An enquiry into the pricing efficiency of exchange-traded funds : are tracking errors priced?

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An Enquiry into the Pricing Efficiency of Exchange-Traded Funds: Are Tracking Errors Priced?

Master Thesis By: Emmet King Supervisor: Dr. Philippe Versijp University of Amsterdam

MSc. Finance: Asset Management Track Date: July 1st_{, 2017}

Abstract: This paper investigates whether the performance (tracking error) of an Exchange-Traded Fund (ETF) affects its pricing efficiency (percentage discount/premium). Several factors were previously reported to impact ETF performance. By quantifying the tracking errors of 131 passive ETFs, we first confirm the effect of these factors and subsequently assess the impact of various measures of tracking error on pricing efficiency.

Our results demonstrate that dividend distribution frequency and dividend yield have a significant effect on tracking error, thereby corroborating previous studies. Additionally, we find significant results for the average number of constituents held scaled by fund size. Interestingly, and contrary to existing literature, this study demonstrates a positive effect of liquidity on tracking error.

Regarding the relationship between pricing efficiency and tracking error, our non-parametric results identify a significant relationship, which indicates a pricing efficiency in ETF relating to tracking error. Our parametric coefficient of interest fails to achieve statistical significance.

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2 Statement of originality:

This document is written by student Emmet King who declares to take full responsibility for the contents of this document.

I declare that the text and the work presented in this document is original and that no sources other than those mentioned in the text and its references have been used in creating it.

The Faculty of Economics and Business is responsible solely for the supervision of completion of the work, not for the contents

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1. Introduction

Exchange Traded Funds(ETFs) are marketable securities that track an index, commodities, bonds, or a basket of assets such as an index fund. ETFs were first introduced in 1993 and have grown substantially since in terms of volume traded and dollar volume. According to a recent article by the Financial Times, ETFs accounted for roughly 30% of all equity trading by volume, and 23% by share value in the US in 2016 (Wigglesworth, 2017).

ETFs and mutual funds are passive investment vehicles that offer investors diversification and exposure to particular asset classes and markets. Instead of diversifying a portfolio with a broad range of individual securities, investors can use investment vehicles such as ETFs or mutual funds to achieve these goals more effectively. According to various authors, ETFs are more suitable as passive investment vehicles than mutual funds. Harper, Madura, and Schnusenberg find that ETFs exhibit higher mean returns and higher Sharpe ratios than closed-ended mutual funds (2006). Agapova argues the superiority of ETFs over mutual funds for tax-sensitive investors (2011). De Freitas and Baker investigate whether ETFs are suitable for use in an optimal portfolio construction as a proxy for the market and find a positive result for this, stating that this is due to their low costs, improved tracking, and higher efficiency (2005).

Passive ETFs are designed to replicate the returns of their benchmark, the index they track. The term “passive” is derived from the fact that there is no active management involved, meaning that the composition of the portfolio is determined by the objective. With the objective being the replication of the returns of the underlying index, the portfolio is composed of securities that, collectively, deliver said index’s returns.

Previous research has confirmed the existence of tracking errors - differences in the returns of the fund’s Net Asset Values (NAV) and the underlying securities’ returns– in ETFs (Rompotis, 2012, Kuak-Kun Chu ,2011, Defuso, Ivanov, and Karels 2009), etc. Several factors have been identified that, in part, explain this tracking error. These include differences in the dividend yield, differences in dividend payment frequencies (Charaput and Miu, 2013), differences in the expense ratios charged by a fund (Kuak-Kun Chu, 2011), and differences in fund sizes (Grinblatt and Titman, 1989).

Roll states that tracking errors are important when examining the performance of an index mutual fund (1992). Due to their similarities, the same is true for ETFs. While previous literature has confirmed the existence of tracking errors, as well as factors that have an influence on its magnitude, this study aims to contribute to the current literature by investigating the combined influence of the various factors, including those mentioned above. We test whether the pricing efficiency of an ETF is influenced by its performance by use of a regression of the percentage discount/premium of an ETF on its tracking error. Furthermore, we use a variation of Pop and Yadav’s third measure for the calculation of tracking errors to construct factor-adjusted tracking errors (1994). This, in order to

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5 investigate whether tracking errors absent the influencing factors have a relationship with an ETF’s pricing efficiency.

With the replication of benchmark returns being the objective of passive ETFs, tracking error can be seen as a measure of their performance. The rational for this paper is to investigate whether performance is priced, as measured by the pricing efficiency. Furthermore, there are instances whereby multiple passive ETFs are replicating the returns of the same underlying benchmark. By the law of one price, if one ETF was to achieve this objective relatively worse than the other, it would logically have to command a discount in order to compensate.

This paper will proceed as follows. In section 2, we will present a theoretical background by discussing the current relevant literature. In section 3 we describe our data sources. Our experimental design will be presented in section 4, and the results thereof, together with its interpretation, will be presented in section 5. In section 6 we perform robustness checks. Finally, section 7 concludes.

2. Literature review

In this section, we will explore the available literature on ETFs and their close substitutes, mutual (index) funds. First, we describe what ETFs are, compare them to mutual funds, and briefly discuss their history. Next, we explain the creation and redemption process of ETF shares, the pricing efficiency of ETFs, and how they relate to one another. After that, we will discuss tracking errors: their existence in previous literature and how they are measured. Finally, we will consider the various factors that have been shown to influence the magnitude of tracking errors.

2.1.1. Exchange Traded Funds

As implied by its name, an ETF is a fund that is traded on an exchange in a similar fashion to a stock (Gallagher & Segara, 2006). ETFs are investment unit trusts that are generally designed to replicate the returns of an index. ETFs have characteristics in common with both open-ended and closed-ended mutual funds. Mutual funds derive their value from a pre-specified bundle of securities held in their portfolio. In open-ended mutual funds, the funds stands ready to buy back shares after market close at their Net Asset value (NAV). Hence, the price that open-ended mutual funds shares commands is always equal to its NAV. In a closed-ended fund, an investor wanting to divest their investment in the fund would need to find another investor to buy his/her shares. Due to the fund not standing ready to buy back shares at NAV, this allows the price of the share to depart from this value.

Similarly to open-ended funds, there is a mechanism whereby ETF shares can be redeemed. However, this is generally done in-kind, and can only be done by so called Authorized Participants (APs). We will further elaborate on this in the next section.

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6 The first modern ETF was launched in the US in 1993. The SPDR (Standard & Poor’s Depository Receipts) was designed the track the S&P 500. Since then, many ETFs have been introduced, tracking various different indices, which can be whole markets, sectors, or other subsets of the market. For this reason ETFs are appealing to investors seeking a certain kind of exposure in their portfolio. Instead of investors having to manage a large and diversified portfolio of securities, investors can instead opt to invest in a single ETF which will provide them with the same exposure.

Prior to the introduction of ETFs, mutual index funds were used to achieve this purpose. However, although similar, ETFs command certain advantages over mutual funds that make them the preferable investment vehicle. Firstly, mutual funds only allow trading after trading closes and the NAV is established, whereas ETFs can be traded continuously (Poterba & Shoven, 2002). Secondly, in relative terms, ETFs are more tax-efficient than mutual funds (Agapova, 2011). While mutual funds must pass through realized gains to their shareholders, ETFs are able to avoid this through in-kind redemption, thereby avoiding tax burden initiated by capital gains (Poterba & Shoven, 2002). This redemption process will be elaborated on in the next section. Finally, in the context of country-specific funds, ETFs provide diversification at lower costs, and with lower tracking errors than mutual funds (Miffre, 2007).

2.1.2. Creation and Redemption process

The creation of an ETF goes as follows: It starts with an ETF company/sponsor that would like to create a new ETF that mimics the returns of an index. After designing a portfolio composition of securities that would deliver a return that is similar to the benchmark, an Authorized Participant (AP) is approached. This AP then purchases and delivers the portfolio of securities in-kind to the ETF sponsor in exchange for creation-units, which most commonly amount to multiples of 50,000 shares of the ETF. This interaction between the ETF sponsor and the AP is referred to as the primary market. The AP will then, together with liquidity providers, act as market markers on the exchange on which the ETF will be traded, the so called the secondary market. Here, investors can trade the ETF shares on exchanges through brokers. The value of an ETF is derived from the values of the underlying securities in the ETF. If the prices of ETF deviate largely from their NAV, the AP can also redeem ETFs in the primary market. This is done by delivering and amount of ETF shares equal to the amount in a creation-unit to the ETF sponsor, and receiving the underlying shares in exchange. In some cases, ETF sponsors allow for redemption to be done for cash rather than in-kind. The creation and/or redemption of creation-units at the ETF sponsor by the AP does however, involve transaction costs. These vary between ETF sponsors, but typically involve a fixed component, ranging from $250 to $3000,- per creation-unit, and a variable component depending on which type of redemption is pursued.

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7 This arbitrage mechanism should create bounds within which the ETF should be traded on in the market relative to its NAV. For example: an ETF for which the redemption transaction costs only include a fixed component of $3,000 per creation unit is currently trading at a price of $10.08. The per-share transaction costs involved in the redemption process are (3,000/50,000) $0.06. Hence, in this example, the bounds established outside which an arbitrage opportunity arises is when the market prices of an ETF share deviates from its NAV by more than 6 cents, or 0.56%.

2.2. Pricing Efficiency

An implication of being traded on two separate markets, primary and secondary, is that an ETF derives its value from two sources. Frist, The value of the portfolio underlying an ETF share, the NAV. Secondly, the quoted price at which an ETF is traded on exchanges, the secondary market. In the latter market, the forces of supply and demand can be of influence on the price at which an ETF is traded.

Put simply, pricing efficiency refers to the price of an ETF share relative to the proportionate value of its holdings, the net asset value. The conventional method of expressing this is the percentage discount/premium a fund is trading at:

%𝐷𝑖𝑠𝑐𝑜𝑢𝑛𝑡/𝑃𝑟𝑒𝑚𝑖𝑢𝑚 = (𝑁𝐴𝑉𝑖,𝑡− 𝐸𝑇𝐹𝑖,𝑡 𝑁𝐴𝑉𝑡

) ∗ 100%

Despite the bounds created by the creation and redemption mechanism outlined in the above section, evidence exists that the prices of ETF can still fluctuate within a band of up to 200 basis points (Petajisto, 2017). However, according to Petajisto, on average, premiums across funds are only 6 basis points, but have a volatility of 49 basis points. Furthermore, this pricing inefficiency is more pronounced for ETFs that contain international holdings, while funds holding liquid domestic securities are priced relatively efficient. A similar result is found by Decloure and Zhong in the context of Exchange-Traded world equity funds, which trade at significant premium up to 50% of the time (2007). One of the reasons put forth by the authors is the issue of stale NAVs. Funds calculate and publish the NAV of their holdings after the exchange on which the fund is traded closes at days-end. However, if a fund holds international securities, it is possible that the exchange on which said security is traded is still open. Hence, instead of using closing prices, and alternative valuation must be determined. However, Petajisto still finds pricing deviations that are economically significant after correcting for the problem of stale NAVs by using an alternative method of measurement (2017).

Finally, Ackert and Tian find that funds that track specific indices, index mutual funds, experience more pricing efficiency, i.e. lower discount/premium percentages (2000). Their findings demonstrate that index mutual funds with the S&P 500 as benchmark do not trade at a statistically significant premium or discount on average. However, Ackert and Tian do rapport significant pricing

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8 deviations for index mutual funds with the Midcap S&P 400 as their benchmark, and attribute this to the higher costs of arbitrage.

2.3.1. Tracking errors: Measurement

In the context of index mutual funds, Chiang (1998) states that a fund will never be able to deliver results identical to its benchmark. He argues that this is due to a fund’s portfolio being subject to market frictions that differ from those faced by an index. Roll (1992) states that tracking errors may be important when evaluating the performance of a fund. Due to the similarities between mutual funds and ETFs, one can argue that tracking error is of equal importance when evaluating the performance of ETFs.

In the article by Pope and Yadav, three different methods for measuring tracking errors of mutual index funds are outlined (1994). This methodology was subsequently applied to measuring ETF’s tracking errors by various authors including, Frino and Gallagher (2002), Gallagher and Segara (2004), and Rompotis (2006, 2011, 2012). These measures of tracking error are, first, the average of the absolute difference between the ETF’s NAV returns and the index’s return. Secondly, the standard deviation of the difference between the ETF’s NAV returns and the index’s return. Thirdly, the Standard Error of Regressions (SER) in an Ordinary Least Squares (OLS) regression of ETF’s NAV on index’s returns. With SER being the estimate of the standard deviation of the error term, it can be seen as the spread around the predicted regression line. A higher SER would indicate more spread in the distribution of the ETF’s NAV returns around those of the index’s returns, meaning a higher tracking error.

If the resulting α from this regression is positive (negative), it would indicate the fund outperforms (underperforms) its benchmark. However, due to the factors listed below, this is generally not the case, and the resulting α is expected to be negative. Whether these measures of tracking error are equal depends on the resulting index’s return coefficient in the regression. If equal to unity, measures two and three will be equal to each other. If this is not the case, measures one and three will be equal. Furthermore, according to Kuak-Ku Chan, a coefficient on index’s returns that is unequal to unity would lead to an overstatement of the magnitude of the tracking error if the OLS regression is not linear (2011). Additionally, Kuak-Ku Chan argues that if an ETF was to consistently outperform or underperform its benchmark with the same magnitude, this would result in the tracking error as measured by the standard deviation being equal to zero. Finally, he argues that this measure is not suitable for calculating daily tracking errors, due to the almost certain presence of serial correlation in daily returns (2011).

Alternatively, instead of using the SER from the OLS regression, one could also use the resulting R2_{of this regression (Cresson, Curdd, & Lipscomb, 2002). They find that when using this,}

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self-9 admittedly, naïve measure of tracking error on daily returns the performance of funds is lower compared to monthly returns.

By applying the above methods, various authors have shown the existence of tracking errors in ETFs. Elton et al were among the first, studying the performance of the SPDR ETFs that track the S&P 500, and comparing its performance to other methods used for indexing (2002). Defuso, Ivanov and Karels show that the pricing deviations of three of the most liquid ETFs from their underlying index are predictable and nonzero (2009). They state that even though consistent premiums or discounts in ETFs trigger the redemption or creation of shares, a pricing deviation persists. Other authors that have found tracking errors include Kuak-Kun Chu (2011) for ETFs traded on the Hong Kong stock exchange, Purohit and Malhotra (2015) in Indian ETFs, and Rompotis (2012) in the German ETF market.

2.3.2. Tracking errors: Sources

Tracking errors can be influenced by a variety of factors. What follows next is a brief exploration of factors identified by previous authors.

Elton et al. (2002) was among the first to identify the relationship between tracking errors and the expense ratios charged by funds. Indeed, due to the reduction of fees in the form of the expenses charged decreasing the return delivered by the ETF relative to its benchmark, this result is intuitive. This relationship has been corroborated by various other authors such as Kuak-Kun Chu (2011), Blitz et al (2012), Rompotis (2006, 2011), Charaput and Miu (2013). In a study on German ETFs, however, no statistically significant results were found for this relationship (Rompotis 2012).

An ETF’s cash holdings relative to total assets is another factor of influence. The reason being that due to cash holdings not making (similar) returns (to the benchmark), they have a negative relationship with tracking error (Charupat & Miu, 2013) (Kostovetsky, 2003) (Elton et. al, 2002). This would imply that, all else equal, in a declining market, the ETF with a relatively larger cash holding would perform better relative to its benchmark than an ETF with a relatively smaller proportion of cash holdings. This, due to the negative returns of the benchmark not being achieved by the cash holdings in the ETF’s portfolio. Hence, if a larger proportion of an ETF’s assets is in cash holdings, a larger proportion of the portfolio would be achieving above-benchmark returns. Conversely, in a rising market, cash holdings would yield returns that are lower than the returns of the benchmark. Therefore, ETFs with a relatively higher cash holding would achieve lower returns than their benchmark, which would result in a higher tracking error.

Due to ETFs being organized as unit investment trusts, dividends earned from constituents are accrued into cash holdings prior to being distributed to ETF shareholders. As outlined above, higher cash holdings relative to total assets have a negative effect on tracking error. Elton et Al (2002) show that the higher dividend yields and the longer the delay before dividend distribution, the higher the

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10 tracking errors are for the SPDR ETF tracking the S&P 500. The same result is verified by Frino et al (2004) for index funds tracking the S&P 500.

Another factor to consider are the transaction costs. Frino and Gallagher argue that the transaction costs incurred in rebalancing portfolios prevent funds from perfectly replicating the returns of their benchmark (2005). Over time, in order to ensure that a fund’s portfolio delivers returns that are similar to those of the benchmark, the portfolio will have to be rebalanced. The transactions costs the fund must incur while buying and selling securities are expenses incurred that are not included in the expense ratio. These additional costs reduce the returns delivered and therefore increase the tracking error.

Related to transaction costs is the replication strategy an ETF uses to replicate the returns of its benchmark. Charaput and Miu outline two different methods (2013). The choice of using a statistical replication strategy rather than a full replication one would result in the underlying assets in an ETF being different from what the index tracks. A statistical replication strategy involves the fund holding only a representative subset of the index’s securities. While this statistical strategy will generally result in lower levels of transactions costs relative to a full replication strategy, Charaput and Miu argue that it will, however, be subject to a relatively larger tracking error (2013).

Due to ETFs being traded on stock exchanges, they are subject to the same forces of supply and demand as any other exchange traded security. As with any other security, prices are more efficient when the security is liquid than when it is illiquid. This pricing efficiency, the price at which an ETF is traded relative to the value of its underlying holdings, then has an implication for the tracking error. Hence, there is a negative relationship between tracking error and liquidity. Or, stated differently, a positive relationship between illiquidity and tracking error. Kuak-Kun Chu, for example, using daily trading volume as a proxy for liquidity identifies a positive relationship in his sample of Hong-Kong traded ETFs (2013). In addition, Buetow and Henderson regress tracking error on two different measures of liquidity, log dollar volume and turnover faction, and find significant coefficients for both measures (2012).

Finally, fund size is reported to have a negative relationship with tracking error. If examined from the perspective of economies of scale, it makes sense that a larger fund would incur relatively lower transaction costs (Grinblatt & Titman, 1989). With fund size measured by the natural logarithm of number of shares outstanding multiplied by the ETFs price, this is has been confirmed for ETFs (Charupat & Miu, 2013) (Buetow & Henderson, 2012).

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3. Data

The following section discusses our sources of data and the operations we performed on them. We aim to relate pricing efficiency with ETF performance, and test whether the hypothesized factors are of influence on the performance. As a measure of performance, we calculate ETF’s tracking errors for which we needed returns on NAV and the ETF’s benchmarks. To establish a measure of pricing efficiency, we needed ETF prices and NAVs. Furthermore, we needed to retrieve data on the various factors that potentially influence the magnitude of the tracking error.

Using the ETF screener tool on etfdb.com, we created a list of US passive equity ETFs that are without leverage and with no inverse holdings in order to avoid complications. We limited our sample to US ETFs in order to avoid the complications caused by stale NAVs (Petajisto, 2017). Additionally, we retrieved the expense ratio charged, and the respective benchmark index of each ETF.

From Datastream, we retrieved the following for each ETF: NAV, shares outstanding, price, trading volume, dividend payment dates, and dividend yields, which is the ratio of dividends received each year to the average price. Additionally, for those ETFs for which it was available, we retrieved the quoted price of the respective benchmark. Some ETFs track a custom index created by the sponsor which is not traded publicly. These ETFs were subsequently excluded from our sample.

Using these data, we calculated the daily return of ETF’s price as well as the daily return of the ETF’s NAV, and of its respective benchmark index. We use these returns in order to calculate the tracking errors. Next, using the NAV and price of each ETF, we calculated its daily percentage discount/premium and its monthly average. Furthermore, we used the dividend payment dates to determine the yearly frequency with which a fund issues dividends to its shareholders. Next, using trading volume and shares outstanding, we calculated an ETF’s turnover, which is the fraction of total shares outstanding that were traded. Finally, in order to calculate the ETF’s size, we multiplied the traded price by the number of shares outstanding, and took its natural logarithm in order to scale it.

Next, in order to determine the amount of holdings an ETF has, we used data retrieved from the ETF Global constituent database. Using this, we calculated the daily amount of constituents, and the monthly average per fund. Originally, we intended to have cash holdings, and changes therein, to be the main variable of investigation in this paper in order to investigate the effect of discretionary management choices on tracking errors. However, due to limitations in the ETF Global database, this was not possible. In the amount of constituents, we corrected for discrepancies such as a fund going from 15 to 2 to 15 constituents in the span of three days, which we assumed to be wrong, and corrected to 15 for all three days.

We observed that some ETFs had daily percentages discount/premiums that exceeded 100%. From this we inferred that, despite screening for firms without leverage on etfdb.com, some ETFs with this characteristic were included. These ETFs were excluded from our sample.

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12 Taken together, our efforts resulted in a dataset of 131 passive US ETFs. Due to Datastream not providing net asset values of ETFs prior to 2014, the period under consideration starts from the first trading day in 2014, namely 02/01/2014. Furthermore, ETF Global does not provide complete data later September 2016. Therefore, the period under consideration will span from 01/01/2014 to 09/09/2016, a total of 33 months.

4. Methodology

Although previous literature has researched tracking error, the contribution of this study is to link tracking error with relative pricing efficiency. We aim to do this in order to investigate whether the performance of ETFs, as measured by tracking errors, is priced. Furthermore, this study will aim to statistically test the contribution of previously identified factors that influence tracking error. Additionally, we attempt to calculate factor-adjusted tracking errors, which are tracking errors in which we control for the effect of the factors of influence, and investigate whether these measures have a relationship with pricing efficiency.

This section will proceed as follows: First, the method for calculating the percentage discount/premium will be outlined, in conjunction with its implications. The percentage discount/premium will be used as our measure of pricing efficiency. Secondly, the various methods for calculating tracking error are discussed. Tracking error will be our measure of ETF performance, thereby enabling us to investigate the possible relationship between pricing efficiency and performance. Third, we will discuss the method by which we will evaluate the contribution of the various factors that influence tracking error, allowing us to identify factors that are of influence to the magnitude of the tracking error. Fourth, we will relate the pricing efficiency to ETF performance. Finally, we will calculate our factor-adjusted tracking errors and relate these to the pricing efficiency of ETFs, while comparing them to the third measure of tracking errors. These factor-adjusted tracking errors are absent the factors that influence the magnitude of tracking error, and therefore allow us the relate ETF performance with pricing efficiency absent their influence.

4.1. Pricing efficiency

As stated in section 2.2, the value of an ETF share is derived from two different sources. Due to an ETF being a portfolio of underlying securities, one source of value is that of said underlying securities divided by the total amount of shares the fund has outstanding. In other words, the Net Asset Value (NAV). Secondly, as ETFs are traded on exchanges, the second source of value is the traded price. In order to evaluate the pricing efficiency, we look at the ratio between an ETF’s traded price and the value of its underlying, the NAV. We use the conventional method to express the pricing efficiency of an ETF as a ratio of the traded ETF price to its NAV:

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13 %𝐷𝑖𝑠𝑐𝑜𝑢𝑛𝑡/𝑃𝑟𝑒𝑚𝑖𝑢𝑚 = (𝑁𝐴𝑉𝑖,𝑡− 𝐸𝑇𝐹𝑖,𝑡

𝑁𝐴𝑉𝑡

) ∗ 100%

After calculating the daily percentage discount/premium, we also calculate monthly averages in order to be able to relate them to the tracking errors. If the NAV is lower than the price at which the ETF is trading, it is said to be trading at a discount. Because the share is worth more than the securities that are underlying the ETF, this would mean that the ETF is underpriced. Conversely, if the NAV is higher than the price at which the ETF is trading, it is said to be trading a premium. In this case, because the value of the underlying securities of the share are higher than the price the ETF share commands, the ETF is overpriced.

As mentioned in section 2.1.2, the creation and redemption process should create bound for the percentage discount/premium. If the percentage discount/premium were to move out of bounds, there would be an arbitrage opportunity that an AP could take advantage of. However, as stated in section 2.2, previous authors have found that economically significant movements out of these bounds do occur, despite these arbitrage opportunities.

4.2. Tracking error

As a measure of the performance of the ETFs, we calculate their tracking error. This is done by applying the methodology first outlined by Pope and Yadav (1994), resulting in three different measures. Firstly, the average of the absolute difference between the ETF’s NAV returns and the index’s return:

𝑇𝐸1,𝑖,𝑡=

∑_𝑡=1𝑛 _{| (𝑅}

𝑁𝐴𝑉,𝑖,𝑡− 𝑅𝑖𝑛𝑑𝑒𝑥,𝑖,𝑡) |

𝑛

Where TE stands for tracking error of ETF i in month t, RNAV is the return on the net asset value of ETF i on day t, Rindex is the return on the ETFs respective index i that it tracks on day t, and n is the number of trading days per month. Secondly, the standard deviation of the difference between the ETF’s returns on its NAV and the return on the index:

𝑇𝐸2,𝑖,𝑡 = √

1

(𝑛 − 1)(𝑒𝑖,𝑡− 𝑒̅) 𝑖 2

Where 𝑒𝑖,𝑡 is the difference in returns of ETF i’s NAV with Index i on day t, and 𝑒̅̅̅̅ is the average 𝑖,𝑡

difference. Finally, the third measure of tracking error will be the Standard Error of Regression (SER) in and OLS regression of NAV’s returns on index’s returns:

𝑇𝐸3,𝑖,𝑡 = SER = √

𝑆𝑆𝑅

𝑁 − 2 𝑖𝑛 𝑡ℎ𝑒 𝑂𝐿𝑆 𝑟𝑒𝑔𝑟𝑒𝑠𝑠𝑖𝑜𝑛 𝑜𝑓: 𝑅𝑁𝐴𝑉,𝑖,𝑡= 𝛼 + 𝛽 ∗ 𝑅𝑖𝑛𝑑𝑒𝑥,𝑖,𝑡+ 𝜖𝑡

Where the independent variable is the return on ETF i’s NAV on trading day t, and the dependent variable is the index i’s return which the ETF tracks on day t. This regression is performed for each ETF and each month in which they are traded.

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14 When comparing the resulting tracking errors, if the coefficient on index returns in the above regression is equal to unity, measures two and three should be similar. As mention in section 2.3.1, the third measure can overstating the tracking error in the case of non-unity. Therefore, we us the average of the three measures of tracking error in calculating the influence of the contributing factors and the relationship with pricing efficiency. Additionally, in order to avoid the complications arising in measure two due to serial correlations in daily returns, we use monthly tracking errors rather than daily ones.

4.3. Factor contributions to tracking error

As mentioned in section 2.3.2, previous authors have identified several factors which influence the tracking error. In order to dissect their contribution to the magnitude of the tracking error, we run a cross-sectional regression of the tracking errors on the various factors. However, due to limited access to relevant databases, and the inaccurate data contained in the ETF Global database, we are unable to include the cash holdings, replication strategy, and the transaction costs incurred of funds. As a proxy for transaction costs and replication strategy, we include a scaled measure of the average number of constituents held by and ETF. We make the assumption that these are correlated, as when rebalancing holdings, the transaction costs should increase if there are more constituents present in the portfolio. Simultaneously, as funds that employ a full-replication strategy will hold relatively more securities, we assume that this variable will also capture the effect of the choice of replication strategy. Additionally, as a proxy for liquidity, we use the natural logarithm of the sum of the trading volume per month, divided by the average shares outstanding, in accordance with the methodology employed by Petajisto (2007).

We perform a cross-sectional regression in order to identify the magnitude of each factor’s contribution to ETF tracking error. In order to control for variables that differ between ETFs but do not vary over time, we add entity-fixed effects, thereby reducing any omitted variable biases. Additionally, we run the cross-sectional regression with both entity-fixed and time-fixed effects in order to control for both omitted variables that are constant over time by vary across ETFS, and those that vary across time but are constant across ETFs. This regression is as follows:

𝑇𝐸𝑖,𝑡= 𝛼 + 𝛽1∗ 𝐸𝑥𝑝𝑒𝑛𝑠𝑒𝑅𝑎𝑡𝑖𝑜 + 𝛽2∗ 𝐷𝑖𝑣𝑖𝑑𝑒𝑛𝑑𝐹𝑟𝑒𝑞𝑢𝑒𝑛𝑐𝑦 + 𝛽3∗ 𝐷𝑖𝑣𝑖𝑑𝑒𝑛𝑑𝑌𝑖𝑒𝑙𝑑 + 𝛽4

∗ ln(𝑇𝑢𝑟𝑛𝑜𝑣𝑒𝑟) + 𝛽5∗ ln(𝐹𝑢𝑛𝑑𝑆𝑖𝑧𝑒) + 𝛽6∗ ScaledConstituentsCount + λ𝑡+ 𝜎𝑖

+ 𝜖𝑡

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15 We hypothesize that expense ratio and the dividend yield will both have a positive relationship with tracking error: an increase in those factors would increase the tracking error. Hence, we expect their respective coefficients to be positive. Formally, stated:

𝐻0∶ 𝛽𝑖 = 0 versus 𝐻1: 𝛽𝑖 >0

The other variables we hypothesize to have a negative relationship with tracking error: an increase in those variables would decrease he tracking error. Therefore, we expect their respective coefficients to be negative. Once again, formally stated:

𝐻0∶ 𝛽𝑖 = 0 versus 𝐻1: 𝛽𝑖 <0

4.3. Relation between pricing efficiency and tracking error

Next we attempt to investigate whether a relation exists between the pricing efficiency of an ETF, as measured by its percentage discount/premium, and the tracking error. In order to do so, we regress the monthly average percentage discount/premium on the tracking errors calculated. In addition to a simple OLS regression, we run a cross-sectional regression that include entity- and time-fixed effects in order to reduce any omitted variable biases.

% 𝐷𝑖𝑠𝑐𝑜𝑢𝑛𝑡/𝑃𝑟𝑒𝑚𝑖𝑢𝑚 = 𝛼 + 𝛽1∗ 𝑇𝐸 + λ𝑡+ 𝜎𝑖+ 𝜖𝑡

Where TE is the tracking error, λt are the time-fixed effects, and σi the entity-fixed effects We test the following hypothesis:

𝐻0∶ 𝛽1= 0 versus 𝐻1: 𝛽1≠0

Under the null hypothesis, the coefficient for tracking error is zero. Implying that tracking error does not have an influence on the percentage discount/premium, and therefore is not priced. With tracking error not being priced, we would infer that there is a pricing inefficiency present. Under the alternative hypotheses, the tracking error does influence the percentage discount/premium, and therefore is priced. Hence, with tracking error priced we would infer pricing efficiency.

4.4 Factor-adjusted tracking errors

Next, we recalculate tracking errors using a variation of the third measure (Pope and Yadav, 1994). We include the factors of influence on the tracking error in the regression in order to control for these effects. 𝑇𝐸3,𝑖,𝑡 = SER = √ 𝑆𝑆𝑅 𝑁 − 𝑘 − 1 𝑖𝑛 𝑡ℎ𝑒 𝑂𝐿𝑆 𝑟𝑒𝑔𝑟𝑒𝑠𝑠𝑖𝑜𝑛 𝑜𝑓: 𝑅𝑁𝐴𝑉,𝑖,𝑡 = 𝛼 + 𝛽 ∗ 𝑅𝑖𝑛𝑑𝑒𝑥,𝑖,𝑡+ 𝛼 + 𝛽2∗ 𝐸𝑥𝑝𝑒𝑛𝑠𝑒𝑅𝑎𝑡𝑖𝑜 + 𝛽3∗ 𝐷𝑖𝑣𝑖𝑑𝑒𝑛𝑑𝐹𝑟𝑒𝑞𝑢𝑒𝑛𝑐𝑦 + 𝛽4 ∗ 𝐷𝑖𝑣𝑖𝑑𝑒𝑛𝑑𝑌𝑖𝑒𝑙𝑑 + 𝛽5∗ ln(𝑇𝑢𝑟𝑛𝑜𝑣𝑒𝑟) + 𝛽6∗ ln(𝐹𝑢𝑛𝑑𝑆𝑖𝑧𝑒) + 𝛽7 ∗ ScaledConstituentsCount + 𝜖𝑡

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16 Where, contrary to the factor contribution regression, these additional factors are daily measures instead of monthly averages. This regression is performed per ETF for monthly periods and, hence, the resulting measure for tracking error shall be a monthly measure.

The resulting SER is our factor-adjusted tracking error. Due to the addition of the control variables, their contribution to the tracking error shall be absent. The standard error of the regression, which is the estimate of the standard deviation of the error term, should decrease as more explanatory variables are included. This measure of tracking error we then use to test whether the tracking error absent the influence of these factors has a relationship with pricing efficiency. We again run the cross-sectional regression of the percentage discount/premium on tracking errors, both for the factor-adjusted tracking error as the third measure in order to ensure that a comparison is possible. We test the following hypothesis:

𝐻0∶ 𝛽1 = 0 versus 𝐻1: 𝛽1≠0

Under the null hypothesis the coefficient on tracking error is equal to zero, and therefore does not influence the percentage discount/premium, implying a pricing inefficiency. Conversely, if the coefficient on the tracking error is significantly different from zero, this would imply that tracking errors does influence the percentage discount/premium, and we infer that prices are efficient.

5. Results

We will now proceed to discuss the results of our experiment. Starting with examining the descriptive statistics of the main variables of interest in our research, we will then proceed to briefly inspect the calculated tracking errors. Next, the results of the regression of tracking errors on potential factors of influence will be discussed. This, in order to test if said factors’ contribution is statistically significant. We then regress the percentage discount/premium on tracking error in order to investigate whether a relation is present. Next, we calculate factor-adjusted tracking errors, which are absent the hypothesized factors of influence, and run a regression of the percentage discount/premium on these factor-adjusted tracking errors, as well as the third measure of tracking error in order to ensure that a comparison is possible. The results thereof will allow us to investigate whether there is a relation between the pricing efficiency and the tracking error after correcting for the factors of influence. 5.1. Descriptive statistics

Table I presents the descriptive statistics of the main variables of interest which are used in our research. We observe a large degree of dispersion in the fund size, turnover, and average number of constituents. Therefore, we scaled the number of constituents with the fund size, and use the natural logarithm of fund size and turnover. The reason for scaling is to ensure that our coefficients resulting from our regression will be of a magnitude that will be presentable in our results tables.

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17 Table I: Descriptive statistics

This table reports the descriptive statistics for the main variables which will be used in this paper. Dividend yield is the ratio of dividends received to average yearly price. Dividend frequency are the amount of times dividends are distributed per year. Shares outstanding are the fund’s outstanding shares in thousands. Fund size is the number of shares outstanding times the ETF price per month and ln[Fund Size] is its natural logarithm. Expense ratio is the ratio of total yearly expenses charged to monthly fund size. Volume is the daily number of shares traded in thousands. Turnover is the ratio of the monthly volume of shares traded to the monthly average of shares outstanding, and ln[Turnover] is its natural logarithm. Average Constituents is the monthly average number of constituents held by a fund, and Scaled Average Constituents is Average Constituents divided by fund size.

Variables Observations Minimum

5th

Percentile 95th

Percentile Maximum Mean

Standard Deviation

ETF Price 87714 10.95 30.27 143.90 398 82.69 39.02

Net Asset Value 87714 10.87 30.28 143.89 3.98E+02 82.69 39.01

Shares Outstanding 87714 50 900 169,104.00 3.28E+05 32153.36 55,120.51

Volume 87714 0 2.2 1,538.30 1.18E+05 571.8115 3,480.63 Dividend Yield 87714 0 0.18 3.16 5.27 1.56 0.85 Benchmark Price 87714 56.51 191.31 6,224.83 10172.33984 1,804.28 1,928.99 NAV Return 87714 -0.13607 -0.01746 0.016759 0.089218 0.00027 0.0107 ETF Return 87714 -0.13311 -0.01744 0.016682 0.088392 0.00027 0.010655 Benchmark Return 87714 -0.13616 -0.01726 0.016658 0.091438 0.000258 0.010632 % Discount/premium 87714 -4.49299 -0.15778 0.15 3.08 -0.00601 0.148074 Dividends Frequency 87714 0 1 4 12 3.723625 1.617289 Expense Ratio 87714 0.000024 0.000066 0.000567 0.050088 0.000275 0.000291

Fund Size 87714 6,980.10 51522.95 13,600,000.00 31,355,700.00 2.87E+06 5.37E+06

Ln[Fund Size] 87714 8.850818 10.84978 16.425583 17.260906 13.50842 1.736665 Average Constituents 87714 15 30 1342.578979 3059.526367 357.451615 512.2246 Scaled Average Constituents 87714 0.000006 0.000019 0.005221 0.035571 0.001161 0.003205 Turnover 87714 0.00319 0.051214 0.70 6.58 0.244835 0.377635 Ln[Turnover] 87714 -5.74759 -2.97174 -0.351039 1.884602 -1.84021 0.840736 Number of ETFs 131 131 131 131 131 131 131 5.2. Tracking errors

The calculated tracking errors are presented in table II, ranging from 0.002% to 2%, with the average close to 1%, which is an economically significant magnitude. We observe that measure one and three are very similar in terms of range, but measures two and three are similar in terms of their distribution. This indicates that the index returns coefficient resulting from the CAPM estimation that was used to calculate TE3 is equal to unity. The fourth measure (TE) is an average of the first three. As mentioned in section 2.3.1, there is a possibility that the third measure will overstate the magnitude of the tracking error. Therefore we take the average of the three measures. This average measure is used in our regression on the contributing factors, and when relating tracking error to pricing efficiency.

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18 Table II: Tracking Errors

This table displays the summary statistics of the monthly tracking errors (TE) calculated. TE1 is the average of the absolute difference between the ETF’s NAV returns and the index’s return. TE2 is the standard deviation of the difference between the ETF’s NAV returns and the index’s return. TE3 the Standard Error of Regressions (SER) in the OLS regression of ETF NAV’s return on index’s return. TE is the average of the prior three methods of calculating tracking error.

Variable Observations Minimum

5th Percentile

95th

Percentile Maximum Mean

Standard Deviation TE1 87714 0.000016 0.000047 0.003246 0.015294 0.000609 0.001358 TE2 87714 0.000021 0.000059 0.004512 0.024796 0.000973 0.001874 TE3 87714 0.000021 0.000059 0.004287 0.014027 0.000912 0.001617 TE 87714 0.000019 0.000056 0.003956 0.015807 0.000831 0.00159 Number of ETFs 131 131 131 131 131 131 131 5.3. Factor contributions

Table III presents the results of our cross-sectional regression of tracking error on the factors hypothesized to contribute to its magnitude. We will now proceed to interpret the direction as well as the magnitude of the coefficients. Following that, we will assess the overall strength of the model. Firstly the expense ratio, the monthly proportion of the yearly expenses incurred by the fund, expressed as a ratio to the fund size. We hypothesized that, due to the expenses incurred by funds reducing the returns it delivers, it would have a positive effect on tracking error. We find this factor to be significantly different from zero in the simple OLS regression. However, due to the observations not being independent from each other, no information regarding the effect van be inferred from the simple OLS. In our cross-sectional regressions with entity-fixed and time-fixed effects, our resulting coefficient is not statistically significantly different from zero. The reported coefficient does however coincide with the direction we hypothesized. Namely, the higher the expense ratio of an ETF, the higher the tracking error. However, due to statistical insignificance, we cannot reject our hypothesis that higher levels of expense ratios leads to higher levels of tracking error.

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19 Table III: Factor Contributions Regression

This table displays the results of the regression of tracking error (TE) on the various factors hypothesized to contribute to it. Dividend yield is the ratio of dividends received to average yearly price. Dividend frequency are the amount of times dividends are paid per year. Ln[Fund size] is the natural logarithm of number of shares outstanding times the ETF price. Expense ratio is the ratio of total yearly expenses charged to monthly fund size. Ln[Turnover] is the natural logarithm of the ratio of the monthly volume of shares traded to the monthly average of shares outstanding. Scaled Average Constituents is the monthly average number of constituents held by a fund scaled by the fund’s size. Column (1) shows a simple OLS regression. Column (2) shows a panel regression with entity-fixed effects. Column (3) shows a panel regression with both entity- and time-fixed effects. Standard errors are clustered by ETF and are reported in brackets, with *, **, and *** denoting statistical significance at the 10%, 5%, and 1% level respectively.

(1) (2) (3) Variables TE TE TE Expense Ratio 1.7548** 0.6246 0.4019 (2.5246) (1.6262) (0.9285) Dividends Frequency 0 -0.0001** -0.0001** (0.892) (-2.3115) (-2.1752) Dividend Yield 0.0003** 0.0004*** 0.0003*** (2.2650) (4.4779) (2.9153) Ln[Turnover] 0 0.0001** 0.0001*** (0.1361) (2.383) (2.6932) Ln[Fund Size] -0.0001* 0.0002 0.0001 (-1.8566) (1.4683) (0.8558) Scaled Average Constituents -0.0404** -0.0941* -0.0973* (-2.5320) (-1.8646) (-1.8779) Constant 0.0011 -0.0015 -0.0009 (1.2913) (-0.9797) (-0.4407) Observations 4,323 4,323 4,323 R-squared 0.072 0.058 0.143 Number of ETFs 131 131 131

The coefficient on the yearly frequency at which dividends are distributed is negative, which coincides with our hypothesis. Namely, that higher dividends frequencies have a negative effect on tracking error, due to the fund accumulating less cash holdings due to more frequent distributions. Although small at only -0.0001, it is highly significant at the 1% level, allowing us to reject our null hypothesis and infer that an increase in the level of the frequency of dividend distributions have a negative effect on the level of the tracking error in our sample.

The coefficient on the dividend yield also coincides with what we hypothesized, having a positive relationship with tracking error. With an increase in the level of dividend yield, this would indicate more dividends are accrued in cash holdings prior to being distributed to the ETF shareholders. As with dividend frequency, this coefficient although small is significant, thus allowing the rejection of our null hypothesis.

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20 The coefficient on the natural logarithm of turnover is positive, and significant at the 5% level. Indicates that a 1% increase in the level of turnover, the fraction of shares traded to total shares outstanding, increases the tracking error by 0.000001 points. With turnover being a measure of liquidity, this result is unexpected and contrary to what we hypothesized and what was found in previous literature. It implies that higher levels of liquidity lead to higher levels of tracking error. Conversely, ETFs with lower level of liquidity have a lower tracking error.

In the OLS regression, we find a significantly negative coefficient for the natural logarithm of fund size, which coincides with what we hypothesized. The larger a firm, the more economies of scale it enjoys which translate into lower transaction costs and therefore lower tracking errors. However, in the cross-sectional regressions with the inclusion of entity-fixed effects and time-fixed effects, this coefficient changes. We see that it becomes positive and insignificantly different from zero, meaning we do not have enough statistical inference in order to reject the null hypothesis.

Lastly, the variable of the average number of constituents held scaled by the fund size. We hypothesized that this variable measures two effects. Firstly, due to ETFs that employ a full replication strategy generally holding a higher number of constituents in their portfolio compared to ETFs that employ a statistical replication strategy, it can be seen as a proxy for this. Secondly, due to funds periodically having to rebalance their portfolio, a higher number of constituents should indicate higher transaction costs. In essence, it captures the tradeoff present in the choice of the replication strategy faced by ETFs. We observe a statistically significant negative coefficient which coincides with what we expected and allows rejection of the null hypothesis. From this we infer that, in our sample, an increase in the level of the number of constituents scaled by funds size has a negative effect on the tracking error.

Overall we see that compared to the OLS, the cross-sectional regression with the inclusion of entity- and time-fixed effects can better explain the variation observed in the tracking errors. With the R2_{, the ratio of explained to total variation, increasing from 7.2% to 14.3%. This implies that the factors} we hypothesized to influence tracking error are able to explain a significant proportion of the tracking error variation we observed. From this observation we can infer that the factors dividend frequency, dividend yield, liquidity, and the scaled average number of constituents have an effect on the magnitude of the tracking error in our sample.

5.3. The effect of tracking error on pricing efficiency

We now proceed by investigating whether the pricing efficiency of an ETF is affected by its tracking error. In Table IV the results of the regressions of pricing efficiency, as measured by the percentage discount/premium, on the tracking error.

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21 A value for the tracking error’s coefficient that is statistically different from zero would result in a rejection of our null hypothesis that tracking error is not priced. The acceptance of the alternative hypothesis then implies that there is pricing efficiency in our sample.

Table IV: Pricing efficiency regression

This table displays the results of the regression of the percentage discount/premium on tracking error (TE). The percentage discount/premium is the monthly average of the daily ratio of ETF price to its net asset value. Tracking error is the average of the following three different measures: The average of the absolute difference between the ETF’s NAV returns and the index’s return, the standard deviation of the difference between the ETF’s NAV returns and the index’s return, and the Standard Error of Regressions (SER) in the regression of ETF’s NAV returns on its benchmark index’s returns. Column (1) shows the results from an OLS regression. Column (2) shows the results from a panel regression with entity-fixed effects. Column (3) shows the results of a panel regression with both entity-fixed effects and time-fixed effects. Standard errors are clustered by ETF and are reported in brackets, with *, **, and *** denoting statistical significance at the 10%, 5%, and 1% level respectively.

Variables % Discount/premium % Discount/premium % Discount/premium

(1) (2) (3) TE -0.7447 -2.7757 -2.6335 (-0.7677) (-1.6019) (-1.3897) Constant -0.0054*** -0.0037** -0.0209*** (-3.7754) (-2.5631) (-3.8043) Observations 4,323 4,323 4,323 R-squared 0.001 0.004 0.091 Number of ETFs 131 131

We observe a coefficient on tracking error that is not statistically different form zero in our OLS regression, as well as our panel regression, both with entity-fixed effects and with entity- and time-fixed effects. From these results, we cannot reject our null hypothesis, indicating that we cannot statistically infer that there is pricing efficiency present in our sample.

5.4 Factor-adjusted tracking error results

In table V the factor-adjusted tracking error are presented together with the third measure of tracking error. We find that controlling for the factors that are of influence on the tracking error, as established in section 5.2, results in a slight decrease in the mean and standard deviation of the tracking errors.

Table V: Descriptive statistics for factor-adjusted tracking error.

This table displays the summary statistics of the monthly tracking errors (TE) calculated. TE3 the Standard Error of Regressions (SER) in the OLS regression of ETF NAV’s return on index’s return. The factor-adjusted tracking error (FA_TE) also is the Standard Error of Regression, but included in the regression are the factors hypothesized to influence tracking error in order to control for them.

Variables Observations Minimum

5th Percentile

95th

Percentile Maximum Mean

Standard Deviation

TE3 87714 0.000021 0.000059 0.004287 0.014027 0.000912 0.001617

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22 Table VI presents the results of our tests of pricing efficiency. Panel A presents the results of the regression of the percentage discount/premium on the third measure of tracking error, and panel B the results the percentage discount/premium on our factor-adjusted tracking error.

Table VI: Pricing efficiency regression with factor corrected standard errors

This table displays the results of the regression of the percentage discount/premium on tracking error (TE). The percentage discount/premium is the monthly average of the daily ratio of ETF price to its net asset value. Panel A’s tracking error is the Standard Error of Regressions (SER) in the regression of ETF’s NAV returns on its benchmark index’s returns. Panel B’s tracking error is the factor-adjusted tracking error. Which again is the SER of the regression of the ETF’s NAV returns on its benchmark index’s returns, but includes as control variables the factors that are hypothesized to influence to the magnitude of the tracking error. Column (1) shows the results from an OLS regression. Column (2) shows the results from a panel regression with fixed effects. Column (3) shows the results of a panel regression with both entity-fixed effects and time-entity-fixed effects. Standard errors are clustered by ETF and are reported in brackets, with *, **, and *** denoting statistical significance at the 10%, 5%, and 1% level respectively.

Panel A

(1) (2) (3)

TE3 -1.0789 -3.0477** -2.6196 (-1.0596) (-2.0664) (-1.5074) Constant -0.0051*** -0.0033*** -0.0208*** (-3.4170) (-2.3995) (-3.7770) Observations 4,323 4,232 4,232 R-squared 0.002 0.005 0.091 Number of ETFs 131 131 131 Panel B (1) (2) (3)

FA_TE -1.0773 -3.0053** -2.5044 (-1.0808) (-2.0639) (-1.5038) Constant -0.0052*** -0.0034*** -0.0209*** (-3.5313) (-2.6264) (-3.7822) Observations 4,323 4,323 4,323 R-squared 0.002 0.005 0.092 Number of ETFs 131 131 131

We find that in our OLS regressions as well as our cross-sectional regression with the inclusion of entity-fixed and time-fixed effects, our coefficient of interest is not statistically significantly different from zero. Therefore, we are unable to reject our null hypothesis and infer that there is a pricing inefficiency present in our sample. However, in column 2, when we only include entity-fixed effects, we do find a statistically significant coefficient. Indicating that an increase in the level of tracking error would lead to a decrease in the percentage discount/premium, possible due to a trend factor that varies across time but is constant across ETFs. In the following section, we will statistically test whether the inclusion of time-fixed effect was warranted.

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23

6. Robustness checks

Next, we will perform various robustness checks in order to assess the validity of our results. Frist we test whether the specification of our models are correct. We employ the Sargan-Hansen and Wald test in order to investigate whether the inclusion of entity- and time-fixed effects were merited. After that, we use a non-parametric test to reevaluate our results regarding the relation between pricing efficiency and performance.

6.1 Model misspecification

In order to test for model misspecification in the factor contribution and the pricing efficiency regressions, we use the test statistics resulting from the Sargan-Hansen test (Arellano, 1993). Under the null hypothesis, the estimators are consistent in both a fixed effects as well as a random effects model. Rejection of the null hypothesis in favor of the alternative would then indicate that the estimators of the random effect model are not consistent, and therefore use of the fixed effects model is appropriate. Our resulting test statistics and their corresponding P-values are reported in the following table:

Table VII: Sargan-Hansen test statistics

This table presents the results of the Sargan-Hansen test on the cross-sectional regressions used in this paper. It tests whether the estimators are consistent in a random effects model as well as a fixed effects model. Rejection of the null hypothesis implies that the estimators of the random effects model are not consistent. Statistical significance is reported with *, **, and *** denoting statistical significance at the 10%, 5%, and 1% level respectively.

Panel A: Factor contributions regression:

Sargan-Hansen statistic: 16.092** Chi-sq(6)

P-value 0.00133

Panel B: Pricing efficiency regression:

Sargan-Hansen statistic: 2.524* Chi-sq(1)

P-value 0.0605

Panel C: Pricing efficiency regression with factor corrected tracking errors:

Panel A:

P-value 0.0227

Panel B:

P-value 0.0248

With a statistical significance level of 5%, we reject the null hypothesis for the factor contributions regression and the pricing efficiency regression with factor corrected tracking errors. This implies that for these models, the use of entity-fixed effects in our models was correct. However, for the pricing efficiency regression in panel B, we are unable to reject the null hypothesis, indicating that the use of a random effects model would be appropriate. In the appendix the results for the pricing efficiency

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24 regression with random effects can be found. However, even after using this correct model, we still find insignificant results for our coefficient of interest. Hence, we cannot reject our null hypothesis indicating a pricing inefficiency in our sample.

Additionally, we use the Wald test to investigate whether the inclusion of time-fixed effects in our models was warranted. In doing so, we test the hypothesis that the coefficient on the dummies for our time variables are jointly equal to zero. Rejection of this null hypothesis would indicate that time-fixed effects should be included in our models. In table VIII we report the corresponding F-statistic along with the p-values for these tests.

Table VIII: Wald test statistics:

This table presents the results of the Wald test, which tests the hypothesis that the included dummies for our monthly time variables in the models used in this paper are jointly equal to zero. Statistical significance is reported with *, **, and *** denoting statistical significance at the 10%, 5%, and 1% level respectively.

Panel A: Factor contributions regression:

F-statistic(32,130) 22.19***

P-value 0

Panel B: Pricing efficiency regression:

F-statistic(32,130) 9.6***

P-value 0

Panel C: Pricing efficiency regression with factor corrected tracking errors: Panel A: F-statistic(32,130) 9.67*** P-value 0 Panel B: F-statistic(32,130) 9.87*** P-value 0

We find that in all our models, the reported p-values call for the rejection of the null hypothesis. From this we conclude that the addition of time-fixed effects in our model was correct.

6.2 Non-parametric tests

In addition to the parametric test found in table IV, table V presents the results of non-parametric tests. We use Spearman’s rank correlation to in order to investigate whether a relation can be found between the percentage discount/premium, and the measures used for tracking error. Under the null hypothesis, Spearman’s rank correlation test states that the variables are independent. Hence, rejection of the null hypothesis would imply that the variables are dependent, and therefore a relationship exists between them.

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25 Table IX: Spearman’s rank correlation:

This table displays the results of the Spearman’s rank correlation test, which is a non-parametric test that measures the strength and the direction of a relation between two variables. We test if the percentage discount/premium, which is the monthly average of the daily ratio of ETF price to its net asset value is related to tracking error. In panel A, tracking error is the average of the following three different measures: The average of the absolute difference between the ETF’s NAV returns and the index’s return, the standard deviation of the difference between the ETF’s NAV returns and the index’s return, and the Standard Error of Regressions (SER) in the regression of ETF’s NAV returns on its benchmark index’s returns. In Panel B, the tracking error is the Standard Error of Regressions (SER) in the regression of ETF’s NAV returns on its benchmark index’s returns. Panel C’s tracking error is the factor-adjusted tracking error. Which again is the SER of the regression of the ETF’s NAV returns on its benchmark index’s returns, but includes as control variables the factors that are hypothesized to influence to the

magnitude of the tracking error. Statistical significance is reported with *, **, and *** denoting statistical significance at the 10%, 5%, and 1% level respectively Panel A: Percentage premium / discount and average tracking error

Number of observations 4323

Spearman's rho -0.0264*

P-value 0.0822

Panel B: Percentage discount/premium and tracking error measure 3

Spearman's rho -0.0311**

P-value 0.0409

Panel C: Percentage discount/premium and factor-adjusted tracking error

Spearman's rho -0.0339**

P-value 0.0279

From the results in table IX we are unable to reject the null hypothesis that the percentage discount/premium is independent of the tracking error at the 10% significance level in panel A, and the 5 % significant level in panels B and C. This is in contrast with the results found in our parametric tests. Namely, that the percentage discount/premium did not have a significant relationship with tracking error. The direction of the coefficient coincides with what we hypothesized, that a larger tracking error should cause the percentage discount/premium to decrease.

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7. Conclusion / Discussion

In this paper, we investigated factors that were previously identified to have an effect on the performance of Exchange-Traded Funds, as measured by their tracking error. By doing so, we corroborated the results of previous authors, and constructed factor-adjusted tracking errors. Secondly, using both parametric and non-parametric tests, we investigated whether various measures of tracking errors were priced. With tracking error being a measure of the performance of an ETF, this would allow us to investigate whether performance is priced, and the degree of pricing efficiency that ETFs experience. By means of a cross-sectional regression with both entity- and time-fixed effects, we found statistically significant coefficients on the following factors: Dividend yield, dividend frequency, and average number of constituents held by a fund. Indicating that these factors have a statically significant relationship with tracking error, and therefore have an influence on the magnitude of the tracking error. These results are in line with results published by previous authors.

Contrary to previous authors, however, we identified a positive relationship between liquidity, as measured by turnover, and tracking error. For the factors expense ratio and fund size, we were unable to achieve statistical significance. These results contradicted the results found by previous authors. These different results could be due to the use of a different time period and different ETFs, or due to the fact that our model excluded certain factors.

Next, we calculated factor-adjusted tracking errors which controlled for the contribution of these factors. To test whether tracking errors were priced, we first calculated a measure of pricing efficiency; the percentage discount/premium of an ETF’s traded price relative to its Net Asset Value. We then regressed this measure of pricing efficiency on tracking errors. Using parametric tests, were unable to reject our null hypothesis that tracking errors were not priced, from which we infer that there is pricing inefficiency present in our sample. Using non-parametric tests, however, we found evidence to the contrary. The results thereof indicated that there in fact was a relationship present in our sample between tracking error and pricing efficiency. Therefore, with contradicting results from parametric and non-parametric tests, our results are inconclusive regarding the pricing efficiency of ETFs with respect to tracking errors. Hence, further research is required to determine whether the performance of ETFs, as measured by their tracking errors, is priced.

Our research was subject to various limitations. First, due to limited access to databases, and the incomplete data contained in the ETF Global database, we were unable to include certain variables that previous authors have identified as having an influence on tracking error. Examples of such variables include cash holdings, transaction costs, and the replication technique employed by ETFs. Although the inclusion of entity- and time-fixed effects should correct for these omitted variables, these variables would have helped in creating a more comprehensive model. Additionally, due to some ETF’s benchmark being indices that are not publicly traded, and therefore do not have data on