trading ambiguity: a tale of two heterogeneities

Transcription

1 trading ambiguity: a tale of two heterogeneities Sujoy Mukerji, Queen Mary, University of London Han Ozsoylev, Koç University and University of Oxford Jean-Marc Tallon, Paris School of Economics, CNRS ICEF Moscow, 28 October 2017

2 introduction: general research question What does ambiguity and ambiguity aversion, on top of risk and risk aversion, add to our understanding of financial and economic arrangements? 2

3 introduction: ambiguity Risk can be expressed using a single probability distribution. For instance, x N(θ, σ), θ and σ known Ambiguity cannot be expressed using a single probability distribution. Agents are uncertain about the true probability. For instance, x N(θ, σ), θ and/or σ unknown (e.g., θ N(µ, ζ), or θ [0, 2]...) Experimental evidence dating back to Ellsberg (1961) documents that a majority of subjects exhibit aversion to ambiguity. 3

4 introduction: related literature Ambiguity models have had some success in explaining (aggregate) asset pricing puzzles Epstein & Wang (1994), Cao, Wang & Zhang (2005), Ju & Miao (2012), Epstein & Schneider (2008), Hansen & Sargent (2015), Collard et al. (2016)... Regarding quantities: Zero position / No trade / Liquidity Dow & Werlang (1992), Mukerji & Tallon (2001), Easley & O Hara (2009), Ozsoylev & Werner (2010), Illeditsch et al. (2015)... Portfolio efficiency Wang (2005), Hara and Honda (2016),... Risk sharing Dana et al. (2000), Rigotti et al. (2008), Strzalecki & Werner (2011),... Home Bias/ Diversification Epstein & Miao (2003), Dimmock et al. (2016),... Speculative trade Werner (2015),... 4

5 introduction: what we do Our focus is on portfolio choice, asset pricing and trade at equilibrium in a model with two kinds of heterogeneities: differently ambiguous assets, differently ambiguity averse agents. Core mechanism: portfolio rebalancing due to heterogeneity in ambiguity and ambiguity aversion. 5

6 introduction: a loose empirical background/motivation 1. Asset allocation puzzle (Canner, Mankiw and Weil, 1997): Why is it that financial advisors propose different funds depending on how conservative/agressive clients are? 2. Documented departures from CAPM (e.g., Fama and French, 1992, 1993): firms of small market cap, or with high book-to-market ratio tend to earn higher returns. 3. Trade following earnings announcements (Kandel and Pearson, 1995): Observe large volume of trade after public announcements (even without any significant price change). difficult to reconcile with usual risk modelling we provide a simple departure from expected utility that provides a theoretical framework that helps thinking about/providing possible explanations for these different problems simultaneously. 6

7 static setting: asset structure - one risk-free asset, two uncertain assets - R: return vector for uncertain assets - M represents model a model can be thought of as fixing the means of uncertain asset returns - R M N(M, Σ R Σ M ): distribution of R conditional on model M. Model uncertainty only affects the conditional mean of the return, not its variance. Model and asset returns are jointly distributed according to: ( ) (( ) ( )) M µ Σ M Σ M N, R µ Σ M Σ R where Σ M = ( (σ1 M)2 σ12 M σ12 M (σ2 M)2 ) and Σ R = ( (σ 1 ) 2 σ 12 σ 12 (σ 2 ) 2 ) 7

8 static setting: preferences - CARA: u(x) = exp( θx) - Overall utility given by smooth ambiguity model of Klibanoff, Marinacci and Mukerji (2005): U(Z) = E [Φ (Eu(Z) M)] with Φ(z) = ( z) γ/θ - Denote η γ θ 1, the coefficient of relative ambiguity aversion of the decision maker. 8

9 maximization problem - a i : monetary holding of asset i = f, 1, 2, equal to q i p i - normalize p f = 1 - Notation a i,n when deal with several agents, n = 1,..., N. Maximization problem Max af,a 1,a 2 U(a f R f + a 1 R 1 + a 2 R 2 ) subject to a f + a 1 + a 2 W. Let a = (a 1, a 2 ). Maximization of U(.) is equivalent to maximization of (see Maccheroni, Marinacci, Ruffino (2013) as well) : Max af,av (a f, a) = a f R f + µ a θ 2 a Σ X a γ θ a Σ M a 2 Note: when γ = θ, i.e., η = 0, then model variance Σ M is not relevant for asset demand and we are back to the standard mean-variance preference. 9

10 portfolio choice Ratio of demand for the two uncertain assets: a i = (µ ( i R f ) σ 2 j + η(σj M ) 2) (µ j R f ) ( σ 12 + ησ12) M a j (µ j R f ) ( σi 2 + η(σi M ) 2) (µ i R f ) ( ) σ 12 + ησ12 M does not depend on risk aversion. Ambiguity aversion, η, affects demand, via the weight put on model variance. Under ambiguity neutrality (η = 0) or homogenous ambiguity aversion (i.e., same η for all), we have the mutual fund theorem. 10

11 comparative statics Theorem 1: Assume µ 1 Rf µ 2 R f σm 1 σ M 2 σ 1 σ 2 σm 1 σ M 2, and If agent n is more ambiguity averse than agent n, i.e., η n > η n, then. a 1,n < a1,n a 2,n a 2,n and q 1,n < q1,n. q 2,n q 2,n 11

12 comparative ambiguity Asset 1 is more ambiguous (I) than asset 2 in the sense of Jewitt-Mukerji (2017) if all ambiguity neutral preferences are indifferent between the two assets and all ambiguity averse preferences in the class (here KMM) prefer 2 to 1. Asset 1 is more ambiguous (I) than asset 2 if µ 1 = µ 2, σ 1 = σ 2 and σ M 1 > σm 2 > 0 Theorem 1 says that if asset 1 is more ambiguous (I) than asset 2, a more ambiguity averse investor will hold proportionally less of it. 12

13 comparative ambiguity Notion of more ambiguous (I) rather restrictive (fixes mean and variance of the unconditional distribution). Jewitt-Mukerji (2017) develop another notion that amounts here to saying that asset 1 is more ambiguous (II) than asset 2 if σ M 1 > σm 2 Theorem 1 says that if asset 1 is sufficiently more ambiguous (II) than asset 2, a more ambiguity averse investor will hold proportionally less of it. 13

14 asset allocation puzzle: canner, mankiw, weil (1997) Table I - Asset Allocations Recommended By Financial Advisors Percent of portfolio Advisor and investor type Cash Bonds Stocks Ratio of bonds to stocks A. Fidelity Conservative Moderate Aggressive B. Merrill Lynch Conservative Moderate Aggressive C. Jane Bryant Quinn Conservative Moderate Aggressive D. New York Times Conservative Moderate Aggressive

15 asset allocation puzzle: a possible resolution? Table II - The Distribution of Annual Real Returns Arithmetic Standard Correlation Correlation Asset mean return (%) deviation (%) w/ bonds w/ stocks Treasury bills Long-term government bonds Common stock If interpret all uncertainty as mere risk, cannot explain financial advice that depends on risk aversion in simple ways. If think that there is also model uncertainty, and it is more accentuated for equity, then room for explaining this advice. 15

16 equilibrium asset pricing - Move now to equilibrium analysis. Assets are in fixed aggregate supply (e 1, e 2 ). Take safe asset to be inside asset. - Let p1e1r1+p2e2r2 p 1e 1+p 2e 2 R Market be the market portfolio return. Theorem 2: Assume a continuum of investors with ambiguity aversion distributed according to some continuous distribution function on [η, η], then there exists an η m [η, η] such that where E[R i ] r f = β i (η m ) (E[R Market ] r f ) β i (η m ) = cov(r Market, R i ) + η m cov M (R Market, R i ) var(r Market ) + η m var M. (R Market ) 16

17 ambiguous capm Under ambiguity neutrality, usual CAPM formula. Ruffino (2014) derives a similar CAPM-like formula but with homogeneously ambiguity averse agents. As shown earlier, the mutual fund theorem holds with homogeneity in ambiguity aversion. Here, we derive a CAPM-like formula with heterogeneity of ambiguity aversion, i.e., without the mutual fund theorem. Different agents still hold different portfolios, and yet, a CAPM-like single-factor pricing model holds. 17

18 link with documented departures from capm Let us now define βi Risk (η m ) = cov(r Market, R i ) var(r Market ) }{{} OLS coefficient estimate and β Amb i (η m ) = cov M (R Market, R i ) var M. (R Market ) Then, where α i E[R i ] r f = α i + β Risk i (η m ) (E[R Market ] r f ) η mvar M (R Market ) var(r Market )+η mvar M (R Market ) ( β Amb i (η m ) βi Risk (η m ) ) (E[R Market ] r f ). could be exploited to address empirical puzzles (e.g., Fama-French factors beyond the market risk premium) 18

19 why? Small firms are followed by fewer equity analysts. So, less information about them, i.e., more ambiguity. Also, small firms are more dependent on bank financing and therefore more likely to default compared to firms which can rely on equity financing. Hence, more susceptible financial distress, associated with heightened uncertainty. High book-to-market ratio is often a sign of financial distress (Fama-French interpretation). Therefore, overall, small caps and high book-to-market firms are more sensitive to macro uncertainty shocks, hence they carry higher systematic ambiguity. 19

20 dynamic extension: modeling earnings announcements - Simple extension of previous model: two trading periods. - Two long-term uncertain assets (in fixed equal supply) and a short-term risk-free asset. - No intermediate consumption. - Trade at t = 0. - At t = 1 get a signal, update beliefs, get the return on risk free-asset, and re-trade while prices of uncertain assets adjust. - At time t = 2, get the exogenous returns on the two uncertain assets, return on the risk-free asset and consume. 20

21 dynamic extension: signal structure and asset returns - Signal structure: M R N S µ µ µ, Σ M Σ M Σ M Σ M Σ R 0 Σ M 0 Σ R - Conditional on a model, R and S are i.i.d. The signal informs about the underlying model, but does not provide direct evidence about the realization of the payoff in the last period. ( R M S M ) N (( M M ), ( Σ R Σ M 0 0 Σ R Σ M )) - Let M S be the posterior for M after observing S. Then M S N (µ S, Σ S ) where µ S is a linear function of S, and Σ S does not depend on realization S. 21

22 trivial vs non-trivial trading the equilibrium entails trivial trading at signal realization S if the composition of the uncertain portfolio stays the same across periods 0 and 1 for all agents, but the split between the uncertain portfolio and the risk-free asset changes for at least one agent the equilibrium entails non-trivial trading at signal realization S if the composition of the uncertain portfolio changes period 0 to 1 for some agent the equilibrium entails no trade at signal realization S if it entails neither trivial nor non-trivial trading 22

23 a trade result Theorem 3: The equilibrium entails non-trivial trading at almost all signal realizations if agents are heterogenous in ambiguity aversion. If agents are homogenous in ambiguity aversion, then the equilibrium entails no trade at any signal realization. If agents are homogeneous in ambiguity aversion, then they do not change their uncertain portfolio composition after receipt of a public signal: they hold the market portfolio as their uncertain portfolio in both periods. If agents are heterogeneous in ambiguity aversion and interim, assets bear different ambiguity then they change their uncertain portfolio composition after receipt of the signal. 23

24 numerical illustration σ1 2 =.04 σ2 2 =.04 risk correlation=.5 µ 1 = 1.12 µ 2 = 1.12 r f = 1.05 (5%) ambiguity correlation =.5 Different scenarios used for the asset ambiguity (model variance): scenario 1 σ1 M =.0007 σm 2 =.0007 scenario 2 σ1 M =.0006 σm 2 =.0008 scenario 3 σ1 M =.0008 σm 2 =.0016 scenario 4 σ1 M =.0006 σm 2 =

25 trade without price movements 8 % 10 % 6 % Trading Volume 4 % Trading Volume 5 % 2 % 0 % 0 % Agent 2's Ambiguity Aversion Agent 2's Ambiguity Aversion Trading volumes for assets 1 & 2: These graphs depict trading volume for asset 1 (left panel) and 2 (right panel) in the four scenarios, for a particular signal realization that leads to no change in prices. The ambiguity aversion of agent 1 is fixed at η 1 = 9. 25

26 likelihood of trade 0 % 5 % 10 % 15 % Trading Volume 0 % 5 % 10 % 15 % Trading Volume Histogram of trading volumes for assets 1 & 2: These graphs depict the histogram of trading volume in asset 1 (left panel) and 2 (right panel) for scenarios 2 (pale grey), 3 (grey) and 4 (black) for the asset ambiguity for a thousand realizations of the signals. The ambiguity aversions are fixed as η 1 = 9 and η 2 = 2. 26

27 discussion All investors start with same (second order) beliefs. Updating upon arrival of interim information. All investors have the same (second order) revised beliefs, that depend on the public information received. change in the mean of second order distribution this depends on realization of S change in ambiguity that is, variance of second order distribution state independent (does not depend on realization of S) How does this lead to trade? Prices could adapt to state dependent change in the mean so that demand remains the same (this is what happens under risk). But change in ambiguity together with heterogeneous ambiguity aversion leads to agent-specific change in the asset demand functions. Prices therefore cannot fully adjust, and trade follows. 27

28 uncertainty shocks Model with only 3 interim states. Signals indicate (H)igh, (I)ntermediate or (L)ow ambiguity. Assume the mean of M, µ, to be independent of S. For each realization S = H, I, L, the posterior M S satisfies: M S N(µ, Σ S ) and X M S N(M S, Σ X Σ S ) where one can think of the following specification: Σ L = λ L Σ, Σ I = λ I Σ, and Σ H = λ H Σ where Σ is some positive definite matrix and λ H > λ I > λ L (in numerical example below take λ H = 2, λ I = 1 and λ L = 1/2). 28

29 trading under uncertainty shocks In this setup, agents hold the market portfolio ex ante. Interim, after the realisation of S = L, I, H, we are as in the static framework and, under heterogeneity of ambiguity aversion will hold different portfolio. trade & portfolio choice implications: (i) The volume of trade is higher, the larger the uncertainty, i.e., it is higher when H is realized, than when I is realized, which is in turn higher than when L is realized. (ii) The distance from the market portfolio is higher, the larger the uncertainty. 29

30 numerical simulation: trading volume 10 % 30 % Trading Volume 5 % Trading Volume 20 % 10 % 0 % 0 % Agent 2's Ambiguity Aversion Agent 2's Ambiguity Aversion Trading volume in assets 1 & 2 with uncertainty shocks: This graph depicts the trading volume in assets 1 (left panel) & 2 (right panel) following an uncertainty shock (H,I,L). The ambiguity aversion of agent 1 is fixed at η 1 = 9. 30

31 numerical simulation: distance from the market portfolio 10 % Distance from Market Portfolio 5 % 0 % Agent 2's Ambiguity Aversion Distance of the equilibrium portfolio to the market portfolio: This graph depicts the distance of agent 1 portfolio from the market portfolio following an uncertainty shock (H,I,L). The ambiguity aversion of agent 1 is fixed at η 1 = 9. 31

32 conclusion Under heterogeneous ambiguity aversion: different investors will optimally hold different uncertain portfolios. There is an ambiguity premium: an asset, whose ambiguity beta is higher than its risk beta, carries a premium compared to its CAPM-predicted return. Provide a derivation of a CAPM-like formula under ambiguity aversion, which holds promises to provide some intuitive explanation to cross-sectional asset pricing anomalies. Trade upon receiving public information whenever investors have different ambiguity aversion. Can generate trade even without (large) price movements. All this with, we hope, an intuitive underlying mechanism based on heterogeneity among uncertain assets and across agents. 32