Interbank Lending - Modelling the Development of Interbank Markets

The modelOptimal policies in t1

Example

Interbank LendingModelling the Development of Interbank Markets during a Liquidity

Shock

Dominik Joos,Karlsruhe Institute of Technology

21st March 2011

Dominik Joos, Karlsruhe Institute of Technology Interbank Lending


Example

Motivation: decoupling of interest rates



Example

The model

We study a model with

three dates t = 0, t1, t2

a [0, 1] continuum of identical banks maximizing expected utility oftheir profit

households which have future liquidity needs and let banks managetheir funds

one risky illiquid asset with uncertain payoff at t = t2

a government bond with certain return at t = t2

costless storage



Example

The model







costless storage



Example

The model







costless storage



Example

The model







costless storage



Example

The model







costless storage



Example

The model







costless storage



Example

Management of funds

We assume that each bank has one unit of the households’ fundsunder management at t = 0 and offers claims that can be withdrawneither at t = t1 or t = t2 worth c1 and c2 respectively.

The demand for liquidity on individual bank level is determined by a[0, 1]-valued random variable Λ describing the fraction of capital,which is withdrawn in t = t1.

Thus banks have the following random cashflows:

t 0 t1 t2

deposit contracts 1 −Λc1 −(1− Λ)c2



Example

Management of funds




t 0 t1 t2




Example

Management of funds




t 0 t1 t2




Example

External investment possibilities

The payoff of the illiquid asset is modelled by an R+-valued process(St)0≤t≤t2 and the price for the bond is denoted by P0 in t = 0 andP2 in t = t2.

Then we have the following assets and financial claims:

t 0 t1 t2

cash −δ0 δ0

−δ1 δ1

stock −α0 0 α0St2

S0

bond −β0 0 β0P2

P0

cash flow 1 −Λc1 −(1− Λ)c2



Example

External investment possibilities

The payoff of the illiquid asset is modelled by an R+-valued process(St)0≤t≤t2 and the price for the bond is denoted by P0 in t = 0 andP2 in t = t2.

Then we have the following assets and financial claims:

t 0 t1 t2

cash −δ0 δ0

−δ1 δ1

stock −α0 0 α0St2

S0

bond −β0 0 β0P2

P0




Example

Interbank markets

In order to smooth out the different liquidity levels in t = t1, we allow forinterbank markets to develop:

Secured lending is modelled by an outright sale of bonds at price P1

The unsecured market consits of borrowing and lending at aninterest rate r

We assume that both markets are anonymous and competitive

Banks are price takers, prices are set by a Walrasian auctioneer

Banks are completely diversified across unsecured interbank loans

We denote the probability that a borrower in the unsecured marketis solvent in t = t2 by p̂



Example

Interbank markets










Example

Interbank markets










Example

Interbank markets










Example

Interbank markets










Example

Interbank markets










Example

Interbank markets










Example

This yields the following assets and financial claims:

t 0 t1 t2

cash −δ0 δ0

−δ1 δ1

stock −α 0 αSt2

S0

bond −β0 β0P1

P0

−β1 β1P2

P1

unsecured ib debt −γ+ γ+p̂(1 + r)

γ− −γ−(1 + r)




Example

The optimization problem in t1

We consider the investment decision from t = 0, (α,β0,δ0) as given.Let λ, r and P1 be fixed, then the optimization problem in t1 is

maxβ1,γ+,γ−,δ1

E [u(V2(β1, γ+, γ−, δ1))]

Constraints are

β1 + γ+ + δ1 = δ0 + β0P1

P0+ γ− − λc1,

β1, γ+, γ−, δ1 ≥ 0,

where

V2(β1, γ+, γ−, δ1) = αSt2

S0+β1

P2

P1+γ+(1+r)p̂−γ−(1+r)+δ1−(1−λ)c2.



Example

Optimal policies in t1

We have to optimize pointwise for every possible liquidity demand andevery price pair on the interbank market, thus we derive an optimalsupply and demand function, i.e. a function

(β1, γ+, γ−, δ1) : supp(Λ)× (−1,∞)× (0,∞)→ R4≥0

(λ, r ,P1) 7→ (βλ1 (r ,P1), γλ+(r ,P1), γλ−(r ,P1), δλ1 (r ,P1))

which (pointwise for every (λ, r ,P1)) solves our optimization problem.



Example

Since expectation is monotone and the utility function is increasing theobjective function

E [u(αSt2

S0+ βλ1

P2

P1+ γλ+(1 + r)p̂ − γλ−(1 + r) + δλ1 − (1− λ)c2)]

becomes

βλ1P2

P1+ γλ+(1 + r)p̂ − γλ−(1 + r) + δλ1 ,

which is an affine function (as well as the constraint functions).Therefore Karush-Kuhn-Tucker conditions are necessary and sufficient foroptimality.



Example

In equilibrium, aggregate demand equals aggregate supply.Aggregate net demand functions are

E [(βΛ1 (r ,P1)− β0

P1

P0)+], E [γΛ

+(r ,P1)],

net supply is

E [(β0P1

P0− βΛ

1 (r ,P1))+], E [γΛ−(r ,P1)].

Thus an equlibrium price pair (r∗,P∗1 ) satisfies

E [γΛ+(r∗,P∗1 )] = E [γΛ

−(r∗,P∗1 )],

E [βΛ1 (r∗,P∗1 )] = β0

P∗1P0.



Example

In equilibrium, aggregate demand equals aggregate supply.Aggregate net demand functions are

E [(βΛ1 (r ,P1)− β0

P1

P0)+], E [γΛ

+(r ,P1)],

net supply is

E [(β0P1

P0− βΛ

1 (r ,P1))+], E [γΛ−(r ,P1)].

Thus an equlibrium price pair (r∗,P∗1 ) satisfies

E [γΛ+(r∗,P∗1 )] = E [γΛ

−(r∗,P∗1 )],

E [βΛ1 (r∗,P∗1 )] = β0

P∗1P0.



Example

For δ0 < E (Λ)c1, there are no equilibrium prices for the interbankmarkets.

For δ0 > E (Λ)c1, there is secured interbank lending. An unsecuredinterbank market develops iff

β0P2

P0+ δ0 < c1ess sup(Λ).

The equilibrium prices are given by

r =1

p̂− 1, P1 = P2.



Example

For δ0 < E (Λ)c1, there are no equilibrium prices for the interbankmarkets.

For δ0 > E (Λ)c1, there is secured interbank lending. An unsecuredinterbank market develops iff

β0P2

P0+ δ0 < c1ess sup(Λ).

The equilibrium prices are given by

r =1

p̂− 1, P1 = P2.



Example

For δ0 = E (Λ)c1, there is secured interbank lending. An unsecuredinterbank market develops iff

β0P2

P0< c1(ess sup(Λ)− E (Λ)).

In this case the equilibrium prices satisfy

r ∈ [1

p̂− 1,∞), P1 =

P2

(1 + r)p̂.

If there is no unsecured lending, the bond price must satisfy

P1 ∈ [P0c1

β0(ess sup(Λ)− E (Λ)),P2].



Example

In order to get a closed form for the optimal policies, we define

Ss := E[(Λc1−δ0)1{ 1c1δ0≤Λ≤ 1

c1(β0

P1P0

+δ0)}]+β0P1

P0P(Λ >

1

c1(β0

P1

P0+δ0)),

the aggregate supply of secured interbank loans

,

Su := E[(Λc1 − (β0P1

P0+ δ0))1{Λ> 1

c1(β0

P1P0

+δ0)}],

the aggregate supply of unsecured interbank loans and

D := E[(δ0 − Λc1)1{Λ< 1c1}]

the excess capital of banks with liquidity surplus, which equals theaggregate demand for interbank loans plus demand for liquid asset.



Example


Ss := E[(Λc1−δ0)1{ 1c1δ0≤Λ≤ 1

c1(β0

P1P0

+δ0)}]+β0P1

P0P(Λ >

1

c1(β0

P1

P0+δ0)),

the aggregate supply of secured interbank loans,

Su := E[(Λc1 − (β0P1

P0+ δ0))1{Λ> 1

c1(β0

P1P0

+δ0)}],

the aggregate supply of unsecured interbank loans

and

D := E[(δ0 − Λc1)1{Λ< 1c1}]




Example


Ss := E[(Λc1−δ0)1{ 1c1δ0≤Λ≤ 1

c1(β0

P1P0

+δ0)}]+β0P1

P0P(Λ >

1

c1(β0

P1

P0+δ0)),

the aggregate supply of secured interbank loans,

Su := E[(Λc1 − (β0P1

P0+ δ0))1{Λ> 1

c1(β0

P1P0

+δ0)}],

the aggregate supply of unsecured interbank loans and

D := E[(δ0 − Λc1)1{Λ< 1c1}]




Example

βλ∗1 =

0 , λ > 1

c1(β0

P1

P0+ δ0)

β0P1

P0+ δ0 − λc1 , 1

c1δ0 ≤ λ ≤ 1

c1(β0

P1

P0+ δ0)

β0P1

P0+ Ss

D (δ0 − λc1) , λ < 1c1δ0

γλ∗+ =

0 , λ ≥ 1c1δ0

Su

D (δ0 − λc1) , λ < 1c1δ0

γλ∗− =

λc1 − (β0P1

P0+ δ0) , λ > 1

c1(β0

P1

P0+ δ0)

0 , λ ≤ 1c1

(β0P1

P0+ δ0)

δλ∗1 =

0 , λ ≥ 1c1δ0

(1− Su+Ss

D )(δ0 − λc1) , λ < 1c1δ0



Example

For δ0 ≥ E (Λ)c1 the strategy (βλ∗1 , γλ∗+ , γλ∗− , δλ∗1 ) is optimal and profit in

t = t2 is given by

V2(α, β0, δ0) = αSt2

S0+ β0

P2

P0− (1− Λ)c2 + (δ0 − Λc1)

P2

P11{Λ≤ 1

c1δ0}+

+ (δ0 − Λc1)P2

P11{ 1

c1δ0<Λ≤ 1

c1(β0

P1P0

+δ0)} + (δ0 − Λc1)P2

P1p̂1{Λ> 1

c1(β0

P1P0

+δ0)}

where P2

P1= (1 + r)p̂.



Example

Allowing for failure in t = t1

Now we allow for banks which are not able to fulfill the deposit claims(δ0 < λc1) to fail in t = t1. The cost of failure is the sum of depositcontracts not fulfilled in t1 and open contracts in t2.

Thus utility of a failed bank is

Fλ := u(δ0 − λc1 − (1− λ)c2).

Banks which fail do not take part in the interbank market.

As in the situation without failure we have to consider three caseslooking for possible equilibria.



Example




Fλ := u(δ0 − λc1 − (1− λ)c2).





Example




Fλ := u(δ0 − λc1 − (1− λ)c2).





Example




Fλ := u(δ0 − λc1 − (1− λ)c2).





Example

For δ0 < E (Λ)c1 we get the following result:

If there is r̃ ∈ [ 1p̂ − 1,∞) s.t.

E(

Λ1{FΛ>E(u(V Λ2 )), Λc1>δ0}

)= E (Λ)− 1

c1δ0,

then (r̃ , P2

(1+r̃)p̂ ) is an equilibrium price pair and banks with

Fλ > E (u(V λ2 (α, β0, δ0))) decide to fail in t = t1.

The remaining banks choose the strategy (βλ∗1 , γλ∗+ , γλ∗− , δλ∗1 ).

Else all banks with λc1 > δ0 fail, there is no interbank trading and theremaining banks hold all their bonds and invest spare liquidity in theliquid asset.



Example

For δ0 = E (Λ)c1 the interest rates can’t be arbitrarily large anymore:

If (α, β0, δ0) satisfy

E

(u(α

St2

S0+ β0

P2

P0− (1− λ)c2 + (δ0 − λc1)

1

p̂)

)≥ Fλ

for all λ > 1c1

(β0P2

P0+ δ0), then we get an upper bound for r

r̄ := inf{r ∈ [1

p̂− 1,∞) : P(FΛ > E (V2(α, β0, δ0)|Λ >

1

c1δ0)) > 0},

else all banks with λc1 > δ0 fail and there is no interbank market.



Example

The optimization problem in t = 0

The objective function (δ0 ≥ E (Λ)c1) is∫ ∞0

∫ 1c1

(β0P2P1

+δ0)

0

u(αs+β0P2

P0+λ(c2−c1

P2

P1)−c2+δ0

P2

P1)PΛ(dλ)P

St2S0 (ds)

+

∫ ∞0

∫ 1

1c1

(β0P2P1

+δ0)

u(αs+β0P2

P0+λ(c2−c1

P2

P1p̂)−c2+δ0

P2

P1p̂)PΛ(dλ)P

St2S0 (ds).

We maximize w.r.t. (α, β0, δ0) under constraints

α, β0 ≥ 0, δ0 ≥ E (Λ)c1

α + β0 + δ0 = 1



Example

CRR model for S , Binomial distribution for Λ

Let λl < λh, d < P2

P0< u and P(Λ = λl) = πl = 1− P(Λ = λh),

P(St2

S0= u) = p = 1− P(

St2

S0= d), with p, πl ∈ (0, 1).

Set λ̄ := πlλl + (1− πl)λh. Assume exponential utilityu(x) := − exp(−κx) with κ > 0.

In this case an unsecured interbank market develops and the optimalsolution is interior iff

1− p

p

P2

P0− d

u − P2

P0

∈ (exp(−(u−d)(1−λ̄)κ), exp(−(u−d)(1−λ̄−πl(λh−λl)P0

P2))κ)



Example




P(St2

S0= u) = p = 1− P(

St2

S0= d), with p, πl ∈ (0, 1).



1− p

p

P2

P0− d

u − P2

P0


P2))κ)



Example




P(St2

S0= u) = p = 1− P(

St2

S0= d), with p, πl ∈ (0, 1).



1− p

p

P2

P0− d

u − P2

P0


P2))κ)



Example

CRR / Binomial

Then the optimal solution is

(α∗, β∗0 , δ∗0 ) =

− log

(1−pp

P2P0−d

u− P2P0

)κ(u − d)

, 1− λ̄c1 +

log

(1−pp

P2P0−d

u− P2P0

)κ(u − d)

, λ̄c1

For κ = 1, λl = 14 , λh = 3

4 , u = 1.2, d = 1, P2

P0= 1.1 and p = 20

39 we get

(α∗, β∗0 , δ∗0 ) = (− log(0.95),

1

2+ log(0.95),

1

2)

≈ (0.0222763947, 0.477723605, 0.5)

An equilibirum price pair is (1.145, 1) and the solvency probability forlenders in the unsecured interbank market is 20

39 .



Example

CRR / Binomial


(α∗, β∗0 , δ∗0 ) =

− log

(1−pp

P2P0−d

u− P2P0

)κ(u − d)

, 1− λ̄c1 +

log

(1−pp

P2P0−d

u− P2P0

)κ(u − d)

, λ̄c1

For κ = 1, λl = 1

4 , λh = 34 , u = 1.2, d = 1, P2

P0= 1.1 and p = 20

39 we get

(α∗, β∗0 , δ∗0 ) = (− log(0.95),

1

2+ log(0.95),

1

2)

≈ (0.0222763947, 0.477723605, 0.5)


39 .



Example

CRR / Binomial


(α∗, β∗0 , δ∗0 ) =

− log

(1−pp

P2P0−d

u− P2P0

)κ(u − d)

, 1− λ̄c1 +

log

(1−pp

P2P0−d

u− P2P0

)κ(u − d)

, λ̄c1

For κ = 1, λl = 1

4 , λh = 34 , u = 1.2, d = 1, P2

P0= 1.1 and p = 20

39 we get

(α∗, β∗0 , δ∗0 ) = (− log(0.95),

1

2+ log(0.95),

1

2)

≈ (0.0222763947, 0.477723605, 0.5)


39 .



Example

References

Heider, F. and Hoerova, A. (2009). Interbank Lending, CreditRisk Premia and Collateral,ECB Working Paper

Freixas, X. and Holthausen, C. (2005). Interbank MarketIntegration under Asymmetric Information,Review of Financial Studies 18, 459-490

Hanson, M. (1981). On Sufficiency of the Kuhn-TuckerConditions,Journal of Mathematical Analysis and Applications 80, 545-550


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Interbank Lending - Modelling the Development of Interbank Markets