The Information Content of Asset Markets

Philip Bond, University of Washington

August 2017

Introduction

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• Often claimed that information is a primary output of financial markets

• What is the value of this information?

Talk overview

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• I’ll explore this question by drawing heavily on three of my own papers

• Many other people have also worked on this topic, of course; and I

am in indebted to my coauthors Itay Goldstein and Diego Garcı́a

1. Value of information in an endowment economy

• “The equilibrium consequences of indexing,” with Diego Garcı́a

2. Value of information in a production economy, no “feedback” to asset

prices

• “Bailouts and the information content of investments”

3. Value of information in a production economy, with feedback

• “Government intervention and information aggregation by prices,”

with Itay Goldstein, JF 2015

Preliminary remarks

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• Focus on competitive economies

• all traders in financial market are price takers

• no individual ability to affect or “manipulate” prices

• Since this is a summer school, I’ll spend some time discussing

analytical techniques

1. Information in an endowment economy

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• Canonical setting (risk sharing)

• Each agent has income eiθ, where ei privately observed

• ei = Z + ui, where Z aggregate

• Financial asset trades, paying risky dividend θ

• Risk sharing: agents with high ei want to sell to agents with low ei

• Agents (differentially) devote resources to privately forecasting θ

• CARA utility, normal distributions for everything

• Note: This is Grossman-Stiglitz (1980)/Hellwig (1980) with exogenous

“noise” trades replaced with endogenous “liquidity” trades motivated

by differential exposures

• Ganguli and Yang (2009), Manzano and Vives (2011)

Welfare

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• How does information production affect welfare?

• Despite canonical setting, we’ve been unable to find prior answer

• Medrano and Vives (2004): “the expressions for the expected utility

of a hedger ... are complicated.”

• Kurlat and Veldkamp (2015): “there is no closed-form expression

for investor welfare.”

Preliminary: price efficiency

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• Price efficiency ≡ the extent to which asset price predicts cash flow θ

• sometimes refer to as price informativness

• This economy has a linear equilibrium with price P = p̄+ pθθ − pZZ

• Info content of P same as info content of P−p̄pθ

= θ −(

pθpZ

)−1Z

• Hence info content of P measured by pθpZ

• (Later in talk: true because Z is normal, more care required in general)

Welfare: Good vs bad price efficiency

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• Bad price efficiency

• Hirshleifer (1971): Less risk-sharing

• eg, no medical insurance after genetic screening

• In this context: Price efficiency prevents agents from trading asset

to share exposure to X efficiently

Good price efficiency

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• Recall: ei = Z + ui

• Z high ↔ lots of cash flow risk θ to share

• But P low, so high ui agents receive little when selling to low ui agents

• Agents dont like this correlation between P and total risk to share

• So agents prefer small exposure pZ to Z in price P = p̄+ pθθ − pZZ

• ie, not only do agents want P unrelated to θ

• they also want P unrelated to Z

• Loosely speaking, price efficiency ↔ small pZ

• Good price efficiency

Model, more detail

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• Unit mass of investors, CARA utility, risk aversion γ

• λI are institutional investors

• privately observe signals Yi = θ + ǫi

• 1− λI are retail investors

• no signal of θ

• Financial asset, pays dividend θ

• mean µθ• each investor has constant endowment s̄ of each asset

• Agent i chooses position Xi

• Agent i also has income eiθ

• ei = Z + ui

• All rvs normal and independent, and (other than θ) mean zero

• use precision notation, ie, τθ =1

var(θ)

Model

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• After trading, agent i has wealth

s̄P +Xi (θ − P ) + eiθ

• Rational expectations equilibrium (REE)

• price function P (θ, Z)

• each agent trades conditional on price, endowment ei, and (if

applicable) signal Yi

• market clearing: agents’ trades clear the market

• As usual, focus on linear equilibrium

P = p̄+ pθθ − pZZ

Portfolio choice

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• CARA-normal framework implies portfolio choice Xi given by

Xi + ei =1

γ

E [θ − P |Yi, P, ei]

var [θ − P |Yi, P, ei]

• Standard normal-normal updating

E [θ|Yi, P, ei] =

(

τθµθ + τǫYi +

(

pθ

pZ

)2

τZP − p̄

pθ+

(

pθ

pZ

)2

τuP − p̄+ pZe

pθ

× var [θ|Yi, P, ei]

• Note: Estimation of θ makes use of endowment ei

• individual endowment has info about aggregate endowment Z

• aggregate endowment Z affects price

Equilibrium property: Price informativeness

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• Price P = p̄+ pθθ − pZZ and price informativeness is ρ ≡ pθpZ

• Consider small shocks of pZδ to θ and pθδ to Z

• By construction, P unchanged

• Institutional investor signals increase by dδ

• aggregate institutional demand ↑ by λI1γτǫpZδ

• Exposure shocks increase by pθδ

• estimation effect: aggregrate demand ↑ by 1× 1γ

(

τupθpZ

)

pθδ

• direct effect: aggregate demand ↓ by pθδ

• Market clearing: λI1γτǫpZ + 1

γ

(

τupθpZ

)

pθ − pθ = 0

• So price info ρ given by:

λIτǫ + τuρ2 − γρ = 0

• Larger root is unstable, so user smaller root

• Eq price info ↑ in # of institutional investors (λI ) and quality (τǫ)

Key step in welfare calculation

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• Link risk premium, E [θ − P ], to average amount of risk to share, s̄

• Easy case: no-one has any information about θ

• Hence

Xi + ei =1

γi

E [θ − P ]

var [θ]

• So integrating over investors, and applying MC

(∫

i

di

γi

)−1E [θ − P ]

var [θ]= s̄

• Turns out that this generalizes ...

Equilibrium property: Risk premium

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• Integrating out all uncertainty, MC =⇒

E [θ − P ]

∫

i

di

γivar [θ|Fi]= s̄

• Under normality, ∂∂θE[

θ̃|Fi (θ)]

= 1−var[θ̃|Fi]var[θ̃]

• Consider a small increase in θ; so MC also =⇒

∫

i

1

γi

1− var[θ|Fi]var[θ] − cov[P,θ]

var[θ]

var [θ|Fi]di = 0

which rewrites as∫

i

di

γivar [θ|Fi]=

∫

i

di

γivar [θ](

1− cov[P,θ]var[θ]

) =1

cov [θ − P, θ]

∫

i

di

γi

• Hence:

E [θ − P ] =

(∫

i

di

γi

)−1

s̄cov [θ − P, θ]

How does better institutional investor information affect retail

investor welfare?

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• Using the risk premium from previous slide, retail investor welfare

conditional on ei is

−

√

var [θ|ei, P ]

var [θ − P |ei]E[

exp(−( cov[P,ei]var[ei]

+ γcov [θ − P, θ])2

2var(θ − P |ei)e2i − γ (s̄+ ei) θ)

]

• In particular, this expression is:

• ↓ in |cov(P, ei)|, ie, ↑ in price efficiency

• ↓ in cov(P, θ), ie, ↓ in price efficiency

• Further substitution: write everything as function of price info ρ

• Retail investor welfare ↓ in ρ , and hence ↓ in

• # of institutional investors, λI• quality of institutional investor info τǫ• bad price efficiency dominates

Institutional investor welfare

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• Conflicting effects

• as τǫ ↑, price efficiency ρ ↑, worse risk sharing (like retail investors)

• but information advantage relative to retail investors may increase

• Information advantage given by

τǫ

τθ + (τZ + τu) ρ2

• For τǫ large enough, information advantage ↓ in τǫ

• In this case, institutional welfare certainly ↓ as τǫ ↑

Retail investors buy too much when price is high

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• Average institutional position = Average retail position + τǫγ(θ − P )

• E [θ − P |P ] is ↓ in P

• Hence an econometrician would observe

• retail share of asset ↑ in P

• subsequent return ↓ in P

• Consistent with evidence of Ben-Rephael, Kandel, Wohl (2012)

Aside: indexing

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• This is all from a paper currently titled “The equilibrium consequences

of indexing”

• What does this have to do with indexing?

• Very quick summary

• Economy with two risky assets

• Indexing ≡ removes “noise” from asset markets

• Price efficiency increases

• Welfare drops

Beyond endowment economies

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• Everything so far concerns an endowment economy

• aggregate cash flows are exogenous

• Next, consider production economies, in which information produced

by financial markets may affect “real” decisions

2. Information in a production economy, no feedback

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• Based on: “Bailouts and the information content of investments”

• In this paper, I explore the following (simple) idea:

• Do government bailouts affect economy by changing the

information content of traded securities?

• ie, if investors in MBS anticipate bailout, price of MBS changes, and

reveals more (or perhaps less) information about housing market

• in turn, distorting investment (eg, construction) decisions

• Challenge: bailouts complicate the assumption of CF normality

• preventing use of standard CARA-normal framework

• (though see next paper ...)

• In this paper, I develop tools to address this issue

Investment problem

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• Investor, makes an investment b ∈ B ⊂ ℜ

• Henceforth, builder

• Objective function V (b, θ)

• eg, V (b, θ) = p (θ) u (W + brb) + (1− p (θ))u (W − b)

• Minimal assumptions on V

• main assumption is: V satisfies single crossing property (SCP)

• SCP (Milgrom-Shannon 1994)

• If b′′ ≥ b′ and θ′′ ≥ θ′ and V (b′′, θ′) ≥ V (b′, θ′) then

V (b′′, θ′′) ≥ V (b′, θ′′)

• SCP implies that, if builder knows θ, then the builder’s optimal

investment is increasing in θ

• Builder is uncertain about θ

Market signals about θ

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• Builder uncertain about θ, so tries to extract information from price of a

traded financial asset (e.g, an MBS)

• write P for price

• Measure 1 of informed investors, each takes price P as given,

chooses to buy an amount x to maximize

q(θ)U (W + x (r − P )) + (1− q(θ))U (W − xP )

• θ is fundamental quality of investment

Bailouts

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• Bailouts for MBS

q(θ) = θ + (1− θ)(1− ψ)

• bailout probability is 1− ψ

• more generally, q (θ, ψ), increasing in θ, decreasing in ψ

• Bailouts for other dimensions of investor portfolio

q (θ) = θ

and

U (W + x (r − P )) = ψu (W − L+ x (r − P ))

+ (1− ψ)u (W + x (r − P ))

Equilibrium price

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• Let x (P, θ, ψ) be solution to investor’s portfolio problem

• ie, demand of each investor

• Assume random supply (Grossman-Stiglitz etc)

• supply is s (t) ∈ [0, s̄]• t random and cont. distributed, wlog uniform between 0 and 1

• Market clearing condition

x (P, θ, ψ) = s (t)

• Equilibrium price is function P (θ, t;ψ) such that MC holds

Informativeness non-trivial to measure

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• In general, no closed-form solution for price function

• Even if closed form solution exists, only in very special cases does the

solution take the convenient form

g(P ) = θ + κt

• And moreover, even in this case, informativeness non-trivial to

measure

• θ + 23 t isn’t more Blackwell informative than θ + t

• Lehmann (1988)

• (though Blackwell ranking does hold if t normal)

Lehmann informativeness

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• How to measure informativeness?

1. Impose sufficient structure that one can simply compute builder’s

E[payoff]

2. Impose sufficient structure on trading game (generating P ) that

Blackwell’s informativeness measure is applicable

• an increase in Blackwell informativeness implies that a signal

leads to better outcomes in an arbitrary decision problem

• signal P ′′ is more Blackwell informative than signal P ′ if for

some function g and rv ε, g (P ′′, ε) and P ′ have the same

distribution conditional on θ.

• Blackwell ranking is very incomplete (see earlier example)

3. Lehmann informativeness

• Minimal structure on builder’s problem (SCP)

• Much less structure required on process generating P

Lehmann informativeness: Definition

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• F (·|θ;ψ) = distribution of P conditional on θ in bailout regime ψ

• For any pair of regimes ψ′ and ψ′′, define the function T (P, θ;ψ′, ψ′′) by

F(

P |θ;ψ′)

= F(

T(

P, θ;ψ′, ψ′′)

|θ;ψ′′)

• Lehmann informativeness: The signal P is more informative in regime

ψ′′ than regime ψ′ if for any P , T (P, θ;ψ′, ψ′′) is increasing in θ.

• Proposition 1 (see also Lehmann 1988, Quah and Strulovici 2009):

• Suppose P more Lehmann-informative in ψ′′ than ψ′

• write P ′ and P ′′ for rvs arising in ψ′ and ψ′′

• let ζ : support(P ′) → B be a weakly increasing function

• Then ∃ φ : support(P ′′) → B such that, ∀θ,

V(

φ(

P ′′)

, θ)

FOSD V(

ζ(

P ′)

, θ)

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• Relative to Lehmann, Quah-Strulovici, my result allows the support of

a to shift with θ and ψ

• eg, support of price P is ⊂ [0, qr]

• Result is independent of prior on θ

Lehmann informativeness: Intuition

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• P ′′ and T (P ′, θ) have same information content

Pr(

P ′′ ≤ z|θ)

= F(

z|θ;ψ′′)

= F(

T−1 (z, θ) |θ;ψ′)

= Pr(

P ′ ≤ T−1 (z, θ) |θ)

= Pr(

T(

P ′, θ)

≤ z|θ)

.

• T measures the extra info that P ′′ contains about θ relative to P ′

• Lehmann informativeness: Is this extra information useful?

• Recall: Builder’s objective has SCP, so want higher actions in higher

states

• By supposition, if builder observes P ′, takes action ζ (P ′)

• Lehmann informativeness: T (P ′, θ) is increasing

• so the extra information P ′′ contains about θ is put to good use

Lehmann informativeness and SCP

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• For any t ∈ [0, 1], consider the (1− t)-percentile of the distribution of P

given state θ and regime ψ.

• This is F−1 (1− t|θ;ψ).

• Proposition 2:

• Suppose: ∀ ψ and θ′′ > θ′, P given θ′′ FOSD P given θ′

• Then Lehmann-informativeness of P ↑ in ψ iff F−1 (1− t|θ;ψ)satisfies the SCP in ((θ, t) ;ψ)

• (where Θ× [0, 1] has the product ordering)

Lehmann informativeness and Spence-Mirrlees single crossing

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• SCP: If θ′′ ≥ θ′, t′′ ≥ t′, ψ′′ ≥ ψ′ and

F−1 (1− t′′|θ′′;ψ′) ≥ F−1 (1− t′|θ′;ψ′), then

F−1 (1− t′′|θ′′;ψ′′) ≥ F−1 (1− t′|θ′;ψ′′)

• Spence-Mirrlees:∂

∂θF−1(1−t|θ;ψ)

| ∂

∂tF−1(1−t|θ;ψ)|

increasing in ψ

• In general, two conditions not equivalent

• But they are equivalent if F−1 (1− t|θ;ψ) is increasing in θ and

decreasing in t, as is the case here

• Proposition 3:

• Suppose: ∀ ψ and θ′′ > θ′, P given θ′′ FOSD P given θ′

• Then Lehmann-informativeness of P ↑ in ψ iff F−1 (1− t|θ;ψ)satisfies Spence-Mirrlees single crossing.

Illustration of Spence-Mirrlees single crossing

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θ

t

Isoquant of F−1 (1− t|θ;ψ)

Slope is −∂

∂θF−1(1−t|θ;ψ)

∂

∂tF−1(1−t|θ;ψ)

Steeper ↔ more Lehmann informative

Justification of linearization

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• Proposition 3 justifies linearization approach

• Linearizing,

F−1 (1− t|θ;ψ) ≈ F−1 (1− t0|θ0;ψ)

+∂

∂θF−1 (1− t|θ;ψ) (θ − θ0)

+∂

∂tF−1 (1− t|θ;ψ) (t− t0)

• Tempting to measure information content by the ratio

∂∂θF−1 (1− t|θ;ψ)

∂∂tF−1 (1− t|θ;ψ)

• Earlier, saw that even when approximation exact, approach not

justified by Blackwell informativeness

• Proposition 3: Approach is justified by Lehmann informativeness

Back to bailouts

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• F−1 (1− t|θ;ψ) is simply the equilibrium price at θ and s(t)

• denote P ∗ (θ, t;ψ)

• So to check Spence-Mirrlees single crossing, need to evaluate how

slope of iso-price lines changes with bailout parameter ψ

• Applying the market clearing condition,

∂P ∗

∂t

∂x

∂P=

∂s

∂t∂P ∗

∂θ

∂x

∂P= −

∂x

∂θ

• So Spence-Mirrlees can be checked solely using individual investor

demand, without solving for the equilibrium price

Bailouts for MBS reduce price informativeness

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Dark line=bailouts. Grey line = no bailouts

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

θ

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

t

• Iso-price lines; horizontal axis ↔ θ; vertical axis ↔ t

• Dark line ↔ bailouts

• The bailout iso-price lines are flatter

↔ bailouts decrease Lehmann informativeness

• (analytical result)

Bailouts for other dimensions of portfolio

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• The bailout iso-price lines are steeper

↔ bailouts increase Lehmann informativeness

• (assuming underlying utility function exhibits prudence)

• Economics: bailouts reduce residual portfolio risk, and make trades

more responsive to expected profits

• Hence prices more responsive to θ

Tying up a loose end

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• The key result that shows that an increase in Lehmann

informativeness → an increase in builder payoffs ...

• ... is predicated on the builder’s investment b increasing in the

observed realization of P

• Loosely, this property should hold

• builder’s objective V has SCP

• as discussed, =⇒ that investment b ↑ in θ

• of course, builder doesn’t see θ, sees P instead

• this is the whole problem!

• but increase in θ shifts distribution of P in FOSD sense

• The problem is that this isn’t enough to guarantee that b increases in P

Monotone probability ratio

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• Usual solution in economics+finance

• impose monotone likelihood ratio (MLR) property

• f(P |θ′′)f(P |θ′) ↑ in P

• But here, P is endogenous, can’t just impose this

• and checking MLR property for endogenous P yields intractable

condition

• Athey (2002): If V satisfies a mild regularity condition, the monotone

probability ratio (MPR) is sufficient to ensure that b increases in P

• F (P |θ′′)F (P |θ′) ↑ in P

• MLR =⇒ MPR =⇒ FOSD

Checking MPR

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• MPR can be checked by examining derivatives of (1− t) percentile

• Lemma 2: MPR is equivalent to

(1− t)∂

∂tln

(

∂∂θF−1 (1− t|θ)

∣

∣

∂∂tF−1 (1− t|θ)

∣

∣

)

+ 1 ≥ 0.

3. Information in a production economy, feedback

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• In economy just discussed, asset prices contain information

• Information is used by economic agents (the “builder”) to make

decisions that affect cash flows

• But asset prices are unaffected by these decisions (no “feedback”)

• The last assumption makes life much easier

• But in many cases is unlikely to hold

• In final part of talk, consider case with feedback

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• Based on “Government intervention and information aggregation by

prices,” with Itay Goldstein, JF 2015

• “Market-based corrective actions,” with Itay Goldstein and Edward S.

Prescott, RFS 2010, is also closely related

• But one of the things we tried to do in the more recent paper is to write

everything in a very standard model

• Back to CARA-normal models based on Grossman-Stiglitz

(1980)/Hellwig (1980)

• Key innovation: cash flow of traded asset is endogenously determined

• specifically, determined by actions of imperfectly informed agents

Model

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• Investors observe noisy signals si = θ + noise

• θ is a state variable

• Noisy supply −Z

• Trade shares in a firm, at (equilibrium) price P

• Govt sees price, also own signal sG = θ + noise, then makes an

intervention decision

• Cash flows realized

• Note

• noise terms are independent, and normal

• investors have CARA utility

Govt objective and firm cash flow

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• Write T for government’s intervention

• Cost of T for government is µT

• Benefit of intervention is v (T − θ) where v is concave

• (more detail in next slide)

• θ state variable representing economic conditions, unobserved by govt

• So govt chooses T to maximize

E [v (T − θ) |sG, P ]− µT

where sG is noisy signal of θ, recall P is price

• Firm cash flow is

T + δ

where δ is exogenous, unforecastable, normal, mean δ̄

Examples

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1. Bank lending

• s (x) is marginal social surplus of xth dollar lent by the bank

• θ is bank funds absent intervention

• bank lends θ + T

• So social surplus associated with injection T is

∫ θ+T

0s (x) dx = v (θ + T )− v (0) .

where v is anti-derivative of s

2. Firm is a bank, financial system fragility, T decreases fragility, value of

decreased fragility is v (T − θ)

3. Firm has externalities (on workers, for example), value of externalities

is v (T − θ)

Other applications

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• Firm decisions

• Central banks and interest rates

• Exchange rates

• Focus here on govt interventions in firms

Equilibrium (REE)

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• An equilibrium is a price mapping P : ℜ2 → ℜ

• price for each (θ, Z)

• In an REE, each investor’s demand for the stock is xi (si, P )

• note: demand based on understanding of how cash flow will

respond to price P in eq

• The equilibrium condition is that the market clears, ie, supply =

demand, ie∫

xi (si, P ) di = −Z

Govt decision

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• Key result: endogenous cash flow is normally distributed

• Govt chooses T according to FOC

E[

v′ (T − θ) |sG, P]

= µ

• Guess and verify that equilibrium price is linear in state variable θ and

supply shock Z

• Given linearity, distribution of θ conditional on sG, P is normal

• Normal distribution fully characterized by mean and variance

• So E [v′ (T − θ) |sG, P ] is a function of E [θ|sG, P ] and var [θ|sG, P ]

G (E [T − θ|sG, P ] , var [T − θ|sG, P ]) = µ

• Consequently, can write FOC as

E [T − θ|sG, P ] = g (var [T − θ|sG, P ])

Govt decision (continued)

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• So

T = E [θ|sG, P ] + g (var [θ|sG, P ])

• So endogenous cash flow T + δ is linear in expectation of θ

• Standard arguments then confirm existence of linear equilibrium,

justifying guess-and-verify

• Intervention T is increasing in govt’s belief about θ (normalization)

Price informativeness

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• Eq price P = p̄+ pθθ + pzz

• As in part 1 of talk, relevant measure of price efficiency is ρ ≡ pθpz

• Note: measures how much info P contains about θ, not about CF

• so this is no longer the standard definition used in empirical finance

• by same argument as before, market clearing implies

ρ =1

γ

∂∂siE [T |si, P ]

var [δ + T |si, P ]

• Reflects aggressiveness of speculator trading

• Numerator: more valuable information → more aggressive trading

• Denominator: more risk → less aggressive trading

• Note fixed point problem: conditional moments of T depend on ρ

Price informativeness

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• Intervention T is a linear function of E [θ|sG, P ]• So by normal-normal updating, can write intervention as

T = wsG + function of price

where w ∈ [0, 1] is weight the govt puts on its own info sG• Without commitment, w derived from Bayes’ rule, depends on price

informativeness ρ

• Equilibrium ρ solves:

ρ =1

γ

w (ρ) ∂∂siE [θ|si, P ]

w (ρ)2 (var [θ|si, P ] + var [εG]) + var [δ]

• Speculators trade because si has info about sG, which affects T

• Trading → price P that aggregates speculator info about θ

• So govt pays attention to P as well as to sG

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• The fact that the govt uses info in prices affects price informativeness

• Would government do better by following market less? Or more?

Price informativeness and weight placed on market prices

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• What happens when the government puts more weight on its own

information, and less on market prices?

• formally, consider commitment to rules(

w,KP , T̄)

• price informativeness still given by

ρ =1

γ

w ∂∂siE [θ|si, P ]

w2 (var [θ|si, P ] + var [εG]) + var [δ]

• rule only affects price informativeness via w

• Fix an equilibrium ρ∗

• w (ρ∗) govt’s weight on own info under Bayes rule

• Definition: Govt follows the market too much if payoff would be higher

if it commited to put marginally more weight on its own information,

w > w (ρ∗)

Price informativeness and weight placed on market prices

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ρ =1

γ

w ∂∂siE [θ|si, P ]

w2 (var [θ|si, P ] + var [εG]) + var [δ]

• Offsetting effects:

• Information importance:

Increase in w makes speculator info more important, increases ρ

• Residual risk :

Increase in w makes intervention less forecastable by price,

increases residual risk speculator is exposed to when he trades,

reduces ρ

• Information importance is dominant effect whenever most risk is

exogenous, ie, due to δ

Implication 1: Committing to underweight the price

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• When information importance is the dominant effect,

• then committing to underweight the price/overweight own info

• increases price informativeness

• This is good for govt welfare: intervention is now more efficient

• Govt shouldn’t completely follow the market

• Don’t take this effect to extreme though: completely ignoring the price

often maximizes price informativeness,

• but if govt ignores price, don’t care about price informativeness

Limits to informativeness

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• Low risk-aversion and/or low supply variance var[Z] make price very

informative

• Most residual risk is then exogenous

• So govt should commit to underweight price precisely when

informative

• Imprecise govt information makes w very small

• So most risk exogenous

• So govt should commit to underweight price and overweight own info

precisely when own info is uninformative

Implication 2: Complementarity of govt info and prices

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• Given the availability of free information in market prices, a natural

conjecture is that this reduces the govt’s benefit from collecting

additional information

• But there is a significant countervailing effect

• Recall that in many cases the govt would like to commit to overweight

its info

• How can it do this?

• By increasing quality of own info, which then raises the weight w

• So govt info and market info are complements

Implication 3: Transparency

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• Example: Should govt publicly release results of a bank stress-test?

• Information corresponds to sG in model

• Model prediction: This type of transparency is bad!

• Why?

• After sG is disclosed, speculators know what the intervention T will

be

• So their signals si are no longer useful in forecasting the cash flow

• So stop trading on information

• Price no longer provides the govt with any useful info

Implication 3: Transparency (continued)

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• In contrast, govt should release dimensions of info in which it has finer

information than speculators

• Model extension: benefit to intervention is v (T − ψ − θ)

• Govt sees a signal σG of ψ

• Speculators see only coarser signals σi of σG

• Examples:

• ψ is govt policy objective, so govt naturally knows more

• ψ is condition of aggregate economy, plausibly govt knows more

than speculators

• Prop 5: In contrast to previous case, govt should release its signal σGof ψ

Transparency about ψ

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• Limit case in which speculators see infinitely noisy signal of σG is easy

• equilibrium takes same form as before

• transparency reduces residual risk

• transparency has no effect on information importance:

T = E[θ|P, sG] +E[ψ|σG] + other terms

• so price informativeness improved

Transparency about ψ

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• Non-limit case is harder

• Now, can show equilibrium is form P = p0 + ρpZθ + ξρpZσG + pZZ

• because govt sees σG, the ratio ρ remains the right measure of

price informativeness

• as in limit case, easy to see that transparency reduces residual risk

• information importance is harder: T = E[θ|P, sG, σG] +E[ψ|σG] + ...

• si and σi both affect speculator’s expectation of E[ψ|σG] ...

• ... and of E[θ|P, sG, σG]

• So transparency affects sensitivity of E [T |si, σi, P ] to si• Does sensitivity increase or decrease? Conflicting effects.

• Result: Transparency increases sensitivity, hence increases

information importance effect

• (proof is bunch of matrix algebra)

• loosely speaking, transparency lets speculator focus on using si to

predict sG

Implication 4: Tradeoff between intervention subsidy and price

informativeness

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• So far, intervention is pure subsidy, ie, CF is T + δ

• Now, suppose instead that CF is (1−R)T + δ

• so R is gross interest rate on cash injection

• No subsidy, R = 1

• traders don’t care what govt does

• so don’t trade on their information

• so price contains no information

• Subsidy, R < 1

• price is now informative

• So govt faces tradeoff between degree of subsidy and price

informativeness

• (under many conditions, this tradeoff extends beyond the region

R = 1)

Implication 5: How to measure price informativeness?

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• A more methodological point

• Need to be careful about correct definition of price informativeness

• Traditional finance defn of price informativeness is

var [cash flow|P ]−1 = var [δ + T |P ]−1

• Our defn: how much does the price P tell the decision-maker

(government) about relevant state variable (θ)

• defn driven by model that says why we care about informativeness

• Important point: two measures do not always move together

Example 1

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• Consider effect of moving to a pure price-based intervention rule

• So price perfectly forecasts govt action

• Traditional measure of price informativeness attains its upper bound,

var[δ]−1

• But the relevant measure attains its lower bound, since from

market-clearing condition price is now given by

1

γ

T (P ) + δ − P

var [δ]+ Z = 0

and hence is function of Z only

Example 2

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• Consider again benefits of transparency about sG• Relevant measure of price informativeness attains its lower bound,

since price now contains no useful info

• But when var[δ] small, traditional measure of price informativeness

approaches its upper bound, since price satisfies

1

γ

T (sG, P ) + δ − P

var [δ]+ Z = 0

and so almost perfectly forecasts T

• Application: don’t evaluate success of stress tests by using traditional

measure of informativeness

Summary

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• Amount and value of information produced by financial markets

• Looked at settings in which financial markets aggregate information

• In benchmark endowment economy, net effect on risk sharing is

negative

• In production economies, information also informs “real” decisions

• characterize conditions under which bailouts increase (decrease)

the information value of asset prices

• analyze information aggregation when real decisions both respond

to and affect asset prices

• “limits to informativeness”: govt should underweight market

information precisely when it would seem most useful

Summary

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• Discussion of analytical techniques

• use of market clearing condition to extract clean equilibrium

characterizations

• Lehmann measure of information content, and its equivalence to

single-crossing

• maintaining cash flow normality in models with endogenous cash

flows