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
Introduction and motivationModel and proofs
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
Jouini and Napp (2006) Brunnermeier Slides on Jouini-Napp
Introduction and motivationModel and proofs
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
Jouini and Napp (2006) Brunnermeier Slides on Jouini-Napp
Introduction and motivationModel and proofs
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
Jouini and Napp (2006) Brunnermeier Slides on Jouini-Napp
Introduction and motivationModel and proofs
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
Jouini and Napp (2006) Brunnermeier Slides on Jouini-Napp
Introduction and motivationModel and proofs
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
Jouini and Napp (2006) Brunnermeier Slides on Jouini-Napp
Introduction and motivationModel and proofs
QuestionsFirst examplePessimism and 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.
Jouini and Napp (2006) Brunnermeier Slides on Jouini-Napp
Introduction and motivationModel and proofs
QuestionsFirst examplePessimism and 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.
Jouini and Napp (2006) Brunnermeier Slides on Jouini-Napp
Introduction and motivationModel and proofs
QuestionsFirst examplePessimism and 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.
Jouini and Napp (2006) Brunnermeier Slides on Jouini-Napp
Introduction and motivationModel and proofs
QuestionsFirst examplePessimism and 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.
Jouini and Napp (2006) Brunnermeier Slides on Jouini-Napp
Introduction and motivationModel and proofs
QuestionsFirst examplePessimism and 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.
Jouini and Napp (2006) Brunnermeier Slides on Jouini-Napp
Introduction and motivationModel and proofs
QuestionsFirst examplePessimism and risk-aversion
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.
Jouini and Napp (2006) Brunnermeier Slides on Jouini-Napp
Introduction and motivationModel and proofs
QuestionsFirst examplePessimism and risk-aversion
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.
Jouini and Napp (2006) Brunnermeier Slides on Jouini-Napp
Introduction and motivationModel and proofs
QuestionsFirst examplePessimism and risk-aversion
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.
Jouini and Napp (2006) Brunnermeier Slides on Jouini-Napp
Introduction and motivationModel and proofs
QuestionsFirst examplePessimism and risk-aversion
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.
Jouini and Napp (2006) Brunnermeier Slides on Jouini-Napp
Introduction and motivationModel and proofs
QuestionsFirst examplePessimism and risk-aversion
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.
Jouini and Napp (2006) Brunnermeier Slides on Jouini-Napp
Introduction and motivationModel and proofs
QuestionsFirst examplePessimism and risk-aversion
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.
Jouini and Napp (2006) Brunnermeier Slides on Jouini-Napp
Introduction and motivationModel and proofs
QuestionsFirst examplePessimism and risk-aversion
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.
Jouini and Napp (2006) Brunnermeier Slides on Jouini-Napp
Introduction and motivationModel and proofs
QuestionsFirst examplePessimism and risk-aversion
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.
Jouini and Napp (2006) Brunnermeier Slides on Jouini-Napp
Introduction and motivationModel and proofs
QuestionsFirst examplePessimism and risk-aversion
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.
Jouini and Napp (2006) Brunnermeier Slides on Jouini-Napp
Introduction and motivationModel and proofs
QuestionsFirst examplePessimism and risk-aversion
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.
Jouini and Napp (2006) Brunnermeier Slides on Jouini-Napp
Introduction and motivationModel and proofs
QuestionsFirst examplePessimism and risk-aversion
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.
Jouini and Napp (2006) Brunnermeier Slides on Jouini-Napp
Introduction and motivationModel and proofs
QuestionsFirst examplePessimism and risk-aversion
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.
Jouini and Napp (2006) Brunnermeier Slides on Jouini-Napp
Introduction and motivationModel and proofs
QuestionsFirst examplePessimism and risk-aversion
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
i
)= πu′
i
(ygood
i
)∀i : (1− πi)u′
i
(ybad
i
)= (1− π)u′
i
(ybad
i
)
Jouini and Napp (2006) Brunnermeier Slides on Jouini-Napp
Introduction and motivationModel and proofs
QuestionsFirst examplePessimism and risk-aversion
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
i
)= πu′
i
(ygood
i
)∀i : (1− πi)u′
i
(ybad
i
)= (1− π)u′
i
(ybad
i
)
Jouini and Napp (2006) Brunnermeier Slides on Jouini-Napp
Introduction and motivationModel and proofs
QuestionsFirst examplePessimism and risk-aversion
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
i
)= πu′
i
(ygood
i
)∀i : (1− πi)u′
i
(ybad
i
)= (1− π)u′
i
(ybad
i
)
Jouini and Napp (2006) Brunnermeier Slides on Jouini-Napp
Introduction and motivationModel and proofs
QuestionsFirst examplePessimism and risk-aversion
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
i
)= πu′
i
(ygood
i
)∀i : (1− πi)u′
i
(ybad
i
)= (1− π)u′
i
(ybad
i
)
Jouini and Napp (2006) Brunnermeier Slides on Jouini-Napp
Introduction and motivationModel and proofs
QuestionsFirst examplePessimism and risk-aversion
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
i
)= πu′
i
(ygood
i
)∀i : (1− πi)u′
i
(ybad
i
)= (1− π)u′
i
(ybad
i
)
Jouini and Napp (2006) Brunnermeier Slides on Jouini-Napp
Introduction and motivationModel and proofs
QuestionsFirst examplePessimism and risk-aversion
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
i
)= πu′
i
(ygood
i
)∀i : (1− πi)u′
i
(ybad
i
)= (1− π)u′
i
(ybad
i
)
Jouini and Napp (2006) Brunnermeier Slides on Jouini-Napp
Introduction and motivationModel and proofs
QuestionsFirst examplePessimism and risk-aversion
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.
Jouini and Napp (2006) Brunnermeier Slides on Jouini-Napp
Introduction and motivationModel and proofs
QuestionsFirst examplePessimism and risk-aversion
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.
Jouini and Napp (2006) Brunnermeier Slides on Jouini-Napp
Introduction and motivationModel and proofs
QuestionsFirst examplePessimism and risk-aversion
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.
Jouini and Napp (2006) Brunnermeier Slides on Jouini-Napp
Introduction and motivationModel and proofs
QuestionsFirst examplePessimism and risk-aversion
Solving continued...
Multiplying:
πθ1θ
1 πθ2θ
2 e−ygood1 +ygood
2θ = πe−
ygood1 +ygood
2θ
But aggregate endowments the same:
π =∏
i πθiθ
i
A corresponding formula 1− π =∏
i(1− πi)θiθ applies to the
bad state.
Jouini and Napp (2006) Brunnermeier Slides on Jouini-Napp
Introduction and motivationModel and proofs
QuestionsFirst examplePessimism and risk-aversion
Solving continued...
Multiplying:
πθ1θ
1 πθ2θ
2 e−ygood1 +ygood
2θ = πe−
ygood1 +ygood
2θ
But aggregate endowments the same:
π =∏
i πθiθ
i
A corresponding formula 1− π =∏
i(1− πi)θiθ applies to the
bad state.
Jouini and Napp (2006) Brunnermeier Slides on Jouini-Napp
Introduction and motivationModel and proofs
QuestionsFirst examplePessimism and risk-aversion
Solving continued...
Multiplying:
πθ1θ
1 πθ2θ
2 e−ygood1 +ygood
2θ = πe−
ygood1 +ygood
2θ
But aggregate endowments the same:
π =∏
i πθiθ
i
A corresponding formula 1− π =∏
i(1− πi)θiθ applies to the
bad state.
Jouini and Napp (2006) Brunnermeier Slides on Jouini-Napp
Introduction and motivationModel and proofs
QuestionsFirst examplePessimism and risk-aversion
Solving continued...
Multiplying:
πθ1θ
1 πθ2θ
2 e−ygood1 +ygood
2θ = πe−
ygood1 +ygood
2θ
But aggregate endowments the same:
π =∏
i πθiθ
i
A corresponding formula 1− π =∏
i(1− πi)θiθ applies to the
bad state.
Jouini and Napp (2006) Brunnermeier Slides on Jouini-Napp
Introduction and motivationModel and proofs
QuestionsFirst examplePessimism and risk-aversion
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.
Jouini and Napp (2006) Brunnermeier Slides on Jouini-Napp
Introduction and motivationModel and proofs
QuestionsFirst examplePessimism and risk-aversion
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.
Jouini and Napp (2006) Brunnermeier Slides on Jouini-Napp
Introduction and motivationModel and proofs
QuestionsFirst examplePessimism and risk-aversion
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.
Jouini and Napp (2006) Brunnermeier Slides on Jouini-Napp
Introduction and motivationModel and proofs
QuestionsFirst examplePessimism and risk-aversion
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.
Jouini and Napp (2006) Brunnermeier Slides on Jouini-Napp
Introduction and motivationModel and proofs
QuestionsFirst examplePessimism and risk-aversion
What about endowments?
We can also determine how endowments differ in the ECBE:
πie−
ygoodiθi =
∏j
π
θjθ
j e−ygood
iθi
Taking the logarithm, arithmetic yield:
ygoodi − ygood
i = θi∑j 6=i
θj
θ
[log(πj)− log(πi)
]Property 3, =⇒ :
ei − ei = θi∑
j 6=iθj
θ
[log(πj)− log(πi)
]= θi
[log(π)− log(πi)
]Thus, heterogeneous beliefs =⇒ heterogeneousendowments.
Jouini and Napp (2006) Brunnermeier Slides on Jouini-Napp
Introduction and motivationModel and proofs
QuestionsFirst examplePessimism and risk-aversion
What about endowments?
We can also determine how endowments differ in the ECBE:
πie−
ygoodiθi =
∏j
π
θjθ
j e−ygood
iθi
Taking the logarithm, arithmetic yield:
ygoodi − ygood
i = θi∑j 6=i
θj
θ
[log(πj)− log(πi)
]Property 3, =⇒ :
ei − ei = θi∑
j 6=iθj
θ
[log(πj)− log(πi)
]= θi
[log(π)− log(πi)
]Thus, heterogeneous beliefs =⇒ heterogeneousendowments.
Jouini and Napp (2006) Brunnermeier Slides on Jouini-Napp
Introduction and motivationModel and proofs
QuestionsFirst examplePessimism and risk-aversion
What about endowments?
We can also determine how endowments differ in the ECBE:
πie−
ygoodiθi =
∏j
π
θjθ
j e−ygood
iθi
Taking the logarithm, arithmetic yield:
ygoodi − ygood
i = θi∑j 6=i
θj
θ
[log(πj)− log(πi)
]Property 3, =⇒ :
ei − ei = θi∑
j 6=iθj
θ
[log(πj)− log(πi)
]= θi
[log(π)− log(πi)
]Thus, heterogeneous beliefs =⇒ heterogeneousendowments.
Jouini and Napp (2006) Brunnermeier Slides on Jouini-Napp
Introduction and motivationModel and proofs
QuestionsFirst examplePessimism and risk-aversion
What about endowments?
We can also determine how endowments differ in the ECBE:
πie−
ygoodiθi =
∏j
π
θjθ
j e−ygood
iθi
Taking the logarithm, arithmetic yield:
ygoodi − ygood
i = θi∑j 6=i
θj
θ
[log(πj)− log(πi)
]Property 3, =⇒ :
ei − ei = θi∑
j 6=iθj
θ
[log(πj)− log(πi)
]= θi
[log(π)− log(πi)
]Thus, heterogeneous beliefs =⇒ heterogeneousendowments.
Jouini and Napp (2006) Brunnermeier Slides on Jouini-Napp
Introduction and motivationModel and proofs
QuestionsFirst examplePessimism and risk-aversion
What about endowments?
We can also determine how endowments differ in the ECBE:
πie−
ygoodiθi =
∏j
π
θjθ
j e−ygood
iθi
Taking the logarithm, arithmetic yield:
ygoodi − ygood
i = θi∑j 6=i
θj
θ
[log(πj)− log(πi)
]Property 3, =⇒ :
ei − ei = θi∑
j 6=iθj
θ
[log(πj)− log(πi)
]= θi
[log(π)− log(πi)
]Thus, heterogeneous beliefs =⇒ heterogeneousendowments.
Jouini and Napp (2006) Brunnermeier Slides on Jouini-Napp
Introduction and motivationModel and proofs
QuestionsFirst examplePessimism and risk-aversion
What about endowments?
We can also determine how endowments differ in the ECBE:
πie−
ygoodiθi =
∏j
π
θjθ
j e−ygood
iθi
Taking the logarithm, arithmetic yield:
ygoodi − ygood
i = θi∑j 6=i
θj
θ
[log(πj)− log(πi)
]Property 3, =⇒ :
ei − ei = θi∑
j 6=iθj
θ
[log(πj)− log(πi)
]= θi
[log(π)− log(πi)
]Thus, heterogeneous beliefs =⇒ heterogeneousendowments.
Jouini and Napp (2006) Brunnermeier Slides on Jouini-Napp
Introduction and motivationModel and proofs
QuestionsFirst examplePessimism and risk-aversion
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.
Jouini and Napp (2006) Brunnermeier Slides on Jouini-Napp
Introduction and motivationModel and proofs
QuestionsFirst examplePessimism and risk-aversion
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.
Jouini and Napp (2006) Brunnermeier Slides on Jouini-Napp
Introduction and motivationModel and proofs
QuestionsFirst examplePessimism and risk-aversion
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.
Jouini and Napp (2006) Brunnermeier Slides on Jouini-Napp
Introduction and motivationModel and proofs
QuestionsFirst examplePessimism and risk-aversion
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.
Jouini and Napp (2006) Brunnermeier Slides on Jouini-Napp
Introduction and motivationModel and proofs
QuestionsFirst examplePessimism and risk-aversion
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
Jouini and Napp (2006) Brunnermeier Slides on Jouini-Napp
Introduction and motivationModel and proofs
QuestionsFirst examplePessimism and risk-aversion
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
Jouini and Napp (2006) Brunnermeier Slides on Jouini-Napp
Introduction and motivationModel and proofs
QuestionsFirst examplePessimism and risk-aversion
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
Jouini and Napp (2006) Brunnermeier Slides on Jouini-Napp
Introduction and motivationModel and proofs
QuestionsFirst examplePessimism and risk-aversion
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.
Jouini and Napp (2006) Brunnermeier Slides on Jouini-Napp
Introduction and motivationModel and proofs
QuestionsFirst examplePessimism and risk-aversion
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.
Jouini and Napp (2006) Brunnermeier Slides on Jouini-Napp
Introduction and motivationModel and proofs
QuestionsFirst examplePessimism and risk-aversion
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.
Jouini and Napp (2006) Brunnermeier Slides on Jouini-Napp
Introduction and motivationModel and proofs
QuestionsFirst examplePessimism and risk-aversion
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.
Jouini and Napp (2006) Brunnermeier Slides on Jouini-Napp
Introduction and motivationModel and proofs
QuestionsFirst examplePessimism and risk-aversion
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.
Jouini and Napp (2006) Brunnermeier Slides on Jouini-Napp
Introduction and motivationModel and proofs
QuestionsFirst examplePessimism and risk-aversion
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.
Jouini and Napp (2006) Brunnermeier Slides on Jouini-Napp
Introduction and motivationModel and proofs
QuestionsFirst examplePessimism and risk-aversion
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.
Jouini and Napp (2006) Brunnermeier Slides on Jouini-Napp
Introduction and motivationModel and proofs
QuestionsFirst examplePessimism and risk-aversion
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.
Jouini and Napp (2006) Brunnermeier Slides on Jouini-Napp
Introduction and motivationModel and proofs
QuestionsFirst examplePessimism and risk-aversion
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?
Jouini and Napp (2006) Brunnermeier Slides on Jouini-Napp
Introduction and motivationModel and proofs
QuestionsFirst examplePessimism and risk-aversion
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?
Jouini and Napp (2006) Brunnermeier Slides on Jouini-Napp
Introduction and motivationModel and proofs
QuestionsFirst examplePessimism and risk-aversion
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?
Jouini and Napp (2006) Brunnermeier Slides on Jouini-Napp
Introduction and motivationModel and proofs
QuestionsFirst examplePessimism and risk-aversion
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?
Jouini and Napp (2006) Brunnermeier Slides on Jouini-Napp
Introduction and motivationModel and proofs
QuestionsFirst examplePessimism and risk-aversion
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?
Jouini and Napp (2006) Brunnermeier Slides on Jouini-Napp
Introduction and motivationModel and proofs
QuestionsFirst examplePessimism and risk-aversion
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?
Jouini and Napp (2006) Brunnermeier Slides on Jouini-Napp
Introduction and motivationModel and proofs
QuestionsFirst examplePessimism and risk-aversion
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?
Jouini and Napp (2006) Brunnermeier Slides on Jouini-Napp
Introduction and motivationModel and proofs
QuestionsFirst examplePessimism and risk-aversion
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?
Jouini and Napp (2006) Brunnermeier Slides on Jouini-Napp
Introduction and motivationModel and proofs
QuestionsFirst examplePessimism and risk-aversion
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?
Jouini and Napp (2006) Brunnermeier Slides on Jouini-Napp
Introduction and motivationModel and proofs
QuestionsFirst examplePessimism and risk-aversion
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?
Jouini and Napp (2006) Brunnermeier Slides on Jouini-Napp
Introduction and motivationModel and proofs
QuestionsFirst examplePessimism and risk-aversion
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?
Jouini and Napp (2006) Brunnermeier Slides on Jouini-Napp
Introduction and motivationModel and proofs
QuestionsFirst examplePessimism and risk-aversion
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
2θ
θ .
Expected marginal utility tomorrow is πe−2θ +(1−π)e−
1θ
θ .Thus the risk free rate:
πe−2θ +(1−π)e−
1θ
e−3
2θ
− 1
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.
Jouini and Napp (2006) Brunnermeier Slides on Jouini-Napp
Introduction and motivationModel and proofs
QuestionsFirst examplePessimism and risk-aversion
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
2θ
θ .
Expected marginal utility tomorrow is πe−2θ +(1−π)e−
1θ
θ .Thus the risk free rate:
πe−2θ +(1−π)e−
1θ
e−3
2θ
− 1
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.
Jouini and Napp (2006) Brunnermeier Slides on Jouini-Napp
Introduction and motivationModel and proofs
QuestionsFirst examplePessimism and risk-aversion
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
2θ
θ .
Expected marginal utility tomorrow is πe−2θ +(1−π)e−
1θ
θ .Thus the risk free rate:
πe−2θ +(1−π)e−
1θ
e−3
2θ
− 1
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.
Jouini and Napp (2006) Brunnermeier Slides on Jouini-Napp
Introduction and motivationModel and proofs
QuestionsFirst examplePessimism and risk-aversion
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
2θ
θ .
Expected marginal utility tomorrow is πe−2θ +(1−π)e−
1θ
θ .Thus the risk free rate:
πe−2θ +(1−π)e−
1θ
e−3
2θ
− 1
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.
Jouini and Napp (2006) Brunnermeier Slides on Jouini-Napp
Introduction and motivationModel and proofs
QuestionsFirst examplePessimism and risk-aversion
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
2θ
θ .
Expected marginal utility tomorrow is πe−2θ +(1−π)e−
1θ
θ .Thus the risk free rate:
πe−2θ +(1−π)e−
1θ
e−3
2θ
− 1
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.
Jouini and Napp (2006) Brunnermeier Slides on Jouini-Napp
Introduction and motivationModel and proofs
QuestionsFirst examplePessimism and risk-aversion
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
2θ
θ .
Expected marginal utility tomorrow is πe−2θ +(1−π)e−
1θ
θ .Thus the risk free rate:
πe−2θ +(1−π)e−
1θ
e−3
2θ
− 1
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.
Jouini and Napp (2006) Brunnermeier Slides on Jouini-Napp
Introduction and motivationModel and proofs
QuestionsFirst examplePessimism and risk-aversion
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
2θ
θ .
Expected marginal utility tomorrow is πe−2θ +(1−π)e−
1θ
θ .Thus the risk free rate:
πe−2θ +(1−π)e−
1θ
e−3
2θ
− 1
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.
Jouini and Napp (2006) Brunnermeier Slides on Jouini-Napp
Introduction and motivationModel and proofs
QuestionsFirst examplePessimism and risk-aversion
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
2θ
θ .
Expected marginal utility tomorrow is πe−2θ +(1−π)e−
1θ
θ .Thus the risk free rate:
πe−2θ +(1−π)e−
1θ
e−3
2θ
− 1
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.
Jouini and Napp (2006) Brunnermeier Slides on Jouini-Napp
Introduction and motivationModel and proofs
QuestionsFirst examplePessimism and risk-aversion
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
2θ
θ .
Expected marginal utility tomorrow is πe−2θ +(1−π)e−
1θ
θ .Thus the risk free rate:
πe−2θ +(1−π)e−
1θ
e−3
2θ
− 1
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.
Jouini and Napp (2006) Brunnermeier Slides on Jouini-Napp
Introduction and motivationModel and proofs
QuestionsFirst examplePessimism and risk-aversion
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
2θ
And in the bad state:
(1− π)e−1θ
(1− π?)e−3
2θ
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.
Jouini and Napp (2006) Brunnermeier Slides on Jouini-Napp
Introduction and motivationModel and proofs
QuestionsFirst examplePessimism and risk-aversion
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
2θ
And in the bad state:
(1− π)e−1θ
(1− π?)e−3
2θ
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.
Jouini and Napp (2006) Brunnermeier Slides on Jouini-Napp
Introduction and motivationModel and proofs
QuestionsFirst examplePessimism and risk-aversion
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
2θ
And in the bad state:
(1− π)e−1θ
(1− π?)e−3
2θ
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.
Jouini and Napp (2006) Brunnermeier Slides on Jouini-Napp
Introduction and motivationModel and proofs
QuestionsFirst examplePessimism and risk-aversion
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
2θ
And in the bad state:
(1− π)e−1θ
(1− π?)e−3
2θ
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.
Jouini and Napp (2006) Brunnermeier Slides on Jouini-Napp
Introduction and motivationModel and proofs
QuestionsFirst examplePessimism and risk-aversion
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
2θ
And in the bad state:
(1− π)e−1θ
(1− π?)e−3
2θ
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.
Jouini and Napp (2006) Brunnermeier Slides on Jouini-Napp
Introduction and motivationModel and proofs
QuestionsFirst examplePessimism and risk-aversion
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
2θ
And in the bad state:
(1− π)e−1θ
(1− π?)e−3
2θ
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.
Jouini and Napp (2006) Brunnermeier Slides on Jouini-Napp
Introduction and motivationModel and proofs
QuestionsFirst examplePessimism and risk-aversion
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
2θ
And in the bad state:
(1− π)e−1θ
(1− π?)e−3
2θ
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.
Jouini and Napp (2006) Brunnermeier Slides on Jouini-Napp
Introduction and motivationModel and proofs
QuestionsFirst examplePessimism and risk-aversion
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?
Jouini and Napp (2006) Brunnermeier Slides on Jouini-Napp
Introduction and motivationModel and proofs
QuestionsFirst examplePessimism and risk-aversion
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?
Jouini and Napp (2006) Brunnermeier Slides on Jouini-Napp
Introduction and motivationModel and proofs
QuestionsFirst examplePessimism and risk-aversion
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?
Jouini and Napp (2006) Brunnermeier Slides on Jouini-Napp
Introduction and motivationModel and proofs
QuestionsFirst examplePessimism and risk-aversion
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.
Jouini and Napp (2006) Brunnermeier Slides on Jouini-Napp
Introduction and motivationModel and proofs
QuestionsFirst examplePessimism and risk-aversion
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.
Jouini and Napp (2006) Brunnermeier Slides on Jouini-Napp
Introduction and motivationModel and proofs
QuestionsFirst examplePessimism and risk-aversion
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.
Jouini and Napp (2006) Brunnermeier Slides on Jouini-Napp
Introduction and motivationModel and proofs
QuestionsFirst examplePessimism and risk-aversion
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.
Jouini and Napp (2006) Brunnermeier Slides on Jouini-Napp
Introduction and motivationModel and proofs
QuestionsFirst examplePessimism and risk-aversion
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
(9
20 + 1100
)w = 20.
w = 100023 = 20 + 10p2 =⇒ p2 = 54
23
Investor 2 buys c12 = 100
207 ≈12 and c2
2 = 53 = 10
3 c12 .
Investor 1 buys c11 = 450
23 ≈ 1912 and c2
1 = 612 > 3
10c11
Jouini and Napp (2006) Brunnermeier Slides on Jouini-Napp
Introduction and motivationModel and proofs
QuestionsFirst examplePessimism and risk-aversion
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
(9
20 + 1100
)w = 20.
w = 100023 = 20 + 10p2 =⇒ p2 = 54
23
Investor 2 buys c12 = 100
207 ≈12 and c2
2 = 53 = 10
3 c12 .
Investor 1 buys c11 = 450
23 ≈ 1912 and c2
1 = 612 > 3
10c11
Jouini and Napp (2006) Brunnermeier Slides on Jouini-Napp
Introduction and motivationModel and proofs
QuestionsFirst examplePessimism and risk-aversion
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
(9
20 + 1100
)w = 20.
w = 100023 = 20 + 10p2 =⇒ p2 = 54
23
Investor 2 buys c12 = 100
207 ≈12 and c2
2 = 53 = 10
3 c12 .
Investor 1 buys c11 = 450
23 ≈ 1912 and c2
1 = 612 > 3
10c11
Jouini and Napp (2006) Brunnermeier Slides on Jouini-Napp
Introduction and motivationModel and proofs
QuestionsFirst examplePessimism and risk-aversion
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
(9
20 + 1100
)w = 20.
w = 100023 = 20 + 10p2 =⇒ p2 = 54
23
Investor 2 buys c12 = 100
207 ≈12 and c2
2 = 53 = 10
3 c12 .
Investor 1 buys c11 = 450
23 ≈ 1912 and c2
1 = 612 > 3
10c11
Jouini and Napp (2006) Brunnermeier Slides on Jouini-Napp
Introduction and motivationModel and proofs
QuestionsFirst examplePessimism and risk-aversion
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
(9
20 + 1100
)w = 20.
w = 100023 = 20 + 10p2 =⇒ p2 = 54
23
Investor 2 buys c12 = 100
207 ≈12 and c2
2 = 53 = 10
3 c12 .
Investor 1 buys c11 = 450
23 ≈ 1912 and c2
1 = 612 > 3
10c11
Jouini and Napp (2006) Brunnermeier Slides on Jouini-Napp
Introduction and motivationModel and proofs
QuestionsFirst examplePessimism and risk-aversion
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
(9
20 + 1100
)w = 20.
w = 100023 = 20 + 10p2 =⇒ p2 = 54
23
Investor 2 buys c12 = 100
207 ≈12 and c2
2 = 53 = 10
3 c12 .
Investor 1 buys c11 = 450
23 ≈ 1912 and c2
1 = 612 > 3
10c11
Jouini and Napp (2006) Brunnermeier Slides on Jouini-Napp
Introduction and motivationModel and proofs
QuestionsFirst examplePessimism and risk-aversion
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
(9
20 + 1100
)w = 20.
w = 100023 = 20 + 10p2 =⇒ p2 = 54
23
Investor 2 buys c12 = 100
207 ≈12 and c2
2 = 53 = 10
3 c12 .
Investor 1 buys c11 = 450
23 ≈ 1912 and c2
1 = 612 > 3
10c11
Jouini and Napp (2006) Brunnermeier Slides on Jouini-Napp
Introduction and motivationModel and proofs
QuestionsFirst examplePessimism and risk-aversion
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
(9
20 + 1100
)w = 20.
w = 100023 = 20 + 10p2 =⇒ p2 = 54
23
Investor 2 buys c12 = 100
207 ≈12 and c2
2 = 53 = 10
3 c12 .
Investor 1 buys c11 = 450
23 ≈ 1912 and c2
1 = 612 > 3
10c11
Jouini and Napp (2006) Brunnermeier Slides on Jouini-Napp
Introduction and motivationModel and proofs
QuestionsFirst examplePessimism and risk-aversion
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.
Jouini and Napp (2006) Brunnermeier Slides on Jouini-Napp
Introduction and motivationModel and proofs
QuestionsFirst examplePessimism and risk-aversion
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.
Jouini and Napp (2006) Brunnermeier Slides on Jouini-Napp
Introduction and motivationModel and proofs
QuestionsFirst examplePessimism and risk-aversion
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.
Jouini and Napp (2006) Brunnermeier Slides on Jouini-Napp
Introduction and motivationModel and proofs
QuestionsFirst examplePessimism and risk-aversion
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.
Jouini and Napp (2006) Brunnermeier Slides on Jouini-Napp
Introduction and motivationModel and proofs
QuestionsFirst examplePessimism and risk-aversion
How does it generalize?
These were nice toy examples. But how do the insightsgeneralize? How do we define concepts in a fully generalrigorous manner?
Jouini and Napp (2006) Brunnermeier Slides on Jouini-Napp
Introduction and motivationModel and proofs
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
Jouini and Napp (2006) Brunnermeier Slides on Jouini-Napp
Introduction and motivationModel and proofs
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
Jouini and Napp (2006) Brunnermeier Slides on Jouini-Napp
Introduction and motivationModel and proofs
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
Jouini and Napp (2006) Brunnermeier Slides on Jouini-Napp
Introduction and motivationModel and proofs
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
Jouini and Napp (2006) Brunnermeier Slides on Jouini-Napp
Introduction and motivationModel and proofs
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
Jouini and Napp (2006) Brunnermeier Slides on Jouini-Napp
Introduction and motivationModel and proofs
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
Jouini and Napp (2006) Brunnermeier Slides on Jouini-Napp
Introduction and motivationModel and proofs
Belief aggregationCCAPM
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.
Jouini and Napp (2006) Brunnermeier Slides on Jouini-Napp
Introduction and motivationModel and proofs
Belief aggregationCCAPM
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.
Jouini and Napp (2006) Brunnermeier Slides on Jouini-Napp
Introduction and motivationModel and proofs
Belief aggregationCCAPM
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.
Jouini and Napp (2006) Brunnermeier Slides on Jouini-Napp
Introduction and motivationModel and proofs
Belief aggregationCCAPM
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.
Jouini and Napp (2006) Brunnermeier Slides on Jouini-Napp
Introduction and motivationModel and proofs
Belief aggregationCCAPM
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
j
Then Q is given by:
Q =[ ∑
i γi(Qi)η] 1
η
Jouini and Napp (2006) Brunnermeier Slides on Jouini-Napp
Introduction and motivationModel and proofs
Belief aggregationCCAPM
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
j
Then Q is given by:
Q =[ ∑
i γi(Qi)η] 1
η
Jouini and Napp (2006) Brunnermeier Slides on Jouini-Napp
Introduction and motivationModel and proofs
Belief aggregationCCAPM
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
j
Then Q is given by:
Q =[ ∑
i γi(Qi)η] 1
η
Jouini and Napp (2006) Brunnermeier Slides on Jouini-Napp
Introduction and motivationModel and proofs
Belief aggregationCCAPM
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
j
Then Q is given by:
Q =[ ∑
i γi(Qi)η] 1
η
Jouini and Napp (2006) Brunnermeier Slides on Jouini-Napp
Introduction and motivationModel and proofs
Belief aggregationCCAPM
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
j
Then Q is given by:
Q =[ ∑
i γi(Qi)η] 1
η
Jouini and Napp (2006) Brunnermeier Slides on Jouini-Napp
Introduction and motivationModel and proofs
Belief aggregationCCAPM
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
j
Then Q is given by:
Q =[ ∑
i γi(Qi)η] 1
η
Jouini and Napp (2006) Brunnermeier Slides on Jouini-Napp
Introduction and motivationModel and proofs
Belief aggregationCCAPM
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
Jouini and Napp (2006) Brunnermeier Slides on Jouini-Napp
Introduction and motivationModel and proofs
Belief aggregationCCAPM
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
Jouini and Napp (2006) Brunnermeier Slides on Jouini-Napp
Introduction and motivationModel and proofs
Belief aggregationCCAPM
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
Jouini and Napp (2006) Brunnermeier Slides on Jouini-Napp
Introduction and motivationModel and proofs
Belief aggregationCCAPM
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
Jouini and Napp (2006) Brunnermeier Slides on Jouini-Napp
Introduction and motivationModel and proofs
Belief aggregationCCAPM
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
Jouini and Napp (2006) Brunnermeier Slides on Jouini-Napp
Introduction and motivationModel and proofs
Belief aggregationCCAPM
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
Jouini and Napp (2006) Brunnermeier Slides on Jouini-Napp
Introduction and motivationModel and proofs
Belief aggregationCCAPM
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
Jouini and Napp (2006) Brunnermeier Slides on Jouini-Napp
Introduction and motivationModel and proofs
Belief aggregationCCAPM
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+1Si
t− 1
If π? is equilibrium price deflator process (under P) thenπ?S is martingale
Thus EPt [Rt+1]− r f
t+1 = −covPt
[π?
t+1EP
t [π?t+1]
, Rt+1
]
Jouini and Napp (2006) Brunnermeier Slides on Jouini-Napp
Introduction and motivationModel and proofs
Belief aggregationCCAPM
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+1Si
t− 1
If π? is equilibrium price deflator process (under P) thenπ?S is martingale
Thus EPt [Rt+1]− r f
t+1 = −covPt
[π?
t+1EP
t [π?t+1]
, Rt+1
]
Jouini and Napp (2006) Brunnermeier Slides on Jouini-Napp
Introduction and motivationModel and proofs
Belief aggregationCCAPM
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+1Si
t− 1
If π? is equilibrium price deflator process (under P) thenπ?S is martingale
Thus EPt [Rt+1]− r f
t+1 = −covPt
[π?
t+1EP
t [π?t+1]
, Rt+1
]
Jouini and Napp (2006) Brunnermeier Slides on Jouini-Napp
Introduction and motivationModel and proofs
Belief aggregationCCAPM
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+1Si
t− 1
If π? is equilibrium price deflator process (under P) thenπ?S is martingale
Thus EPt [Rt+1]− r f
t+1 = −covPt
[π?
t+1EP
t [π?t+1]
, Rt+1
]
Jouini and Napp (2006) Brunnermeier Slides on Jouini-Napp
Introduction and motivationModel and proofs
Belief aggregationCCAPM
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+1Si
t− 1
If π? is equilibrium price deflator process (under P) thenπ?S is martingale
Thus EPt [Rt+1]− r f
t+1 = −covPt
[π?
t+1EP
t [π?t+1]
, Rt+1
]
Jouini and Napp (2006) Brunnermeier Slides on Jouini-Napp