Optimal Deposit Insurance Eduardo D avila and Itay Goldstein - - PowerPoint PPT Presentation

Optimal Deposit Insurance

Eduardo D´ avila and Itay Goldstein

Yale/NYU Stern/NBER and UPenn Wharton

Diamond/Dybvig@36 Conference 3/30/2019

1 / 19

Motivation

• Deposit insurance: main explicit financial guarantee
• Significant effects
• 4,000 bank failures only in 1933
• 4,000 bank failures between 1934 and 2014

Question

What is the optimal level of deposit insurance?

• Are existing coverage levels (✩250,000 or e100,000) optimal?

Figure 2 / 19

Motivation

• Deposit insurance: main explicit financial guarantee
• Significant effects
• 4,000 bank failures only in 1933
• 4,000 bank failures between 1934 and 2014

Question

What is the optimal level of deposit insurance?

• Are existing coverage levels (✩250,000 or e100,000) optimal?

Figure

• This paper
• 1. Characterize welfare impact of changes in the level of DI coverage dW

dδ

• Applies broadly
• 2. As a function of a small number of sufficient statistics
• Connects theory and measurement

2 / 19

Main results

• 1. Welfare impact of change in level of coverage

dW dδ = A × B − C × D

• Marginal benefit
• A -Sensitivity of bank failure probability to DI change
• B Utility gain of preventing marginal failure
• Marginal cost
• C Probability of bank failure
• D Expected marginal social cost of intervention in case of bank failure

3 / 19

Main results

• 1. Welfare impact of change in level of coverage

dW dδ = A × B − C × D

• Marginal benefit
• A -Sensitivity of bank failure probability to DI change
• B Utility gain of preventing marginal failure
• Marginal cost
• C Probability of bank failure
• D Expected marginal social cost of intervention in case of bank failure

3 / 19

Main results

• 1. Welfare impact of change in level of coverage

dW dδ = A × B − C × D

• Marginal benefit
• A -Sensitivity of bank failure probability to DI change
• B Utility gain of preventing marginal failure
• Marginal cost
• C Probability of bank failure
• D Expected marginal social cost of intervention in case of bank failure
• 2. Insights
• Sufficient statistics
• If C → 0, unlimited DI is optimal (DD83)

3 / 19

Main results

• 1. Welfare impact of change in level of coverage

dW dδ = A × B − C × D

• Marginal benefit
• A -Sensitivity of bank failure probability to DI change
• B Utility gain of preventing marginal failure
• Marginal cost
• C Probability of bank failure
• D Expected marginal social cost of intervention in case of bank failure
• 2. Insights
• Sufficient statistics
• If C → 0, unlimited DI is optimal (DD83)
• 3. Ex-ante regulation
• Focus on marginal fiscal externalities (not fairly-priced DI)
• Both asset- and liability-side regulation are needed

3 / 19

Main results

• 1. Welfare impact of change in level of coverage

dW dδ = A × B − C × D

• Marginal benefit
• A -Sensitivity of bank failure probability to DI change
• B Utility gain of preventing marginal failure
• Marginal cost
• C Probability of bank failure
• D Expected marginal social cost of intervention in case of bank failure
• 2. Insights
• Sufficient statistics
• If C → 0, unlimited DI is optimal (DD83)
• 3. Ex-ante regulation
• Focus on marginal fiscal externalities (not fairly-priced DI)
• Both asset- and liability-side regulation are needed
• 4. Quantitative implications
• Direct measurement
• Model simulation

3 / 19

Outline

• 1. Basic framework
• Positive analysis
• Welfare analysis ⇒ dW

dδ (main result)

• 2. Extensions
• 3. Measurement
• 4. Conclusion
• Theoretical contribution: rich cross-section of depositors

Literature 4 / 19

Environment

• t = 0, 1, 2
• Aggregate state (profitability) s ∈ [s, s], known at date 1, cdf F(·)

5 / 19

Environment

• t = 0, 1, 2
• Aggregate state (profitability) s ∈ [s, s], known at date 1, cdf F(·)
• Depositors
• Double continuum of depositors, mass D0i ∼ G(·) (cdf),
• Fraction λ of early types
• Endowments Y1i(s) (early), Y2i(s) (late)
• Ex-ante expected utility

Es [λU (C1i (s)) + (1 − λ) U (C2i (s))]

• Depositors choose D1i(s) ∈ [0, R1D0i]

5 / 19

Environment

• t = 0, 1, 2
• Aggregate state (profitability) s ∈ [s, s], known at date 1, cdf F(·)
• Depositors
• Double continuum of depositors, mass D0i ∼ G(·) (cdf),
• Fraction λ of early types
• Endowments Y1i(s) (early), Y2i(s) (late)
• Ex-ante expected utility

Es [λU (C1i (s)) + (1 − λ) U (C2i (s))]

• Depositors choose D1i(s) ∈ [0, R1D0i]
• Banks technology
• −1 → ρ1(s) (date 1) → ρ2(s) > 0 (date 2)
• Returns ρ1(s) > 0 and ρ2(s) > 0 increasing in s

5 / 19

Environment

• t = 0, 1, 2
• Aggregate state (profitability) s ∈ [s, s], known at date 1, cdf F(·)
• Depositors
• Double continuum of depositors, mass D0i ∼ G(·) (cdf),
• Fraction λ of early types
• Endowments Y1i(s) (early), Y2i(s) (late)
• Ex-ante expected utility

Es [λU (C1i (s)) + (1 − λ) U (C2i (s))]

• Depositors choose D1i(s) ∈ [0, R1D0i]
• Banks technology
• −1 → ρ1(s) (date 1) → ρ2(s) > 0 (date 2)
• Returns ρ1(s) > 0 and ρ2(s) > 0 increasing in s
• Deposit contract
• Banks offer noncontingent deposit rate R1
• Pro-rata distribution after failure

5 / 19

Environment

• t = 0, 1, 2
• Aggregate state (profitability) s ∈ [s, s], known at date 1, cdf F(·)
• Depositors
• Double continuum of depositors, mass D0i ∼ G(·) (cdf),
• Fraction λ of early types
• Endowments Y1i(s) (early), Y2i(s) (late)
• Ex-ante expected utility

Es [λU (C1i (s)) + (1 − λ) U (C2i (s))]

• Depositors choose D1i(s) ∈ [0, R1D0i]
• Banks technology
• −1 → ρ1(s) (date 1) → ρ2(s) > 0 (date 2)
• Returns ρ1(s) > 0 and ρ2(s) > 0 increasing in s
• Deposit contract
• Banks offer noncontingent deposit rate R1
• Pro-rata distribution after failure
• Deposit insurance
• Government guarantees δ dollars
• Fiscal shortfall is T (s); Cost of public funds κ (T (s))
• DWL 1 − χ(s) after bank failure

5 / 19

Environment

• Taxpayers

Vτ (δ, R1) = Es [U (Yτ (s) − T (s) − κ (T (s)))]

6 / 19

Environment

• Taxpayers

Vτ (δ, R1) = Es [U (Yτ (s) − T (s) − κ (T (s)))]

• Timeline

Deposit insurance δ determined Deposit rate R1 determined Depositors choose depositholdings D1i s is realized t = 0 t = 1 t = 2

6 / 19

Environment

• Taxpayers

Vτ (δ, R1) = Es [U (Yτ (s) − T (s) − κ (T (s)))]

• Timeline

Deposit insurance δ determined Deposit rate R1 determined Depositors choose depositholdings D1i s is realized t = 0 t = 1 t = 2

• Two possibilities at date 1
• 1. Bank failure
• 2. No bank failure

C1i (s) =

• min {D0iR1, δ} + αF (s) max {D0iR1 − δ, 0} + Y1i (s) ,

Bank Failure D0iR1 + Y1i (s) , No Failure, C2i (s) =

• min {D0iR1, δ} + αF (s) max {D0iR1 − δ, 0} + Y2i (s) ,

Bank Failure αN (s) D0iR1 + Y2i (s) , No Failure,

6 / 19

Equilibrium: Definition

• Equilibrium: depositors choose D1i (s) optimally, given other

depositors choices and given values of R1 and δ

• Symmetric equilibria
• Sunspot π ∈ [0, 1]

7 / 19

Equilibrium: Definition

• Equilibrium: depositors choose D1i (s) optimally, given other

depositors choices and given values of R1 and δ

• Symmetric equilibria
• Sunspot π ∈ [0, 1]
• Key assumptions
• 1. Restriction to deposit contract (noncontingent and demandable)
• 2. Single policy instrument (noncontingent deposit insurance with full

commitmment)

7 / 19

Equilibrium: Definition

• Equilibrium: depositors choose D1i (s) optimally, given other

depositors choices and given values of R1 and δ

• Symmetric equilibria
• Sunspot π ∈ [0, 1]
• Key assumptions
• 1. Restriction to deposit contract (noncontingent and demandable)
• 2. Single policy instrument (noncontingent deposit insurance with full

commitmment)

• Three scenarios
• 1. R1 predetermined (baseline)
• 2. R1 chosen by competitive banks
• 3. R1 chosen by the planner (perfect regulation)

7 / 19

Equilibrium: Depositors’ behavior

• Three types of depositor
• 1. Early depositors: withdraw all deposits

8 / 19

Equilibrium: Depositors’ behavior

• Three types of depositor
• 1. Early depositors: withdraw all deposits
• 2. Full insured late depositors: leave all deposits

8 / 19

Equilibrium: Depositors’ behavior

• Three types of depositor
• 1. Early depositors: withdraw all deposits
• 2. Full insured late depositors: leave all deposits
• 3. Partially insured late depositors: leave δ or all deposits (indeterminacy)

8 / 19

Equilibrium: Depositors’ behavior

• Three types of depositor
• 1. Early depositors: withdraw all deposits
• 2. Full insured late depositors: leave all deposits
• 3. Partially insured late depositors: leave δ or all deposits (indeterminacy)

Deposits State (s) D−

1 (δ, R1) = (1 − λ)

D

0 min {D0iR1, δ} dF (i)

D+

1 (R1) = (1 − λ) D0R1

˜ D1 (s) = (R1−ρ1(s))D0

1−

1 ρ2(s)

s s s∗(δ, R1) ˆ s(R1) ρ−1

1 (R1)

ρ−1

2 (1)

Fundamental Failures Panic Failures Unique (Failure) Equilibrium Multiple Equilibria Unique (No Failure) Equilibrium ↑ δ ⇒ ↓ Multiplicity Region 8 / 19

Equilibrium: Regions

• Failure probability

qF (δ, R1) = F (ˆ s (R1)) + π [F (s∗ (δ, R1)) − F (ˆ s (R1))]

δ s s s DR1 s∗ (δ, R1) ˆ s (R1) ρ−1

1

(R1) Unique (No Failure) Equilibrium Multiple Equilibria Unique (Failure) Equilibrium

9 / 19

Welfare

W (δ) =

• Vj (δ, R1) d

j =

• Vi (δ, R1) dG (i)
• Depositors

+ Vτ (δ, R1)

• Taxpayers
• Same pareto weight
• Scope for reweighting

10 / 19

Main result: Directional Test for level of DI

11 / 19

Main result: Directional Test for level of DI

Marginal change in DI (first order effects)

dW dδ = − ∂qF ∂δ U

• CN

j (s∗)

• − U
• CF

j (s∗)

• d

j+ + qF EF

s

• U ′

CF

j

∂CF

j

∂δ d j

• 11 / 19

Main result: Directional Test for level of DI

Marginal change in DI (first order effects)

dW dδ = − ∂qF ∂δ U

• CN

j (s∗)

• − U
• CF

j (s∗)

• d

j+ + qF EF

s

• U ′

CF

j

∂CF

j

∂δ d j

• Marginal benefit

∂qF ∂δ

• Change in Failure Probability

U

• CN

j (s∗)

• − U
• CF

j (s∗)

• d

j

• Aggregate Consumption Drop

11 / 19

Main result: Directional Test for level of DI

Marginal change in DI (first order effects)

dW dδ = − ∂qF ∂δ U

• CN

j (s∗)

• − U
• CF

j (s∗)

• d

j+ + qF EF

s

• U ′

CF

j

∂CF

j

∂δ d j

• Marginal benefit

∂qF ∂δ

• Change in Failure Probability

U

• CN

j (s∗)

• − U
• CF

j (s∗)

• d

j

• Aggregate Consumption Drop

CN

j

• s∗

− CF

j

• s∗

d j =

• ρ2
• s∗

− 1 ρ1

• s∗

− λR1

• D0
• Net Return Loss

+

• 1 − χ
• s∗

ρ1

• s∗

D0

• Bank Failure

Deadweight Loss

+ κ

• T
• s∗
• Total Net Cost
• f Public Funds

11 / 19

Main result: Directional Test for level of DI

Marginal change in DI (first order effects)

dW dδ = − ∂qF ∂δ U

• CN

j (s∗)

• − U
• CF

j (s∗)

• d

j+ + qF EF

s

• U ′

CF

j

∂CF

j

∂δ d j

• Marginal cost

∂CF

j

∂δ d j = −

• Mg. Cost
• f Public Funds
• κ′ (T (s))

Mass of Partially Insured

• D

δ R1

dG (i) qF ≡ Probability of failure

11 / 19

Main result: Directional Test

Preference free approximation

dW dδ ≈ −∂qF ∂δ CN

j (s∗) − CF j (s∗)

• d

j + qF EF

s

∂CF

j

∂δ d j

• ,

12 / 19

Endogenous Deposit Rate and Ex-ante Regulation

• 1. Regulated deposit rate
• (As if) government chooses R1
• Same equation holds (!)

13 / 19

Endogenous Deposit Rate and Ex-ante Regulation

• 1. Regulated deposit rate
• (As if) government chooses R1
• Same equation holds (!)
• 2. Directional test without regulation (R1 chosen by competitive banks)

dW dδ = −∂qF ∂δ U

• CN

j (s∗)

• − U
• CF

j (s∗)

• d

j + qF EF

s

• U ′

CF

j

∂CF

j

∂δ d j

• +

∂Vτ ∂R1 dR1 dδ

• Fiscal Externality

13 / 19

Endogenous Deposit Rate and Ex-ante Regulation

• 1. Regulated deposit rate
• (As if) government chooses R1
• Same equation holds (!)
• 2. Directional test without regulation (R1 chosen by competitive banks)

dW dδ = −∂qF ∂δ U

• CN

j (s∗)

• − U
• CF

j (s∗)

• d

j + qF EF

s

• U ′

CF

j

∂CF

j

∂δ d j

• +

∂Vτ ∂R1 dR1 dδ

• Fiscal Externality
• 3. Pigouvian correction (wedge): marginal fiscal externality

τR1 = − ∂Vτ ∂R1 ≈ ∂Es [T (s) + κ (T (s))] ∂R1

• Exact measure of moral hazard

13 / 19

Implications for regulation

• 1. Non-unique implementation (deposit premium/deposit caps)
• 2. Optimal corrective policy vs. fairly priced deposit insurance
• 3. Double dividend of DI fund (corrective vs. revenue raising)
• 4. Return of insurance fund is relevant (interaction with fiscal cost)

14 / 19

Extensions

• 1. General portfolio and investment decisions (richer forms of moral

hazard) ∂Vτ ∂R1 dR1 dδ

• Liability-side regulation

+

• j

∂Vτ ∂ψj dψj dδ .

• Asset-side regulation
• 2. Global game (information structure)
• 3. Macro effects (spillovers)

15 / 19

Extensions

• 1. General portfolio and investment decisions (richer forms of moral

hazard) ∂Vτ ∂R1 dR1 dδ

• Liability-side regulation

+

• j

∂Vτ ∂ψj dψj dδ .

• Asset-side regulation
• 2. Global game (information structure)
• 3. Macro effects (spillovers)
• Further extensions
• 1. Lender of last resort
• 2. Multiple deposit accounts
• 3. Equityholders/Debtholders/Liquidity benefits
• 4. Departures from bank value maximization: imperfect competition and

agency frictions

• 5. Unregulated sector

15 / 19

Quantitative Implications

Model Primitives ⇒

• (2)

Sufficient Statistics ⇒

• (1)

Welfare

• Dual approach
• 1. Direct measurement
• Local test for whether to change δ∗
• 2. Model simulation

16 / 19

Quantitative Implications: Direct Measurement

• Normalized approximation

dWk dδ

Gk ≈ −∂qF

k

∂δ CN

j,k (s∗) − CF j,k (s∗)

• d

j Gk − qF

k EF s [κ′ (·)]

D

δ R1 dGk (i)

Gk

✩

17 / 19

Quantitative Implications: Direct Measurement

• Normalized approximation

dWk dδ

Gk ≈ −∂qF

k

∂δ CN

j,k (s∗) − CF j,k (s∗)

• d

j Gk − qF

k EF s [κ′ (·)]

D

δ R1 dGk (i)

Gk

• Measured marginal gain: dWk

dδ

≈ $11.5

• Gain of ✩100,000 increase in limit is 0.06% of assets

17 / 19

Quantitative Implications: Model Simulation

100 200 300 400 500 600 700

• 1.135
• 1.13
• 1.125
• 1.12
• 1.115
• 1.11
• 1.105

10-6

• Asymmetric welfare function

18 / 19

Conclusion

• Characterization of optimal deposit insurance
• As a function of a few sufficient statistics
• For a wide range of environments
• New insights regarding optimal ex-ante regulation
• Direct implications for measurement

19 / 19

Literature

• 1. Bank Runs: Diamond and Dybvig 83, Cooper and Ross 98, Rochet

and Vives 05, Goldstein and Pauzner 05, Allen and Gale 07, Uhlig 10, Keister 12

• 2. Deposit Insurance/Government Guarantees: Merton 77, Kareken

and Wallace 78, Calomiris 90, Chan, Greenbaum and Thakor 92, Freixas and Rochet 98, Freixas and Gabillon 99, Cooper and Ross 02, Allen, Carletti, Goldstein and Leonello 14

• 3. Quantitative Work:
• Microeconometric: Demirguc-Kunt and Detragiache 02, Ioannidou

and Penas 10, Iyer and Puri 12

• Structural: Gertler and Kiyotaki 13, Egan, Hortacsu and Matvos 14,

Kashyap, Tsomocos and Vardoulakis 14

• 4. Sufficient Statistics/PF: Diamond 98, Saez 01, Chetty 09, Matvos

13, Davila 14, Weyl 15

Back to text 1 / 4

Equilibrium: R∗

1(δ)

• When banks choose R1
• 1. Banks internalize effect of R1 on failure probability
• 2. Banks do not internalize effect of R1 on fiscal shortfall T(s)

3.

dR∗

1

dδ > 0

2 / 4

Main result: Optimal Exemption

• dW

dδ = 0 (under regularity conditions) ⇒ δ∗

3 / 4

Main result: Optimal Exemption

• dW

dδ = 0 (under regularity conditions) ⇒ δ∗

Optimal Deposit Insurance

δ∗ = εq

δ

U

• CF

j (s∗)

• − U
• CN

j (s∗)

• d

j qF EF

s

• U ′
• CF

j

∂CF

j

∂δ d

j

• Intuition
• εq

δ ≡ ∂q(δ) ∂ log δ < 0

• Partial derivative
• Numerator: δ∗ increases with |εq

δ|, CF j (s) − CN j (s)

• Denominator: δ∗ decreases with qF , κ
• If qF → 0, expected marginal cost → 0, so δ∗ → ∞ (DD83)

3 / 4

Main result: Optimal Exemption

• dW

dδ = 0 (under regularity conditions) ⇒ δ∗

Optimal Deposit Insurance

δ∗ = εq

δ

U

• CF

j (s∗)

• − U
• CN

j (s∗)

• d

j qF EF

s

• U ′
• CF

j

∂CF

j

∂δ d

j

• Interpretation
• Variables are endogenous and observable
• Sufficient statistic logic (CAPM, q-theory analogy)
• Similar to optimal taxation problems
• Exact expression
• Non-marginal changes W (δ1) − W (δ0) =

δ1

δ0 dW dδ

• ˜

δ

• d˜

δ

3 / 4

Example W(δ)

100 200 300 400 500 600 700

• 1.135
• 1.13
• 1.125
• 1.12
• 1.115
• 1.11
• 1.105

10-6

Parameters

• Convexity not-guaranteed

4 / 4