FSA in an ETF world

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.

Appendix

1.1 Lemma 1

Derive the price and value of information in the no-ETF economy as follows. Substitute in the mean and variance values from (2) and (4) into the demand functions in (5) and then substitute these demands into the market clearing condition in (6), and then solve for P_0j to get

$$ {\displaystyle \begin{array}{l}{P}_{0j}=\Big\{\lambda \left[\eta +\frac{h{Y}_j^s}{h+s}\right]\left[g+\frac{h{Q}_0}{h+{Q}_0}\right]+\left(1-\lambda \right)\left[\eta +\frac{h\left({Y}_j^s-{K}_0{X}_j\right)}{h+{Q}_0}\right]\left[g+\frac{hs}{h+s}\right]\\ {}\kern2em -\frac{\left(g+\frac{hs}{h+s}\right)\left(g+\frac{h{Q}_0}{h+{Q}_0}\right){X}_j}{\rho}\Big\}\\ {}\kern2em \div \left\{\lambda \left(g+\frac{hs}{h+s}\right)+\left(1-\lambda \right)\left(g+\frac{h{Q}_0}{h+{Q}_0}\right)\right\}.\end{array}} $$

(40)

Note that (40) is linear in ${Y}_j^s$ and X_j. The equilibrium price is determined by equating the coefficients in (40) with the coefficients in our conjectured price, ${P}_{0j}={a}_0+{a}_1{Y}_j^s-{a}_2{X}_j$. More precisely, recalling that K₀ = a₂/a₁, we need to equate the ratio of coefficients on X_j ${Y}_j^s$ in (40) to K₀. Defining G = gh + gs + hs, this gives

$$ {K}_0=\frac{\left(1-\lambda \right){hK}_0G+G\left( gh+{gQ}_0+{hQ}_0\right)/\rho }{\lambda h\left( gh+{gQ}_0+{hQ}_0\right)+\left(1-\lambda \right) hG}. $$

(41)

Substitute in ${Q}_0=s+{K}_0^2w$ from (4) and simplify to get

$$ {K}_0^3-{K}_0^2\left[\frac{G}{\lambda ph}\right]+{K}_0\left[\frac{G}{\left(g+h\right)w}\right]-\frac{G^2}{\lambda ph\left(g+h\right)w}=0. $$

(42)

The solution to this cubic defines the equilibrium K₀, and thus the equilibrium coefficients in the price equation. Factoring the cubic yields the unique solution

$$ {K}_0=\left(\frac{a_2}{a_1}\right)=\frac{G}{\lambda \rho h}. $$

(43)

1.2 Lemma 3

The key to taking the limits on the demand functions given in (19)–(22) as N goes to infinity is to note three things. First,

$$ \underset{N\to \infty }{\mathit{\lim}} Det\left({V}_{1I}\right)=\frac{ghs}{h+s} and\underset{N\to \infty }{\mathit{\lim}} Det\left({V}_{1U}\right)=\frac{ghQ_1}{h+{Q}_1}. $$

(44)

Second, (29) establishes that lim $\underset{N\to \infty }{\mathit{\lim}}{K}_1=\frac{s}{\lambda \rho}$ And third, (1 − J) ∈ (0,1).

With this, for the informed traders

$$ \underset{N\to \infty }{\mathit{\lim}}{D}_{1I}\left( firm\kern0.17em asset\right)=\frac{\rho \left(h+s\right)}{ghs}\left[g\left(\eta +\frac{hY_j^s}{h+s}-{P}_{1j}\right)-g\left(\eta -{P}_{ETF}\right)\right], and $$

(45)

$$ \underset{N\to \infty }{\mathit{\lim}}{D}_{1I}\left( ETF asset\right)=\frac{\rho \left(h+s\right)}{ghs}\left[-g\left(\eta +\frac{hY_j^s}{h+s}-{P}_{1j}\right)+\left(g+\frac{hs}{h+s}\right)\left(\eta -{P}_{ETF}\right)\right] $$

(46)

$$ =-\underset{N\to \infty }{\mathit{\lim}}{D}_{1I}\left( firm\ asset\right)+\frac{\rho }{g}\left(\eta -{P}_{ETF}\right). $$

(47)

The same steps yield the same result for the uninformed trader demands.

1.3 Lemma 4

All we need from the equilibrium price vector is the value of K₁ written in terms of the exogenous parameters and to verify the initial assumption that the ratio of coefficients b4/b3 in the conjectured pricing equation equals K₁. To derive K₁ in ETF economy, begin with the demand functions given in Eqs. (19)–(22). To find K₁ express the demands generally as linear expressions of the information variables (${\mathrm{Y}}_j^s and{\hat{P}}_{ETF}={\overline{Y}}^s-{K}_1\overline{X}$ for the informed and ${\hat{P}}_{1j}={Y}_j^s-{K}_1{X}_j\ and\ {\hat{P}}_{ETF}$ for the uninformed) and prices to get

$$ {D}_{1I}\left( firm\ asset\right)={\beta}_0+{\beta}_y{Y}_j^s+{\beta}_{\overline{Y}}{\overline{Y}}^s+{\beta}_{\overline{X}}\overline{X}+{\beta}_j{P}_{1j}+{\beta}_{ETF}{P}_{ETF} $$

(48)

$$ {D}_{1U}\left( firm\kern0.17em asset\right)={\gamma}_0+{\gamma}_y{Y}_j^s+{\gamma}_x{X}_j+{\gamma}_{\overline{Y}}{\overline{Y}}^s+{\gamma}_{\overline{X}}\overline{X}+{\gamma}_j{P}_{1j}+{\gamma}_{ETF}{P}_{ETF} $$

(49)

Market clearing for the N firm assets, j=1 to N requires

$$ \lambda {D}_{1I}\left( firm\ asset\right)+\left(1-\lambda \right){D}_{1U}\left( firm\ asset\right)={X}_j. $$

(50)

Substituting the demands into the market clearing equation and solving for P_1j gives

$$ {\displaystyle \begin{array}{l}{P}_{1j}=\left[{X}_j-\lambda \left({\beta}_0+{\beta}_y{Y}_j^s+{\beta}_x{X}_j+{\beta}_{\overline{Y}}{\overline{Y}}^s+{\beta}_X\overline{X}+{\beta}_{ETF}{P}_{ETF}\right)-\left(1-\lambda \right)\right)\Big({\gamma}_0+{\gamma}_y{Y}_j^s+\\ {}\kern2.75em {\gamma}_x{X}_j+{\gamma}_{\overline{Y}}{\overline{Y}}^s+{\gamma}_{\overline{X}}\overline{X}+{\gamma}_{ETF}{P}_{ETF}\left)\right]\div \left[\lambda {\beta}_j+\left(1-\lambda \right){\gamma}_j\right].\end{array}} $$

(51)

Note that (51) is linear in (${Y}_j^s$, X_j, ${\overline{Y}}^s$, $\overline{X}$) and P_ETF, and, because P_ETF is the average of the individual P_1j’s, it is linear in (${\overline{Y}}^s$, $\overline{X}$); consequently, the conjectured linear form of P_1j in (9) is validated. From (51) we can pick out the coefficients on (${Y}_j^s$, X_j, ${\overline{Y}}^s$, $\overline{X}$)

$$ coefficient on{Y}_j^s:\frac{-{\lambda \beta}_y-\left(1-\lambda \right){\gamma}_y}{{\lambda \beta}_j+\left(1-\lambda \right){\gamma}_j}, $$

(52)

$$ coefficient on{X}_j:\frac{1-{\lambda \beta}_x-\left(1-\lambda \right){\gamma}_x}{{\lambda \beta}_j+\left(1-\lambda \right){\gamma}_j}, $$

(53)

$$ coefficient on{\overline{Y}}^s:\frac{-{\lambda \beta}_{\overline{Y}}-\left(1-\lambda \right){\gamma}_{\overline{Y}}}{{\lambda \beta}_j+\left(1-\lambda \right){\gamma}_j}, and $$

(54)

and

$$ coefficient on\overline{X}:\frac{-{\lambda \beta}_{\overline{X}}-\left(1-\lambda \right){\gamma}_{\overline{X}}}{{\lambda \beta}_j+\left(1-\lambda \right){\gamma}_j}. $$

(55)

In the equilibrium conjecture, the ratio of the negative of the coefficient on Xj to the coefficient on ${Y}_j^s$ equals K₁= b2/b1; that is,

$$ {K}_1=\frac{1-{\lambda \beta}_x-\left(1-\lambda \right){\gamma}_x}{{\lambda \beta}_y+\left(1-\lambda \right){\gamma}_y}. $$

(56)

From the demand functions in (19)–(22), we can find the (β_x, β_y, γ_x, γ_y) written in terms of K₁. These are

$$ {\beta}_x=0, $$

(57)

$$ {\beta}_y=\frac{\rho }{Det\left({V}_{1I}\right)}\left\{\left[g+\frac{hs+{h}^2\left(1-J\right)}{N\left(h+s\right)}\right]\left[\frac{h}{\left(h+s\right)}\right]-\left[g+\frac{hs}{N\left(h+s\right)}\right]\left[\frac{hK_1^2w}{\left(h+s\right)I}\right]\right\}, $$

(58)

$$ {\gamma}_x=-{\gamma}_y{K}_1. $$

(59)

$$ {\gamma}_y=\frac{\rho }{Det\left({V}_{1U}\right)}\left\{\left[g+\frac{hQ_1}{N\left(h+{Q}_1\right)}\right]\left[\frac{h}{\left(h+{Q}_1\right)}\right]\right\}, and $$

(60)

Plugging (β_x, γ_y, γ_x) into the ratio for K₁ gives the intermediate expression K₁λβ_y = 1. Substituting in β_y and simplifying gives the cubic in K₁, as shown in (28) in the text.

Also from (19)–(22), we can find (${\beta}_{\overline{Y}}$, ${\beta}_{\overline{X}}$, ${\gamma}_{\overline{Y}}$, ${\gamma}_{\overline{X}}$) to compute the ratio of the coefficients on $\overline{X}$ and ${\overline{Y}}^s$ given in (54) and (55):

$$ {\beta}_{\overline{Y}}=-\frac{\rho }{Det\left({V}_{1I}\right)}\left\{\left[g+\frac{hs}{N\left(h+s\right)}\right]\left[\frac{h\left(N-1\right)}{I}\right]\right\}, $$

(61)

$$ {\beta}_{\overline{X}}=-{\beta}_{\overline{Y}}{K}_1, $$

(62)

$$ {\gamma}_{\overline{Y}}=-\frac{\rho }{Det\left({V}_{1U}\right)}\left\{\left[g+\frac{hQ_1}{\left(h+{Q}_1\right)}\right]\left[\frac{h}{\left(h+{Q}_1\right)}\right]\right\}, and $$

(63)

$$ {\gamma}_{\overline{X}}=-{\gamma}_{\overline{Y}}{K}_1. $$

(64)

It is easy to see that the ratio of the negative of the coefficient on $\overline{X}$ to ${\overline{Y}}^s$ also equals K₁, as originally conjectured in (10).

1.4 Proof of Theorem 2

The sign of the difference between Φ₁ and Φ₀ is determined by the difference in the ratio of the determinants: ${\Phi}_1>{\Phi}_0\ when\ \frac{Det\left({V}_{U1}\right)}{Det\left({V}_{I1}\right)}>\frac{Det\left({V}_{U0}\right)}{Det\left({V}_{I0}\right)}$ and visa versa. Label $\Delta =\frac{Det\left({V}_{U1}\right)}{Det\left({V}_{I1}\right)}-\frac{Det\left({V}_{U0}\right)}{Det\left({V}_{I0}\right)}$. The limiting values of these determinants as N goes to infinity are given in (32) and (44). Plugging them in gives

$$ \Delta =\left[\frac{G}{h+s}\right]\left[\frac{ghQ_1}{h+{Q}_1}\right]-\left[g+\frac{hQ_0}{h+{Q}_0}\right]\left[\frac{ghs}{h+s}\right] $$

(65)

$$ =\frac{gh}{h+s}\left\{\frac{GQ_1\left(h+{Q}_0\right)-\left(h+{Q}_1\right)\left[ hs\left(h+{Q}_0+{hsQ}_0\right)\right]}{\left(h+{Q}_1\right)\left(h+{Q}_0\right)}\right\}. $$

(66)

The sign of Δ is determined by the sign of the numerator in the braces term.

Substitute in ${Q}_0=s+{K}_0^2w\ and\ {Q}_1=s+{K}_1^2w$ into the numerator of the braces term and collect terms around w to get the following quadratic in w:

$$ {K}_0^2{K}_1^2{ghw}^2-\left({h}^2{sK}_0^2-{GhK}_1^2\right)w=0. $$

(67)

A trivial root is at w=0; when there is no exogenous noise, there is no price noise, and so all private information is conveyed to the market through price and the uninformed effectively have the same information as the informed. The quadratic is then negative for small values of w (meaning that Φ₀ > Φ₁). After substituting in for K₁ and K₀, the second root can be found at

$$ {w}^{\ast }=\frac{h\left(h+s\right){\left({\lambda}_j\rho \right)}^2}{Gs} $$

(68)

and the quadratic is increasing in w beyond this point (meaning that Φ₁ > Φ₀).

1.5 Theorem 3

Consider the case where N is finite, but w goes to infinity. Referring to (8) for the value of information in the no-ETF economy and taking the limit with respect to w gives

$$ \underset{w\to \infty }{\mathit{\lim}}{\Phi}_0\frac{\rho }{2}\ast \mathit{\ln}\left(\frac{G+{h}^2}{G}\right). $$

(69)

For the ETF economy, refer to (16) for Det(V_1I) and note that the only place w appears is as part of J, so that we can compute $\underset{w\to \infty }{\mathit{\lim}}\left(1-J\right)=\frac{N-1}{N}$

With this

$$ \underset{w\to \infty }{\mathit{\lim}} Det\left({V}_{1I}\right)=\frac{\left[G+\left(n-1\right) gs\right]h\left(N-1\right)}{N^2\left(h+s\right)}. $$

(70)

Refer to (18) for Det(V_1U) and note that w only appears as part of Q₁. Substituting in $s+{K}_1^2w$ for Q₁ and taking the limit gives

$$ \underset{w\to \infty }{\mathit{\lim}} Det\left({V}_{1U}\right)=\frac{\left( gN+h\right)h\left(N-1\right)}{N^2}. $$

(71)

Putting these together gives the value of information in the ETF market with infinite liquidity noise:

$$ \underset{w\to \infty }{\mathit{\lim}}{\Phi}_1=\frac{\rho }{2}\ast \mathit{\ln}\left(\frac{\left( gN+h\right)\left(h+s\right)}{G+\left(N-1\right) gs}\right). $$

(72)

The difference between the arguments inside the log functions determine the difference between Φ₁ and .Φ₀. This difference is increasing in N and is zero when N=1. Thus Φ₁ > Φ₀ for N > 1.