understanding ``consensus' - the jouini-napp model of ...markus/teaching/eco525/21...n...

Introduction and motivationModel and proofs

Understanding “Consensus"The Jouini-Napp Model of Belief Aggregation

Markus Brunnermeier E. Glen Weyl

Department of EconomicsPrinceton University

Financial Economics IFall 2006

Jouini and Napp (2006) Brunnermeier Slides on Jouini-Napp

QuestionsFirst examplePessimism and risk-aversion

Why study belief aggregation?

Market prices embed consensus set of beliefs amonginvestors?What information do we need to extract beliefs?Standard model assumes“market consensus". What islost?What happens if we us standard model to understandworld with heterogeneous beliefs?False beliefs, pessimism and first-order risk-aversion

A simple example of belief aggregation

Two-periods, two states of the world ωgood and ωbad in thesecond period.

N investors with CRRA utility (i.e. ui(w) = −e−wθi , θi is

individual i ’s constant absolute risk-tolerance), nodiscounting.Complete markets (two securities, each paying off 1 unit ofconsumption in each state).Investors have different beliefs; investor i believes the goodstate happens with probability πi 6= πj .Both investors endowed with 2 in the good state, 1 in thebad state and 1.5 today.

Solution concept

Solve for equivalent common belief equilibrium (ECBE). ECBEmeans:

1 Investors have a common set of beliefs, as in standardmodel.

2 All prices (i.e. investor marginal valuations andbehavior)same as in the original equilibrium.

3 Trading is the same.4 Aggregate, but not individual, endowment the same.

Heterogeneous beliefs =⇒ heterogeneous endowments.

Solution concept

The old way to solve

We could solve by brute force:Solve the consumers’ optimization problem given anyprices.Find the prices that clear the market.Find a probability measure leading to a ECBE.Adjust individual endowments to match the equilibrium

BUT tedious! Instead, we use a quicker and simpler method.

An easier way

Directly use condition of consistent marginal valuations.

Let y ji be the wealth that i buys in state j in the original

equilibrium.Let y j

i be the wealth she buys in the ECBE.Also, let π represent the common belief of the probability ofthe good state.

∀i : πiu′i

(ygood

)= πu′

(ygood

)∀i : (1− πi)u′

)= (1− π)u′

An easier way

equilibrium.Let y j

∀i : πiu′i

(ygood

)= πu′

(ygood

)∀i : (1− πi)u′

)= (1− π)u′

An easier way

equilibrium.Let y j

∀i : πiu′i

(ygood

)= πu′

(ygood

)∀i : (1− πi)u′

)= (1− π)u′

An easier way

equilibrium.Let y j

∀i : πiu′i

(ygood

)= πu′

(ygood

)∀i : (1− πi)u′

)= (1− π)u′

An easier way

equilibrium.Let y j

∀i : πiu′i

(ygood

)= πu′

(ygood

)∀i : (1− πi)u′

)= (1− π)u′

An easier way

equilibrium.Let y j

∀i : πiu′i

(ygood

)= πu′

(ygood

)∀i : (1− πi)u′

)= (1− π)u′

Solving

By simple differentiation we have that u′i (x) = e

− xθi

θi. Property 4

=⇒ aggregate endowment the same. We exploit this:Define θ ≡

∑i θi .

We raise the equations for individual i to the power θiθ

toput the argument of their marginal utility in “commoncurrency". We obtain:

∀i , πθiθ

i e−ygoodi

θ = πθiθ e−

ygoodiθ

Corresponding equation for bad state.

Solving

− xθi

θi. Property 4

∑i θi .

∀i , πθiθ

i e−ygoodi

θ = πθiθ e−

ygoodiθ

Solving

− xθi

θi. Property 4

∑i θi .

∀i , πθiθ

i e−ygoodi

θ = πθiθ e−

ygoodiθ

Solving continued...

Multiplying:

πθ1θ

1 πθ2θ

2 e−ygood1 +ygood

2θ = πe−

ygood1 +ygood

But aggregate endowments the same:

π =∏

i πθiθ

A corresponding formula 1− π =∏

i(1− πi)θiθ applies to the

bad state.

Multiplying:

πθ1θ

1 πθ2θ

2 e−ygood1 +ygood

2θ = πe−

ygood1 +ygood

π =∏

i πθiθ

bad state.

Multiplying:

πθ1θ

1 πθ2θ

2 e−ygood1 +ygood

2θ = πe−

ygood1 +ygood

π =∏

i πθiθ

bad state.

Multiplying:

πθ1θ

1 πθ2θ

2 e−ygood1 +ygood

2θ = πe−

ygood1 +ygood

π =∏

i πθiθ

bad state.

Interpreting the solution

Note:pi is not a probability measure. Suppose that there are justtwo investors, θ1 = θ2 = 1 and π1 = 1− π2 = .9. Thenpi = 1− pi = .18 .5. Not clear how we should interpretthese “common beliefs".Effect on common belief proportional to fraction of totalrisk-bearing; those willing to bet more on their beliefsshape market beliefs more.Common belief is weighted geometric mean of theindividual beliefs.

What about endowments?

We can also determine how endowments differ in the ECBE:

πie−

ygoodiθi =

j e−ygood

Taking the logarithm, arithmetic yield:

ygoodi − ygood

i = θi∑j 6=i

[log(πj)− log(πi)

]Property 3, =⇒ :

ei − ei = θi∑

j 6=iθj

]= θi

[log(π)− log(πi)

]Thus, heterogeneous beliefs =⇒ heterogeneousendowments.

πie−

ygoodiθi =

j e−ygood

ygoodi − ygood

i = θi∑j 6=i

]Property 3, =⇒ :

ei − ei = θi∑

j 6=iθj

]= θi

πie−

ygoodiθi =

j e−ygood

ygoodi − ygood

i = θi∑j 6=i

]Property 3, =⇒ :

ei − ei = θi∑

j 6=iθj

]= θi

πie−

ygoodiθi =

j e−ygood

ygoodi − ygood

i = θi∑j 6=i

]Property 3, =⇒ :

ei − ei = θi∑

j 6=iθj

]= θi

πie−

ygoodiθi =

j e−ygood

ygoodi − ygood

i = θi∑j 6=i

]Property 3, =⇒ :

ei − ei = θi∑

j 6=iθj

]= θi

πie−

ygoodiθi =

j e−ygood

ygoodi − ygood

i = θi∑j 6=i

]Property 3, =⇒ :

ei − ei = θi∑

j 6=iθj

]= θi

Interpreting endowment shift

Change in belief proportional to: difference fromconsensus and risk-tolerance.If individual has higher probability on a state than thecommon belief, she tends to trade towards this state.Rationalized by individual having a lower endowment inthis state(hedging an idiosyncratic risk).Thus standard model will interpret trading due toheterogeneous beliefs as resulting from idiosyncratic risks.

Endowments continued...

Adjustment also proportional to risk tolerance.Those with large risk tolerance will bet a lot on beliefs:large endowment adjustment necessary to rationalizeStandard model: risk-tolerant person with idiosyncraticbeliefs trading because of large idiosyncratic risks

A motivating example of pessimism

Can beliefs generate first-order risk-aversion?Would you consider risk-averse someone who was afraid to fly?This cannot be (economic) “risk-aversion"

If arrives earns uIf he drives he earns disutility from inconvenience vIf he crashes he earns utility 0Two people, both with the same inconvenience from notflying and value to flying but one chooses not to fly, whilethe other chooses to fly.Is this plausible? It seems to happen a lot.

Pessimism example explained

Risk-aversion is a property of utility functions.Parameters equal between the two, economic risk-aversioncannot do it. What is?First individual thinks that crashing is more likely, assigningprobability q, second assigns p < qThen if qu − v > 0, but pu − v < 0 the “risk-averse"individual chooses not to fly, but the “risk-seeking"individual chooses to fly.Thus “risk-aversion" in standard parlance is about beliefs,not utility. It is a sort of pessimism.“Luck is out to get me" or “nothing ever goes my way."Can we incorporate into finance?

Pessimism in the model

A try:Suppose objective probability π? of the good stateoccurring.Individual i “first-order pessimistic" if πi < π?. Thisdefinition quite obvious here, but more general later.Does pessimism affect the risk-free rate in the same wayas risk-aversion? Market price of risk?

Solving the model with pessimism

Sssume there only one investor.Risk-bearing capacity θ and belief about the probability ofthe good state π.

Marginal utility of consumption today is e−3

Expected marginal utility tomorrow is πe−2θ +(1−π)e−

θ .Thus the risk free rate:

πe−2θ +(1−π)e−

e−2θ < e−

1θ , so risk free rate is clearly decreasing in π.

If π < π? then the risk-free rate is higher than if beliefs arecorrect.Thus pessimism increases risk-free rate.

πe−2θ +(1−π)e−

e−2θ < e−

πe−2θ +(1−π)e−

e−2θ < e−

πe−2θ +(1−π)e−

e−2θ < e−

πe−2θ +(1−π)e−

e−2θ < e−

πe−2θ +(1−π)e−

e−2θ < e−

πe−2θ +(1−π)e−

e−2θ < e−

πe−2θ +(1−π)e−

e−2θ < e−

πe−2θ +(1−π)e−

e−2θ < e−

More on pessimism

What effect does pessimism have on the pricing kernel underobjective measure? The value (objective) value of the pricingkernel in the good state is:

πe−2θ

π?e−3

And in the bad state:

(1− π)e−1θ

(1− π?)e−3

Kernel in good state increasing in π.Kernel in bad state is decreasing.Pessimism decreases the kernel in the good state,increases it in the bad state.

More on pessimism

πe−2θ

π?e−3

(1− π)e−1θ

(1− π?)e−3

More on pessimism

πe−2θ

π?e−3

(1− π)e−1θ

(1− π?)e−3

More on pessimism

πe−2θ

π?e−3

(1− π)e−1θ

(1− π?)e−3

More on pessimism

πe−2θ

π?e−3

(1− π)e−1θ

(1− π?)e−3

More on pessimism

πe−2θ

π?e−3

(1− π)e−1θ

(1− π?)e−3

More on pessimism

πe−2θ

π?e−3

(1− π)e−1θ

(1− π?)e−3

Interpreting pessimism

Leads to a greater “separation" of the price kernel for thetwo states. This is an increase in the “risk-premium", Thus,similar effect on market prices to risk aversion.Optimism has opposite effect; in fact, enough optimism canlead the good state kernel to be higher than the low-statekernel, i.e. risk-seeking.Are there differences?

A problem with pessimism

Suppose two individualsEach has utility u(c) = log cIf coin comes up heads, first endowed with 18, second with2If tails, first endowed with 9, second with 1.Suppose fair coin. Person 1 knows this; person 2 isdrastically pessimistic: believes that probability is .9 thatcomes up tails.

Solving the problematic pessimism example

Normalize price of consumption in good state to 1.First individual spends 1

2 of his wealth on consumption ingood state; has 9

10 of total perfectly correlated wealth.

Second spends 110 and has 1

10 .Let w ≡ 20 + 10p2, where ptails is price of consumption inthe tails state.Market clearing implies

20 + 1100

)w = 20.

w = 100023 = 20 + 10p2 =⇒ p2 = 54

Investor 2 buys c12 = 100

207 ≈12 and c2

2 = 53 = 10

3 c12 .

23 ≈ 1912 and c2

1 = 612 > 3

20 + 1100

)w = 20.

w = 100023 = 20 + 10p2 =⇒ p2 = 54

207 ≈12 and c2

2 = 53 = 10

3 c12 .

23 ≈ 1912 and c2

1 = 612 > 3

20 + 1100

)w = 20.

w = 100023 = 20 + 10p2 =⇒ p2 = 54

207 ≈12 and c2

2 = 53 = 10

3 c12 .

23 ≈ 1912 and c2

1 = 612 > 3

20 + 1100

)w = 20.

w = 100023 = 20 + 10p2 =⇒ p2 = 54

207 ≈12 and c2

2 = 53 = 10

3 c12 .

23 ≈ 1912 and c2

1 = 612 > 3

20 + 1100

)w = 20.

w = 100023 = 20 + 10p2 =⇒ p2 = 54

207 ≈12 and c2

2 = 53 = 10

3 c12 .

23 ≈ 1912 and c2

1 = 612 > 3

20 + 1100

)w = 20.

w = 100023 = 20 + 10p2 =⇒ p2 = 54

207 ≈12 and c2

2 = 53 = 10

3 c12 .

23 ≈ 1912 and c2

1 = 612 > 3

20 + 1100

)w = 20.

w = 100023 = 20 + 10p2 =⇒ p2 = 54

207 ≈12 and c2

2 = 53 = 10

3 c12 .

23 ≈ 1912 and c2

1 = 612 > 3

20 + 1100

)w = 20.

w = 100023 = 20 + 10p2 =⇒ p2 = 54

207 ≈12 and c2

2 = 53 = 10

3 c12 .

23 ≈ 1912 and c2

1 = 612 > 3

What does this show us about pessimism?

Thus pessimistic individual has less balanced portfolio!Very different from risk aversion...bets on bad outcomes.Is this interesting?Alternatively, could have individuals that believe things goagainst them. Might generate behavior more like riskaversion, but like non-expected utility.

How does it generalize?

These were nice toy examples. But how do the insightsgeneralize? How do we define concepts in a fully generalrigorous manner?

Belief aggregationCCAPM

Assumptions of model

N agents, T periodsDifferentiable, concave, increasing utilityEndowments ei

t bounded below w.p.1 and aboveIndividual probability Qi

Complete markets: optimal individual consumption y?i

We look for ECBE with common “characteristic" M,endowments ei and consumption y i such that:

1 Qtu′i (t , y i) = Qi

tu′i (t , y?i

) ∀t , i2 y?i − ei = y i − ei ∀i

Can be shown that this always exists and is unique.

tu′i (t , y?i

) ∀t , i2 y?i − ei = y i − ei ∀i

tu′i (t , y?i

) ∀t , i2 y?i − ei = y i − ei ∀i

tu′i (t , y?i

) ∀t , i2 y?i − ei = y i − ei ∀i

Some formulae

Already derived formula with exponential utility. Ageneralization:

Suppose utility HARA with common slope:fracu′u′′ = θi + ηxLet λi be the lagrange multiplier in i ’s max

Let γi ≡λ−η

iPj λ−η

Then Q is given by:

Q =[ ∑

i γi(Qi)η] 1

Some formulae

Let γi ≡λ−η

iPj λ−η

Then Q is given by:

Q =[ ∑

i γi(Qi)η] 1

Some formulae

Let γi ≡λ−η

iPj λ−η

Then Q is given by:

Q =[ ∑

i γi(Qi)η] 1

Some formulae

Let γi ≡λ−η

iPj λ−η

Then Q is given by:

Q =[ ∑

i γi(Qi)η] 1

Some formulae

Let γi ≡λ−η

iPj λ−η

Then Q is given by:

Q =[ ∑

i γi(Qi)η] 1

Some formulae

Let γi ≡λ−η

iPj λ−η

Then Q is given by:

Q =[ ∑

i γi(Qi)η] 1

Interpretation

Weighted “mean" of beliefsWeight proportional to wealth (λ−η) as this determines riskbearingCloser to geometric mean as η → 0 (exponential case),arithmetic mean if η = 1, “superarithmetic" if η > 1Therefore η < 1 =⇒ Q(Ω) < 1, η = 1 =⇒ Q(Ω = 1),η > 1 =⇒ Q(Ω) > 1This is aggregation “bias", cumulates over time

Define Bt by B0 = 1, Bt = Bt−1Et−1[Qt ]

Qt−1; then B is a sort of

deflatorThen Q ≡ Q

B is a belief

Interpretation

B is a belief

Interpretation

B is a belief

Interpretation

B is a belief

Interpretation

B is a belief

Interpretation

B is a belief

Interpretation

B is a belief

Setting up the CCAPM

Only skimmed over asset pricing in our example. Let’s solve itgenerally here:

Objective measure P

Riskless asset S0 with rate r ft+1 ≡

S0t+1S0

t− 1 and risky assets

with prices Sit and return R i

t+1 ≡Si

t− 1

If π? is equilibrium price deflator process (under P) thenπ?S is martingale

Thus EPt [Rt+1]− r f

t+1 = −covPt

t [π?t+1]

, Rt+1

Objective measure P

S0t+1S0

t+1 ≡Si

t− 1

t+1 = −covPt

t [π?t+1]

, Rt+1

Objective measure P

S0t+1S0

t+1 ≡Si

t− 1

t+1 = −covPt

t [π?t+1]

, Rt+1

Objective measure P

S0t+1S0

t+1 ≡Si

t− 1

t+1 = −covPt

t [π?t+1]

, Rt+1

Objective measure P

S0t+1S0

t+1 ≡Si

t− 1

t+1 = −covPt

t [π?t+1]

, Rt+1

understanding ``consensus' - the jouini-napp model of ...markus/teaching/eco525/21...n...

Documents