Three Period Rollover Crisis Model

Lawrence J. Christiano Husnu Dalgic,with Xiaoming Li

September 21, 2019

Background• Banks:

– borrow short term and acquire long-term assets.– term mismatch.

• Vulnerability:– Banks with term mismatch need creditors to roll over short

term loans (‘deposits’), or else they could be forced to sell theirassets at fire sale prices (see, e.g., Shleifer-Vishny JEP2011).

– Central Banks have solved this problem for commercial banks(deposit insurance, liquidity backstops).

• Many financial institutions lie outside protective umbrella of thecentral bank: Shadow Banks.

• Roll-over Crisis scenario:– All creditors refuse to roll over their short term loans, forcing

banks into asset fire sales.

– If fire sale prices are low enough, banking system could becomeinsolvent and collapse, with damaging consequences for rest ofthe economy.

Zoltan Pozsar, Tobias Adrian, Adam Ashcraft, and Hayley Boesky, ‘Shadow Banking’, FederalReserve Bank of New York Economic Policy Review , December 2013

Background• View that run on Shadow Banks was trigger (or, massive

amplifier) of Great Recession:– Gary Gorton Slapped by the Invisible Hand, the Panic of 2007,

Oxford University Press, 2010.– Bernanke, statement before the Financial Crisis Inquiry

Commission, September 2010.– Formal models: Gertler-Kiyotaki, Banking, Liquidity, and Bank

Runs in an Infinite Horizon Economy, in AER2015.

• Here: provide a three period version of Gertler-Kiyotaki model.– Exploit simplicity of setup to explore number of equilibria and

impact of macro-prudential policy.

• Model is as simple as possible– Minimize agent heterogeneity.– Minimize number of periods: need at least three to have

maturity mismatch.– Style of Diamond-Dybvig (JPE1983), Chang-Velasco

(QJE2001).

Outline: Baseline Equilibrium

t = 0 t = 1 t = 2

Roll-Over

Annihilation

Partial-Run

Outline: Baseline Equilibrium

t = 0 t = 1 t = 2

No-run

Annihilation

Partial-Run

In period 0, banks issue one-period deposits, d0.Limited by amount of banker net worth, N0.

In period 0, banks use N0+d0 to purchasetwo-period lived capital.

In period 1, two possible equilibrium outcomes:

(i) No-run. Banks roll over their liabilities.

(ii) Annihilation run. Banks cannot roll over.

Bankers, Workers and Households

• Workers earn exogenous income in each period.

• Bankers acquire long-lived assets financed by their own networth and deposits.

• Workers and bankers live in identical households.

– Simplifies welfare analysis.

• The household instructs bankers to maximize the presentdiscounted value of profits.

– They can do this being good bankers and playing by the rules.

– Or, if bankers have an opportunity to bring home higherpresent value of profits by exploiting opportunities to steal,that’s ok too.

Bankers• At start of period t banker has assets, kb

t−1, and liabilities,Rt−1dt−1, acquired in previous period:

capital︷︸︸︷kb

t−1 ,

gross interest rate︷︸︸︷Rt−1 ×

deposits︷︸︸︷dt−1 .

• During period t = 0, 1:– Bankers combine deposits, dt, with net worth, Nt, and

purchase assets,Qtkb

t = Nt + dt,

Nt = (Zt + Qt) kbt−1 − Rt−1dt−1.

• In t = 2, Q2 = d2 = 0.• Here,

Zt ∼ exogenous productivity of capital.

Qt ∼ period t market price of capital.

Bankers, Scaling

• Later, will find it useful to scale variables.

• Banker net worth:

Nt = (Zt + Qt) kbt−1 − Rt−1dt−1

=

Rkt︷ ︸︸ ︷(

Zt + Qt

Qt−1

) φt−1︷ ︸︸ ︷Qt−1kb

t−1Nt−1

Nt−1 − Rt−1

φt−1−1︷ ︸︸ ︷dt−1

Nt−1Nt−1

=[(

Rkt − Rt−1

)φt−1 + Rt−1

]Nt−1

• Here,

φt ∼ banker leverage in period t

Banker Problem• Objective:

maxdt≥0

Vt,

where Vt denotes the value of banking, t = 0, 1, subject to:

Qtkbt = Nt + dt.

• Banker has the option to ‘run away’ with θQtkbt .

– Assets are diverted to banker’s household and (1− θ)Qtkbt are

destroyed, depositors get nothing.

• Banker must announce dt in advance, so depositors knowwhether banker intends to run away or not.

– As a result, bankers only consider dt for which

θQtkbt ≤ Vt.

• Must ensure banker problem has well-defined solution.

Rollover Crisis in period t = 1• Suppose all bankers receive zero deposits in t = 1 :

d∗1 = 0.

• Then, to pay period t = 1 liabilities, R0d0, the bankers mustsell capital, kb

0.

• With all bankers selling capital, the only buyers are households.– Households assign relatively low value to capital.– Capital sold at fire sale price, Q∗1.– Suppose the fire sale is sufficiently severe,

(Z1 + Q∗1) kb0 − R0d0 < 0→ N∗1 = 0

then households only get:

R0xd0,recovery ratio︷︸︸︷

x =(Z1 + Q∗1) kb

0R0d0

< 1.

Possible Events in Period t = 1• Previous slide suggests that collapse of the banking system

could be an equilibrium.– We will parameterize the model so that this is in fact the case.

• There is another equilibrium in period t = 1 when householdsprovide deposits and bankers pay off R0d0 without having tofire sale assets.

• So, have two equilibria:– Run in which bank net worth is wiped out (annihilation run).– No-run equilibrium in which d1 > 0 and Q1 > Q∗1.

• Later, will explore possibility of a third, partial run, equilibrium.

• For now, follow Gertler-Kiyotaki and suppose there are twothings that can happen in period 1:

P = prob[annihilation run]= 1− x1− P = prob[no-run equilibrium]= x.

Banker Problem in Period t = 0• Problem:

V0 = maxd0≥0

βm1 (1− P) [N1 + V1]+P×value of banking in run state︷︸︸︷

0 ,

s.t.: θQ0kb0 ≤ V0.

• Here,

P ∼ prob of rollover crisis in period 1

βm1 ∼ discount factor for no-run period 1 state

• Banker belongs to representative household and householdrequires discounting by intertemporal marginal utility ofconsumption, βm1:

βmt =βu′ (ct)

u′ (ct−1).

Scaled Banker Problem in t = 0• Period 0 problem:

V0 = maxd0≥0

βm1 (1− P) [N1 + V1]

s.t.: θQ0kb0 ≤ V0.

• Divide objective by N0 (ψt ≡ Vt/Nt):

ψ0 = maxd0≥0

βm1 (1− P)[

N1

N0+

V1

N1

N1

N0

]= max

φ0≥1βm1 (1− P) [1 + ψ1]

[(Rk

1 − R0

)φ0 + R0

]subject to participation constraint:

θφ0 ≤ ψ0.

• We only consider examples in which Rk1 > R0

– so banker sets φ0 to maximum allowed by participationconstraint.

Banker Problem in No-run Period t = 1• Period 1 problem:

V1 = maxd1≥0

βm2N2 = maxd1≥0

βm2

[Z2kb

1 − R1d1

],

s.t.: θQ1kb1 ≤ V1

• Scale by N1:

ψ1 = β maxφ1≥1

m2

[(Rk

2 − R1

)φ1 + R1

]subject to

θφ1 ≤ ψ1.

• We only consider examples where Rk2 > R1

– so, banker sets φ1 to the max allowed by participationconstraint.

Ensuring Banker Problem Well-defined inNo-run Period t = 1

• Because we assume Rkt+1 > Rt, bankers always go to boundary

of participation constraint:

θφt = ψt

• Boundary must be finite for equilibrium to exist.– Infinite leverage incompatible with loan market clearing.

• To ensure ψ1 < ∞ must have (see figure on next slide):

βm2

(Rk

2 − R1

)< θ

• Discussion of household problem below ensures that, inequilibrium

βm2R1 = 1, m2 = u′ (c2) /u′ (c1)

Figure: Condition for Finite Leverage in No-run Period t = 1

Leverage, !1

Value of bank, ψ1

Slope = "#$ %$& − %(

Value of bank (per unit of net worth), as a function of leverage.

If "m2(Rk2-R1) > )then infinite leverage consistent with participation constraint.

Slope = )

Participation constraint requires that bank have at least this value, ψ1=)!1 , for given leverage, !1.

1

"#$%$& = %$&/%(

)

Restrictions in no-run t = 1 forParticipation Constraint to be Binding• Previous slide showed must have

βm2

(Rk

2 − R1

)< θ

to ensure ψ1 < ∞.

• Since βm2R1 = 1 this corresponds to

Rk2 < (1 + θ)R1,

so banker selecting leverage so that θφ1 = ψ1 implies

θφ1 = βm2

[(Rk

2 − R1

)φ1 + R1

]or,

φ1 =βm2R1

θ − βm2(Rk

2 − R1) =

R1

(1 + θ)R1 − Rk2< ∞.

Restrictions in t = 1 Annihilation Run forParticipation Constraint to be Binding• The value of the bank in the period 1 run state (N∗1 = 0):

V∗1 = maxd1

βm∗2N2 = maxd1

βm∗2[Z2kb

1 − R∗1d1

]

= βm∗2

R∗,k2

=d∗1︷︸︸︷Q∗1kb

1−R∗1d∗1

.

• Participation constraint:

θd∗1 ≤ βm∗2[R∗,k2 − R∗1

]d∗1 .

• For this to represent a restriction on d1, need

(1 + θ)R∗1 > R∗,k2 ,

in which case d∗1 = 0.

Restrictions in t = 0 for ParticipationConstraint to be Binding

• Period 0 problem - maximize value by choice of φ0 :

ψ0 = maxφ0≥1

βm1 (1− P) [1 + ψ1][(

Rk1 − R0

)φ0 + R0

]subject to θφ0 ≤ ψ0.

• Assume Rk1 > R0, so banker wants φ0 big as possible.

• Need slope of firm value in φ0 smaller than θ:

βm1 (1− P) [1 + ψ1](

Rk1 − R0

)< θ

in which case, going to boundary of part. const. implies:

φ0 =βθ m1 (1− P) [1 + ψ1]R0

1− βθ m1 (1− P) [1 + ψ1]

(Rk

1 − R0) < ∞

We also require, φ0 > 1 (i.e., value at φ0 = 1 greater than θ).

Figure: Condition for Finite Leverage in Period t = 0

Leverage, !0

Value of bank, ψ0

ψ0 = "m1(1-P)(1+ψ1) [(Rk1-R0)!0+R0]

Value of bank (per unit of net worth), as a function of leverage.

If "m1(1-P)(1+ψ1) (Rk1-R0) > #

then infinite leverage consistent with participation constraint.

Slope = #

Value of bank required byparticipation constraint, as afunction of leverage, ψ0=#!0.

1

"m1(1-P)(1+ψ1)R1k

#

Households

• In period 0, inherit assets from the past, R−1d−1, kh−1.

• Choose deposits, consumption and capital holdings in period 0,period 1 (with probability, P, it is a run state).

• Consume in period 2 and then disappear.

• Bankers live in the households and so social welfare function isjust the utility of the typical household.

Table: Balance Sheets and Budget Constraints

Bank t Household

Q0kb0 = (Z0 + Q0) kb

−1 − R−1d−1︸ ︷︷ ︸N0

+d0 0 Q0

(kh

0 − kh−1

)+ c0 + d0 + f

(kh

0

)≤ R−1d−1 + Z0kh

−1 + y0

Q1kb1 = (Z1 + Q1) kb

0 − R0d0︸ ︷︷ ︸=N1=

[Rk

1φ0−R0(φ0−1)]N0

+d1 1 Q1

(kh

1 − kh0

)+ c1 + d1 + f

(kh

1

)≤ R0d0 + Z1kh

0 + y1

π2 = Z2kb1 − R1d1︸ ︷︷ ︸

=N2=[Rk

2φ1−R1(φ1−1)]N1

2 c2 ≤ R1d1 + Z2kh1 + π2

,

Households• utility at date 0:

u (c0) + βP [u (c∗1) + βu (c∗2)] + β (1− P) [u (c1) + βu (c2)]

• first order conditions for deposits (no deposits in run state ofperiod 1):

u′ (c0) = β[(1− P) u′ (c1) + Pu′ (c∗1) x

]R0

u′ (c1) = βu′ (c2)R1

• first order condition for capital decision in period 0:

u′ (c0) = β (1− P) u′ (c1)

(Z1 + Q1

Q0 + f ′(kh

0))

+ βPu′ (c∗1)

(Z1 + Q∗1

Q0 + f ′(kh

0))+ µu′ (c0)

µkh0 = 0, µ, kh

0 ≥ 0

Households

• first order condition for capital decision in no-run period 1:

u′ (c1) = βu′ (c2)Z2

Q1 + f ′(kh

1

) + ν× u′ (c1)

νkh1 = 0, ν, kh

1 ≥ 0

• first order condition for capital decision in run-state in period 1(when households hold kh

1 = 1) :

u′ (c∗1) = βu′ (c∗2)Z2

Q∗ + f ′ (1)

Market Clearing and other AggregateConditions

• Capital market:kh

t + kbt = 1

• Resource constraints obtained by combining household budgetconstraint with bank budget constraint in table above.

– Periods t = 0, 1 : ct + f(kh

t)= Zt + yt.

– Period t = 2 : c2 = c∗2 = Z2.• Probability of a run in period 1:

P = 1−min {x, 1} , x =(Z1 + Q∗) kb

0R0d0

.

• Discounting:

m1 =u′ (c1)

u′ (c0), m2 =

u′ (c2)

u′ (c1),

taken as exogenous by banks.

Solving the Model

• Exogenous variables:

R−1d−1, Z0kb−1, Z0kh

−1, y0 = y1 = y, Z0 = Z1 = Z, Z2

and parameters:α, β, σ, θ

where

u (c) =c1−σ

1− σ, f

(kh)=

α

2

(kh)2

.

• Solving simple in some cases, e.g., c2 = c∗2 = Z2.

• Other equations are set equal to zero, enforcing non-negativityconstraints and constraints about variables lying inside unitinterval (e.g., kh

t ).

Baseline Parameter Values

Parametersβ 0.9097σ 1.4951α 0.0626θ 0.3626Z 0.1263Z2 0.0908y 0.0878

kh−1 0.1738

Rd−1 0.3502

Baseline Equilibrium

Baseline Baselineφ0 5.95 x 0.90φ1 3.09 P 0.10ψ0 2.16 R0 1.08ψ1 1.12 R1 0.30c0 0.21 R∗1 0.39

kb0 0.98 Rk

1 1.11

kb1 0.82 Rk

2 0.32c1 0.21 d0 0.30c∗1 0.18 d1 0.16c2 0.09 N0 0.06Q0 0.37 N1 0.08Q1 0.28 σC 4.68Q∗ 0.17 Risk Premium 1.43

Macro Prudential Analysis

• Let φ̃0 denote leverage in the baseline equilibrium (= 5.95).

– Impose a restriction, φ0 ≤ φ̃0τ

– So, banker chooses φ0 ≤ min{

φ̃0τ, ψθ

}– The best equilibrium is one associated with τ = 0.98.

• Leverage restriction forces banks to internalize impact on P ofhigher leverage.

Baseline and Leverage Restriction

Baseline Policy Change (%) Baseline Policy Change (%)

φ0 5.95 5.83 -2.00 x 0.90 0.91 1.67

φ1 3.09 3.10 0.58 P 0.10 0.09 -14.54

ψ0 2.16 2.32 7.57 R0 1.08 1.08 0.15

ψ1 1.12 1.13 0.58 R1 0.30 0.31 0.06

c0 0.21 0.21 -0.12 R∗1 0.39 0.39 0.00

kb0 0.98 0.91 -7.56 Rk

1 1.11 1.12 1.24

kb1 0.82 0.82 -0.98 Rk

2 0.32 0.32 0.25

c1 0.21 0.21 -0.04 d0 0.30 0.28 -9.23

c∗1 0.18 0.18 0.00 d1 0.16 0.16 -0.95

c2 0.09 0.09 0.00 N0 0.06 0.06 -6.99

Q0 0.37 0.37 -1.40 N1 0.08 0.07 -1.80

Q1 0.28 0.28 -0.25 σC 4.68 4.36 -7.04

Q∗ 0.17 0.17 0.00 Risk P. 1.43 1.0474 -26.62

Welfare 0.0636Note: ‘welfare’ is the percent increase in period t = 0 consumption in the baseline equilibrium,which makes household indifferent between baseline equilibrium and leverage restrictedequilibrium.

Implementation Problem

• When we imposed the leverage constraint, we found that therewas another equilibrium:

– The other equilibrium is one with lower welfare (bank capitalfalls a lot).

– In the model, need to do more than just announce a leveragerestriction, to get good results.

• Example:

– Tax capital holdings, τ ×(kh

0 − k̄)

, where k̄ denotes capitalholdings in the desired equilibrium.

Looking for other Period t = 1 Equilibria

• Two equilibria: ‘no-run’ and ‘annihilation’ equilibrium.

• Can think about the two equilibria using Diamond-Dybvig logic.

– An equilibrium is the fixed point of a best response function.– Each bank chooses its d1 based on a conjecture about D1,

what all the other banks do:

d1 = f (D1) .

Looking for other Period t = 1 Equilibria

• Annihilation equilibrium:

– Each bank conjectures other banks set D1 = 0.– Each bank understands the other banks are not rolling over

their liabilities, so they must fire-sale their assets:

(Z + Q∗1) kb0 < R0d0.

– In this case, each bank chooses d1 = 0 knowing that it iswiped out:

0 = f (0) ,

fixed point!– This is why D1 = 0 is an equilibrium.

Looking for other Period t = 1 Equilibria• No-run equilibrium:

– Each bank expects the other banks will set high deposits, D1.– Each bank knows that bankers will be able to roll over their

liabilities and not fire-sale assets.– So, Q1 high and all banks solvent:

(Z + Q1) kb0 > R0d0.

– Each bank responds with d1 = D1, or,

D1 = f (D1)

– Another equilibrium!

• Can search for all equilibria by fixing the date 0 equilibriumallocations and graphing the function, f .

– An equilibrium is a D1 where f crosses the 45 degree line.

Best Response Function

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.160

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

No run equilibrium

Annihilationequilibrium

Partial run equilibrium

Annihilation, partial run and no-run equilibrium.

Observations

• There are three equilibria: annihilation, partial run and no-runequilibrium.

• One expects these three equilibria to exist, as long as x < 1,i.e., depositors lose money in an annihilation run equilibrium.

• As D1 rises above zero, Q1 rises because banks don’t have tosell so many assets to pay off debt.

– But, initially, N1 remains stuck at zero because x < 1:

N1 = max{0, (Z + Q1) kb0 − R0d0}

– So, f (D1) remains stuck at zero too.– Eventually, D1 rises enough that N1 becomes positive.

• Then, f rises rapidly and cuts 45 degree line from below.• That’s because banks earn a lot per dollar of deposits when net

worth is low (recall Gertler-Karadi).

Three Period 1 Equilibria

t = 0 t = 1 t = 2

Roll-Over

Annihilation

Partial-Run

Relationship to Gertler-Kiyotaki (AER2014)

• GK have Infinite horizon model.

• Must drain equity from bankers, or they’ll accumulate toomuch.

– A fraction of bankers die each period.– An equal fraction is born, with a small amount of net worth.

• Newborn bankers must be kept out of business during anannihilation run, because otherwise it would not be anequilibrium.

• If we assume that new-born bankers stay out during anannihilation run and enter in small numbers as D1 rises abovezero, we find the same three equilibria in GK.

Deviation in State s = t, from GKEquilibrium

Aggregate Best Response, GK Model

xOne interpretation of GK: aggregate best response functiondiscontinuous at zero.

Aggregate Best Response, GK Model

xObtain indicated best response function in GK model if assumenewborns coming in very slowly at low levels of D1

Conclusion• Described simple three-period model with maturity mismatch in

banks.

• Three findings.

1 Limitations on deposits reduces probability of crisis andimproves welfare.

2 Macro prudential policy requires solving a non-trivialimplementation problem.

– Can have multiple equilibria.– Showed how supplementing leverage constraint with a tax on

household capital could solve implementation problem.

3 Gertler-Kiyotaki run/no-run result is fragile.– Under reasonable changes in assumptions about the arrival

rate of new banks, there is a qualitative change in the set ofequilibria.

– One change results in a unique equilibrium with no runs.– Another change introduces a third, partial-run, equilibrium.