liquidity provision, ambiguous asset returns and the financial crisis by willem spanjers

Liquidity Provision Ambiguous Asset Returns and the Financial Crisis

by

Willem Spanjers

(Kingston University andRimini Centre for Economic Analysis)

15th May 2013

University of Bologna at Rimini

Rimini Italy

1

Content1 Introduction

2 Ambiguity

3 Updating

4 The Basic Model

5 Second Best Efficiency

6 Financial Sector

7 Regulation

8 The Impact of Ambiguity

9 Conclusions

Liquidity Provision Ambiguous Asset Returns and the Crisis 2

1 IntroductionBasic questionsbull Could the financial crisis be caused by a failure

to recognized the presence of incalculable riskbull Should financial regulation be tightened

Answersbull Updating ambiguous beliefs may lead to an

endogenous loss of confidence causing a crisisbull Crises should be dealt with if and when they

arisebull This policy should be public knowledge

preventing excessive caution by investors


Related literaturebull Liquidity provision as in

- Jacklin and Bhattacharya (1988) and - Allen and Gale (JoF 1998)

bull Intuition of ambiguity as in - Knight (1921) and Keynes (1937)

bull Modelling of ambiguity- in the tradition of Schmeidler (19821989) - E(llsberg)-capacities as in Eichberger and Kelsey (1999)


Keynes (1937) gives a description of what is meant by ambiguity

ldquoBy lsquouncertainrsquo knowledge let me explain I do not mean merely to distinguish what is known for certain from what is only probable The game of roulette is not subject in this sense to uncertainty [] The sense in which I am using the term is that [] there is no scientific basis on which to form any calculable probability whatever We simply do not knowrdquo [pp 113-114]

5Liquidity Provision Ambiguous Asset Returns and the Crisis

To Keynes these implications are not without consequences for financial economics

ldquo[T]he fact that our knowledge of the future is fluctuating vague and uncertain renders wealth a peculiarly unsuitable subject for the methods of the classical economic theory This theory might work very well in a world in which economic goods are necessarily consumed within a short interval of their being produced But it requires I suggest considerable amendment if it is to be applied to a world in which the accumulation of wealth for an indefinitely postponed future is an important factor and the greater the proportionate part played by such wealth accumulation the more essential does such amendment becomerdquo [p 113]


2 Ambiguity

We consider

bull Calculable vs incalculable risk

bull Sure Thing Principle

bull (Subjective) Expected Utility

bull Choquet Expected Utility


Calculable vs Incalculable Risk

bull Uncertainty can be distinguished in - (calculable) risk and - (incalculable) ambiguity

bull Risk may fail to be calculable because - one cannot make a reasonable probability

estimate for the relevant states of natureandor - one does not know the outcome that is

obtained for the specific states of nature


Investors who face ambiguity tend tobull hope for the best (optimism) andorbull fear the worst (pessimism)

Examples for situations with ambiguity arebull after the terrorist attacks of 911

(prob known outcomes unknown pessimism)bull BSE crisis

(prob unknown outcomes known pessimism)bull Dotcom bubble

(prob unknown outcomes known optimism)


Sure Thing Principlebull For some consumers the trade-off between the

amounts of coffee and amounts of tea they consume may well depend on the amount of milk they have

bull For comparisons between income in different states of nature the counterpart of this situation seems less plausible

bull The sure thing principle states that the trade-off of levels of income (or consumption) in two states is independent of the level of income (or consumption) in a third state


(Subjective) Expected Utilitybull Preferences that satisfy (amongst others) the sure

thing principle can be represented as if they arise from an expected utility function U ℝS rarr ℝ with

U(x1xSπ1 πS) = π1 u(x1) + + πS u(xS)

where ndash (π

1 π

S) is a probability distribution

ndash u ℝ rarr ℝ is a von Neumann-Morgenstern

utility index


Choquet Expected Utilitybull Choquet Expected Utility extends the expected

utility to deal with ambiguitybull In a simplified form it describes beliefs by - a probability estimate π - a level of confidence γ ϵ [01] in the probabilty

estimate 1 ndash γ being the level of ambiguity - an optimism parameter β ϵ [01] reflecting the

ambiguity attitudebull An outcome (x(s))

sϵS is evaluated as

γ Eu(x) + (1-γ) β maxsєS u(x(s)) + (1-γ) (1-β) minsєS u(x(s))


E(llsberg) Capacitiesbull For some situations with ambiguity the

ambiguity is present in some parts of the state space S but not in all

bull For example in the case of liquidity provision the individual liquidity preference may be represented by an (additive) probability distribution whereas there is ambiguity regarding asset returns

bull E(llsberg) capacities provide a generalization of Simple Capacities that allow for such structures


bull An (E)llsberg capacity is described by - a probability assessment π - a level of confidence γ ϵ [01]

- an additive partition E1E

n of the state

space S in additive components bull Let I(AE) be the indicator function with

I(AE) = 1 if A is contained in E and I(AE) = 0 otherwise

bull The capacity v is now defined by v(E) = Σj=1

m [γ π(EcapEj) + (1-γ) π(Ej) I(EjE)]


bull For state contingent payouts x define m(Ej) as

the minimum of u(x(s)) over all s ϵ E

j

bull The Choquet Expected Utility for a pessimistic consumer now is obtained as

U(x) = γ Eπu(x(s)) + (1 - γ) Σj=1

m π(Ej) m(Ej)


3 Updating of Ambiguous Beliefs

For this we considerbull The multiple prior representation

bull Bayesian updating

bull Updating capacities


Main questionDoes the updating in the presence of ambiguity leadto the same type of results as applying Bayesrsquo Ruleto probability distributions

AnswerNo when updating in the presence of ambiguity onetends to encounter dynamic inconsistency ieconsumers deviate from their initial statecontingent plans when the contingency actuallyarises


The Multiple Prior Representationbull In applications the multiple prior approach

(Maxmin Expected Utility) is often preferred over the CEU approach

bull Advantages of Maxmin Expected Utility approach compared to CEU are

- it is more intuitive

- it allows for more general beliefs

bull The big disadvantage is that ambiguity attitudes cannot be included in a natural way


(001)

(100)

(010)

bullpmin

Π


Capacities and multiple priorsbull Consider a capacity v P(S)rarr[01]bull The core of the capacity v is the set

C(v) = p є ΔS | for each E є P(S) p(E) ge v(E)bull A capacity v is convex if for all A and B in P(S)

v(AںB) ge v(A) + v(B) ndash v(AcapB)bull The core of a capacity is non-empty if and only if

the capacity is convex


(001)

(100)

(010)

Core of v

p3 ge v3

p1 ge v1

p2 ge v2

p3 le 1 - v12p1 le 1 - v23

p2 le 1 - v13


(001)

(100)

(010)

p3 ge γπ3

p1 ge γπ1 p2 ge γπ2

The core ofa simplecapacity

p3 le 1 - γπ12

p1 le 1 - γπ23

p2 le 1 - γπ13


(001)

(100)

(010)

p3 le 1 - v12

p3 ge v3

p2 = 1 - v13

p1 le 1 - v23

p1 ge v1

Core of anE-capacity

The core ofan E-Capacity with E1=13and E2=2


Bayesian Updatingbull We consider updating of the probabilities of the

states of nature over a state space

bull We assume that if the information A is received the states that can still be attained is restricted to the set A є P(S) thus ruling out all states in SA

bull Bayeslsquo rule now gives the conditional probability of an event B є P(S) as

PB|A = PBcapA PA


(001)

(100)

(010)

bull

bull

Bayesian Updatingafter receiving the Information 23


Updating CapacitiesWhen updating capacities one would want todistinguish between

bull updating a capacity that describes the ambiguity experienced by a decision maker and

bull updating a capacity that is obtained from preferences by the Choquet Expected Utility approach which describe the interaction of ambiguity and ambiguity attitude


The Dempster-Shafer Rulebull The main updating rule for capacities suggests to

restrict attention to those probability distributions in the set of priors for which the information received was most likely to occur

bull Therefore this updating rule is sometimes referred to as a ldquomaximum likelihoodrdquo updating rule

bull This rule is the Dempster-Shafer rulebull There also exists an ldquoaxiomaticrdquo justification by

Gilboa and Schmeidler (1993) for applying the Dempster-Shafer rule for pessimistic consumers


bull When the information A is received the Dempster-Shafer Update of v ie the conditional capacity value v(B|A) for B є P(S) is obtained as

vDS(B|A) = [v((BcapA) ں (SA)) ndash v(SA)] [1 ndash v(SA)]

bull For the additive case this gives Bayesrsquo Rule since [PBcapA + (1 ndash PA) ndash (1 ndash PA)] PA

= PBcapA PA


(001)

(100)

(010)

Dempster-Shafer Updating of a convex capacity forthe information 23


(001)

(100)

(010)

bull

bullbull

Dempster-Shafer Updating of an E-Capacity for theinformation 12

bull


(001)

(100)

(010)

bull

bull

Dempster-Shafer Updating of an E-Capacity for the information 23


bull The updating of the E-capacity for the information 12 lead to an updated set of probability distributions that is ldquolargerrdquo than the initial set

bull This suggests that the amount of ambiguity as experienced by the consumer has increased which may lead to dynamically inconsistent behaviour

bull The updating of the E-capacity for the information 23 leads to a single point

bull Here updating made the ambiguity disappear


Concluding Remarks on Updatingbull The modelling of ambiguity as Choquet Expected

Utility is consistent both with the interpretation of

- ldquounknown outcomesrdquo and of

- ldquounknown probabilitiesrdquo

bull Updating of ambiguous beliefs may lead to dynamic inconsistency in decision making

bull Simple capacities and their variations provide a workable simplification that retains the full intuition of decision making under ambiguity


4 The Basic Modelbull Continuum [01] of ex-ante identical investorsbull Investment opportunities

- zero interest money holdings

- illiquid assets which pay out

α1 lt 1 when liquidated prematurely

αh gt 1 when matured and successful

αℓ = 0 when matured and failure

34

Liquidity Provision Ambiguous Asset Returns and the Crisis

Information regarding assets

Signal σ Return ρ Probability π σρ

b h δ b ℓ ε

g h 1 - (δ + ε)

g ℓ 0


TimingPeriod 0 - investment decisions

Period 1 - individual liquidity preference t isinHL becomes privately known

- signal σ isinbg becomes publicly known

- interaction

- possibility for liquidation of assets - consumption

Period 2 - return ρ isinhℓ occurs

- consumption

36


Beliefsbull ex-ante (calculable) risk with respect to

- individual liquidity preference t isin HL

bull ex-ante (incalculable) ambiguity with respect to- joint probability distribution of (σρ)

bull beliefs over HLtimesbgtimeshℓ consist ofan additive probability distribution P- an additive partition of the state space with components

FH = (Hbh) (Hbℓ) (Hgh) (Hgℓ)

FL = (Lbh) (Lbℓ) (Lgh) (Lgℓ)- a level of confidence γ isin [01] in P


Preferencesbull risk-neutral with vNM-utility index

u(x1(σ) x2(σρ)t) = βtmiddotx1(σ) + x2(σρ)

where βH gt βL gt 1

bull ex-ante Choquet expected utilityγ times expected utility wrt Pplus(1-γ) times

πH times minimum utility over FH

plus πL times minimum utility over FL


Updating the ambiguous beliefs leads tobull Bayesian updating of the probability distribution Pbull an endogenous decrease in the level of confidence

- after a good signal σ = g the level of confidence becomes

γg = γmiddotπg(1-γmiddotπb) = γmiddot[(1-(δ+ε)) (1-γmiddot(δ+ε))] lt γ

- after a bad signal σ = b the level of confidence becomes

γb = γmiddotπb(1-γmiddotπg) = γmiddot[(δ+ε) (1-γmiddot(1-(δ+ε)))]

= γmiddot[(δ+ε) (1-γ + γmiddot(δ+ε))] lt γ


4 Second Best EfficiencyLow reserves bull such that the incentive constraint of L-types holds even after

a bad signal

bull consequence little welfare enhancing ad interim redistribution to H-types

bull the money holdings are

μb(γ) = πH γb πhb αh (πL βL + πH γb πh

b αh)


High reservesbull such that incentive constraint of L-types only holds after a

good signalbull consequence high amount of welfare enhancing ad interim

redistribution to H-typesbull the money holdings are

μg(γ) = πH γg πhg αh (πL βL + πH γg πh

g αh)

= πH γg middot αh (πL βL + πH γg αh)

Assumption on parametersbull the high reserves are second best efficient


ICLb

ICLb γ = 1

γ lt 1 y2

y1 μE(γ) πH

α2(1-μE(γ)) πL

Second BestEfficiency

bull

VEff


ICLg

ICLg

VEff

μE(1) πH

α2(1-μE(1)) πL

42

5 The Financial Sectorbull Modelled as an unregulated competitive banking

sector offering deposit contracts bull Period 0 - deposit contracts are offered

- investment decisions Period 1 - withdrawal decisions

- possible liquidation of assets

bull A deposit contract specifies- promised repayments in Periods 1 and 2

- fraction of the deposits held as money reserves

- withdrawals in Period 1 have priority over withdrawals in Period 2


bull Only ldquofundamentalrdquo bank runs are taken into account

bull High reserves - a bank run occurs after a bad signal- not second best efficient because assets are

liquidated after a bad signal

bull Low reserves - no bank run after either signal

- not second best efficient because reserves are too

low by assumption


ICLb(γ)

VEff VBank

y2

y1 μB(γ) πH

bull

bull

UnregulatedBanks withγ le 1 andμB(γ) = μb(γ)


ICLg(γ)

μE(γ) πH

α2(1-μB(γ)) πL

α2(1-μE(1)) πL

45

6 Regulationbull In the case of a bank run the regulator should

interfere by- making immediate withdrawal less attractive and- making waiting more attractive

bull This can be done by a revenue neutral scheme of- taxes on withdrawals and - subsidies on waiting

bull Investors and banks should be made aware of this policy which- would NOT create moral hazard but - stop the banksrsquo being overly cautious


7 The Impact of Ambiguitybull If recognized ex-ante

- reduces efficient reserve holdings- no further impact

bull If not recognized ex-ante- a bad signal unexpectedly affects the incentive constraint- investor seem to ldquoover-reactrdquo on bad news- ldquopanickingrdquo investors cause a bank run even for the ldquosaferdquo (low) reserve holdings


8 Concluding RemarksThe model suggestsbull a competitive banking sector with a regulatory

commitment to an appropriate tax-and-subsidize scheme implements the second best efficient liquidity allocation

bull the ldquopanicrdquo and ldquoover-reactionsrdquo of markets during the crises may be due to a failure to recognize the ambiguity experienced by professional and institutional investors

bull reducing ambiguity as suggested by the ldquoSoziale Marktwirtschaftrdquo based on the ldquoFreiburger Schulerdquo reduces ldquoover-reactionsrdquo and enhances welfare


In the financial crisis bull on aggregate the economy faces the choice between either

liquidating illiquid assets or changing the trade-off current and future income

bull this problem is similar to that faced by the aggregated unregulated banking sector

bull the combined policy of flooding the market with liquidity and bailing out distressed banks captures the mechanism of the second best efficient tax-and-subsidy scheme where

- the provision of liquidity is the counterpart of a tax on immediate consumption and

- providing guarantees and bailing out distressed banks is the counterpart to subsidizing future consumption


Content1 Introduction

2 Ambiguity

3 Updating

4 The Basic Model

5 Second Best Efficiency

6 Financial Sector

7 Regulation

8 The Impact of Ambiguity

9 Conclusions




















2 Ambiguity

We consider


























where ndash (π

1 π



utility index
















n of the state








j


U(x) = γ Eπu(x(s)) + (1 - γ) Σj=1

m π(Ej) m(Ej)

















(001)

(100)

(010)

bullpmin

Π







(001)

(100)

(010)

Core of v

p3 ge v3

p1 ge v1

p2 ge v2

p3 le 1 - v12p1 le 1 - v23

p2 le 1 - v13


(001)

(100)

(010)

p3 ge γπ3



p3 le 1 - γπ12

p1 le 1 - γπ23

p2 le 1 - γπ13


(001)

(100)

(010)

p3 le 1 - v12

p3 ge v3

p2 = 1 - v13

p1 le 1 - v23

p1 ge v1








PB|A = PBcapA PA


(001)

(100)

(010)

bull

bull
















= PBcapA PA


(001)

(100)

(010)



(001)

(100)

(010)

bull

bullbull


bull


(001)

(100)

(010)

bull

bull





















34




b h δ b ℓ ε

g h 1 - (δ + ε)

g ℓ 0





- interaction



- consumption

36
























a bad signal




b αh)






g αh)




ICLb

ICLb γ = 1

γ lt 1 y2

y1 μE(γ) πH

α2(1-μE(γ)) πL


bull

VEff


ICLg

ICLg

VEff

μE(1) πH

α2(1-μE(1)) πL

42














low by assumption


ICLb(γ)

VEff VBank

y2

y1 μB(γ) πH

bull

bull



ICLg(γ)

μE(γ) πH

α2(1-μB(γ)) πL

α2(1-μE(1)) πL

45








































2 Ambiguity

We consider


























where ndash (π

1 π



utility index
















n of the state








j


U(x) = γ Eπu(x(s)) + (1 - γ) Σj=1

m π(Ej) m(Ej)

















(001)

(100)

(010)

bullpmin

Π







(001)

(100)

(010)

Core of v

p3 ge v3

p1 ge v1

p2 ge v2

p3 le 1 - v12p1 le 1 - v23

p2 le 1 - v13


(001)

(100)

(010)

p3 ge γπ3



p3 le 1 - γπ12

p1 le 1 - γπ23

p2 le 1 - γπ13


(001)

(100)

(010)

p3 le 1 - v12

p3 ge v3

p2 = 1 - v13

p1 le 1 - v23

p1 ge v1








PB|A = PBcapA PA


(001)

(100)

(010)

bull

bull
















= PBcapA PA


(001)

(100)

(010)



(001)

(100)

(010)

bull

bullbull


bull


(001)

(100)

(010)

bull

bull





















34




b h δ b ℓ ε

g h 1 - (δ + ε)

g ℓ 0





- interaction



- consumption

36
























a bad signal




b αh)






g αh)




ICLb

ICLb γ = 1

γ lt 1 y2

y1 μE(γ) πH

α2(1-μE(γ)) πL


bull

VEff


ICLg

ICLg

VEff

μE(1) πH

α2(1-μE(1)) πL

42














low by assumption


ICLb(γ)

VEff VBank

y2

y1 μB(γ) πH

bull

bull



ICLg(γ)

μE(γ) πH

α2(1-μB(γ)) πL

α2(1-μE(1)) πL

45










































where ndash (π

1 π



utility index
















n of the state








j


U(x) = γ Eπu(x(s)) + (1 - γ) Σj=1

m π(Ej) m(Ej)

















(001)

(100)

(010)

bullpmin

Π







(001)

(100)

(010)

Core of v

p3 ge v3

p1 ge v1

p2 ge v2

p3 le 1 - v12p1 le 1 - v23

p2 le 1 - v13


(001)

(100)

(010)

p3 ge γπ3



p3 le 1 - γπ12

p1 le 1 - γπ23

p2 le 1 - γπ13


(001)

(100)

(010)

p3 le 1 - v12

p3 ge v3

p2 = 1 - v13

p1 le 1 - v23

p1 ge v1








PB|A = PBcapA PA


(001)

(100)

(010)

bull

bull
















= PBcapA PA


(001)

(100)

(010)



(001)

(100)

(010)

bull

bullbull


bull


(001)

(100)

(010)

bull

bull





















34




b h δ b ℓ ε

g h 1 - (δ + ε)

g ℓ 0





- interaction



- consumption

36
























a bad signal




b αh)






g αh)




ICLb

ICLb γ = 1

γ lt 1 y2

y1 μE(γ) πH

α2(1-μE(γ)) πL


bull

VEff


ICLg

ICLg

VEff

μE(1) πH

α2(1-μE(1)) πL

42














low by assumption


ICLb(γ)

VEff VBank

y2

y1 μB(γ) πH

bull

bull



ICLg(γ)

μE(γ) πH

α2(1-μB(γ)) πL

α2(1-μE(1)) πL

45




































n of the state








j


U(x) = γ Eπu(x(s)) + (1 - γ) Σj=1

m π(Ej) m(Ej)

















(001)

(100)

(010)

bullpmin

Π







(001)

(100)

(010)

Core of v

p3 ge v3

p1 ge v1

p2 ge v2

p3 le 1 - v12p1 le 1 - v23

p2 le 1 - v13


(001)

(100)

(010)

p3 ge γπ3



p3 le 1 - γπ12

p1 le 1 - γπ23

p2 le 1 - γπ13


(001)

(100)

(010)

p3 le 1 - v12

p3 ge v3

p2 = 1 - v13

p1 le 1 - v23

p1 ge v1








PB|A = PBcapA PA


(001)

(100)

(010)

bull

bull
















= PBcapA PA


(001)

(100)

(010)



(001)

(100)

(010)

bull

bullbull


bull


(001)

(100)

(010)

bull

bull





















34




b h δ b ℓ ε

g h 1 - (δ + ε)

g ℓ 0





- interaction



- consumption

36
























a bad signal




b αh)






g αh)




ICLb

ICLb γ = 1

γ lt 1 y2

y1 μE(γ) πH

α2(1-μE(γ)) πL


bull

VEff


ICLg

ICLg

VEff

μE(1) πH

α2(1-μE(1)) πL

42














low by assumption


ICLb(γ)

VEff VBank

y2

y1 μB(γ) πH

bull

bull



ICLg(γ)

μE(γ) πH

α2(1-μB(γ)) πL

α2(1-μE(1)) πL

45
























j


U(x) = γ Eπu(x(s)) + (1 - γ) Σj=1

m π(Ej) m(Ej)

















(001)

(100)

(010)

bullpmin

Π







(001)

(100)

(010)

Core of v

p3 ge v3

p1 ge v1

p2 ge v2

p3 le 1 - v12p1 le 1 - v23

p2 le 1 - v13


(001)

(100)

(010)

p3 ge γπ3



p3 le 1 - γπ12

p1 le 1 - γπ23

p2 le 1 - γπ13


(001)

(100)

(010)

p3 le 1 - v12

p3 ge v3

p2 = 1 - v13

p1 le 1 - v23

p1 ge v1








PB|A = PBcapA PA


(001)

(100)

(010)

bull

bull
















= PBcapA PA


(001)

(100)

(010)



(001)

(100)

(010)

bull

bullbull


bull


(001)

(100)

(010)

bull

bull





















34




b h δ b ℓ ε

g h 1 - (δ + ε)

g ℓ 0





- interaction



- consumption

36
























a bad signal




b αh)






g αh)




ICLb

ICLb γ = 1

γ lt 1 y2

y1 μE(γ) πH

α2(1-μE(γ)) πL


bull

VEff


ICLg

ICLg

VEff

μE(1) πH

α2(1-μE(1)) πL

42














low by assumption


ICLb(γ)

VEff VBank

y2

y1 μB(γ) πH

bull

bull



ICLg(γ)

μE(γ) πH

α2(1-μB(γ)) πL

α2(1-μE(1)) πL

45





































(001)

(100)

(010)

bullpmin

Π







(001)

(100)

(010)

Core of v

p3 ge v3

p1 ge v1

p2 ge v2

p3 le 1 - v12p1 le 1 - v23

p2 le 1 - v13


(001)

(100)

(010)

p3 ge γπ3



p3 le 1 - γπ12

p1 le 1 - γπ23

p2 le 1 - γπ13


(001)

(100)

(010)

p3 le 1 - v12

p3 ge v3

p2 = 1 - v13

p1 le 1 - v23

p1 ge v1








PB|A = PBcapA PA


(001)

(100)

(010)

bull

bull
















= PBcapA PA


(001)

(100)

(010)



(001)

(100)

(010)

bull

bullbull


bull


(001)

(100)

(010)

bull

bull





















34




b h δ b ℓ ε

g h 1 - (δ + ε)

g ℓ 0





- interaction



- consumption

36
























a bad signal




b αh)






g αh)




ICLb

ICLb γ = 1

γ lt 1 y2

y1 μE(γ) πH

α2(1-μE(γ)) πL


bull

VEff


ICLg

ICLg

VEff

μE(1) πH

α2(1-μE(1)) πL

42














low by assumption


ICLb(γ)

VEff VBank

y2

y1 μB(γ) πH

bull

bull



ICLg(γ)

μE(γ) πH

α2(1-μB(γ)) πL

α2(1-μE(1)) πL

45



























(001)

(100)

(010)

Core of v

p3 ge v3

p1 ge v1

p2 ge v2

p3 le 1 - v12p1 le 1 - v23

p2 le 1 - v13


(001)

(100)

(010)

p3 ge γπ3



p3 le 1 - γπ12

p1 le 1 - γπ23

p2 le 1 - γπ13


(001)

(100)

(010)

p3 le 1 - v12

p3 ge v3

p2 = 1 - v13

p1 le 1 - v23

p1 ge v1








PB|A = PBcapA PA


(001)

(100)

(010)

bull

bull
















= PBcapA PA


(001)

(100)

(010)



(001)

(100)

(010)

bull

bullbull


bull


(001)

(100)

(010)

bull

bull





















34




b h δ b ℓ ε

g h 1 - (δ + ε)

g ℓ 0





- interaction



- consumption

36
























a bad signal




b αh)






g αh)




ICLb

ICLb γ = 1

γ lt 1 y2

y1 μE(γ) πH

α2(1-μE(γ)) πL


bull

VEff


ICLg

ICLg

VEff

μE(1) πH

α2(1-μE(1)) πL

42














low by assumption


ICLb(γ)

VEff VBank

y2

y1 μB(γ) πH

bull

bull



ICLg(γ)

μE(γ) πH

α2(1-μB(γ)) πL

α2(1-μE(1)) πL

45






















(001)

(100)

(010)

p3 ge γπ3



p3 le 1 - γπ12

p1 le 1 - γπ23

p2 le 1 - γπ13


(001)

(100)

(010)

p3 le 1 - v12

p3 ge v3

p2 = 1 - v13

p1 le 1 - v23

p1 ge v1








PB|A = PBcapA PA


(001)

(100)

(010)

bull

bull
















= PBcapA PA


(001)

(100)

(010)



(001)

(100)

(010)

bull

bullbull


bull


(001)

(100)

(010)

bull

bull





















34




b h δ b ℓ ε

g h 1 - (δ + ε)

g ℓ 0





- interaction



- consumption

36
























a bad signal




b αh)






g αh)




ICLb

ICLb γ = 1

γ lt 1 y2

y1 μE(γ) πH

α2(1-μE(γ)) πL


bull

VEff


ICLg

ICLg

VEff

μE(1) πH

α2(1-μE(1)) πL

42














low by assumption


ICLb(γ)

VEff VBank

y2

y1 μB(γ) πH

bull

bull



ICLg(γ)

μE(γ) πH

α2(1-μB(γ)) πL

α2(1-μE(1)) πL

45






















(001)

(100)

(010)

p3 le 1 - v12

p3 ge v3

p2 = 1 - v13

p1 le 1 - v23

p1 ge v1








PB|A = PBcapA PA


(001)

(100)

(010)

bull

bull
















= PBcapA PA


(001)

(100)

(010)



(001)

(100)

(010)

bull

bullbull


bull


(001)

(100)

(010)

bull

bull





















34




b h δ b ℓ ε

g h 1 - (δ + ε)

g ℓ 0





- interaction



- consumption

36
























a bad signal




b αh)






g αh)




ICLb

ICLb γ = 1

γ lt 1 y2

y1 μE(γ) πH

α2(1-μE(γ)) πL


bull

VEff


ICLg

ICLg

VEff

μE(1) πH

α2(1-μE(1)) πL

42














low by assumption


ICLb(γ)

VEff VBank

y2

y1 μB(γ) πH

bull

bull



ICLg(γ)

μE(γ) πH

α2(1-μB(γ)) πL

α2(1-μE(1)) πL

45


























PB|A = PBcapA PA


(001)

(100)

(010)

bull

bull
















= PBcapA PA


(001)

(100)

(010)



(001)

(100)

(010)

bull

bullbull


bull


(001)

(100)

(010)

bull

bull





















34




b h δ b ℓ ε

g h 1 - (δ + ε)

g ℓ 0





- interaction



- consumption

36
























a bad signal




b αh)






g αh)




ICLb

ICLb γ = 1

γ lt 1 y2

y1 μE(γ) πH

α2(1-μE(γ)) πL


bull

VEff


ICLg

ICLg

VEff

μE(1) πH

α2(1-μE(1)) πL

42














low by assumption


ICLb(γ)

VEff VBank

y2

y1 μB(γ) πH

bull

bull



ICLg(γ)

μE(γ) πH

α2(1-μB(γ)) πL

α2(1-μE(1)) πL

45






















(001)

(100)

(010)

bull

bull
















= PBcapA PA


(001)

(100)

(010)



(001)

(100)

(010)

bull

bullbull


bull


(001)

(100)

(010)

bull

bull





















34




b h δ b ℓ ε

g h 1 - (δ + ε)

g ℓ 0





- interaction



- consumption

36
























a bad signal




b αh)






g αh)




ICLb

ICLb γ = 1

γ lt 1 y2

y1 μE(γ) πH

α2(1-μE(γ)) πL


bull

VEff


ICLg

ICLg

VEff

μE(1) πH

α2(1-μE(1)) πL

42














low by assumption


ICLb(γ)

VEff VBank

y2

y1 μB(γ) πH

bull

bull



ICLg(γ)

μE(γ) πH

α2(1-μB(γ)) πL

α2(1-μE(1)) πL

45



































= PBcapA PA


(001)

(100)

(010)



(001)

(100)

(010)

bull

bullbull


bull


(001)

(100)

(010)

bull

bull





















34




b h δ b ℓ ε

g h 1 - (δ + ε)

g ℓ 0





- interaction



- consumption

36
























a bad signal




b αh)






g αh)




ICLb

ICLb γ = 1

γ lt 1 y2

y1 μE(γ) πH

α2(1-μE(γ)) πL


bull

VEff


ICLg

ICLg

VEff

μE(1) πH

α2(1-μE(1)) πL

42














low by assumption


ICLb(γ)

VEff VBank

y2

y1 μB(γ) πH

bull

bull



ICLg(γ)

μE(γ) πH

α2(1-μB(γ)) πL

α2(1-μE(1)) πL

45






















(001)

(100)

(010)



(001)

(100)

(010)

bull

bullbull


bull


(001)

(100)

(010)

bull

bull





















34




b h δ b ℓ ε

g h 1 - (δ + ε)

g ℓ 0





- interaction



- consumption

36
























a bad signal




b αh)






g αh)




ICLb

ICLb γ = 1

γ lt 1 y2

y1 μE(γ) πH

α2(1-μE(γ)) πL


bull

VEff


ICLg

ICLg

VEff

μE(1) πH

α2(1-μE(1)) πL

42














low by assumption


ICLb(γ)

VEff VBank

y2

y1 μB(γ) πH

bull

bull



ICLg(γ)

μE(γ) πH

α2(1-μB(γ)) πL

α2(1-μE(1)) πL

45






















(001)

(100)

(010)

bull

bullbull


bull


(001)

(100)

(010)

bull

bull





















34




b h δ b ℓ ε

g h 1 - (δ + ε)

g ℓ 0





- interaction



- consumption

36
























a bad signal




b αh)






g αh)




ICLb

ICLb γ = 1

γ lt 1 y2

y1 μE(γ) πH

α2(1-μE(γ)) πL


bull

VEff


ICLg

ICLg

VEff

μE(1) πH

α2(1-μE(1)) πL

42














low by assumption


ICLb(γ)

VEff VBank

y2

y1 μB(γ) πH

bull

bull



ICLg(γ)

μE(γ) πH

α2(1-μB(γ)) πL

α2(1-μE(1)) πL

45






















(001)

(100)

(010)

bull

bull





















34




b h δ b ℓ ε

g h 1 - (δ + ε)

g ℓ 0





- interaction



- consumption

36
























a bad signal




b αh)






g αh)




ICLb

ICLb γ = 1

γ lt 1 y2

y1 μE(γ) πH

α2(1-μE(γ)) πL


bull

VEff


ICLg

ICLg

VEff

μE(1) πH

α2(1-μE(1)) πL

42














low by assumption


ICLb(γ)

VEff VBank

y2

y1 μB(γ) πH

bull

bull



ICLg(γ)

μE(γ) πH

α2(1-μB(γ)) πL

α2(1-μE(1)) πL

45








































34




b h δ b ℓ ε

g h 1 - (δ + ε)

g ℓ 0





- interaction



- consumption

36
























a bad signal




b αh)






g αh)




ICLb

ICLb γ = 1

γ lt 1 y2

y1 μE(γ) πH

α2(1-μE(γ)) πL


bull

VEff


ICLg

ICLg

VEff

μE(1) πH

α2(1-μE(1)) πL

42














low by assumption


ICLb(γ)

VEff VBank

y2

y1 μB(γ) πH

bull

bull



ICLg(γ)

μE(γ) πH

α2(1-μB(γ)) πL

α2(1-μE(1)) πL

45
























b h δ b ℓ ε

g h 1 - (δ + ε)

g ℓ 0





- interaction



- consumption

36
























a bad signal




b αh)






g αh)




ICLb

ICLb γ = 1

γ lt 1 y2

y1 μE(γ) πH

α2(1-μE(γ)) πL


bull

VEff


ICLg

ICLg

VEff

μE(1) πH

α2(1-μE(1)) πL

42














low by assumption


ICLb(γ)

VEff VBank

y2

y1 μB(γ) πH

bull

bull



ICLg(γ)

μE(γ) πH

α2(1-μB(γ)) πL

α2(1-μE(1)) πL

45

























- interaction



- consumption

36
























a bad signal




b αh)






g αh)




ICLb

ICLb γ = 1

γ lt 1 y2

y1 μE(γ) πH

α2(1-μE(γ)) πL


bull

VEff


ICLg

ICLg

VEff

μE(1) πH

α2(1-μE(1)) πL

42














low by assumption


ICLb(γ)

VEff VBank

y2

y1 μB(γ) πH

bull

bull



ICLg(γ)

μE(γ) πH

α2(1-μB(γ)) πL

α2(1-μE(1)) πL

45












































a bad signal




b αh)






g αh)




ICLb

ICLb γ = 1

γ lt 1 y2

y1 μE(γ) πH

α2(1-μE(γ)) πL


bull

VEff


ICLg

ICLg

VEff

μE(1) πH

α2(1-μE(1)) πL

42














low by assumption


ICLb(γ)

VEff VBank

y2

y1 μB(γ) πH

bull

bull



ICLg(γ)

μE(γ) πH

α2(1-μB(γ)) πL

α2(1-μE(1)) πL

45


























g αh)




ICLb

ICLb γ = 1

γ lt 1 y2

y1 μE(γ) πH

α2(1-μE(γ)) πL


bull

VEff


ICLg

ICLg

VEff

μE(1) πH

α2(1-μE(1)) πL

42














low by assumption


ICLb(γ)

VEff VBank

y2

y1 μB(γ) πH

bull

bull



ICLg(γ)

μE(γ) πH

α2(1-μB(γ)) πL

α2(1-μE(1)) πL

45






















ICLb

ICLb γ = 1

γ lt 1 y2

y1 μE(γ) πH

α2(1-μE(γ)) πL


bull

VEff


ICLg

ICLg

VEff

μE(1) πH

α2(1-μE(1)) πL

42














low by assumption


ICLb(γ)

VEff VBank

y2

y1 μB(γ) πH

bull

bull



ICLg(γ)

μE(γ) πH

α2(1-μB(γ)) πL

α2(1-μE(1)) πL

45



































low by assumption


ICLb(γ)

VEff VBank

y2

y1 μB(γ) πH

bull

bull



ICLg(γ)

μE(γ) πH

α2(1-μB(γ)) πL

α2(1-μE(1)) πL

45






















ICLb(γ)

VEff VBank

y2

y1 μB(γ) πH

bull

bull



ICLg(γ)

μE(γ) πH

α2(1-μB(γ)) πL

α2(1-μE(1)) πL

45






















liquidity provision, ambiguous asset returns and the financial crisis by willem spanjers

Documents

ambiguous asset returns

utilityliquidity provision

ambiguous beliefs

conclusionsliquidity

incalculable ambiguity

intuition of ambiguity

modelling of ambiguity

calculable probability