1
Liquidity Provision, Ambiguous Asset
Returns and the Financial Crisis
by
Willem Spanjers
(Kingston University and
Rimini Centre for Economic Analysis)
15th May 2013
University of Bologna at Rimini
Rimini, Italy
Liquidity Provision, Ambiguous Asset Returns and the Crisis
Content
1. 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
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Liquidity Provision, Ambiguous Asset Returns and the Crisis
1. Introduction
Basic questions:
• Could the financial crisis be caused by a failure
to recognized the presence of incalculable risk?
• Should financial regulation be tightened?
Answers:
• Updating ambiguous beliefs may lead to an
endogenous loss of confidence, causing a crisis.
• Crises should be dealt with if and when they
arise.
• This policy should be public knowledge,
preventing excessive caution by investors.
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Liquidity Provision, Ambiguous Asset Returns and the Crisis
Related literature:
• Liquidity provision as in
- Jacklin and Bhattacharya (1988) and
- Allen and Gale (JoF, 1998).
• Intuition of ambiguity as in
- Knight (1921) and Keynes (1937).
• Modelling of ambiguity
- in the tradition of Schmeidler (1982/1989)
- E(llsberg)-capacities as in Eichberger and
Kelsey (1999).
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Liquidity Provision, Ambiguous Asset Returns and the Crisis
Keynes (1937) gives a description of what is
meant by ambiguity:
“By ‘uncertain’ 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 know.”
[pp. 113-114]
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Liquidity Provision, Ambiguous Asset Returns and the Crisis
To Keynes, these implications are not without
consequences for financial economics:
“[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 become.” [p. 113]
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Liquidity Provision, Ambiguous Asset Returns and the Crisis
2. Ambiguity
We consider:
• Calculable vs incalculable risk
• Sure Thing Principle
• (Subjective) Expected Utility
• Choquet Expected Utility
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Liquidity Provision, Ambiguous Asset Returns and the Crisis
Calculable vs Incalculable Risk
•
•
Uncertainty can be distinguished in
- (calculable) risk and
- (incalculable) ambiguity.
Risk may fail to be calculable because
- one cannot make a reasonable probability
estimate for the relevant states of nature
and/or
- one does not know the outcome that is
obtained for the specific states of nature.
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Liquidity Provision, Ambiguous Asset Returns and the Crisis
Investors who face ambiguity tend to:
• hope for the best (optimism) and/or
• fear the worst (pessimism).
Examples for situations with ambiguity are:
• after the terrorist attacks of 9/11.
(prob. known, outcomes unknown pessimism)
• BSE crisis
(prob. unknown, outcomes known, pessimism)
• Dot.com bubble
(prob. unknown, outcomes known, optimism).
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Liquidity Provision, Ambiguous Asset Returns and the Crisis
Sure Thing Principle
•
•
•
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.
For comparisons between income in different
states of nature, the counterpart of this situation
seems less plausible.
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.
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Liquidity Provision, Ambiguous Asset Returns and the Crisis
(Subjective) Expected Utility
•
Preferences that satisfy (amongst others) the sure
thing principle can be represented as if they arise
from an expected utility function U: ℝS → ℝ
with
U(x1,...,xS;π1,..., πS) = π1 u(x1) + ... + πS u(xS)
where
–
–
(π1,..., πS) is a probability distribution
u: ℝ → ℝ is a von NeumannMorgenstern
utility index
1
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Liquidity Provision, Ambiguous Asset Returns and the Crisis
Choquet Expected Utility
•
•
•
Choquet Expected Utility extends the expected
utility to deal with ambiguity.
In a simplified form, it describes beliefs by:
- a probability estimate π
- a level of confidence γ ϵ [0,1] in the probabilty
estimate; 1 – γ being the level of ambiguity
- an optimism parameter β ϵ [0,1] reflecting the
ambiguity attitude.
An outcome (x(s))sϵS is evaluated as:
γ E{u(x)} + (1-γ) β maxsєS u(x(s))
+ (1-γ) (1-β) minsєS u(x(s)).
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Liquidity Provision, Ambiguous Asset Returns and the Crisis
E(llsberg) Capacities
•
•
•
For some situations with ambiguity, the
ambiguity is present in some parts of the state
space S, but not in all.
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.
E(llsberg) capacities provide a generalization of
Simple Capacities that allow for such structures.
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Liquidity Provision, Ambiguous Asset Returns and the Crisis
•
•
•
An (E)llsberg capacity is described by:
- a probability assessment π
- a level of confidence γ ϵ [0,1]
- an additive partition {E1,...,En} of the state
space S in additive components.
Let I(A;E) be the indicator function with I(A;E)
:= 1 if A is contained in E and
I(A;E) := 0 otherwise.
The capacity v is now defined by
v(E) := Σj=1m [γ π(E∩Ej) + (1-γ) π(Ej) I(Ej;E)].
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Liquidity Provision, Ambiguous Asset Returns and the Crisis
•
•
For state contingent payouts x define m(Ej) as
the minimum of
u(x(s)) over all s ϵ Ej.
The Choquet Expected Utility for a pessimistic
consumer now is obtained as
U(x) := γ Eπ{u(x(s))} + (1 - γ) Σj=1m π(Ej) m(Ej).
1
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Liquidity Provision, Ambiguous Asset Returns and the Crisis
3. Updating of Ambiguous Beliefs
For this we consider:
• The multiple prior representation.
• Bayesian updating.
• Updating capacities.
1
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Liquidity Provision, Ambiguous Asset Returns and the Crisis
Main question:
Does the updating in the presence of ambiguity lead
to the same type of results as applying Bayes’ Rule
to probability distributions?
Answer:
No, when updating in the presence of ambiguity one
tends to encounter dynamic inconsistency; i.e.
consumers deviate from their initial state
contingent plans when the contingency actually
arises!
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Liquidity Provision, Ambiguous Asset Returns and the Crisis
The Multiple Prior Representation
• In applications the multiple prior approach
(Maxmin Expected Utility) is often preferred over
the CEU approach.
• Advantages of Maxmin Expected Utility approach
compared to CEU are:
- it is more intuitive
- it allows for more general beliefs.
• The big disadvantage is that ambiguity attitudes
cannot be included in a natural way.
1
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Liquidity Provision, Ambiguous Asset Returns and the Crisis
(0,0,1)
Π
•
pmin
(1,0,0)
(0,1,0)
1
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Liquidity Provision, Ambiguous Asset Returns and the Crisis
Capacities and multiple priors
• Consider a capacity v: P(S)→[0,1].
• The core of the capacity v is the set
C(v) := {p є ΔS | for each E є P(S): p(E) ≥ v(E)}.
• A capacity v is convex if for all A and B in P(S):
v(A‫ں‬B) ≥ v(A) + v(B) – v(A∩B).
• The core of a capacity is non-empty if and only if
the capacity is convex.
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Liquidity Provision, Ambiguous Asset Returns and the Crisis
(0,0,1)
p1 ≥ v1
p1 ≤ 1 - v23
p3 ≤ 1 - v12
Core
of v
p3 ≥ v3
(1,0,0)
p2 ≥ v2
p2 ≤ 1 - v13
(0,1,0)
2
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Liquidity Provision, Ambiguous Asset Returns and the Crisis
The core of
a simple
capacity
(0,0,1)
p1 ≥ γπ1
p2 ≥ γπ2
p3 ≤ 1 - γπ12
p2 ≤ 1 - γπ13
p3 ≥ γπ3
(1,0,0)
p1 ≤ 1 - γπ23
(0,1,0)
2
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Liquidity Provision, Ambiguous Asset Returns and the Crisis
The core of
an E-Capacity
with E1={1,3}
and E2={2}
(0,0,1)
p1 ≥ v1
p1 ≤ 1 - v23
(1,0,0)
p2 = 1 - v13
p3 ≤ 1 - v12
Core of an
E-capacity
p3 ≥ v3
(0,1,0)
2
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Liquidity Provision, Ambiguous Asset Returns and the Crisis
Bayesian Updating
• We consider updating of the probabilities of the
states of nature over a state space.
• 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 S\A.
• Bayes‘ rule now gives the conditional probability
of an event B є P(S) as
P{B|A} = P{B∩A} / P{A}.
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Liquidity Provision, Ambiguous Asset Returns and the Crisis
Bayesian Updating
after receiving the
Information {2,3}
(0,0,1)
•
•
(1,0,0)
(0,1,0)
2
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Liquidity Provision, Ambiguous Asset Returns and the Crisis
Updating Capacities
When updating capacities one would want to
distinguish between:
• updating a capacity that describes the ambiguity
experienced by a decision maker and
• updating a capacity that is obtained from
preferences by the Choquet Expected Utility
approach, which describe the interaction of
ambiguity and ambiguity attitude.
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Liquidity Provision, Ambiguous Asset Returns and the Crisis
The Dempster-Shafer Rule
• 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.
• Therefore, this updating rule is sometimes referred
to as a “maximum likelihood” updating rule.
• This rule is the Dempster-Shafer rule.
• There also exists an “axiomatic” justification by
Gilboa and Schmeidler (1993) for applying the
Dempster-Shafer rule for pessimistic consumers.
2
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Liquidity Provision, Ambiguous Asset Returns and the Crisis
• When the information A is received, the DempsterShafer Update of v, i.e. the conditional capacity
value v(B|A) for B є P(S) is obtained as
vDS(B|A) :=
[v((B∩A) ‫( ﮞ‬S\A)) – v(S\A)] / [1 – v(S\A)].
• For the additive case, this gives Bayes’ Rule since
[P{B∩A} + (1 – P{A}) – (1 – P{A})] / P{A}
= P{B∩A} / P{A}.
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Liquidity Provision, Ambiguous Asset Returns and the Crisis
Dempster-Shafer
Updating of a
convex capacity for
the information {2,3}
(1,0,0)
(0,0,1)
(0,1,0)
2
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Liquidity Provision, Ambiguous Asset Returns and the Crisis
(0,0,1)
Dempster-Shafer
Updating of an
E-Capacity for the
information {1,2}
•
•
•
(1,0,0)
•
(0,1,0)
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Liquidity Provision, Ambiguous Asset Returns and the Crisis
Dempster-Shafer
Updating of an
E-Capacity for the
information {2,3}
(0,0,1)
•
•
(1,0,0)
(0,1,0)
3
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Liquidity Provision, Ambiguous Asset Returns and the Crisis
• The updating of the E-capacity for the information
{1,2} lead to an updated set of probability
distributions that is “larger” than the initial set.
• This suggests that the amount of ambiguity as
experienced by the consumer has increased, which
may lead to dynamically inconsistent behaviour.
• The updating of the E-capacity for the information
{2,3} leads to a single point.
• Here updating made the ambiguity disappear.
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Liquidity Provision, Ambiguous Asset Returns and the Crisis
Concluding Remarks on Updating
• The modelling of ambiguity as Choquet Expected
Utility is consistent both with the interpretation of
- “unknown outcomes” and of
- “unknown probabilities”.
• Updating of ambiguous beliefs may lead to
dynamic inconsistency in decision making.
• Simple capacities and their variations provide a
workable simplification that retains the full
intuition of decision making under ambiguity.
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Liquidity Provision, Ambiguous Asset Returns and the Crisis
4. The Basic Model
• Continuum [0,1] of ex-ante identical investors.
• Investment opportunities:
- zero interest money holdings
- illiquid assets which pay out
α1 < 1, when liquidated prematurely
αh > 1, when matured and successful
αℓ = 0, when matured and failure.
3
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Liquidity Provision, Ambiguous Asset Returns and the Crisis
Information regarding assets
Signal σ
b
b
g
g
Return ρ Probability π σρ
h
δ
ℓ
ε
h
1 - (δ + ε)
ℓ
0
3
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Liquidity Provision, Ambiguous Asset Returns and the Crisis
Timing
Period 0: - investment decisions
Period 1: - individual liquidity preference t ∈{H,L} becomes
privately known
- signal σ ∈{b,g} becomes publicly known
- interaction
- possibility for liquidation of assets
- consumption
Period 2: - return ρ ∈{h,ℓ} occurs
- consumption
3
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Liquidity Provision, Ambiguous Asset Returns and the Crisis
Beliefs
• ex-ante (calculable) risk with respect to:
- individual liquidity preference t ∈ {H,L}.
• ex-ante (incalculable) ambiguity with respect to:
- joint probability distribution of (σ,ρ).
• beliefs over {H,L}×{b,g}×{h,ℓ} consist of:
- an additive probability distribution P
- an additive partition of the state space with components
FH = {(H,b,h), (H,b,ℓ), (H,g,h), (H,g,ℓ)}
FL = {(L,b,h), (L,b,ℓ), (L,g,h), (L,g,ℓ)}
- a level of confidence γ ∈ [0,1] in P.
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Liquidity Provision, Ambiguous Asset Returns and the Crisis
Preferences:
• risk-neutral with vNM-utility index
u(x1(σ), x2(σ,ρ),t) = βt·x1(σ) + x2(σ,ρ)
where βH > βL > 1.
• ex-ante Choquet expected utility
γ
times expected utility w.r.t. P
plus
(1-γ) times
πH times minimum utility over FH
plus
πL times minimum utility over FL
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Liquidity Provision, Ambiguous Asset Returns and the Crisis
Updating the ambiguous beliefs leads to:
• Bayesian updating of the probability distribution P.
• an endogenous decrease in the level of confidence:
- after a good signal σ = g:
the level of confidence becomes
γg := γ·πg/(1-γ·πb) = γ·[(1-(δ+ε)) / (1-γ·(δ+ε))] < γ
- after a bad signal σ = b:
the level of confidence becomes
γb := γ·πb/(1-γ·πg) = γ·[(δ+ε) / (1-γ·(1-(δ+ε)))]
= γ·[(δ+ε) / (1-γ + γ·(δ+ε))] < γ.
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Liquidity Provision, Ambiguous Asset Returns and the Crisis
4. Second Best Efficiency
Low reserves:
• such that the incentive constraint of L-types holds even after
a bad signal.
• consequence: little welfare enhancing ad interim
redistribution to H-types.
• the money holdings are
μb(γ) := πH · γb · πhb · αh / (πL · βL + πH · γb · πhb · αh).
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Liquidity Provision, Ambiguous Asset Returns and the Crisis
High reserves:
• such that incentive constraint of L-types only holds after a
good signal.
• consequence: high amount of welfare enhancing ad interim
redistribution to H-types.
• the money holdings are
μg(γ) := πH · γg · πhg · αh / (πL · βL + πH · γg · πhg · αh)
= πH · γg · αh / (πL · βL + πH · γg · αh).
Assumption on parameters:
• the high reserves are second best efficient.
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Liquidity Provision, Ambiguous Asset Returns and the Crisis
y2
γ=1
γ<1
Second Best
Efficiency
ICLb
ICLb
ICLg
α2(1-μE(γ))
/ πL
α2(1-μE(1))
/ πL
ICLg
•
VEff
VEff
μE(γ) / πH μE(1) / πH
y1
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Liquidity Provision, Ambiguous Asset Returns and the Crisis
5. The Financial Sector
• Modelled as an unregulated competitive banking
sector offering deposit contracts.
• Period 0: - deposit contracts are offered
- investment decisions
Period 1: - withdrawal decisions
- possible liquidation of assets
• 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.
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Liquidity Provision, Ambiguous Asset Returns and the Crisis
• Only “fundamental” bank runs are taken into
account.
• High reserves:
- a bank run occurs after a bad signal.
- not second best efficient because assets are
liquidated after a bad signal.
• Low reserves:
- no bank run after either signal
- not second best efficient because reserves are too
low by assumption.
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Liquidity Provision, Ambiguous Asset Returns and the Crisis
y2
Unregulated
Banks with
γ ≤ 1 and
μB(γ) = μb(γ)
ICLb(γ)
○
α2
/ πL
(1-μB(γ))
α2(1-μE(1))
/ πL
•
○
ICLg(γ)
•
VBank
VEff
μB(γ) / πH μE(γ) / πH
y1
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Liquidity Provision, Ambiguous Asset Returns and the Crisis
6. Regulation
• In the case of a bank run the regulator should
interfere by:
- making immediate withdrawal less attractive and
- making waiting more attractive.
• This can be done by a revenue neutral scheme of:
- taxes on withdrawals and
- subsidies on waiting.
• Investors and banks should be made aware of this
policy, which:
- would NOT create moral hazard, but
- stop the banks’ being overly cautious.
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Liquidity Provision, Ambiguous Asset Returns and the Crisis
7. The Impact of Ambiguity
• If recognized ex-ante:
- reduces efficient reserve holdings
- no further impact.
• If not recognized ex-ante:
- a bad signal unexpectedly affects the incentive
constraint
- investor seem to “over-react” on bad news
- “panicking” investors cause a bank run, even for
the “safe” (low) reserve holdings.
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Liquidity Provision, Ambiguous Asset Returns and the Crisis
8. Concluding Remarks
The model suggests:
• a competitive banking sector with a regulatory
commitment to an appropriate tax-and-subsidize scheme
implements the second best efficient liquidity allocation.
• the “panic” and “over-reactions” of markets during the
crises may be due to a failure to recognize the ambiguity
experienced by professional and institutional investors.
• reducing ambiguity as suggested by the “Soziale
Marktwirtschaft” based on the “Freiburger Schule” reduces
“over-reactions” and enhances welfare.
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Liquidity Provision, Ambiguous Asset Returns and the Crisis
In the financial crisis:
• on aggregate, the economy faces the choice between either
liquidating illiquid assets or changing the trade-off current
and future income.
• this problem is similar to that faced by the aggregated
unregulated banking sector.
• 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.
49