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Review of Accounting Studies

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08-03-2021

FSA in an ETF world

Author: Russell J. Lundholm

Published in: Review of Accounting Studies | Issue 4/2021

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Abstract

This paper models the value of conducting financial statement analysis (FSA) in the presence of an electronically traded fund (ETF) that gives exposure to the firm’s systematic value. FSA is characterized as a costly process that yields a private signal about the idiosyncratic portion of a firm’s future payoffs. The value of this signal depends on how efficiently price transmits information to uninformed traders. A popular argument is that ETFs are attracting noise traders away from the underlying firm, making prices more informative and private information less valuable. While I find that prices are more informative after the introduction of an ETF, I show that this isn’t because of a change in the amount or location of noise trading. Holding noise trading constant, ETFs allow informed investors to hedge out exposure to the portion of firm value that they are uninformed about, which causes them to place larger bets on their private information. This is what causes firm prices to be more informative. The introduction of an ETF into an economy thus presents two competing forces on the value of conducting FSA. On the one hand, prices are more informative after the arrival of an ETF, making private information less valuable, but on the other, informed traders can use the ETF to hedge, making private information more valuable. I characterize how these forces trade off as a function of the exogenous noise in the economy. These results are unavailable in previous theoretical papers about ETFs, because they modeled investors as being risk neutral, thus eliminating their desire to hedge out uncertainty.

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The Israeli et al. (2017) results may appear at odds with Glosten et al. (2016). However, Israeli et al. (2017) show that the Glosten et al. results apply to the concurrent pricing of information about the common component of the news, while their results apply to the pricing of future information and is largely driven by the idiosyncratic component of the news.

Another branch of the ETF literature considers how the ETF aids trading in assets that are otherwise illiquid. Bhattacharya and O’Hara (2017) consider the case where there is no market for the underlying asset, and so informed traders must use the ETF as an indirect means of trading on their information.

To see this, note, in Subrahmanyam (1991), that the derived demands for the firm asset and the ETF asset are always in the same direction (compare equation 5 on page 22 and equation 8 on page 24).

Because the individual firm asset markets are segmented, without loss of generality, every parameter in the model could be firm-specific (that is, indexed by ‘j’). I supress this detail to avoid unnecessary notation.

There are two common options for describing the signal error’s covariance across investors—either the errors are independent across all agents, or they are perfectly correlated, as in my model. Neither assumption is ideal. In my model, every analyst who conducts FSA reaches exactly the same conclusion. However, assuming independent errors is also unrealistic, implying that the analysts collectively know the value of the payoff perfectly. As shown later, there are some practical modeling advantages to assuming all informed agents see exactly the same signal.

There is little tension in a model where investors have a signal about the common factor and then an ETF is added to the market. The ETF allows the factor-informed trader the perfect mechanism to use her information, and so the value of the factor information increases.

By the commutative law of addition, a positive supply shock on the RHS of the market clearing condition is equivalent to a negative demand shock on the LHS.

The general expressions for the conditional mean and variance of normal random variables are as follows. Let F be an n-dimensional vector of payoffs with prior mean vector μ and let Y be an m- dimensional vector of signals about those payoffs, with no particular covariance structure and mean vector u. The covariance matrix of F ∪ Y is n+m dimensional and symmetric. Partition this matrix into an nxn matrix of the top left corner, labeled ∑₁₁, an mxm matrix in the bottom right corner labeled ∑₂₂, an nxm matrix of the top right corner labeled ∑₁₂, and an mxn matrix in the bottom left corner, labeled \( {\sum}_{12}^{\prime } \) The nx1 posterior mean vector is then given by \( E=\mu +{\sum}_{12}{\sum}_{22}^{-1}\left(Y-\upsilon \right) \), and the nxn posterior covariance matrix is given by \( V={\sum}_{11}-{\sum}_{12}{\sum}_{22}^{-1}{\sum}_{12}^{\prime } \) (Welch 2014).

There is no additional information to be gained by conditioning beliefs on the prices of other firm assets j^′ ≠ j because \( {\hat{P}}_{0j},={Y}_{j,}^s-{K}_0\ast {X}_j, \) is uncorrelated with the payoff \( {F}_j={F}^c+{F}_j^s \) and with firm j’s price signal \( {\hat{P}}_{0j}={Y}_j^s-{K}_0\ast {X}_j \)

Recall that the mean of the liquidity demand is the per capita supply of the risky asset.

Admati (1985) provides a solution to the multi-asset noisy rational expectations equilibrium with a reasonably general information structure, but her model does not accommodate the case where information is only about one component of the asset’s final payoff.

With negative exponential utility functions and normal random variables, demand functions do not depend on the level or distribution of the known supply of assets; consequently, assumptions about these values only impact the constant in the price function and have no impact on the informational properties of the model. For instance, we could assume that a fraction μ of each firm asset’s known supply is traded as a basket of securities (as created by the ETF sponsor), while 1-μ of the supply is traded in individual asset markets, without changing the coefficients on the random variables in the pricing function. Importantly, with or without the ETF, the realized economy-wide payoff is the sum of the individual firm payoffs and the variance of the random liquidity noise is the same in both economies.

Arbitraging between the ETF and the basket of constituent securities whenever the ETF price deviates from the value of the basket is a primary function of the Authorized Participant (AP). The AP can accomplish this either by trading in the secondary market for the ETF and the constituent securities or by trading in the primary market with the ETF sponsor. There is a rich literature regarding the microstructure of ETFs—how they are created, how they are priced, and why they offer extremely low transaction costs. Lettau and Madhavan (2018) detail the inner workings of ETFs and related research into it. My model relies only on the facts that an ETF is a basket of securities that collectively provide exposure to some commonly held attribute of the constituent firms, that they can be longed or shorted at very low cost, and that there is no arbitrage between the individual asset prices and the ETF asset price.

Israeli et al. (2017) discuss how liquidity noise trading is leaving individual firm markets and moving to ETF markets. Note, however, that, because \( {P}_{ETF}=\frac{1}{N}{\sum}_{j=1}^N{P}_j={\overline{P}}_j \), the liquidity noise in the firm assets necessarily affects the noise on ETF asset in the same direction; a shift up in one market and down in another market is not possible without a second source of noise. For our purposes, an additional exogenous source of noise in the ETF market would have no impact on the informational properties of the ETF economy. To see this, change the model to allow another noise term in the ETF market, so that \( {P}_{ETF}={b}_0+{b}_1\overline{\mathrm{Y}}\hbox{-} {\mathrm{b}}_2\overline{\mathrm{X}}\hbox{-} {\mathrm{b}}_3{X}_{ETF} \) If this was the case, then \( {\overline{P}}_1 \) would also have to have a b₃X_ETF term (because of the no arbitrage condition), and to do this each of the N individual P_1j’s would therefore have a b₃X_ETF/N term. The investor could then use the P_ETF and P_1j to back out the new X_ETF term, making it irrelevant to the information environment.

There is no additional information to be gleaned from observing other firm asset prices j^′ ≠ j, because \( {\hat{P}}_{1j},={Y}_j^s-{K}_1{X}_j, \) is uncorrelated with the firm payoff \( {F}_j={F}^c+{F}_j^s \) and is a noisy of the ETF price signal \( {\hat{P}}_{ETF}={\overline{\mathrm{Y}}}^s-{K}_1\overline{\mathrm{X}} \) for inferences about the ETF payoff \( {F}_{ETF}={F}^c+{\overline{F}}^s \). Plugging \( {\hat{P}}_{1j} \), into the covariance matrix with the firm price signal and the ETF price signal has no impact on the posterior means or variances.

The fact that the ETF asset can be used as an indirect means of investing based on the private signal comports with a relatively new phenomenon in the financial press. Yahoo!Finance sends targeted messages to users suggesting which ETFs have significant exposure to the individual firms in the user’s portfolio, suggesting that a way to invest in the firm is through the ETF.

In the next section, we fully develop the equilibrium with an infinite number of firms in the ETF; here our purpose is only to cleanly illustrate how hedge plays out in the demand functions.

Perhaps the easiest way to see the diminishing impact of price noise is to note that, at some point, price becomes completely uninformative and additional liquidity noise has no impact on beliefs.

One might be tempted to generalize further and predict that market-wide ETFs are the most valuable of all. However, this would only be true if there were a common component that was being isolated by diversification in the ETF holdings; the more stocks in the ETF, the harder it is to assume that each firm’s payoff has exactly the same common component. In addition, Theorem 3 relies on very high liquidity noise or on a behavioral assumption that traders do not infer information from price; absent this, we are back to Theorem 2.

Assuming N goes to infinity is probably overly restrictive for Theorem 4; technically all that is

needed is that Φ₁(λ) is decreasing in λ.

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Title: FSA in an ETF world
Author: Russell J. Lundholm
Publication date: 08-03-2021
Publisher: Springer US
Published in: Review of Accounting Studies / Issue 4/2021
Print ISSN: 1380-6653
Electronic ISSN: 1573-7136
DOI: https://doi.org/10.1007/s11142-020-09571-8

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