’CH’, ’EM’, ’EU’, ’IF’, ’IS’, ’JA’, ’PC’ and ’XJ’. International small-cap funds with objective code ’IS’ are excluded from the estimation stage since the late appearance of the first ETF in that style leaves us with only two years of valid observations. 38The retail or institutional fund status is unknown for the some funds, so they do not sum up to the total number of funds. The number of index funds might be biased downward as the index fund identifier is only available for funds alive on or after June 2008. 41 3.4.3 ETF premiums Exchange traded funds are similar to traditional index mutual funds in purpose. While open- end mutual funds shares can only be bought or sold at the end of the day at the net asset value, ETFs trade continuously during the day, like closed-end mutual funds, at a price that can be dierent from NAV. In order to maintain prices close to fair value, creation and redemption of shares is allowed in kind – exchanging underlying shares for shares in the ETF – by registered participants. Furthermore, this feature makes short selling easier and cheaper for ETFs than stocks. When borrowing costs become to high, it generates a create-to-lend market where registered participants create new shares for the sole purpose of lending.39 The intuition behind the use of premiums as a source of information lies in the nature of the input used in its computation. The premium or discount on an ETF is computed as the log dierence between price and net asset value: premium= log(price)≠ log(NAV ). While the market price fluctuates freely with trading, the net asset value is an estimate com- puted following a specific formula and using a wide array of data as input. Thus, the premium can be thought of as a measure of the misalignment between the NAV and what the market believes to be the true value. Summary statistics for observed premiums are presented in Table 3.2. Since ETF NAV is computed the same way as unadjusted mutual fund NAV, ETF pre- miums provide an estimate of NAV mispricing for related mutual funds. In order to trade on the information reflected by ETF premiums, one would need an up-to-date (intraday) estimate of NAV. Exchanges provide a measure called the Indicative Optimized Portfolio Value (IOPV) throughout the day that acts as a proxy for NAV. 3.4.4 Matching mutual funds with ETFs and predictors In order to test for a relationship between ETF premiums and future mutual fund NAV returns, I first match each mutual fund with an ETF within the same style. Matches are re-evaluated monthly for each mutual fund. At each matching date, I generate a list of candidate ETFs for which there is at least one year of historical observations prior to the current month. I then pick the match for which the correlation between same-day mutual fund NAV returns and ETF NAV returns is the highest during the previous year (excluding current month observations). The premiums on the matched ETFs are used for estimating the models (using past data excluding the current month) and the state variables (for the current month). The median correlation of mutual fund and ETF monthly matches is presented by year and style in Panel A of Table 3.3. The quality of the matches, as measured by correlation, increases over time, as is expected since the diversity of ETFs increases. 39Ackert and Tian (2000) find that premiums can persist in some ETFs because of limits to arbitrage in the underlying securities. They note however that the resulting premiums are of smaller magnitude than those observed in closed-end funds. 42 Table 3.2. Summary Statistics of Daily Premiums on International ETFs Style Obs. Mean Std. Dev. Min Max China Region Funds 13,548 0.18% 1.72% -11.3% 38.4% Emerging Markets Funds 29,984 0.33 1.60 -15.4 139.2 European Region Funds 47,411 0.11 1.01 -15.0 39.5 International Funds 30,044 0.21 1.10 -19.1 41.2 International Small-Cap Funds 2,650 0.42 1.30 -7.3 10.8 Japanese Funds 11,708 -0.03 1.50 -12.7 43.9 Pacific Region Funds 4,023 0.12 1.18 -9.5 16.5 Pacific Ex Japan Funds 18,300 0.03 1.61 -37.1 18.6 Notes: This table presents summary statistics for premiums computed using daily observations of international ETFs matched from the CRSP Stock Database and the CRSP Mutual Fund Database. Statistics are presented by Lipper Objective for international equity ETFs. The daily premium is defined as premium = log(price)≠ log(NAV ). All values are in %. The same strategy is applied to the selection of the other predictor. Since the dierent specifications for managers action in Section 3.3.2 have dierent assumptions regarding the relationship between the predictor and the observed NAV return, I use a dierent procedure to pick the predictor used for each specification. Candidate predictors are equity indices (S&P 500, Russell 2000 and Nikkei 225, all USD-based) as well as US-listed index futures on the same indices, whenever both open and close data is available from Datastream. I also include same-style ETFs price returns as candidate predictors. Under the assumption that there is no adjustment to NAV by the fund manager and prices are stale, mutual fund returns should be correlated with lagged predictor returns. In the other case, I assume the manager adjusts NAV returns using the predictor, so predictor returns should be correlated with contemporaneous adjusted NAV returns. For each mutual fund, on each month, I compute the correlation of NAV returns with candidate predictor returns and lagged candidate predictor returns using one year of observations prior to the current month. For each fund and month pair, I pick the predictor that is most appropriate for each model based on the observed historical correlations. The median correlation of mutual fund with predictor and lagged predictor matches are presented by year and style in Panels B and C of Table 3.3. The pool of candidate predictors gets larger every year with the appearance of new ETFs. Over the sample period the contemporaneous correlation is increasing, which is consistent with an increase in active adjustment by managers, but could also be explained by the presence of “better” predictors. However, the median correlation with the lagged predictors goes down over time, suggesting that there is less predictability in mutual funds returns, even with the wider pool of candidates. This is consistent with fund managers adjusting NAV for stale pricing later in the sample. 43 Table 3.3. Median Correlation of Returns for Monthly Mutual Fund Matches Style 2000 2001 2002 2003 2004 2005 2006 2007 2008 2009 2010 Panel A: Median correlation of returns for monthly mutual fund and ETF matches China Region 0.78 0.80 0.85 0.80 0.80 0.77 0.79 0.81 0.87 0.93 0.89 Emerging Markets 0.92 0.91 0.94 0.97 0.98 0.99 0.98 European Region 0.81 0.85 0.92 0.96 0.87 0.82 0.88 0.94 0.97 0.98 0.98 International 0.62 0.60 0.82 0.84 0.89 0.91 0.95 0.98 0.98 International Small-Cap 0.67 0.89 0.87 Japanese 0.87 0.87 0.90 0.86 0.80 0.70 0.69 0.61 0.56 0.86 0.80 Pacific Region 0.86 0.86 0.89 0.94 0.97 0.93 Pacific Ex Japan 0.60 0.63 0.77 0.81 0.78 0.74 0.74 0.66 0.54 0.88 0.90 Panel B: Median correlation of returns for monthly mutual fund and predictor matches China Region 0.41 0.49 0.53 0.51 0.63 0.68 0.71 0.78 0.88 0.95 0.92 Emerging Markets 0.43 0.54 0.54 0.50 0.67 0.69 0.83 0.88 0.93 0.96 0.96 European Region 0.32 0.77 0.81 0.82 0.77 0.78 0.85 0.91 0.94 0.97 0.96 International 0.53 0.53 0.59 0.57 0.70 0.70 0.89 0.93 0.95 0.97 0.96 International Small-Cap 0.40 0.47 0.54 0.55 0.57 0.50 0.59 0.71 0.87 0.92 0.92 Japanese 0.68 0.79 0.82 0.81 0.84 0.88 0.91 0.89 0.90 0.92 0.88 Pacific Region 0.69 0.75 0.77 0.80 0.78 0.69 0.81 0.86 0.90 0.95 0.91 Pacific Ex Japan 0.48 0.50 0.59 0.60 0.67 0.75 0.74 0.78 0.86 0.91 0.89 Panel C: Median correlation of returns for monthly mutual fund and lagged predictor matches China Region 0.45 0.45 0.36 0.37 0.29 0.23 0.24 0.14 0.13 0.11 0.09 Emerging Markets 0.44 0.38 0.34 0.37 0.31 0.24 0.28 0.17 0.10 0.07 0.18 European Region 0.41 0.33 0.38 0.37 0.23 0.15 0.19 0.15 0.11 0.04 0.06 International 0.48 0.37 0.41 0.40 0.30 0.17 0.18 0.14 0.11 0.05 0.09 International Small-Cap 0.49 0.46 0.45 0.47 0.36 0.25 0.29 0.23 0.12 0.05 0.07 Japanese 0.38 0.35 0.33 0.27 0.23 0.11 0.08 0.07 0.13 0.05 0.07 Pacific Region 0.50 0.44 0.41 0.43 0.34 0.17 0.14 0.11 0.12 0.07 0.03 Pacific Ex Japan 0.50 0.46 0.38 0.42 0.31 0.23 0.28 0.20 0.11 0.08 0.10 Notes: This table presents the median correlation of returns for monthly mutual fund matches, by year and style. For each monthly mutual fund observation, the chosen match is the one with the highest correlation in daily returns during the previous year (excluding the current month). Panel A presents ETF matches based on the correlation between same-day mutual fund returns and ETF NAV returns among ETFs in the same style. Panel B presents predictors matches based on the correlation between same-day mutual fund returns and predictor returns. Panel C presents predictors matches based on the correlation between mutual fund returns and lagged predictor returns. 3.4.5 Estimation For every monthly mutual fund observation, I estimate the two specifications presented in Section 3.3.2. In addition, I estimate the first specification with the additional assumption that arbitragers do not look at ETF premiums. Each specification is calibrated independently using matched ETF premiums and predictors. Estimation of parameters is done by maximum likelihood optimization using one year of historical daily data. Estimation of state variables for the previous year and the current month is done using the estimated optimal parameters. For each model and each monthly fund observations, I keep all the estimated state variables (previous year and current month) which are used for trading strategies presented in Section 3.5 and 3.6. The initial parameters for optimization are set to the previous month optimal parameters when available. For the remainder of the text, I will refer to the specifications as 44 follow: • No NAV adjustment: Assumes that the fund manager reports the NAV as observed; corresponds to specification 1 in Section 3.3.1. • NAV adjustment: Assumes that the fund manager adjusts the NAV based on the predic- tor; corresponds to specification 2 in Section 3.3.1. • No NAV adjustment, no ETF premiums: Same as no NAV adjustments, but further assumes that arbitragers don’t look at ETF premiums. • Best: For each monthly fund observation, chooses either “No NAV adjustment” or “NAV adjustment” according to the estimated likelihood. Figure 3.1 show the median estimated staleness parameter ÷ by month for the specification with no NAV adjustment. The ÷ for all styles are decreasing over time, indicating that staleness is decreasing, which is consistent with an increase in active adjustments to NAV by mutual fund managers. Table 3.4 presents the median estimated ÷ by year and style for each specification. In the results for the best specification, there is a temporary decrease in 2008 and 2009, but no discernible trend which further supports the hypothesis that mutual fund managers are actively adjusting for stale pricing. Table 3.5 further decomposes the result for the specification with no NAV adjustment in dierent subsamples: funds alive during the full period, institutional funds, retail funds and index funds. The decline in apparent staleness is present in all subsamples, so it does not appear that the results are driven by just a small subset of funds. The next section looks at return predictability in mutual fund returns due to stale pricing. 3.5 Predictability To evaluate the extent of mutual fund predictability, I devise a simple yet unrealistic trading strategy40 that an hypothetical arbitrager could use. The objective of this section is to illustrate the extent of predictability. A more realistic strategy is presented in the next section to evaluate the economic significance of that predictability. Every day, for every pair of matched ETF and mutual fund, I follow the following rule: • If E[vMFt ]>E[nMFút ] , sell the ETF short and buy the mutual fund. • If E[vMFt ]<= E[nMFút ], take no position. where E[vMFt ] and E[nMFút ] are the estimates for each monthly fund observation. If the matched ETF for a fund changes, then the ETF position remains the same but the ETFs are swapped, meaning if there is a short position on the old match, then that position is closed and a new short position is taken on the new match. 40Most mutual funds won’t allow daily transactions or will limit them with the use of fees. 45 Figure 3.1. Median Estimated Staleness by Month for the Specification with no NAV Adjustment Eta 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 Year 2000 2001 2002 2003 2004 2005 2006 2007 2008 2009 2010 2011 Lipper Objective Name China Region Funds Emerging Markets Funds European Region Funds International Funds Japanese Funds Pacific Ex Japan Funds Pacific Region Funds Notes: This figure presents the median estimated ÷ (staleness) by month for the specification with no NAV adjustment, by style. Values for ÷ range from 0 (no staleness) to 1 (fully stale). 46 Table 3.4. Median Estimated Staleness for each Specification Style 2000 2001 2002 2003 2004 2005 2006 2007 2008 2009 2010 Panel A: Best China Region 0.64 0.60 0.54 0.21 0.51 0.46 0.50 0.38 0.04 0.04 0.42 Emerging Markets 0.47 0.59 0.38 0.19 0.02 0.10 0.59 European Region 0.52 0.60 0.43 0.42 0.47 0.46 0.38 0.23 0.01 0.14 0.51 International 0.35 0.41 0.48 0.51 0.30 0.08 0.01 0.03 0.45 Japanese 0.01 0.02 0.05 0.28 0.44 0.34 0.28 0.30 0.33 0.14 0.10 Pacific Ex Japan 0.23 0.58 0.51 0.64 0.48 0.43 0.48 0.40 0.05 0.04 0.28 Pacific Region 0.64 0.39 0.28 0.38 0.32 0.01 0.01 0.41 Panel B: No NAV adjustment China Region 0.64 0.60 0.55 0.65 0.51 0.44 0.44 0.36 0.04 0.04 0.16 Emerging Markets 0.34 0.33 0.25 0.17 0.02 0.06 0.20 European Region 0.52 0.43 0.40 0.41 0.46 0.32 0.33 0.23 0.01 0.14 0.15 International 0.45 0.41 0.42 0.34 0.28 0.13 0.01 0.02 0.02 Japanese 0.72 0.41 0.42 0.44 0.43 0.22 0.06 0.24 0.18 0.15 0.19 Pacific Ex Japan 0.68 0.55 0.48 0.67 0.47 0.44 0.47 0.38 0.05 0.03 0.27 Pacific Region 0.64 0.34 0.25 0.34 0.21 0.01 0.01 0.06 Panel C: NAV adjustment China Region 0.14 0.15 0.67 0.21 0.59 0.44 0.78 0.62 0.01 0.26 0.44 Emerging Markets 0.90 0.69 0.71 0.68 0.60 0.60 0.61 European Region 0.15 0.61 0.81 0.55 0.71 0.71 0.48 0.96 0.29 0.33 0.57 International 0.68 0.78 0.79 0.94 0.44 0.01 0.01 0.50 0.46 Japanese 0.01 0.01 0.02 0.07 0.29 0.34 0.28 0.31 0.35 0.01 0.09 Pacific Ex Japan 0.11 0.06 0.68 0.64 0.50 0.48 0.51 0.43 0.40 0.65 0.48 Pacific Region 0.70 0.46 0.43 0.44 0.72 0.63 0.84 0.47 Panel D: No NAV adjustment, no ETF premiums China Region 0.80 0.66 0.50 0.63 0.53 0.44 0.42 0.34 0.01 0.04 0.20 Emerging Markets 0.35 0.35 0.41 0.18 0.01 0.20 0.31 European Region 0.55 0.42 0.38 0.39 0.44 0.30 0.32 0.24 0.01 0.15 0.21 International 0.44 0.41 0.42 0.34 0.25 0.17 0.01 0.15 0.15 Japanese 0.81 0.40 0.40 0.40 0.39 0.22 0.16 0.20 0.19 0.14 0.20 Pacific Ex Japan 0.76 0.60 0.48 0.65 0.48 0.42 0.48 0.35 0.01 0.03 0.24 Pacific Region 0.61 0.38 0.34 0.32 0.21 0.01 0.05 0.09 Notes: This table presents the median estimated ÷ (staleness) by style and year. Values for ÷ range from 0 (no staleness) to 1 (fully stale). Each entry represents the median estimated ÷ from all the monthly fund observations in a given style during the year. Panel A presents results for the best specification (highest likelihood), Panel B for the specification with no NAV adjustment, Panel C for the specification with NAV adjustment and Panel D for the specification with no NAV adjustment and no ETF premiums. 47 Table 3.5. Median Estimated Staleness for the Specification with no NAV Adjustment Style 2000 2001 2002 2003 2004 2005 2006 2007 2008 2009 2010 Panel A: Full period China Region 0.64 0.60 0.55 0.65 0.51 0.43 0.43 0.37 0.04 0.03 0.11 Emerging Markets 0.35 0.34 0.25 0.18 0.02 0.06 0.20 European Region 0.52 0.43 0.42 0.43 0.46 0.31 0.34 0.24 0.01 0.13 0.14 International 0.45 0.41 0.42 0.33 0.29 0.13 0.01 0.02 0.02 Japanese 0.68 0.38 0.39 0.40 0.38 0.03 0.03 0.24 0.13 0.14 0.18 Pacific Ex Japan 0.71 0.56 0.48 0.70 0.44 0.49 0.49 0.38 0.05 0.04 0.27 Pacific Region 0.54 0.34 0.31 0.34 0.21 0.01 0.01 0.06 Panel B: Retail China Region 0.64 0.60 0.55 0.65 0.51 0.44 0.44 0.37 0.04 0.04 0.16 Emerging Markets 0.34 0.33 0.24 0.17 0.03 0.07 0.20 European Region 0.52 0.43 0.40 0.41 0.47 0.33 0.34 0.24 0.01 0.13 0.17 International 0.44 0.41 0.42 0.34 0.28 0.12 0.01 0.02 0.02 Japanese 0.72 0.40 0.40 0.40 0.43 0.22 0.05 0.24 0.18 0.16 0.19 Pacific Ex Japan 0.68 0.55 0.48 0.67 0.47 0.45 0.47 0.38 0.06 0.04 0.27 Pacific Region 0.60 0.34 0.18 0.01 0.01 0.06 Panel C: Institutional China Region 0.48 0.40 0.45 0.35 0.02 0.03 0.17 Emerging Markets 0.35 0.32 0.26 0.17 0.02 0.05 0.20 European Region 0.54 0.44 0.41 0.42 0.43 0.31 0.30 0.19 0.01 0.15 0.08 International 0.46 0.42 0.42 0.33 0.28 0.13 0.01 0.02 0.02 Japanese 0.72 0.45 0.44 0.46 0.41 0.23 0.16 0.20 0.16 0.05 0.19 Pacific Ex Japan 0.70 0.58 0.53 0.75 0.53 0.35 0.48 0.37 0.05 0.03 0.27 Pacific Region 0.64 0.34 0.25 0.34 0.29 0.01 0.01 0.05 Panel D: Index funds China Region 0.01 0.02 0.01 Emerging Markets 0.36 0.34 0.24 0.08 0.02 0.02 0.19 European Region 0.56 0.58 0.34 0.31 0.45 0.31 0.23 0.17 0.01 0.16 0.13 International 0.45 0.40 0.41 0.32 0.21 0.10 0.01 0.01 0.01 Japanese 0.72 0.45 0.41 0.37 0.03 0.03 0.10 0.24 0.21 0.14 0.21 Pacific Region 0.54 0.34 0.25 0.36 0.12 0.01 0.01 0.02 Notes: This table presents the median estimated ÷ (staleness) by style and year for the specification with no NAV adjustment. Values for ÷ range from 0 (no staleness) to 1 (fully stale). Each entry represents the median estimated ÷ from all the monthly fund observations in a given style during the year. Panel A presents results for funds that are alive during the full period (2000-2010), Panel B for funds open to retail investors, Panel C for funds open to institutional investors and Panel D for index funds. Note that some index funds are likely omitted as the index fund identifier is only available for funds alive on or after June 2008. 48 This result in a zero-cost strategy with risk being partly hedged, the quality of the hedge depending on the proximity of assets under managements between the ETF and the mutual fund. The goal being to illustrate the predictive power and the extent of the NAV mispricing, I do not control for front/rear-load fees, trading restrictions, transaction costs or borrowing cost for shorting; only management fees and expenses are accounted for. The eects of fees and trading restriction are discussed in Section 3.6. Equal-weighted portfolios with daily rebalancing are formed for each specification for the aggregate of funds and by style groups. There is strong evidence of predictability in international funds returns. Table 3.6 presents annualized abnormal returns and Fama-French four factors alphas of the simple trading strategy using signals from the four specifications. Results of Table 3.6 show positive abnormal returns and Fama-French four factor alphas for all years and specifications except for one observation. Furthermore, most of the results for the best specification are statistically significant at the 1% level. The additional information present in ETF premiums matters; returns and – for the specification with no NAV adjustment are all larger than those for the specification with no NAV adjustment and no ETF premiums, some by over twice as much. Looking at the results obtained for the best specification, the first two year of the sample yield larger abnormal returns and –, while afterwards there is no apparent trend. This is consistent with an increase in NAV adjustments around the year 2003, when the problem of stale pricing became more widely known and at least one year prior the rule changes by the SEC. Nonetheless, while it has decreased, there is still statistically significant predictability in international mutual funds in the later part of the sample. Breaking it down by style, Table 3.7 shows that the strategy based on the best specification yields highly statistically signification positive abnormal returns and – in most cases. The few negative returns and – are all not statistically significant at the 10% level, except for one observation which is not significant at the 5% level. Table 3.8 presents annualized returns for the full period and for two sub periods: 2001 to 2004, when the SEC changed the disclosure rule, and 2005 to 2010. The strategy based on the best specification yields annual returns of 33.33% from 2001 to 2004 and returns of 15.85% from 2005 to 2010. While this represents a decrease of more than half, it still remains large and statistically significant. Over the full period, the best specification, which account for the action of managers, yields returns on average 2% higher than the specification with no NAV adjustment. ETF premiums contribute a lot in predicting returns. Compared to the specification with no NAV adjustment and no ETF premiums, adding the ETF premium in the information for filtering increases returns on average by more than 7%. 3.6 Fees and trading restrictions Besides adjusting for fair value pricing, mutual funds can circumvent market timers by imposing fees and trading restrictions. Rear-load fees, or redemption fees, usually get smaller with time and disappear after a set period. They can eectively penalize short-term investing without harming long-run investors, aside from reducing the liquidity of their investment. However, in 49 Table 3.6. Simple Trading Strategy, for Each Specification Specification 2001 2002 2003 2004 2005 2006 2007 2008 2009 2010 Panel A: Returns Best 61.50% 48.00% 9.10% 20.60% 20.50% 18.00% 19.60% 10.60% 8.10% 19.00% (5.88***) (4.07***) (0.95) (3.24***) (5.19***) (5.08***) (4.82***) (1.69*) (1.67*) (5.00***) No NAV adjustment 45.80% 44.80% 10.40% 19.60% 18.20% 17.20% 19.50% 7.20% 8.40% 14.80% (4.53***) (3.71***) (1.05) (3.00***) (4.55***) (4.69***) (4.73***) (1.20) (1.67*) (3.87***) NAV adjustment 50.30% 40.70% 12.10% 16.40% 19.90% 14.70% 13.10% 15.70% 14.40% 18.70% (5.52***) (3.64***) (1.47) (3.00***) (5.31***) (4.61***) (3.85***) (2.09**) (2.99***) (4.95***) No NAV adjustment, 18.00% 22.40% 9.30% 16.50% 11.70% 13.70% 15.90% 1.90% 2.70% 11.30% no ETF premiums (1.95*) (1.93*) (0.93) (2.7***) (3.36***) (3.89***) (3.92***) (0.28) (0.59) (3.28***) Panel B: Fama-French 4 factors – Best 59.40% 42.50% 9.80% 23.70% 21.10% 22.20% 19.00% 6.10% 10.50% 21.00% (6.15***) (4.31***) (0.99) (4.17***) (6.47***) (7.04***) (5.44***) (1.28) (3.07***) (6.68***) No NAV adjustment 42.80% 39.60% 11.30% 23.00% 18.60% 21.30% 19.70% 2.90% 10.60% 17.20% (4.50***) (3.87***) (1.10) (3.92***) (5.79***) (6.54***) (5.31***) (0.62) (3.07***) (6.07***) NAV adjustment 48.60% 34.50% 14.10% 19.00% 20.40% 18.50% 12.60% 9.60% 16.00% 20.60% (5.78***) (3.69***) (1.64) (3.85***) (6.43***) (6.59***) (4.21***) (1.70*) (4.14***) (6.57***) No NAV adjustment, 14.90% 17.20% 10.00% 19.40% 12.20% 16.50% 16.00% -2.90% 5.10% 12.80% no ETF premiums (1.73*) (1.80*) (0.96) (3.41***) (4.14***) (5.13***) (4.15***) (-0.59) (1.46) (4.48***) Notes: This table presents annualized equal-weighted returns and FF4 alphas obtained with a simple zero- investment strategy based on pairs of matched international ETFs and open-end mutual funds, for each spec- ification. The daily trading rule is the following: when the estimated fundamental value of the mutual fund is higher than the expected NAV, buy a share in the mutual fund and short sell the ETF. Otherwise, take no position. These results assume no borrowing fees, trading fees or trading restrictions. T-stats are presented in parenthesis with *, **, *** indicating significance at 10%, 5% and 1% level respectively. an investment universe that includes highly liquid ETFs, this reduction in liquidity is a factor that put mutual funds at a disadvantage. Front-load fees aren’t a significant deterrent to market timers because they are usually waved for large transactions. Table 3.9 presents a breakdown of the number of mutual funds in the sample that charge front-load and rear-load fees. The most frequently used trading restriction is a limit on the number of round trips (buying then selling shares) an investor can make during a year. In this section, I investigate if the predicting power of exchange traded funds is still economically significant after controlling for trading restrictions and redemption fees. Management fees and expenses are also taken into account as in the previous section, but transaction fees and borrowing fees are not. 3.6.1 Trading restrictions The most common trading restriction imposed by mutual funds is a limit on the number of round trips by an investor in one year, a round trip being defined as buying and then selling shares. While one can argue that a market timer could use dierent strategies to mask her identity, it is nonetheless interesting to study the eectiveness of such a rule. To illustrate the eect of this restriction, I modify the simple trading strategy, borrowing inspiration from by the wildcard option methodology introduced by Chalmers, Edelen, and Kadlec (2001). Let st = E[vMFt ]≠E[nMFút ] be a daily trading signal. For every fund, I identify the 2.5% and 50 Table 3.7. Simple Trading Strategy for the Best Specification, by Style Style 2001 2002 2003 2004 2005 2006 2007 2008 2009 2010 Panel A: Returns China Region 120.30% 124.20% 71.00% 47.00% 31.70% 40.40% 52.40% 26.80% 4.20% 15.60% (4.44***) (5.79***) (3.67***) (4.14***) (4.55***) (3.37***) (3.72***) (2.08**) (0.39) (1.24) Emerging Markets 18.50% 29.00% 39.60% 47.90% 49.60% 22.20% 26.70% (1.93*) (3.53***) (3.90***) (4.22***) (2.67***) (2.51**) (5.41***) European Region 51.80% 47.60% 43.50% 19.40% 26.30% 16.10% 12.30% 13.20% 7.40% 28.10% (4.87***) (3.64***) (4.41***) (3.33***) (5.60***) (4.31***) (3.64***) (2.32**) (1.71*) (5.69***) International -0.50% 19.70% 18.70% 13.30% 12.10% 2.20% 5.20% 17.10% (-0.05) (2.74***) (4.90***) (4.41***) (2.98***) (0.37) (1.06) (4.48***) Japanese 56.40% 35.30% 46.20% 25.40% 19.00% 6.30% 3.60% 0.40% 16.40% -12.80% (3.99***) (3.35***) (5.12***) (4.19***) (4.41***) (1.19) (0.93) (0.04) (1.12) (-1.19) Pacific Ex Japan 74.50% 58.40% 73.70% 35.50% 7.00% 49.90% 70.50% 20.60% -14.50% 29.80% (3.58***) (2.81***) (3.26***) (2.54**) (0.72) (4.27***) (5.76***) (1.34) (-1.35) (4.90***) Pacific Region 12.20% 0.20% 15.80% 22.00% -7.20% 11.50% 15.70% (1.11) (0.03) (2.88***) (4.77***) (-1.03) (1.95*) (3.88***) Panel B: Fama-French 4 factors – China Region 111.90% 115.10% 87.90% 51.50% 31.50% 54.30% 58.90% 20.10% 13.00% 23.00% (4.30***) (5.71***) (5.34***) (4.88***) (4.81***) (5.05***) (4.39***) (1.74*) (1.32) (1.94*) Emerging Markets 26.50% 31.30% 48.20% 49.40% 37.00% 27.50% 29.10% (2.91***) (4.38***) (5.25***) (5.87***) (2.49**) (4.00***) (6.72***) European Region 52.30% 42.40% 46.00% 22.80% 26.20% 19.20% 12.00% 10.50% 8.00% 30.00% (5.17***) (3.85***) (5.30***) (4.44***) (6.28***) (5.32***) (3.55***) (2.08**) (2.08**) (6.52***) International -1.10% 22.70% 19.00% 16.50% 11.00% -0.70% 7.00% 18.90% (-0.09) (3.36***) (5.63***) (5.45***) (2.75***) (-0.14) (1.78*) (5.71***) Japanese 49.40% 31.50% 50.60% 27.10% 19.80% 8.20% 3.50% 1.70% 16.10% -9.00% (3.7***) (3.43***) (6.36***) (4.62***) (4.76***) (1.48) (0.88) (0.15) (1.09) (-0.85) Pacific Ex Japan 68.90% 52.40% 106.30% 43.50% 8.50% 59.50% 66.70% 16.70% -13.20% 33.30% (3.36***) (2.68***) (4.63***) (3.30***) (0.95) (5.29***) (6.13***) (1.12) (-1.30) (6.23***) Pacific Region 15.40% 1.00% 18.50% 22.50% -10.90% 11.90% 17.10% (1.43) (0.14) (3.57***) (4.80***) (-1.78*) (2.13**) (4.46***) Notes: This table presents annualized equal-weighted returns and FF4 alphas obtained with a simple zero- investment strategy based on pairs of matched international ETFs and open-end mutual funds for the best specification, by style. The daily trading rule is the following: when the estimated fundamental value of the mutual fund is higher than the expected NAV, buy a share in the mutual fund and short sell the ETF. Otherwise, take no position. These results assume no borrowing fees, trading fees or trading restrictions. T-stats are presented in parenthesis with *, **, *** indicating significance at 10%, 5% and 1% level respectively. 97.5% quantiles from the historical empirical distribution of each signal. Cut-o values are re-estimated every month using the previous 12 months estimates for each model.41 Those estimated thresholds are used to limit transactions to days when the signals are the strongest and when the expected return is the largest. The new strategy is then • If the signal is greater than the 97.5% threshold and you have no position, sell the ETF short and buy the mutual fund. • If the signal is smaller than the 2.5% threshold and you have a position, close your position. • Otherwise, keep the current position. 41For every fund and every month, I use the estimates of vMFt and nMFút generated from the calibrated models for that fund month pair. 51 Table 3.8. Simple Trading Strategy, Comparison Between Specifications and Time Periods Period Specification 2001/2010 2001/2004 2005/2010 2005/2010 - 2001/2004 Best 22.61% 33.33% 15.85% -17.48% (10.39***) (6.97***) (8.55***) (-3.14***) No NAV adjustment 20.21% 29.82% 14.10% -15.72% (9.18***) (6.16***) (7.57***) (-2.82***) NAV adjustment 21.17% 29.15% 16.05% -13.10% (10.40***) (6.81***) (8.45***) (-2.58**) No NAV adj., no ETF prem. 12.08% 16.19% 9.39% -6.80% (5.61***) (3.48***) (5.09***) (-1.30) Best - No NAV adj. 1.99% 2.67% 1.53% (4.23***) (2.68***) (3.72***) No NAV adj. - 7.22% 11.69% 4.29% No NAV adj., no ETF prem. (9.95***) (8.56***) (5.41***) Notes: This table presents annualized equal-weighted returns obtained with a simple zero-investment strategy based on pairs of matched international ETFs and open-end mutual funds, for each specification. The daily trading rule is the following: when the estimated fundamental value of the mutual fund is higher than the expected NAV, buy a share in the mutual fund and short sell the ETF. Otherwise, take no position. These results assume no borrowing fees, trading fees or trading restrictions. The dierence between the two time periods is computed as the dierence of the mean daily returns for each time period. The dierence between specifications is computed by taking a long/short position in strategies associated with each specification. T-stats are presented in parenthesis with *, **, *** indicating significance at 10%, 5% and 1% level respectively. Table 3.9. Count and TNA of International Mutual Funds by Fee Structure Full Sample Front-Load Rear-Load R-L > 1m R-L > 2m R-L > 6m Style N TNA N TNA N TNA N TNA N TNA N TNA China Region 44 9,335 15 2,638 28 7,223 20 6,848 16 5,970 7 525 Emerging Markets 178 87,767 34 23,573 86 33,388 74 27,211 60 26,445 29 18,800 European Region 59 18,451 13 2,405 34 4,564 22 3,981 14 732 11 368 International 632 428,013 139 88,288 286 95,338 196 77,679 145 47,915 84 11,867 Japanese 18 2,035 3 24 10 515 9 511 6 494 1 9 Pacific Region 29 8,796 5 478 16 4,042 11 3,828 5 458 3 90 Pacific Ex Japan 29 14,408 6 970 20 9,081 16 7,008 12 6,877 4 302 Total 989 568,805 215 118,376 480 154,151 348 127,066 258 88,891 139 31,960 Notes: This table presents the count and total net assets of international mutual funds by Lipper Objective and fee structure according to the CRSP Survivor-Bias-Free US Mutual Fund Database as of December 31, 2010. Sample consists of mutual funds with at least one year of returns prior to December 2010, with at least $5 millions and with information on fee structure. Total net assets are in $ millions. Fee structure is broken down in funds that charge front-load fees and rear-loads fees. The latter are further broken down by maturity of the fees, in months. 52 Table 3.10. Complex Trading Strategy, for Each Specification Specification 2001 2002 2003 2004 2005 2006 2007 2008 2009 2010 Panel A: Returns Best 11.70% 4.70% 2.80% 3.80% 5.40% 4.70% 4.10% 1.60% -0.80% 1.50% (1.38) (0.49) (0.37) (1.09) (1.71*) (1.14) (1.05) (0.29) (-0.21) (0.35) No NAV adjustment 12.20% 4.10% 3.00% 3.10% 5.20% 6.30% 4.40% 1.60% -1.00% 1.80% (1.46) (0.43) (0.37) (1.09) (1.63) (1.54) (1.16) (0.29) (-0.29) (0.61) NAV adjustment 7.80% 4.90% 0.60% 2.40% 4.80% 2.50% 3.70% 1.90% -1.20% 1.30% (0.94) (0.51) (0.15) (0.62) (1.51) (0.60) (0.92) (0.28) (-0.20) (0.30) No NAV adjustment, 8.10% 1.60% 2.80% 3.10% 3.10% 5.60% 3.60% 1.40% -2.30% 0.80% no ETF premiums (1.11) (0.17) (0.35) (0.96) (0.94) (1.39) (0.89) (0.23) (-0.47) (0.23) Panel B: Fama-French 4 factors – Best 8.40% -0.70% 2.70% 5.40% 6.00% 9.40% 4.10% -2.80% 0.90% 4.30% (1.05) (-0.09) (0.35) (1.62) (2.21**) (2.76***) (1.11) (-0.70) (0.36) (1.29) No NAV adjustment 10.00% -1.00% 3.00% 4.30% 5.70% 10.60% 4.50% -2.90% 0.40% 3.30% (1.24) (-0.14) (0.35) (1.57) (2.15**) (3.07***) (1.26) (-0.75) (0.17) (1.67*) NAV adjustment 3.50% 0.20% 0.90% 4.40% 5.60% 7.20% 3.70% -3.30% 2.70% 4.10% (0.46) (0.03) (0.24) (1.25) (2.06**) (2.09**) (1.00) (-0.62) (0.71) (1.23) No NAV adjustment, 4.60% -3.40% 2.80% 4.90% 3.70% 9.40% 3.50% -3.40% 0.90% 2.50% no ETF premiums (0.67) (-0.44) (0.33) (1.65) (1.40) (2.76***) (0.89) (-0.81) (0.26) (0.95) Notes: This table presents annualized equal-weighted returns and FF4 alphas obtained with a simple zero- investment strategy based on pairs of matched international ETFs and open-end mutual funds, for each speci- fication. The daily trading rule is the following: when the dierence between the estimated true value and the expected NAV is larger than the higher threshold, buy a share in the mutual fund and short sell the ETF (or keep that position). When the dierence between the estimated true value and the expected NAV is smaller than the lower threshold, close the position (or keep no position). Otherwise, keep current position. Signal thresholds are evaluated monthly using daily filtered estimations from the previous year and are set to the 2.5% and 97.5% percentiles of the empirical distribution. These results assume no borrowing fees or trading fees. T-stats are presented in parenthesis with *, **, *** indicating significance at 10%, 5% and 1% level respectively. Table 3.10 presents annualized abnormal returns and Fama-French four factor alphas of the more realistic trading strategy using signals from the four specifications. On average there are 7.2 transactions per year (fewer than 4 round-trips). Two important results can be extracted from using this restricted trading strategy. First, even thought round-trip restrictions reduce abnormal returns, using the best specification as a signal still yields economically (but not statistically) significant abnormal returns in all years but one and positive – in all but two years. This suggests that while managers are adjusting NAV for stale pricing, their shareholders still face the risk of dilution. However, the actual dilution is likely now lower than the 0.5% to 1.15% reported by Greene and Hodges (2002) and Zitzewitz (2003). Results are similar when breaking down by style as illustrated by Table 3.11. Second, as further evidence that the additional information in ETF premiums matters, abnormal returns for the strategy relying on the specification with no NAV adjustment are all larger than those for the specification with no NAV adjustment and no ETF premiums. This means that the few trading signals generated by the 2.5% and 97.5% threshold give better predictions when ETF premiums enter the filtering problem. Looking at Table 3.12, we can see that even after restricting the number of trades, an arbitrage using the trading strategy 53 Table 3.11. Complex Trading Strategy for the Best Specification, by Style Specification 2001 2002 2003 2004 2005 2006 2007 2008 2009 2010 Panel A: Returns China Region 9.80% 22.30% 6.30% 0.80% 2.10% 9.40% 4.20% 9.70% 2.30% -2.40% (0.53) (1.32) (0.38) (0.15) (0.33) (0.83) (0.36) (1.03) (0.16) (-0.16) Emerging Markets 3.10% 10.80% 9.50% 12.90% 8.40% 0.30% 0.10% (0.40) (1.37) (0.73) (1.43) (0.54) (0.06) (0.02) European Region 7.90% 2.20% 1.50% 1.90% 6.50% 7.10% 0.90% 2.00% 1.50% 3.10% (1.01) (0.19) (0.23) (0.41) (1.93*) (2.27**) (0.36) (0.38) (0.34) (0.53) International 2.50% 4.20% 5.00% 2.60% 1.70% -0.30% -1.60% 1.10% (0.28) (1.07) (1.63) (0.82) (0.39) (-0.05) (-0.42) (0.28) Japanese 10.50% 5.20% 10.40% 3.00% 6.10% -2.20% 1.70% -9.00% 2.10% -4.90% (0.74) (0.50) (1.18) (1.09) (1.71*) (-0.44) (0.49) (-0.81) (0.18) (-0.42) Pacific Ex Japan 17.60% 10.10% 2.80% -0.30% -8.10% 24.70% 22.20% 11.70% -6.70% 3.90% (0.80) (0.56) (0.21) (-0.06) (-0.98) (1.97*) (1.87*) (0.86) (-0.53) (0.58) Pacific Region -2.30% -13.00% 8.10% 9.50% -4.00% 6.20% 4.10% (-0.63) (-1.49) (1.34) (2.38**) (-0.57) (0.95) (0.88) Panel B: Fama-French 4 factors – China Region 5.50% 15.10% 17.20% 2.80% 2.30% 24.60% 11.70% 5.10% 11.50% 9.70% (0.32) (1.01) (1.34) (0.56) (0.40) (2.4**) (1.08) (0.60) (0.95) (0.76) Emerging Markets 7.70% 13.70% 25.70% 15.60% -2.40% 1.70% 3.20% (1.04) (1.98**) (2.36**) (2.09**) (-0.20) (0.51) (0.71) European Region 4.80% -3.10% 3.10% 4.40% 6.60% 9.80% 0.60% -0.50% 2.20% 5.90% (0.63) (-0.35) (0.53) (1.03) (2.19**) (3.30***) (0.22) (-0.12) (0.54) (1.16) International 1.10% 5.70% 5.10% 4.80% 0.90% -3.30% 0.10% 3.30% (0.12) (1.53) (1.80*) (1.59) (0.20) (-0.70) (0.02) (1.07) Japanese 4.00% 1.20% 13.60% 2.50% 6.60% -0.50% 2.30% -8.10% 3.50% -0.50% (0.30) (0.14) (1.76*) (0.94) (1.93*) (-0.09) (0.64) (-0.71) (0.28) (-0.05) Pacific Ex Japan 17.10% 4.50% 18.30% 1.80% -6.10% 36.60% 22.50% 7.40% -5.70% 8.00% (0.77) (0.27) (1.38) (0.36) (-0.77) (3.23***) (2.07**) (0.57) (-0.48) (1.47) Pacific Region -1.80% -10.00% 13.30% 9.50% -6.70% 5.00% 5.60% (-0.49) (-1.22) (2.32**) (2.36**) (-1.03) (0.82) (1.27) Notes: This table presents annualized equal-weighted returns and FF4 alphas obtained with a simple zero- investment strategy based on pairs of matched international ETFs and open-end mutual funds for the best specification, by style. The daily trading rule is the following: when the dierence between the estimated true value and the expected NAV is larger than the higher threshold, buy a share in the mutual fund and short sell the ETF (or keep that position). When the dierence between the estimated true value and the expected NAV is smaller than the lower threshold, close the position (or keep no position). Otherwise, keep current position. Signal thresholds are evaluated monthly using daily filtered estimations from the previous year and are set to the 2.5% and 97.5% percentiles of the empirical distribution. These results assume no borrowing fees or trading fees. T-stats are presented in parenthesis with *, **, *** indicating significance at 10%, 5% and 1% level respectively. 54 could have earned annual returns of 3.92% over the 2001 to 2010 period or 2.73% in the shorter period of 2005 to 2010. Table 3.12. Complex Trading Strategy, Comparison Between Specifications and Time Periods Period Specification 2001/2010 2001/2004 2005/2010 2005/2010 - 2001/2004 Best 3.92% 5.70% 2.73% -2.97% (2.16**) (1.53) (1.59) (-0.71) No NAV adjustment 4.04% 5.58% 3.01% -2.57% (2.27**) (1.49) (1.90*) (-0.62) NAV adjustment 2.84% 3.87% 2.16% -1.71% (1.58) (1.17) (1.07) (-0.44) No NAV adj., no ETF prem. 2.71% 3.74% 2.01% 1.73% (1.48) (1.01) (1.13) (-0.42) Best - No NAV adj. -0.12% 0.10% -0.27% (-0.26) (0.10) (-0.60) No NAV adj. - 1.27% 1.75% 0.95% No NAV adj., no ETF prem. (2.07**) (1.58) (1.35) Notes: This table presents annualized equal-weighted returns obtained with a simple zero-investment strategy based on pairs of matched international ETFs and open-end mutual funds, for each specification. The daily trading rule is the following: when the dierence between the estimated true value and the expected NAV is larger than the higher threshold, buy a share in the mutual fund and short sell the ETF (or keep that position). When the dierence between the estimated true value and the expected NAV is smaller than the lower threshold, close the position (or keep no position). Otherwise, keep current position. Signal thresholds are evaluated monthly using daily filtered estimations from the previous year and are set to the 2.5% and 97.5% percentiles of the empirical distribution. The dierence between the two time periods is computed as the dierence of the mean daily returns for each time period. The dierence between specifications is computed by taking a long/short position in strategies associated with each specification. These results assume no borrowing fees or trading fees. T-stats are presented in parenthesis with *, **, *** indicating significance at 10%, 5% and 1% level respectively. 3.6.2 Fees Some funds use rear-load fees to deter market timing. In most cases, those fees disappear with time. One might expect funds that do not use rear-load fees as a deterrent to use some more eective form of fair pricing adjustment. To study the dierences between funds that impose rear-load fees and those that do not, I separate the sample in two, the funds imposing any level of rear-load fees in the first subsample and the funds with no rear-load fees in the second subsample.42 Table 3.13 is a split of Panel C of Table 3.3 and presents the median correlation of returns for monthly mutual fund and lagged predictor matches for each subsample. The correlation 42The sum of both is not the full sample since fee structure information is not available for all funds. 55 of matches is very similar in both subsamples with no major dierences. Table 3.14 presents the median estimated ÷ for the specification with no NAV adjustment, by year and style. The median annual ÷ is very similar for both subsample with no noticeable dierences. Table 3.15 presents annualized abnormal returns and Fama-French four factor alphas of the simple trading strategy on those two subsamples for all specifications. While this is not an implementable strategy, it should indicate if one group diers from the other in terms of predictability due to fair value adjustment. As before, results are similar for both subsamples. Table 3.13. Median Correlation of Returns for Monthly Mutual Fund Matches with Lagged Predictors, by Year and Style Style 2000 2001 2002 2003 2004 2005 2006 2007 2008 2009 2010 Panel A: No rear-load fees China Region 0.48 0.53 0.38 0.36 0.31 0.21 0.23 0.16 0.14 0.11 0.11 Emerging Markets 0.44 0.38 0.34 0.37 0.31 0.24 0.28 0.18 0.10 0.06 0.19 European Region 0.40 0.33 0.37 0.36 0.24 0.15 0.19 0.16 0.10 0.04 0.07 International 0.49 0.37 0.41 0.40 0.30 0.18 0.18 0.14 0.10 0.04 0.09 International Small-Cap 0.49 0.46 0.45 0.48 0.38 0.26 0.31 0.24 0.12 0.05 0.08 Japanese 0.38 0.34 0.34 0.31 0.24 0.11 0.08 0.10 0.13 0.04 0.08 Pacific Region 0.50 0.44 0.40 0.44 0.35 0.17 0.13 0.11 0.11 0.07 0.03 Pacific Ex Japan 0.50 0.46 0.38 0.40 0.28 0.19 0.25 0.17 0.11 0.08 0.11 Panel B: With rear-load fees China Region 0.41 0.41 0.35 0.38 0.28 0.24 0.24 0.14 0.13 0.10 0.09 Emerging Markets 0.44 0.37 0.34 0.35 0.31 0.25 0.28 0.16 0.10 0.06 0.18 European Region 0.41 0.32 0.38 0.38 0.22 0.17 0.19 0.15 0.10 0.04 0.06 International 0.48 0.37 0.40 0.40 0.29 0.16 0.17 0.14 0.11 0.05 0.09 International Small-Cap 0.50 0.47 0.44 0.47 0.35 0.26 0.29 0.24 0.12 0.04 0.07 Japanese 0.39 0.37 0.33 0.25 0.23 0.12 0.07 0.06 0.14 0.07 0.08 Pacific Region 0.52 0.45 0.41 0.42 0.29 0.14 0.14 0.11 0.12 0.07 0.03 Pacific Ex Japan 0.50 0.44 0.38 0.43 0.33 0.26 0.30 0.22 0.12 0.08 0.10 Notes: This table presents the median correlation of returns for monthly mutual fund matches with lagged predictors, by year and style. For each monthly mutual fund observation, the chosen match is the one with the highest correlation during the previous year (excluding the current month) between mutual fund returns and lagged predictor returns. Panel A presents results for funds without rear-load fees while Panel B presents results for funds with rear-load fees. Results presented in this subsection fail to show any notable dierence between the funds with rear-load fees and without rear-load fees. This suggests that the decision to impose rear- load fees is independent from the eort of adjusting NAV for stale pricing. 3.7 Concluding remarks In this paper I look at the aftermath of the mutual fund market timing scandal and investigate the impact it had on the predictability of mutual fund returns aected by stale pricing. The main conclusion is that there is strong evidence that fund managers do adjust NAV for fair valuation, and that predictability has been reduced by about half since the introduction of new rules by the SEC in 2004. However, there still remains economically significant predictability. 56 Table 3.14. Median Estimated Staleness for Funds With and Without Rear-Load Fees, by Year and Style Style 2000 2001 2002 2003 2004 2005 2006 2007 2008 2009 2010 Panel A: No rear-load fees China Region 0.64 0.61 0.57 0.68 0.52 0.41 0.41 0.38 0.04 0.05 0.21 Emerging Markets 0.34 0.33 0.25 0.17 0.02 0.05 0.20 European Region 0.53 0.42 0.40 0.41 0.47 0.32 0.33 0.22 0.01 0.14 0.17 International 0.45 0.41 0.43 0.33 0.29 0.13 0.01 0.01 0.02 Japanese 0.73 0.41 0.44 0.48 0.43 0.23 0.20 0.24 0.19 0.16 0.19 Pacific Ex Japan 0.68 0.55 0.47 0.66 0.46 0.35 0.43 0.36 0.06 0.05 0.26 Pacific Region 0.54 0.34 0.32 0.34 0.20 0.01 0.01 0.02 Panel B: With rear-load fees China Region 0.63 0.59 0.54 0.63 0.50 0.45 0.46 0.37 0.03 0.04 0.17 Emerging Markets 0.35 0.34 0.23 0.17 0.03 0.07 0.20 European Region 0.51 0.43 0.40 0.41 0.46 0.33 0.34 0.25 0.01 0.15 0.12 International 0.45 0.41 0.43 0.34 0.28 0.12 0.01 0.02 0.02 Japanese 0.69 0.41 0.41 0.40 0.44 0.22 0.03 0.22 0.18 0.16 0.19 Pacific Ex Japan 0.68 0.55 0.49 0.69 0.48 0.48 0.49 0.38 0.05 0.03 0.27 Pacific Region 0.64 0.35 0.24 0.33 0.22 0.01 0.01 0.09 Notes: This table presents the median estimated ÷ (staleness) by style and year. Values for ÷ range from 0 (no staleness) to 1 (fully stale). Each entry represents the median estimated ÷ from all the monthly fund observations in a given style during the year, for the specification with no NAV adjustment. Panel A presents results for funds without rear-load fees while Panel B presents results for funds with rear-load fees. The main methodological contribution of this paper is the way the mutual fund manager’s filtering problem is embedded in the arbitrager’s filtering problem. This comes at a cost: the model is restricted to be linear, which excludes some alternative solutions to the manager’s problem. For example, it is plausible that fund managers adjust NAV only when the unadjusted value is too far o. This solution would lead to a non-linear filtering problem for the arbitrager, which would require more advanced computing resources than those currently available to me. However, this could only improve my current results and yield more predictability, so the conclusion that there still is economically significant predictability would remain unchanged. The other important contribution of this paper is to show that premiums on ETFs contain valuable information and to present a methodology for exploiting this additional information. Mutual fund managers wanting to eliminate completely the predictability illustrated in this paper could do so by implementing the optimal filtering solution to their problem as described in Appendix B.2. 57 Table 3.15. Simple Trading Strategy Using Funds With and Without Rear-Load Fees Specification 2001 2002 2003 2004 2005 2006 2007 2008 2009 2010 Panel A: No rear-load fees (Returns) Best 60.90% 50.10% 8.20% 21.10% 21.10% 17.90% 19.60% 12.30% 10.10% 19.60% (5.86***) (4.20***) (0.83) (3.20***) (5.25***) (4.98***) (4.88***) (1.90*) (1.98**) (5.18***) No NAV adjustment 46.80% 45.00% 9.80% 19.70% 18.70% 17.20% 19.80% 8.10% 10.50% 15.30% (4.62***) (3.71***) (0.97) (2.93***) (4.59***) (4.69***) (4.83***) (1.31) (1.99**) (4.03***) NAV adjustment 50.60% 43.00% 11.20% 16.60% 20.40% 14.10% 12.90% 17.50% 15.50% 19.10% (5.62***) (3.78***) (1.33) (2.96***) (5.37***) (4.40***) (3.81***) (2.30**) (3.12***) (5.09***) No NAV adjustment, 17.90% 21.00% 9.10% 16.90% 11.70% 13.80% 16.30% 2.20% 4.00% 11.80% no ETF premiums (1.94*) (1.82*) (0.89) (2.67***) (3.30***) (3.90***) (4.02***) (0.33) (0.81) (3.41***) Panel B: With rear-load fees (Returns) Best 63.00% 50.10% 12.80% 21.50% 20.60% 19.30% 21.50% 10.90% 6.70% 20.40% (5.70***) (4.02***) (1.41) (3.51***) (5.10***) (5.24***) (5.16***) (1.80*) (1.40) (5.05***) No NAV adjustment 48.30% 47.50% 13.70% 20.80% 18.40% 18.40% 21.10% 8.00% 6.90% 16.50% (4.52***) (3.72***) (1.47) (3.32***) (4.54***) (4.83***) (4.97***) (1.37) (1.39) (4.06***) NAV adjustment 49.30% 42.80% 15.90% 17.80% 20.10% 16.90% 15.50% 14.50% 13.40% 20.20% (5.18***) (3.65***) (2.00**) (3.31***) (5.22***) (5.07***) (4.30***) (1.93*) (2.80***) (5.00***) No NAV adjustment, 19.70% 25.40% 10.90% 17.00% 11.90% 13.90% 16.90% 1.30% 1.80% 12.00% no ETF premiums (2.01**) (2.06**) (1.18) (2.92***) (3.34***) (3.81***) (4.06***) (0.19) (0.38) (3.26***) Panel C: No rear-load fees (FF4 –) Best 58.50% 44.30% 8.70% 24.30% 21.70% 22.20% 19.00% 7.60% 12.30% 21.70% (6.11***) (4.49***) (0.85) (4.07***) (6.54***) (6.92***) (5.48***) (1.57) (3.49***) (6.99***) No NAV adjustment 43.80% 39.60% 10.40% 23.10% 19.10% 21.30% 19.80% 3.60% 12.60% 17.70% (4.59***) (3.88***) (0.98) (3.78***) (5.82***) (6.48***) (5.37***) (0.76) (3.54***) (6.35***) NAV adjustment 48.30% 36.50% 13.00% 19.30% 20.90% 18.10% 12.40% 11.40% 16.90% 21.10% (5.84***) (3.87***) (1.48) (3.76***) (6.51***) (6.36***) (4.17***) (1.98**) (4.25***) (6.85***) No NAV adjustment, 14.80% 15.80% 9.60% 19.80% 12.20% 16.60% 16.20% -2.40% 6.40% 13.40% no ETF premiums (1.71*) (1.66*) (0.90) (3.35***) (4.08***) (5.08***) (4.22***) (-0.48) (1.74*) (4.68***) Panel D: With rear-load fees (FF4 –) Best 60.80% 44.10% 14.40% 24.80% 21.00% 23.60% 21.10% 6.50% 9.20% 22.80% (5.98***) (4.23***) (1.54) (4.64***) (6.39***) (7.32***) (5.97***) (1.42) (2.79***) (6.89***) No NAV adjustment 44.70% 41.70% 15.40% 24.40% 18.80% 22.60% 21.50% 3.70% 9.30% 19.20% (4.49***) (3.86***) (1.60) (4.44***) (5.84***) (6.81***) (5.67***) (0.83) (2.78***) (6.47***) NAV adjustment 47.30% 36.10% 18.70% 20.50% 20.60% 20.70% 15.20% 8.40% 15.20% 22.50% (5.45***) (3.69***) (2.31**) (4.34***) (6.36***) (7.13***) (4.85***) (1.50) (4.07***) (6.72***) No NAV adjustment, 16.20% 19.70% 12.30% 19.90% 12.30% 16.80% 17.10% -3.70% 4.20% 13.80% no ETF premiums (1.77*) (1.95*) (1.28) (3.76***) (4.11***) (5.13***) (4.39***) (-0.75) (1.21) (4.68***) Notes: This table presents annualized equal-weighted returns and FF4 alphas obtained with a simple zero- investment strategy based on pairs of matched international ETFs and open-end mutual funds, for each spec- ification. The daily trading rule is the following: when the estimated fundamental value of the mutual fund is higher than the expected NAV, buy a share in the mutual fund and short sell the ETF. Otherwise, take no position. These results assume no borrowing fees, trading fees or trading restrictions. Panel A and B (C and D) present returns (FF4 alphas) for funds without rear-load fees and with rear-load fees respectively. T-stats are presented in parenthesis with *, **, *** indicating significance at 10%, 5% and 1% level respectively. 58 Chapter 4 Conclusion This thesis is about the eects and implications of the recent increase in index-linked investing. The essay presented in Chapter 2 introduces a general equilibrium model with active investors and indexers, which is used to study how index-linked comovement arises. In the model, comovement is due to market segmentation, which in turn is due to presence of indexers. The wealthier the indexers are relative to the whole economy, the greater the resulting excess comovement is. The subsequent empirical investigation of indexing in the S&P 500 finds results consistent with the model predictions. The second essay revisits the issue of international mutual fund predictability due to stale prices. The simple trading strategy presented yields annual returns of 33% from 2001 to 2004 and 16% from 2005 to 2010. The results suggest that mutual fund managers are more diligent and tend to adjust NAV for stale prices; however the problem is not fully resolved and some eco- nomically significant predictability remains. Results also suggest that US-based international ETFs provide more information than previously known predictors of mutual fund returns. The main methodological contribution of this essay is the filtering approach based on a state-space model that embeds the fund manager problem, thus accounting for unobserved actions of fund managers. 4.1 Limitations I recognize that, as with any research, the work described in this thesis presents some limitations. In developing the asset pricing model presented in Chapter 2, some simplifications were necessary as a trade-o for tractability. The main assumption in the model is that the indexing constraint is exogenous. While this is a common assumption in indexing models, it does leave open the question of why some investors would choose indexing in the first place. The model is even counter-factual on this point; it is non-stationary and the indexer eventually disappears over time. The rise of indexing could be explained by features omitted from the model, such as the dierent costs of each investing strategy. Another limitation stems from the comparative statics analysis, which does not account for possible feedback eects. The empirical investiga- 59 tion of index-linked comovement is limited mainly by data availability on passive assets under management. The limited sample of less than 30 annual observations does not allow rejecting all alternative explanations, such as the concurrent rise in index-based benchmarking of fund managers. Furthermore, the rise in popularity of other indices happened much later than in the S&P 500, leaving insucient observations for the use of these other indices. There are two main limitations to the research presented in Chapter 3. First, the method- ology constrains the alternative specifications to be of the linear type. This rule out alternative strategies on the part of the manager such as adjusting the NAV only when the required adjust- ment is above a certain threshold. While there are filtering techniques that could accommodate non-linear specifications, none is computationally feasible with the tools available to me. The computations required for this project already required over a month of full-time use a rela- tively powerful workstation. The second limitation is in the scope and interpretation of the results. While I conclude that there remains predictability in international mutual funds, I cannot conclude that arbitragers are still actively exploiting it. This would require the careful study of daily funds flow data and would represent an interesting extension; unfortunately the required data is only available from private vendors and quite onerous. 4.2 Future work Many interesting questions related to the rise of index-linked investing remain to this day. First and foremost, the causes of this rise in indexing – and the reason it took so long after the development of the CAPM – are yet to be fully uncovered. 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The index and market baskets therefore also pay dividend streams with dynamics as described in (2.1), with the exception that their variance parameters have the form: ‡DI = 5 1≠ 2s1s2(s1+s2)2 (1≠flD) 6 ‡2D, (A.1) ‡DM = [1≠2(s1s2+s1s3+s2s3)(1≠flD)]‡2D, (A.2) where si is the weight of share of dividends of asset i: si = Di D1+D2+D3 , i œ {1,2,3}. (A.3) Let Êi,t denote the market weight of stock i at time t such that q3 i=1Êi,t = 1 and let ÊIi,t = Êi,t/(Ê1,t+ Ê2,t) denote the weight of asset i œ {1,2} in the index. Then the index return moments are µI,t = ÊI1,tµ1,t+ÊI2,tµ2,t, (A.4) ‡2I,t = (ÊI1,t)2‡21,t+(ÊI2,t)2‡22,t+2ÊI1,tÊI2,tcorr(dZ1,t,dZ2,t)‡1,t‡2,t. (A.5) 66 A.1.2 Agents’ problem Agent j’s optimization problem at time t is to maximize her time additive utility: Uj,t = Et 5⁄ Œ t e≠”(s≠t) logcj,sds 6 (A.6) subject to her budget constraint. Formally, this gives: maxUj,t subject to Et C⁄ Œ 0 ›j,s ›j,t cj,sds D ÆWj,t, (A.7) where ›j,t is the marginal utility of agent j at time t. The first order condition is: Ÿj ›j,s ›j,t = e≠”(s≠t)c≠1j,s , (A.8) where Ÿj is the Lagrange multiplier on the budget constraint and ›j,t is a process given by: d›j,t ›j,t =≠rj,tdt≠◊Õj,tdZt. (A.9) where ◊j,t is the price of risk process for agent j. Note that the process can also be written with respect to the dividend basis and the market basis43 as: d›j,t ›j,t =≠rj,tdt≠◊Õj,tdZD,t =≠rj,tdt≠◊Õj,tdZt. (A.10) The rationale for using two dierent bases, in addition to the initial Brownian motions Z, is that each of the two new bases simplifies the derivation of the solution for a part of the problem and involves independent Brownian motions, which are easier to deal with. It is simpler to solve for optimal portfolios and market clearing under the market basis. However, the market basis transformation depends on stock return covariances, so it is not appropriate to solve for equilibrium price dynamics. The dividend basis is much useful for that purpose. Since both agents trade in the bond, in equilibrium they should have the same riskless rate (i.e. rI,t = rA,t = rt.) However their dierent investment opportunity sets means they will face dierent market price of risk. Following the convex duality methodology approach of CvitaniÊ and Karatzas (1992), I define a fictitious market which the indexer views as complete. In the current setup with log ulity, the market price of risk in the fictitious market is the same as in the incomplete market (see Example 7.2 on p.304 Karatzas and Shreve (1998) for more details.) The idea is to create a fictitious market for agent I by replacing the expected return on asset i by µi(Â) = µi+Âi such that in equilibrium she chooses not to hold the unavailable asset, and 43For a definition of the dierent bases, see Appendix A.2. 67 to hold the index assets according to index weights. In the present setup,  = argminÂ Ë (µ1(Â)≠ r,µ2(Â)≠ r,µ3(Â)≠ r)≠1(µ1(Â)≠ r,µ2(Â)≠ r,µ3(Â)≠ r)Õ È1/2 . (A.11) Substituting the  obtained in (A.11) in the shadow market price of risk of the indexer I obtain, under the market basis: ◊I = „I‡≠1I SWWU ‡1ÊI1+fl12‡2ÊI2Ò 1≠fl212‡2ÊI2 0 TXXV , (A.12) where „I = µI≠r‡I is the Sharpe ratio of the index. Since (‡1Ê I 1+fl12‡2ÊI2)2+( Ò 1≠fl212‡2ÊI2)2 = ‡2I , in scalar form ◊I = „I . The result in (A.12) has the same form if working under the dividend basis following (A.10): ◊I = „I‡≠1I SWWU ÊI1‡11+ÊI2‡21 ÊI1‡12+ÊI2‡22 ÊI1‡13+ÊI2‡23 TXXV . (A.13) Agent A is unconstrained and faces complete markets, so her market price of risk under the market and dividend bases are given by: ◊A = ‡≠1(µ1≠ r, µ2≠ r, µ3≠ r)Õ = SWWWWU „1 „1≠fl12„2Ô 1≠„212 „3(1≠fl212)≠„1(fl13≠fl12fl23)≠„2(fl23≠fl12fl13)Ô 1≠fl212 Ô 1≠fl212≠fl213≠fl223+2fl12fl13fl23 TXXXXV , (A.14) ◊A = ‡≠1(µ1≠ r, µ2≠ r, µ3≠ r)Õ = 1 c SWWU x1(‡23‡32≠‡22‡33)+x2(‡12‡33≠‡13‡32)+x3(‡13‡22≠‡12‡23) x1(‡21‡33≠‡23‡31)+x2(‡13‡31≠‡11‡33)+(x3‡11‡23≠‡13‡21) x1(‡22‡31≠‡21‡32)+x2(‡11‡32≠‡12‡31)+x3(‡12‡21≠‡11‡22) TXXV , (A.15) where c= ‡13(‡22‡31≠‡21‡32)+‡12(‡21‡33≠‡23‡31)+‡11(‡23‡32≠‡22‡33), and xi = µi≠ r is the excess return on asset i. A.1.3 Optimal portfolios Agent A is unconstrained, so her optimal portfolio proportions are given by fiA,t = ≠1t (µt≠ r1). (A.16) 68 Under the market basis the covariance matrix is t = ‡t‡Õt, so fiA = SWWWWU „1(1≠fl223)≠„2(fl12≠fl13fl23)≠„3(fl13≠fl12fl23) ‡1(1≠fl212≠fl213≠fl223+2fl12fl13fl23) „2(1≠fl213)≠„1(fl12≠fl13fl23)≠„3(fl23≠fl12fl13) ‡2(1≠fl212≠fl213≠fl223+2fl12fl13fl23) „3(1≠fl212)≠„1(fl13≠fl12fl23)≠„2(fl23≠fl12fl13) ‡3(1≠fl212≠fl213≠fl223+2fl12fl13fl23) TXXXXV . (A.17) As for agent I, I know from CvitaniÊ and Karatzas (1992) that fiI,t coincides with the optimal portfolio in the incomplete market: fiI = SWWU fiIIÊI1 fiIIÊI2 0 TXXV , (A.18) where fiII,t = (µI,t≠ r)/‡2I,t, so fiI = SWWU ÊI1 „I ‡I ÊI2 „I ‡I 0 TXXV = SWWWWU ÊI1(x1ÊI1+x2ÊI2) ‡21(ÊI1)2+2fl12‡1‡2ÊI1ÊI2+‡22(ÊI2)2 ÊI2(x1ÊI1+x2ÊI2) ‡21(ÊI1)2+2fl12‡1‡2ÊI1ÊI2+‡22(ÊI2)2 0 TXXXXV . (A.19) A.1.4 Proof of Proposition 1 The market clearing condition imposes that: Êt = fiA,t‹A,t+fiI,t‹I,t = SWWWWWU ‹A(x3(fl13≠fl12fl23)‡1‡2+(x2(fl12≠fl13fl23)‡1+x1(≠1+fl223)‡2)‡3) (≠1+fl212+fl213≠2fl12fl13fl23+fl223)‡21‡2‡3 ≠ (≠1+‹A)Ê1(x1Ê1+x2Ê2) ‡21Ê 2 1+2fl12‡1‡2Ê1Ê2+‡22Ê22 ‹A(x3(≠fl12fl13+fl23)‡1‡2+(x2(≠1+fl213)‡1+x1(fl12≠fl13fl23)‡2)‡3) (≠1+fl212+fl213≠2fl12fl13fl23+fl223)‡1‡22‡3 ≠ (≠1+‹A)Ê2(x1Ê1+x2Ê2) ‡21Ê 2 1+2fl12‡1‡2Ê1Ê2+‡22Ê22 ‹A(x3(≠1+fl212)‡1‡2+(x2(≠fl12fl13+fl23)‡1+x1(fl13≠fl12fl23)‡2)‡3) (≠1+fl212+fl213≠2fl12fl13fl23+fl223)‡1‡2‡23 TXXXXXV , (A.20) 69 where xi = µi≠ r are excess returns. Solving for x1, x2 and x3, I get: xú1 = (‡1 (‡2‡3Ê2Ê3 (fl12fl13‡1Ê1≠fl23‡1Ê1+fl13‡2Ê2≠fl12fl23‡2Ê2) +‹A (‡1Ê1+fl12‡2Ê2) 1 ‡21Ê 2 1+2fl12‡1‡2Ê1Ê2+fl13‡1‡3Ê1Ê3+‡22Ê22+fl23‡2‡3Ê2Ê3 222 / 1 ‹A 1 ‡21Ê 2 1+2fl12‡1‡2Ê1Ê2+‡22Ê22 22 = (Ê1‡1+Ê2fl12‡2) A 1≠ Ê3‡3 ‡2I (Ê1fl13‡1+Ê2fl23‡2 B + Ê2Ê3‡2‡3 ‹A‡2I [Ê1‡1(fl12fl13≠fl23)≠Ê2‡2(fl12fl23≠fl13)] . (A.21) I can also write xú1 in terms of xúI : xú1 = 1 ‹A‡2IÊ 2 I {‡1 (‡2‡3Ê2Ê3 (fl12fl13‡1Ê1≠fl23‡1Ê1+fl13‡2Ê2≠fl12fl23‡2Ê2) +‹A (‡1Ê1+fl12‡2Ê2)(xúIÊI))} = 1 ‡2IÊI [xúI(‡1Ê1+fl1,2‡2Ê2) + Ê2Ê3 ‹A 3 Ê1 ÊI [cov(R1,R2)cov(R1,R3)≠‡21cov(R2,R3)] ≠Ê2 ÊI [cov(R1,R2)cov(R2,R3)≠‡22cov(R1,R3)] 46 . (A.22) For xú3, I get xú3 = 1 ‡3 1 ‹Afl13‡31Ê 3 1+‡21Ê21 1 ‹A (2fl12fl13+fl23)‡2Ê2+ 1 1+(≠1+‹A)fl213 2 ‡3Ê3 2 +‡22Ê22 1 ‹Afl23‡2Ê2+ 1 1+(≠1+‹A)fl223 2 ‡3Ê3 2 +‡1‡2Ê1Ê2 (2fl12 (‹Afl23‡2Ê2+‡3Ê3)+fl13 (‹A‡2Ê2+2(≠1+‹A)fl23‡3Ê3)))) / 1 ‹A 1 ‡21Ê 2 1+2fl12‡1‡2Ê1Ê2+‡22Ê22 22 = ÊIcov(RI ,R3)+Ê3‡23 5 1+ ‹I ‹A (1≠fl2I,3) 6 , (A.23) where xúI = ‡21Ê 2 1+2fl12‡1‡2Ê1Ê2+‡22Ê22+fl13‡1‡3Ê1Ê3+fl23‡2‡3Ê2Ê3 Ê1+Ê2 = ‡2IÊI +Ê3cov(RI ,R3), (A.24) with ÊI = Ê1+Ê2. Results for x2 are omitted as they are symmetric to x1. 70 A.1.5 Proof of Proposition 2 Following Cuoco and He (1994), I can still use a social planner to derive equilibrium prices, but the weight ⁄t will be stochastic: Ut = Et ⁄ Œ t e≠”(s≠t) (logcA,s+⁄s logcI,s)ds. (A.25) The consumption sharing rule is given by: 1 = c≠1A,t ⁄tc ≠1 I,t . (A.26) I define Agent j’s equilibrium share of world consumption as ‹j,t = cj,tDM,t . In equilibrium the two agents must consume the aggregate dividend: cA,t+ cI,t =DM,t. Thus, ‹A,t = 1 1+⁄t , ‹I,t = ⁄t 1+⁄t . (A.27) As in Basak and Cuoco (1998), the equilibrium state-price density ›t is given by the state-price density of the unconstrained agent A: ›t = ›A,t = ŸAe≠”t(‹A,tDM,t)≠1. (A.28) To solve for equilibrium prices, I need to derive an expression ⁄t and the related process ‹A,t. Sustituting cA and cI from (A.8) in (A.26), I get: ⁄t = ŸA›A,t/›A,0 ŸI›I,t/›I,0 . (A.29) Solving (A.10), agent j’s state-price density under the dividend basis, gives: ›j,t = ›j,0e≠ s t 0 (rs+ 1 2 ◊ 2 j,s)ds≠ s t 0 ◊ Õ j,sdZD,s (A.30) where ◊j,s = ◊Õj,s1 and 1 is a vector of ones. Substitution (A.30) in (A.29) gives: ⁄t = ŸA ŸI e≠ s t 0 1 2 (◊ 2 A,s≠◊2I,s)ds≠ s t 0 (◊A,s≠◊I,s)ÕdZD,s . (A.31) Applying Itô’s Lemma gives: d⁄t ⁄t = µ⁄,tdt+‡Õ⁄,tdZD,t, (A.32) where µ⁄,t = ◊ Õ I,t(◊I,t≠◊A,t), (A.33) ‡⁄,t = (◊I,t≠◊A,t). (A.34) 71 Rewriting as a scalar process, I get: d⁄t ⁄t = µ⁄,tdt+‡⁄,tdZ⁄,t, (A.35) where ‡⁄,t = Ò (◊I,t≠◊A,t)Õ(◊I,t≠◊A,t), (A.36) dZ⁄,t = ‡≠1⁄,t‡Õ⁄,tdZD,t. (A.37) Remember that: ◊I = xI ‡2I ‡Õ SWWU ÊI1 ÊI2 0 TXXV , ◊A = ‡≠1 SWWU x1 x2 x3 TXXV . Therefore, ◊I ≠◊A = xI ‡2I ‡Õ SWWU ÊI1 ÊI2 0 TXXV≠‡≠1 SWWU x1 x2 x3 TXXV , (A.38) ‡(◊I ≠◊A) = xI ‡2I  SWWU ÊI1 ÊI2 0 TXXV≠ SWWU x1 x2 x3 TXXV = SWWWU ÊI2 ‡2I [x2ÊI2(ÊI1‡21+ÊI2fl12‡1‡2)≠x1ÊI2(ÊI2‡22+ÊI1fl12‡1‡2)] ÊI1 ‡2I [x1ÊI1(ÊI2‡22+ÊI1fl12‡1‡2)≠x2ÊI1(ÊI1‡21+ÊI2fl12‡1‡2)] xI—I,3≠x3 TXXXV , (A.39) where —I,3 = flI,3‡3/‡I = (ÊI1fl13‡1‡3+ÊI2fl23‡2‡3)/‡2I . One can easily see that ◊ Õ I◊I = x2I ‡2I and that ◊ÕI◊A = x2I ‡2I . Note that those results are basis invariant. I obtain: µ⁄ = ◊ Õ I(◊I ≠◊A) = 0. (A.40) Similarly, ‡2⁄ = (◊I ≠◊A)Õ(◊I ≠◊A) =≠x 2 I ‡2I +◊ÕA◊A, (A.41) ∆ ‡⁄ = Û [x1 x2 x3]≠1[x1 x2 x3]Õ≠ x 2 I ‡2I . (A.42) 72 Using the definition of ‹A in (A.27) and applying Itô Lemma gives: d‹A = µ‹Adt+‡Õ‹AdZD, (A.43) where µ‹A = ‹A‹2I‡2⁄, (A.44) ‡‹A = ‹A‹I‡⁄. (A.45) In scalar notation this becomes: d‹A = µ‹Adt+‡‹AdZ⁄, (A.46) ‡‹A =≠‹A‹I‡⁄. (A.47) Applying Itô’s Lemma to (A.28), I obtain: d› › =≠ 5 ”+µDM ≠‡2DM + fl‹ADM‡‹A‡DM ‹A 6 dt ≠ C ‡ÕDM + ‡Õ‹A ‹A D dZD. (A.48) Equaling the terms to those in (A.10), I get: rf = ”+µDM ≠‡2DM + fl‹ADM‡‹A‡DM ‹A , (A.49) ◊ = ‡DM + ‡‹A ‹A . (A.50) A.1.6 Proof of Corollary 1 From (A.28) I can assert that ◊ = ◊A. Thus, from (A.34), (A.45) and (A.50), ◊A = ‡DM + ‡‹A ‹A = ‡DM ≠‹I‡⁄ = ‡DM ≠‹I(◊I ≠◊A), (A.51) ∆ ◊ = ‡DM ‹A ≠ ‹I ‹A ◊I , (A.52) ◊ = ‡DM + ‹I ‹A 1 ‡DM ≠◊I 2 . (A.53) Note here that ‡DM is exogenous to the model (when defined relative to the dividend basis), ‹I and ‹A = 1≠‹I are state variables and the other quantities are determined endogenously in 73 equilibrium. Denoting ◊ı = ‡DM the price of risk when there are no indexers (‹A = 1, ‹I = 0), ◊ = ◊ı+ ‹I ‹A 1 ◊ ı≠◊I 2 ∆ 1 ◊ ı≠◊A 2 =≠ ‹I ‹A 1 ◊ ı≠◊I 2 . (A.54) A.1.7 Proof of Proposition 3 In this section I derive the dynamics of each stock’s price process. The price Si,t of stock i at time t is the expected value of future dividends discounted using the stochastic discount factor of the representative agent › defined in (A.48): Si,t = Et 5⁄ Œ t ›· ›t Di,·d· 6 . (A.55) Using the results from equations (A.8) and (A.28), I have Si,t = Et SU⁄ Œ t e≠”(·≠t) A cA,· cA,t B≠1 Di,·d· TV . (A.56) From (A.27), I have: cA,t = DM,t 1+⁄t , (A.57) thus cA,t cA,· = DM,t DM,· 1+⁄· 1+⁄t . (A.58) Substituting this last result in (A.56), I obtain: Si,t =DM,tEt 5⁄ Œ t e≠”(·≠t) 1+⁄· 1+⁄t si,·d· 6 =DM,tfi,t, (A.59) where fi,t = Et 5⁄ Œ t e≠”(·≠t) 1+⁄· 1+⁄t si,·d· 6 = 11+⁄t¸ ˚˙ ˝ ‹A,t Et 5⁄ Œ t e≠”(·≠t)si,·d· 6 ¸ ˚˙ ˝ fAi,t + ⁄t1+⁄t¸ ˚˙ ˝ ‹I,t Et 5⁄ Œ t e≠”(·≠t) ⁄· ⁄t si,·d· 6 ¸ ˚˙ ˝ fIi,t (A.60) = ‹A,tfAi,t+‹I,tfIi,t. (A.61) Note that in a world without constraints, ⁄t is constant and we thus have fi,t = fAi,t. Alterna- tively, I can get this result by setting ‹A,t = 1 and ‹I,t = 0. 74 A.1.8 Solving for fAi,t fAi,t depends on the relative share of the aggregate dividend of each stock, si,t as defined in (A.3). Therefore, si,t = Di,t DM,t . (A.62) To fully characterize the relative weights of each dividend stream two of those si are sucient, so I need two state variables. Using Itô’s Lemma, I obtain: dsMi sMi = Ë ‡ÕDM (‡DM ≠‡Di) È dt +(‡Di≠‡DM )ÕdZD, (A.63) which after simplification yields dsi = µsidt+‡ÕsidZD, (A.64) where µsi = sis≠i Ë ≠si‡2D+s≠i‡2D≠i +(si≠s≠i)flDiD≠i‡D‡D≠i È , (A.65) ‡si = sis≠i(‡Di≠‡D≠i), (A.66) and D≠i represents the dividend stream of the other two stocks combined. Defining xi,t = log si,ts≠i,t , it follows from Itô’s Lemma that: dxi = µxidt+‡ÕxidZD (A.67) where µxi = 5 µDi≠ 1 2‡ 2 Di 6 ≠ 5 µD≠i≠ 1 2‡ 2 D≠i 6 , (A.68) ‡xi = ‡Di≠‡D≠i . (A.69) In scalar form, dxi = µxidt+‡xidZxi , (A.70) where ‡xi = Ò (‡Di≠‡D≠i)Õ‡Di≠‡D≠i = Ò ‡2Di +‡ 2 D≠i≠2flDiD≠i‡Di‡D≠i , (A.71) Zxi = ‡≠1xi ‡ Õ xidZD. (A.72) From Cochrane, Longsta, and Santa-Clara (2008), I know there is a closed-form expression 75 for fAi,t if xi is the only relevant state variable (‹A is irrelevant for fAi,t). In the present case the moments of the dividend process of portfolio ≠i also depend on the relative dividend of the two stocks in that portfolio, i.e. x1 depends on D2/D3. So fAi,t depends on two state variables representing the relative dividend processes. Let’s use x1 and x2 as the state variables. Note also that since q3i=1 fAi,t = 1” , we only need to solve for two i to get the third one. We’ll solve for i= 1,2 so the functions will be symmetric. Here I show the derivation of fA1,t. Note from (A.64) that si = 0 and si = 1 are absorbing states, so we obtain the following boundary conditions: lim x1æ≠Œ fA1,t = 0, (A.73) lim x1æŒf A 1,t = 1 ” , (A.74) lim x2æŒf A 1,t = 0. (A.75) The boundary condition limx2æ≠Œ fA1,t is less obvious because in that case asset 2 becomes irrelevant, so fA1,t converges to the Cochrane, Longsta, and Santa-Clara (2008) case. From the Feynman-Kac theorem, we can transform the problem to a PDE representation: 1 2‡ 2 x1 ˆ2fA1 ˆx21 + 12‡ 2 x2 ˆ2fA1 ˆx22 +‡Õx1‡x2 ˆ2fA1 ˆx1ˆx2 +µx1 ˆfA1 ˆx1 +µx2 ˆfA1 ˆx2 ≠flfA1 + 1 1+e≠x1 = 0, (A.76) where µx1 =≠ C s2≠s1s2≠s22 (1≠s1)2 D (1≠flD)‡2D, µx2 =≠ C s1≠s1s2≠s21 (1≠s2)2 D (1≠flD)‡2D, ‡2x1 = C 2≠ 2(s2≠s1s2≠s 2 2) (1≠s1)2 D (1≠flD)‡2D, ‡2x2 = C 2≠ 2(s1≠s1s2≠s 2 1) (1≠s2)2 D (1≠flD)‡2D, ‡Õx1‡x2 = C! 1+s1 (≠3+2s1)≠3s2+2s1s2+2s22 " (1≠s1)(1≠s2) D (1≠flD)‡2D. Following Bhamra (2007), I use a perturbation expansion of the form: fA1 = fA1,0+ ‘fA1,1+ ‘2fA1,2+ . . . (A.77) 76 Defining flD = 1≠2‘2, I get: fA1,0 = 1 ”+e≠x1” , fA1,1 = 0, fA1,2 = ex1 ! 1≠ex1 (≠1+s1)2+s21+2s1 (≠1+s2)+2(≠1+s2)s2 " ‡2D (1+ex1)3 (≠1+s1)2”2 , fA1,3 = 0. After simplification, I obtain: fA1 = s1 ” ≠ s1 ! 1≠3s1+2s21≠2s2+2s1s2+2s22 " (≠1+flD)‡2D 2”2 +O(‘ 4). (A.78) A.1.9 Solving for fIi,t Remember that fIi,t = Et 5⁄ Œ t e≠”(·≠t) ⁄· ⁄t si,·d· 6 , (A.79) which depends on x1,t, x2,t and ‹A,t= 11+⁄t . Note that ⁄ is a local martingale and that assuming ‡⁄ is bounded, then it is an exponential martingale. I can then define a new measure:44 PÕ(AT ) = Et [1AT ⁄T ] , ’t, T œ [0,Œ) tÆ T. (A.80) With this change of measure, fIi,t = EP Õ t 5⁄ Œ t e≠”(·≠t)si,·d· 6 . (A.81) From (A.81), it follows that fIi,t satisfies a BSDE. The coecients of the BSDE will depend on ‹Ai,t, which satisfies a FSDE. Together they form a FBSDE. The Feynman-Kac theorem still applies thus fIi,t satisfies the following inhomogeneous elliptic PDE: µP Õ x1 ˆfI1 ˆx1 +µPÕx2 ˆfI1 ˆx2 +µPÕ‹A ˆfI1 ˆ‹A + 12‡ 2 x1 ˆ2fI1 ˆx21 + 12‡ 2 x2 ˆ2fI1 ˆx22 + 12‡ 2 ‹A ˆ2fI1 ˆ‹2A +‡Õx1‡x2 ˆ2fI1 ˆx1ˆx2 +‡Õx1‡‹A ˆ2fI1 ˆx1ˆ‹A +‡Õx2‡‹A ˆ2fI1 ˆx2ˆ‹A ≠flfI1 + 1 1+e≠x1 = 0, (A.82) 44See pages 28-29 of Karatzas and Shreve (1998) for details. 77 where µP Õ x1 = µx1 +‡ Õ x1‡⁄, µP Õ x2 = µx2 +‡ Õ x2‡⁄, µP Õ ‹A = µ‹A+‡ Õ ‹A‡⁄ =≠‹2A(1≠‹A)‡2⁄, ‡Õx1‡‹A =≠‹A(1≠‹A)‡Õx1‡⁄, ‡Õx2‡‹A =≠‹A(1≠‹A)‡Õx2‡⁄, ‡Õx1‡⁄ = ‡ Õ D1‡⁄≠ 3 s2 1≠s1 4 ‡ÕD2‡⁄≠ 3 1≠ s21≠s1 4 ‡ÕD3‡⁄, ‡Õx2‡⁄ = ‡ Õ D2‡⁄≠ 3 s1 1≠s2 4 ‡ÕD1‡⁄≠ 3 1≠ s11≠s2 4 ‡ÕD3‡⁄. Note that ‡2‹A also depends on ‡2⁄ and that ‡⁄ (and ‡2⁄) depends on the endogenously determined ‡. Boundary conditions The required boundary conditions are the following: lim x1æ≠Œ fI1,t = 0, (A.83) lim x1æŒf I 1,t = 1 ” , (A.84) lim x2æŒf I 1,t = 0, (A.85) lim ‹Aæ1 ‹IfI1,t = 0, (A.86) ˆfI1,t ˆ‹A ---- ‹A=0 = 0. (A.87) Finally, when x2æ≠Œ, then the second dividend tree becomes irrelevant and fI1,t converges to the case of Bhamra (2007). The other boundary conditions are justified as follows: 1. limx1æ≠Œ fI1,t = 0 and limx2æŒ fI1,t = 0: When x2æŒ, I must be that x1æ≠Œ. When x1æ≠Œ, the first dividend stream becomes irrelevant so investors aren’t willing to pay anything to own the stock. 2. limx1æŒ fI1,t = 1” : In this case there is a single dividend tree and complete markets (the constraint becomes irrelevant), so: S1 = D1 ” =DM (‹A,tfA1,t+‹I,tfI1,t), ∆ 1 ” = ‹A,t 31 ” 4 +(1≠‹A,t)fI1,t = fI1,t. 78 3. lim‹Aæ1 ‹IfI1,t = 0: When ‹A = 1, agent A, which faces no constraint, consumes all div- idends so markets are complete. Therefore f1,t -- ‹A=1 = f A 1,t so this boundary condition must hold. 4. ˆf I 1,t ˆ‹A ---- ‹A=0 = 0: As ‹Aæ 0, indexers consume all dividends. However, they have a worst investment opportunity set than active investors, so this can’t hold for more than an instant. Therefore this boundary condition must be a reflecting boundary condition. A.1.10 Matching moments I now have expressions for both fAi,t and fIi,t. I have a closed form expression for fAi,t that depends on exogenous parameters and state variables, which is easy to evaluate numerically. For fIi,t, I have a PDE that can be approximated. However, the current form of that solution depends on the endogenously determined ‡ because of the dependence on ‡⁄. I have that Si,t =DM,tfi,t, so: dSi =DMdfi+fidDM +dfidDM , dSi Si = dfi fi + dDM DM + dfi fi dDM DM , (A.88) where dDM DM = µDdt+‡ÕDdZ, and ‡D = s1‡D1+s2‡D2+(1≠s1≠s2)‡D3 . I know that fi,t is a function of exogenous parameters and state processes s1, s2 and ‹A, therefore from Itô’s Lemma I get: dfi = C µ‹A ˆfi ˆ‹A +µs1 ˆfi ˆs1 +µs2 ˆfi ˆs2 + 12 A ‡2‹A ˆ2fi ˆ‹2A +‡2s1 ˆ2fi ˆs21 +‡2s2 ˆ2fi ˆs22 +2‡Õ‹A‡s1 ˆ2fi ˆ‹Aˆs1 +2‡Õ‹A‡s2 ˆ2fi ˆ‹Aˆs2 +2‡Õs1‡s2 ˆ2fi ˆs1ˆs2 BD dt + 5 ‡‹A ˆfi ˆ‹A +‡s1 ˆfi ˆs1 +‡s2 ˆfi ˆs2 6Õ dZ. (A.89) From the definition of stock return process, I also have that: dSi Si = 5 µi≠Di Si 6 dt+‡ÕidZ, (A.90) 79 where µi = 1 fi C µ‹A ˆfi ˆ‹A +µs1 ˆfi ˆs1 +µs2 ˆfi ˆs2 + 12 A ‡2‹A ˆ2fi ˆ‹2A +‡2s1 ˆ2fi ˆs21 +‡2s2 ˆ2fi ˆs22 +2‡Õ‹A‡s1 ˆ2fi ˆ‹Aˆs1 +2‡Õ‹A‡s2 ˆ2fi ˆ‹Aˆs2 +2‡Õs1‡s2 ˆ2fi ˆs1ˆs2 B + 3 ‡‹A ˆfi ˆ‹A +‡s1 ˆfi ˆs1 +‡s2 ˆfi ˆs2 4Õ ‡D D +µD, (A.91) ‡i = 1 fi 5 ‡‹A ˆfi ˆ‹A +‡s1 ˆfi ˆs1 +‡s2 ˆfi ˆs2 6 +‡D. (A.92) Note that the expression I have for ‡⁄ from (A.34) is a function of both ‡ and the equilibrium price ratio f1/f2, since ÊI1 = 1+ f1/f2 and ÊI2 = 1+ f2/f1. I first use the definitions of ‡ and ‡⁄ to create perturbation expansions of these moments as a function of f1, f2, f3 and their own expansions. Substituting these expansions in the PDE (A.82), I create a perturbation expansion of the PDE, and then solve by equating terms in the dierent powers of ‘. The result is the closed-form approximation fI1 = fA1 + 1 2(s1+s2)‹A”2 s1 1 2(≠1+s1+s2) 1 ≠s2+2 1 s21+s1 (≠1+s2)+s22 22 +(s1+s2) 1 1+2s21+2(≠1+s2)s2+s1 (≠3+2s2) 2 ‹A 2 (1≠flD)‡2D+O(‘4). (A.93) As in the unconstrained economy, I find fI2 by symmetry and fA3 by fI3 = 1” ≠ fI1 ≠ fI2 . A drawback of the use of a perturbation expansion is that it is impossible to guarantee that the boundary conditions will be satisfied. It is easy to see that in this case (A.87) is not satisfied, which means that the approximation will not be valid in the neighbourhood of ‹A=0. Since this region is not economically important for the current analysis,45 this does not pose a problem as long as the analysis focuses on values of ‹A that are away from that boundary. A.2 Vector notation This section introduces the two dierent vector bases I use in the proofs. While not a necessary read, this section is a useful appendix for understanding the proofs. The reason for using dierent bases is to simplify certain steps of the proof. Steps involving stock returns are easier to solve under the market basis. However, when solving for equilibrium stock return dynamics, the dividend basis is more appropriate. The dividend processes in (2.1) can be represented as a vector: dDt Dt = µD1dt+‡D1ÕdZDt , (A.94) 45‹A = 0 corresponds to the case where the aggregate wealth is fully owned by the indexer, and the remaining active investor still has to hold the share of the non-index stock. The realization of such a scenario seems highly unlikely. 80 where dDtDt is a vector with dDi,t Di,t as the i-th element and dZDt is a vector with dZDi,t as the i-th element. Since the dZDi,t can be correlated, we can represent the correlation matrix of dZDt as CDt = SWWU 1 flD flD flD 1 flD flD flD 1 TXXV . Stock returns in (2.4) can also be represented in vector notation: dRt = µtdt+‡tdZt, where dRt, µt and dZt are vectors with dRi,t, µi,t and dZi,t as the i-th element and ‡t is a diagonal matrix with ‡i,t as the i-th diagonal element. The dZt BM are correlated with correlation matrix: Ct = SWWU 1 flt,12 flt,13 flt,12 1 flt,23 flt,13 flt,23 1 TXXV . A.2.1 Rotation matrix It is often easier to deal with independent Brownian motions (BM) than correlated ones. It is possible to transform a multivariate BM to a vector of independent BM using a rotation matrix. Under that transformation, drifts, variances and covariances of Itô processes are invariant. Consider the three-dimensional multivariate BM Z = [Z1 Z2 Z3]Õ with correlation matrix: C = SWWU 1 fl12 fl13 fl12 1 fl23 fl13 fl23 1 TXXV . Using the Cholesky decomposition, we can construct a rotation matrix K to transform Z into a three-dimensional vector of independent BM. From the Cholesky decomposition, we get the lower triangular matrix L such that LLÕ =C. The matrix L is often used to generate correlated BM from independent ones such that Z = LX. In this case, I am interested in the inverse process: X =KZ where K = L≠1. Applying the Cholesky decomposition to the matrix C, K =SWWWU 1 0 0 ≠ fl12Ô 1≠fl212 1Ô 1≠fl212 0 fl13≠fl12fl23 (1≠fl212)(1≠fl212≠fl213+2fl12fl13fl23≠fl223) ≠fl12fl13+fl23 (1≠fl212)(1≠fl212≠fl213+2fl12fl13fl23≠fl223) 1Ò 1+ fl213≠2fl12fl13fl23+fl 2 23 ≠1+fl212 TXXXV . (A.95) 81 Changing the set of BMs using a rotation matrix is called a change of basis. Drift terms, total variances and covariances between processes are invariant under a change of basis. Note that if the initial BM are uncorrelated (correlation terms in C all equal to 0), then the rotation matrices L and K collapse to the identity matrix. A.2.2 Dividend basis The BM driving the dividend processes described in (A.94) are correlated. Consider LDt , the lower triangular matrix from the Cholesky decomposition of CDt , and it’s inverse KDt . Then I can rewrite (A.94) as: dDt Dt = µDdt+‡DdZDt = µDdt+‡DLDZDt = µDdt+‡DZDt , where ‡D = ‡DLD and ZDt =KDZDt . This transformation yields a new basis that I call the dividend basis. The variance matrix under the dividend basis can be written as: ‡D = Qccca 1 0 0 flD Ò 1≠fl2D 0 flD Ô1≠flDflDÔ1+flD Ò 3≠2flD≠ 21+flD Rdddb . (A.96) A.2.3 Market basis Similarly, the BM driving the market return processes in (2.4) might be correlated as they are determined endogenously. Consider Lt, the lower triangular matrix from the Cholesky decomposition of Ct, and it’s inverse Kt. Then I can write: dRt = µtdt+‡tdZt = µtdt+‡tLtdZt = µtdt+‡tdZt, where ‡t = ‡tLt and Zt =KtZt. This transformation yields a new basis that I call the market basis. Under this basis, ‡ = SWWWWU ‡1 0 0 fl12‡2 Ò 1≠fl212‡2 0 fl13‡3 ≠fl12fl13+fl23Ô 1≠fl212 ‡3 Ú 1+ fl 2 13≠2fl12fl13fl23+fl223 ≠1+fl212 ‡3 TXXXXV . (A.97) 82 Note that the return process can also be written under the dividend basis as: dRt = µtdt+‡tdZDt , where ‡tdZDt = ‡tdZt = ‡tdZt. ‡t has the generic form: ‡t = SWWU ‡11 ‡12 ‡13 ‡21 ‡22 ‡23 ‡31 ‡32 ‡33 TXXV . (A.98) However, this leaves 9 unknowns to solve for in ‡t (it is a 3◊ 3 matrix), whereas the known structure of ‡t leaves only 6 unknowns to solve for, namely ‡1, ‡2, ‡3, fl12, fl13 and fl23. 83 Appendix B Supporting Materials for Chapter 3 B.1 Kalman filter Let there be a linear dynamic system where the true state at time t is evolved from the state at time t≠1 following: xt = Ftxt≠1t+Btut+wt, wt ≥N (0,Qt), (B.1) where Ft is the state transition matrix, Qt is the covariance of the process noise, ut is the control vector and Bt is the control-input model. At time t, a noisy observation zt of the true state is made: zt =Htxt+‹t, ‹t ≥N (0,Rt), (B.2) where Ht is the mapping from the state space to the observed space and Rt is the covariance of the observation noise. The Kalman filter is a recursive estimator that can be represented as a prediction stage followed by an updating stage. First, a priori estimates of the state and of the covariance are generated from the system dynamics: x̂t|t≠1 = Ftx̂t≠1|t≠1+Btut, (B.3) Pt|t≠1 = FtPt≠1|t≠1F Õt +Qt. (B.4) 84 Then, the estimate is updated with the observations ỹt = zt≠Htx̂t|t≠1, (B.5) St =HtPt|t≠1H Õt+Rt, (B.6) Kt = Pt|t≠1H ÕtS≠1t , (B.7) x̂t|t = x̂t|t≠1+Ktỹt, (B.8) Pt|t = (I≠KtHt)Pt|t≠1. (B.9) B.2 The mutual fund managers’ problem This section presents the details about the Kalman filter used to solve the problem presented in Section 3.3.1. The state dynamics are:C vMFt vMFt≠1 D ¸ ˚˙ ˝ xMt = C 1 0 1 0 D ¸ ˚˙ ˝ FM C vMFt≠1 vMFt≠2 D ¸ ˚˙ ˝ xMt≠1 +wMt , wMt ≥N (0,QM ), (B.10) wMt = C 1 0 D ÁMFt , Q M = C ‡2MF 0 0 0 D . (B.11) The observation equations can then be written in terms of xMt :SWWWWWU nMFt rI≠t ≠ bI≠3 nMFt rI+t ≠ bI+3 nMFt vETFt ≠nETFt +“nMFt TXXXXXV ¸ ˚˙ ˝ zMt = SWWWWWU (1≠÷) ÷ bI≠1 b I≠ 2 bI+1 b I+ 2 “ 0 TXXXXXV ¸ ˚˙ ˝ HM xMt +‹Mt , ‹Mt ≥N (0,RM ), (B.12) ‹Mt = SWWWWWU Ánt ÁI≠t ÁI+t ÁETFt TXXXXXV , RM = SWWWWWU ‡2n 0 0 0 0 (1≠µ)‡2I 0 0 0 0 µ‡2I 0 0 0 0 ‡2ETF TXXXXXV , (B.13) where the values for the bI+/≠j depend on —, µ and ÷ following (3.7) and (3.8): 85 µ= ÷ µ > ÷ µ < ÷ bI≠1 0 0 1— ÷≠µ ÷ bI≠2 ≠ 1— ≠ 1— 1≠µ1≠÷ ≠ 1— bI≠3 1 — 1 — 1≠µ 1≠÷ 1 — µ ÷ bI+1 1 — 1 — 1 — µ ÷ bI+2 0 ≠ 1— µ≠÷1≠÷ 0 bI+3 ≠ 1— ≠ 1— 1≠µ1≠÷ ≠ 1— µ÷ B.2.1 Specification 2 In the suboptimal solution, it is assumed that fund managers do not look at ETF premiums. The solution diers from the optimal solution in the following way: zM2t = SWWU nMFt rI≠t ≠ bI≠3 nMFt rI+t ≠ bI+3 nMFt TXXV , HM2 = SWWU (1≠÷) ÷ bI≠1 b I≠ 2 bI+1 b I+ 2 TXXV , (B.14) ‹M2t = SWWU Ánt ÁI≠t ÁI+t TXXV , RM2 = SWWU ‡2n 0 0 0 (1≠µ)‡2I 0 0 0 µ‡2I TXXV . (B.15) B.3 Arbitragers’ problem This section presents the details about the Kalman filter used to solve the problems presented in Section 3.3.2. All following specifications are estimated by maximum likelihood optimization. The parameters to estimate for each specification are “, µ, ÷, —, ‡ETF , ‡n, ‡I and ‡MF . B.3.1 Specification 1 To the state variable vMFt , I add one lag vMFt≠1 that appears in the observation equations, as well as nMFt , the true NAV at time (which is observed with a lag). Note that nMFt = (1≠ ÷)vMFt + ÷vMFt≠1 + Án,MFt , which can be rewritten as nMFt = vMFt≠1 +(1≠ ÷)ÁMFt + Ánt . The state dynamics become:SWWU vMFt vMFt≠1 nMFt TXXV ¸ ˚˙ ˝ xA1t = SWWU 1 0 0 1 0 0 1 0 0 TXXV ¸ ˚˙ ˝ FA1 SWWU vMFt≠1 vMFt≠2 nMFt≠1 TXXV ¸ ˚˙ ˝ xA1t≠1 +wA1t , wA1t ≥N (0,QA1), (B.16) 86 wA1t = SWWU 1 0 0 0 (1≠÷) 1 TXXV C ÁMFt Ánt D , QA1 = SWWU ‡2MF 0 (1≠÷)‡2MF 0 0 0 (1≠÷)‡2MF 0 ‡2n+(1≠÷)2‡2MF TXXV , (B.17) assuming that ÁMFt and Ánt are independant. The observation equations can then be written in terms of xA1t :SWWWWWU vETFt ≠nETFt nMFt≠1 ≠÷vMFt≠2 rI≠t rI+t TXXXXXV ¸ ˚˙ ˝ zA1t = SWWWWWU “ 0 ≠“ 0 (1≠÷) 0 bI≠1 b I≠ 2 b I≠ 3 bI+1 b I+ 2 b I+ 3 TXXXXXV ¸ ˚˙ ˝ HA1 xA1t +‹A1t , ‹A1t ≥N (0,RA1), (B.18) ‹A1t = SWWWWWU ÁETFt Ánt≠1 ÁI≠t ÁI+t TXXXXXV , RA1 = SWWWWWU ‡2ETF 0 0 0 0 ‡2n 0 0 0 0 (1≠µ)‡2I 0 0 0 0 µ‡2I TXXXXXV . (B.19) Note that vMFt≠2 is not part of the state space, so it is subtracted in the second element of the observation. The value used for this subtraction comes from the previous period estimate xA1t≠1. B.3.2 Arbitragers’ problem: embedded filter This section presents the details about the Kalman filter used to solve the problem presented in Section 3.3.2 for which the manager is assumed to solve his problem using a Kalman filter. Let KM , HM and FM denote matrices of the mutual fund manager’s problem defined in Section B.2. Since I assume that the observed nMFút≠1 for specifications 2 is the result of the mutual fund manager’s problem solved with a Kalman filter, we have vúMFt≠1 = nMFút≠1 where vúMFt is the arbitrager’s belief about the manager’s belief of the true value. 87 B.3.3 Specification 2 The state dynamics are: SWWWWWWWU vMFt vMFt≠1 nMFt nMFút nMFút≠1 TXXXXXXXV ¸ ˚˙ ˝ xA2t = SWWWWWWWU 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 bM2t 0 0 cM2t 0 0 0 bM2t≠1 0 cM2t≠1 TXXXXXXXV ¸ ˚˙ ˝ FA2t SWWWWWWWU vMFt≠1 vMFt≠2 nMFt≠1 nMFút≠1 nMFút≠2 TXXXXXXXV ¸ ˚˙ ˝ xA2t≠1 + SWWWWWWWWU 0 0 0 0 0 0 0 0 0 0 0 0 KM2t,(1,2) K M2 t,(1,3) 0 0 0 0 KM2t≠1,(1,2) KM2t≠1,(1,3) TXXXXXXXXV ¸ ˚˙ ˝ BA2t SWWWWWU rI≠t rI+t rI≠t≠1 rI+t≠1 TXXXXXV ¸ ˚˙ ˝ uA2t +wA2t , wA2t ≥N (0,QA2), (B.20) where cM2t = [1≠KM2t,(1,1)≠KM2t,(1,2)(bI≠1 + bI≠2 )≠KM2t,(1,3)(bI+1 + bI+2 )], bM2t =KM2t,(1,1)≠ bI≠3 KM2t,(1,2)≠ bI+3 KM2t,(1,3), and wA2t = SWWWWWWWU 1 0 0 0 (1≠÷) 1 bM2t (1≠÷) bM2t 0 0 TXXXXXXXV C ÁMFt Ánt D , QA2 = SWWWWWWWU ‡2MF 0 (1≠÷)‡2MF bM2t (1≠÷)‡2MF 0 0 0 0 0 0 (1≠÷)‡2MF 0 ‡2n+(1≠÷)2‡2MF bM2t (‡2n+(1≠÷)2‡2MF ) 0 bM2t (1≠÷)‡2MF 0 bM2t (‡2n+(1≠÷)2‡2MF ) (bM2t )2(‡2n+(1≠÷)2‡2MF ) 0 0 0 0 0 0 TXXXXXXXV . (B.21) 88 The observation equations are:SWWWWWU vETFt ≠nETFt nMFút≠1 rI≠t rI+t TXXXXXV ¸ ˚˙ ˝ zA2t = SWWWWWU “ 0 ≠“ 0 0 0 0 0 0 1 bI≠1 b I≠ 2 b I≠ 3 0 0 bI+1 b I+ 2 b I+ 3 0 0 TXXXXXV ¸ ˚˙ ˝ HA2 xA2t +‹A2t , ‹A2t ≥N (0,RA2), (B.22) ‹A2t = SWWWWWU ÁETFt Áúnt≠1 ÁI≠t ÁI+t TXXXXXV , RA2 = SWWWWWU ‡2ETF 0 0 0 0 ‡2ún 0 0 0 0 (1≠µ)‡2I 0 0 0 0 µ‡2I TXXXXXV . (B.23) B.3.4 Information set without ETF premiums Solving the problem with specification 1 while assuming that ETF premiums are not in the arbitrager’s information set is the same as in B.3 with modified observation equations where the ETF premiums are removed:SWWU nMFt≠1 ≠÷vMFt≠2 rI≠t rI+t TXXV ¸ ˚˙ ˝ zA1 Õ t = SWWU 0 (1≠÷) 0 bI≠1 b I≠ 2 b I≠ 3 bI+1 b I+ 2 b I+ 3 TXXV ¸ ˚˙ ˝ HA1Õ xA1t +‹A1 Õ t , ‹ A1Õ t ≥N (0,RA1 Õ), (B.24) ‹A1 Õ t = SWWU Ánt≠1 ÁI≠t ÁI+t TXXV , RA1Õ = SWWU ‡2n 0 0 0 (1≠µ)‡2I 0 0 0 µ‡2I TXXV . (B.25) 89

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