Indexing and Price Informativeness

Transcription

1 Indexing and Price Informativeness Hong Liu Washington University in St. Louis Yajun Wang University of Maryland IFS SWUFE August 3, 2017 Liu and Wang Indexing and Price Informativeness 1/25

2 Motivation There is a significant drop in both individual equity ownership and institutional money that is actively managed (e.g., Stambaugh (2014)). Many active mangers switch to lower fees and more index-like investing. There are very few research on the impact of the rise of indexing on asset pricing and price informativeness. Main difference from Bond and Garcia (2016): Indexing is exogenous Passive investors and some active investors cannot acquire information The mass of informed investors does not change as the mass of passive investors changes Consequently, many results are qualitatively different Liu and Wang Indexing and Price Informativeness 2/25

3 Overview We develop an equilibrium model to study the impact of endogenous index investing on information acquisition and asset prices. We show that price informativeness first increases and then decreases in the mass of indexers. Market risk premium and market volatility are also non-monotonic in the mass of indexers. In addition, when more traders become indexers, the welfare of both active traders and indexers decrease in most cases. Liu and Wang Indexing and Price Informativeness 3/25

4 The model A continuum of investors with mass 1 can trade at time 0 one risk free and two risky assets to maximize their expected utility at time 1. The first risky asset m is the market portfolio and the second risky asset s is the spread portfolio. The risk-free asset and the spread portfolio has a net supply of zero, while the market portfolio has a net supply of 1 share. Interest rate is normalized to be zero. The time 1 payoff V m of the market portfolio is distributed as N(µ V, τ 1 V ), the time 1 payoff V s of the spread portfolio is distributed as N(0, τ 1 V ), and V m is independent of V s, where µ V and τ V are constants. Liu and Wang Indexing and Price Informativeness 4/25

5 The model (Cont.) All investors have CARA preferences with a risk aversion coefficient of γ and an initial endowment of 1 share of the market portfolio, but no risk-free asset or the spread portfolio. A trader can pay a cost of k just before time 0 to trade in the spread portfolio ( an active a investor ), otherwise the trader invests only in the market portfolio ( a passive p investor ). A trader i can also pay a cost of c(τ im, τ is ) to observe a private signal about the risky assets just before time 0: Y ij = V j + ε ij, j {m, s}, where ε ij s are independently distributed as N(0, τ 1 ij ). Noise traders have a random supply of Z j shares of the j portfolio, where Z j N(0, τz 1 ), j {m, s}. Liu and Wang Indexing and Price Informativeness 5/25

6 Investor s problem at time 0 and market clearing condition Let P m and P s be the time 0 equilibrium prices of the market portfolio and the spread portfolio respectively, I i be time 0 information set of type i investors, and Θ ij be the number of shares of the j portfolio held by type i investors from time 0 to time 1, for j {m, s} and i {a, p}. Type i investors then choose Θ im and Θ is to solve max E[ e γ W i I i ] subject to the budget constraint W i = P m +Θ im (V m P m )+Θ is (V s P s ) c(τ im, τ is ) k1 {Θis 0}. Market clearing condition: Θ ij di = Z j, j {m, s}. i Liu and Wang Indexing and Price Informativeness 6/25

7 Investors problem just before time 0 Investors choose whether to pay the cost k to trade in the spread portfolio. Investors then choose the precisions of their private signals about the payoffs of market portfolio and spread portfolio at the corresponding costs. The equilibrium mass η of indexers is such that investors are indifferent between being indexers and being active investors. Liu and Wang Indexing and Price Informativeness 7/25

8 Equilibrium price and aggregate precision We focus on the linear equilibrium where P m = a m + b m V m d m Z m and P s = b s V s d s Z s, where a m, b m, d m, b s, and d s are constants to be determined. Let τ j denote the aggregate precision of private information for stock j: τ j := ητ pj + (1 η)τ aj, j {m, s}. (1) Liu and Wang Indexing and Price Informativeness 8/25

9 Benchmark Case: All investors trade in both risky assets Given the signal precisions τ ij, i {a, p}, j {m, s}, we obtain (1) the equilibrium price coefficients are a m = γ 2 (µ v τ v γ) γ 2 τ m + τ γ 2 (τ m + τ v ) + τmτ 2, b m = mτ 2 z z γ 2 (τ m + τ v ) + τmτ 2, d m = γ b m, (2) z τ m b s = γ 2 τ s + τ 2 s τ z γ 2 (τ s + τ v ) + τ 2 s τ z, d s = γ b s τ s. (3) (2) The equilibrium holdings in portfolio j {m, s} of trader i {p, a} are Θ im = γ2 (τ im + τ v ) + τ im τ m τ z γ 2 (τ m + τ v ) + τmτ 2 (Z m τ m z γ V m) + τ im γ Y im + γµ v (τ m τ im )τ v + γ 2 (τ im + τ v ) + τ 2 mτ z γ 2 (τ m + τ v ) + τ 2 mτ z, (4) Θ is = γ2 (τ is + τ v ) + τ is τ s τ z γ 2 (τ s + τ v ) + τs 2 (Z s τ s τ z γ V s) + τ is γ Y is. (5) Liu and Wang Indexing and Price Informativeness 9/25

10 Market risk premium and price informativeness The risk premium of market portfolio m is E[V m P m ] = γ 3 γ 2 (τ v + τ m ) + τ 2 mτ z. (6) The informativeness of portfolio j {m, s} is measured by (Var[V j P j ]) 1 = τ v + (τj 2/γ2 )τ z. For given values of τ v, τ z, and γ, the informativeness of portfolio j is measured by the aggregate precision of private information τ j for portfolio j. Liu and Wang Indexing and Price Informativeness 10/25

11 Equilibrium when passive investors only trade the market portfolio Given the signal precisions τ ij, i {a, p}, j {m, s}, we obtain (1) the equilibrium price coefficients a m, b m, and d m are given as a m = γ 2 (µ v τ v γ) γ 2 τ m + τ γ 2 (τ m + τ v ) + τmτ 2, b m = mτ 2 z z γ 2 (τ m + τ v ) + τmτ 2, d m = γ b m, (7) z τ m and coefficients b s and d s are b s = γ 2 τ as + τ 2 as(1 η) 2 τ z γ 2 (τ as + τ v ) + τ 2 as(1 η) 2 τ z, b s d s = γ. (8) (1 η)τ as (2) The equilibrium holding in market portfolio m of trader i, Θ im is given as Θ im = γ2 (τ im + τ v ) + τ im τ m τ z γ 2 (τ m + τ v ) + τmτ 2 (Z m τ m z γ V m) + τ im γ Y im + γµ v (τ m τ im )τ v + γ 2 (τ im + τ v ) + τ 2 mτ z γ 2 (τ m + τ v ) + τ 2 mτ z, (9) Θ as = τ as γ (Y as V s ) + Z s 1 η, Θ ps = 0. (10) Liu and Wang Indexing and Price Informativeness 11/25

12 Numerical results To analyze the impact of indexing, we use the following default parameter values: τ v = τ z = 1, γ = 0.5, µ v = 0.5, and c(τ im, τ is ) = c im τim 2 + c isτis 2 + c imsτ im τ is, for i = a, p. Liu and Wang Indexing and Price Informativeness 12/25

13 Equilibrium mass of indexers η * τ v Figure: The equilibrium fraction of passive investors η against prior precision τ v. The default parameter values are: τ v = τ z = 1, γ = 0.5, µ v = 0.5, c im = c is = 0.1, and c ims = 0.2. Liu and Wang Indexing and Price Informativeness 13/25

14 Equilibrium mass of indexers η * k Figure: The equilibrium fraction of passive investors η against k. The default parameter values are: τ v = τ z = 1, γ = 0.5, µ v = 0.5, c im = c is = 0.1, and c ims = 0.2. Liu and Wang Indexing and Price Informativeness 14/25

15 Price informativeness τ m τ s η η Figure: Informativeness of the market portfolio (τ m) and the spread portfolio (τ s ) against the equilibrium fraction of passive investors η while changing k. The default parameter values are: τ v = τ z = 1, γ = 0.5, µ v = 0.5, c im = c is = 0.1, and c ims = 0.2. Liu and Wang Indexing and Price Informativeness 15/25

16 Equilibrium precisions for the market portfolio τ am τ pm η η Figure: Optimal precisions of the active investors signal (τ am) and passive investors signal (τ pm) about the market portfolio payoff against the equilibrium fraction of passive investors η while changing k. The default parameter values are: τ v = τ z = 1, γ = 0.5, µ v = 0.5, c im = c is = 0.1, and c ims = 0.2. Liu and Wang Indexing and Price Informativeness 16/25

17 Equilibrium precisions for the spread portfolio * * τ as and τ ps 0.20 * τ as for cims= * τ as for cims= * τ ps η * Figure: Optimal precisions of the active investors signal (τ as) and the passive investors signal (τ ps) about the spread portfolio s payoff against the equilibrium fraction of passive investors η while changing k. The default parameter values are: τ v = τ z = 1, γ = 0.5, µ v = 0.5, c im = c is = 0.1, and c ims = 0.2 or 0.1. Liu and Wang Indexing and Price Informativeness 17/25

18 Market risk premium Market Risk Premium η * Figure: Risk premium of market portfolio against the equilibrium fraction of passive investors η while changing k. The default parameter values are: τ v = τ z = 1, γ = 0.5, µ v = 0.5, c im = c is = 0.1, and c ims = 0.2. Liu and Wang Indexing and Price Informativeness 18/25

19 Market volatility Var(V -P )+Var(P ) η Var(V s -P s )+Var(P s ) η Figure: Total variance of market portfolio Var(V m P m ) + Var(P m ) and total variance of spread portfolio Var(V s P s ) + Var(P s ) against the equilibrium fraction of passive investors η while changing k. The default parameter values are: τ v = τ z = 1, γ = 0.5, µ v = 0.5, c im = c is = 0.1, and c ims = 0.2. Liu and Wang Indexing and Price Informativeness 19/25

20 Certainty equivalent wealth for investors CEforTradersaandp η * Figure: Ex-ante expected certainty equivalent wealth of active investors and passive investors (CE for traders a and p) against the equilibrium fraction of passive investors η while changing k. The default parameter values are: τ v = τ z = 1, γ = 0.5, µ v = 0.5, c im = c is = 0.1, and c ims = 0.2. Liu and Wang Indexing and Price Informativeness 20/25

21 Certainty equivalent wealth for investors: exogenous indexing CEforTradersaandp CE for a CE for p η Figure: Ex-ante expected certainty equivalent wealth of active investors and passive investors (CE for traders a and p) against exogenously given fraction of passive investors η. The default parameter values are: τ v = τ z = 1, γ = 0.5, µ v = 0.5, k = 0, c im = c is = 0.1, and c ims = 0.2. Liu and Wang Indexing and Price Informativeness 21/25

22 Expected trading size E Θ am η E Θ pm η Figure: Expected trading size of active investors and passive investors (E Θ am and E Θ pm ) against the equilibrium fraction of passive investors η while changing k. The default parameter values are: τ v = τ z = 1, γ = 0.5, µ v = 0.5, k = 0, c im = c is = 0.1, and c ims = 0.2. Liu and Wang Indexing and Price Informativeness 22/25

23 Expected trading size (Cont.) E Θ as η Figure: Expected trading size of active investors and passive investors (E Θ as ) against the equilibrium fraction of passive investors η while changing k. The default parameter values are: τ v = τ z = 1, γ = 0.5, µ v = 0.5, k = 0, c im = c is = 0.1, and c ims = 0.2. Liu and Wang Indexing and Price Informativeness 23/25

24 Model implications In highly transparent markets, the fraction of indexers tends to be high and therefore, the information acquisition tends to decrease and market risk premium tends to increase when more traders become indexers. Lowering the cost of participation in actively managed funds would increase the information production and reduce the market risk premium. In more opaque markets, the fraction of indexers tends to be low and therefore, the information acquisition tends to increase and market risk premium tends to decrease when more traders become indexers. Discouraging the participation in actively managed funds would help increase the information production for the index portfolio and reduce the market risk premium. Liu and Wang Indexing and Price Informativeness 24/25

25 Conclusion We develop an equilibrium model to study the impact of endogenous index investing on information acquisition and asset prices. We show that price informativeness first increases and then decreases in the mass of indexers. Market risk premium and market volatility are also non-monotonic in the mass of indexers. In addition, when more traders become indexers, the welfare of both active traders and indexers decrease in most cases. Liu and Wang Indexing and Price Informativeness 25/25