Beauty Contests and Iterated Expectations in Asset Markets Franklin Allen Stephen Morris Hyun Song Shin 1
professional investment may be likened to those newspaper competitions in which the competitors have to pick out the six prettiest faces from a hundred photographs, the prize being awarded to the competitor whose choice most nearly corresponds to the average preferences of the competitors as a whole; [] We have reached the third degree where we devote our intelligences to anticipating what average opinion expects the average opinion to be And there are some, I believe, who practise the fourth, fifth and higher degrees Keynes (1936) Questions To what extent is this metaphor valid for asset prices? If valid, what are the implications? 2
Iterated Average Expectations N y, 1 α i observes x i = + ε i, ε i N ³ 0, 1 E i () = αy + x i α + αy + E () = α + E i E () = αy + E i () α + = = EE () = αy + ³ αy+xi µ α + ³ 1 2 y + µ 2 1 y + ³ ³ 2 xi ³ 2 E k () y as k Bias towards shared information y 3
Markov chain E i y Ē y = = 1 0 α 1 0 α y x i y Markov chain over {y, } y is absorbing state Ē k y = = 1 0 " α 1 y y k y 1 0 ³ k ³ as k k # y 4
Fundamentals date 0 1 2 T T +1 y N y, 1 α liquidation at T +1 asset traded at dates 1, 2,,T noisy supply s t N ³0, 1γt 5
Traders date 0 1 2 t T T +1 x it y overlapping generations: trade when young, consume when old ³ private signal x it N, 1 exponential utility e 1 τ c information set of trader i at date t I it = {y, x it,p 1,p 2,,p t } 6
Equilibrium prices date T demand of trader i τ Var it () (E it () p T ) date T aggregate demand τ Var T () market clearing ĒT () p T p T = ĒT () Var T() τ s T 7
date T 1 price p T 1 = ĒT 1 (p T ) Var T 1(p T ) s τ T 1 = ĒT 1ĒT () Var T 1(p T ) τ date t price s T 1 p t = Ēt (p t+1 ) Var t(p t+1 ) τ s t = ĒtĒt+1 (p t+2 ) Var t(p t+1 ) τ s t = ĒtĒt+1 ĒT () Var t(p t+1 ) τ s t 8
Equilibrium prices date 0 1 2 T T +1 E t () p t q t y 1 Mean of time paths of p t and E t () p t and Ē t () deviate systematically Ēt () systematically closer to fundamentals, for t<t Price exhibits sluggishness to shift in drift momentum underreaction 9
Argument Fix REE p 1 = λ 1 y +(1 λ 1 ) + φ 11 s 1 p 2 = λ 2 y +(1 λ 2 ) + φ 21 s 1 + φ 22 s 2 p t = λ t y +(1 λ t ) + φ t1 s 1 + φ t2 s 2 + + φ tt s t Construct price signals ξ t = + ψ t s t ξ t is linear combination from I it Precision of ξ t is ρ t 10
Markov chains again B t = y ξ 1 ξ t ξ t+1 ξ T I ξ 1 ξ t ξ t+1 R t r t ξ T y R t = α t ρ 1t ρ tt α t ρ 1t ρ tt α t ρ 1t ρ tt, r t = t t t α t = α α+ P t i=1 ρ i+, ρ kt = ρ k α+ P t i=1 ρ i+, t = α+ P t i=1 ρ i+ 11
z it = y ξ 1 ξ T x it z = y ξ 1 ξ T E it z = B t z it Ē t z = B t z E it 1 Ē t z = B t B t 1 z it Ē t 1 Ē t z = B t B t 1 z (note reversal of order) Ē t Ē t+1 Ē T z = B T B t+1 B t z 12
B T B T 1 B t = α t ρ 1t ρ tt 0 t B t = α t ρ 1t ρ tt 0 t Mean path of price is determined by B T B T 1 B t Mean path of Ē t () is determined by B t y is sole absorbing state of (nonhomogeneous) Markov chain α t > α t is transient state ξ t is temporary absorbing state ρ st > ρ st 13
Inertia of forward-looking expectations Price is forward-looking expectation, but is sluggish In a dynamic context, the older the information, the more public it is too much weight on y too much weight on past prices Even forward-looking expectations look like adaptive expectations Inertia is worse for long duration assets (technology stocks?) low cost of capital 14