Liquidity Provision, Ambiguous Asset Returns and the Financial Crisis by Willem Spanjers

1. 1 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

2. Liquidity Provision, Ambiguous Asset Returns and the Crisis 2 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

3. Liquidity Provision, Ambiguous Asset Returns and the Crisis 3 • 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.

4. Liquidity Provision, Ambiguous Asset Returns and the Crisis 4 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).

5. Liquidity Provision, Ambiguous Asset Returns and the Crisis 5 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]

6. Liquidity Provision, Ambiguous Asset Returns and the Crisis 6 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]

7. Liquidity Provision, Ambiguous Asset Returns and the Crisis 7 2. Ambiguity We consider: • Calculable vs incalculable risk • Sure Thing Principle • (Subjective) Expected Utility • Choquet Expected Utility

8. Liquidity Provision, Ambiguous Asset Returns and the Crisis 8 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.

9. Liquidity Provision, Ambiguous Asset Returns and the Crisis 9 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).

10. Liquidity Provision, Ambiguous Asset Returns and the Crisis 10 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.

11. Liquidity Provision, Ambiguous Asset Returns and the Crisis 11 (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 Neumann-Morgenstern utility index

12. Liquidity Provision, Ambiguous Asset Returns and the Crisis 12 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: γE{u(x)} + (1-γ) β maxsєSu(x(s)) + (1-γ) (1-β) minsєSu(x(s)).

13. Liquidity Provision, Ambiguous Asset Returns and the Crisis 13 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.

14. Liquidity Provision, Ambiguous Asset Returns and the Crisis 14 • An (E)llsberg capacity is described by: - a probability assessmentπ - a level of confidenceγ 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 [γπ(E∩Ej) + (1-γ) π(Ej) I(Ej;E)].

15. Liquidity Provision, Ambiguous Asset Returns and the Crisis 15 • For state contingent payouts x define m(Ej) as the minimum of u(x(s)) over all sϵ Ej. • The Choquet Expected Utility for a pessimistic consumer now is obtained as U(x) := γ Eπ{u(x(s))} + (1 - γ) Σj=1mπ(Ej) m(Ej).

16. Liquidity Provision, Ambiguous Asset Returns and the Crisis 16 3. Updating of Ambiguous Beliefs For this we consider: • The multiple prior representation. • Bayesian updating. • Updating capacities.

17. Liquidity Provision, Ambiguous Asset Returns and the Crisis 17 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!

18. Liquidity Provision, Ambiguous Asset Returns and the Crisis 18 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.

19. Liquidity Provision, Ambiguous Asset Returns and the Crisis 19 (0,0,1) Π • pmin (0,1,0) (1,0,0)

20. Liquidity Provision, Ambiguous Asset Returns and the Crisis 20 Capacities and multiple priors • Consider a capacity v: P(S)→[0,1]. • The core of the capacity v is the set C(v) := {pєΔS | for each EєP(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.

21. Liquidity Provision, Ambiguous Asset Returns and the Crisis 21 (0,0,1) p1≥ v1 p3≤ 1 - v12 p1≤ 1 - v23 Core of v p3≥ v3 p2≥ v2 p2≤ 1 - v13 (0,1,0) (1,0,0)

22. Liquidity Provision, Ambiguous Asset Returns and the Crisis 22 (0,0,1) The core of a simple capacity p1 ≥ γπ1 p2 ≥ γπ2 p3≤ 1 - γπ12 p2≤ 1 - γπ13 p3 ≥ γπ3 p1≤ 1 - γπ23 (0,1,0) (1,0,0)

23. Liquidity Provision, Ambiguous Asset Returns and the Crisis 23 (0,0,1) The core of an E-Capacity with E1={1,3} and E2={2} p1≥ v1 p3≤ 1 - v12 p1≤ 1 - v23 Core of an E-capacity p3≥ v3 (0,1,0) (1,0,0) p2= 1 - v13

24. Liquidity Provision, Ambiguous Asset Returns and the Crisis 24 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 AєP(S), thus ruling out all states in S\A. • Bayes‘ rule now gives the conditional probability of an event BєP(S) as P{B|A} = P{B∩A} / P{A}.

25. Liquidity Provision, Ambiguous Asset Returns and the Crisis 25 (0,0,1) Bayesian Updating after receiving the Information {2,3} • • (0,1,0) (1,0,0)

26. Liquidity Provision, Ambiguous Asset Returns and the Crisis 26 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.

27. Liquidity Provision, Ambiguous Asset Returns and the Crisis 27 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.

28. Liquidity Provision, Ambiguous Asset Returns and the Crisis 28 • When the information A is received, the Dempster-Shafer Update of v, i.e. the conditional capacity value v(B|A) for B єP(S) is obtained as vDS(B|A) := [v((B∩A) ﮞ (S\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}.

29. Liquidity Provision, Ambiguous Asset Returns and the Crisis 29 (0,0,1) Dempster-Shafer Updating of a convex capacity for the information {2,3} (0,1,0) (1,0,0)

30. Liquidity Provision, Ambiguous Asset Returns and the Crisis 30 (0,0,1) Dempster-Shafer Updating of an E-Capacity for the information {1,2} • • • • (0,1,0) (1,0,0)

31. Liquidity Provision, Ambiguous Asset Returns and the Crisis 31 (0,0,1) Dempster-Shafer Updating of an E-Capacity for the information {2,3} • • (0,1,0) (1,0,0)

32. Liquidity Provision, Ambiguous Asset Returns and the Crisis 32 • 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.

33. Liquidity Provision, Ambiguous Asset Returns and the Crisis 33 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.

34. Liquidity Provision, Ambiguous Asset Returns and the Crisis 34 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.

35. Liquidity Provision, Ambiguous Asset Returns and the Crisis 35 Information regarding assets Signal σ Return ρ Probability π σρ b h δ b ℓ ε g h 1 - (δ + ε) g ℓ 0

36. Liquidity Provision, Ambiguous Asset Returns and the Crisis 36 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

37. Liquidity Provision, Ambiguous Asset Returns and the Crisis 37 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.

38. Liquidity Provision, Ambiguous Asset Returns and the Crisis 38 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

39. Liquidity Provision, Ambiguous Asset Returns and the Crisis 39 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.

40. Liquidity Provision, Ambiguous Asset Returns and the Crisis 40 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).

41. Liquidity Provision, Ambiguous Asset Returns and the Crisis 41 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.

42. Liquidity Provision, Ambiguous Asset Returns and the Crisis 42 Second Best Efficiency γ < 1 y2 γ = 1 ICLb ICLb ICLg ICLg α2(1-μE(γ)) / πL • α2(1-μE(1)) / πL VEff VEff μE(γ)/ πH μE(1)/ πH y1

43. Liquidity Provision, Ambiguous Asset Returns and the Crisis 43 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.

44. Liquidity Provision, Ambiguous Asset Returns and the Crisis 44 • 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 efficientbecause reserves are too low by assumption.

45. Liquidity Provision, Ambiguous Asset Returns and the Crisis 45 Unregulated Banks with γ≤ 1 and μB(γ) = μb(γ) y2 ICLb(γ) ○ ICLg(γ) α2(1-μB(γ)) / πL • ○ • α2(1-μE(1)) / πL VBank VEff μB(γ) / πH μE(γ)/ πH y1

46. Liquidity Provision, Ambiguous Asset Returns and the Crisis 46 6. Regulation • In the case of a bank run the regulator should interfere by:- making immediate withdrawalless 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.

47. Liquidity Provision, Ambiguous Asset Returns and the Crisis 47 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.

48. Liquidity Provision, Ambiguous Asset Returns and the Crisis 48 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 ambiguityas suggested by the “Soziale Marktwirtschaft” based on the “Freiburger Schule” reduces “over-reactions” and enhances welfare.

49. Liquidity Provision, Ambiguous Asset Returns and the Crisis 49 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 efficienttax-and-subsidy scheme where: - the provision of liquidityis the counterpart of a tax on immediate consumption and - providing guarantees and bailing outdistressed banks is the counterpart to subsidizing future consumption.