A theory of nonperforming loans and debt restructuring · A theory of nonperforming loans and debt restructuring Keiichiro Kobayashi1 Tomoyuki Nakajima2 1Keio University 2University

A theory of nonperforming loans and

debt restructuring

Keiichiro Kobayashi1 Tomoyuki Nakajima2

1Keio University

2University of Tokyo

January 19, 2018

OAP-PRI Economics Workshop Series

Bank, Corporate and Sovereign Debt

Kobayashi and Nakajima Non-performing loans and debt restructuring 1 / 30

Introduction

Non performing loans

IMF definition:

A loan is non-performing when

payments of interest and/or principal are past due by 90 days or more, or

interest payments equal to 90 days or more have been capitalized, refinanced, or

delayed by agreement, or

payments are less than 90 days overdue, but there are other good reasons such

as a debtor filing for bankruptcy to doubt that payments will be made in full.

After a loan is classified as nonperforming, it (and/or any replacement

loans(s)) should remain classified as such until written off or payments of

interest and/or principal are received on this or subsequent loans that replace

the original.


Introduction

Non performing loans in Euro area and Japan

0

1

2

3

4

5

6

7

8

9

1997 1998 1999 2000 2001 2002 2003 2004 2005 2006 2007 2008 2009 2010 2011 2012 2013 2014

Euro Area Japan

Notes: Fraction of non-performing loans in total gross loans. Source: World Bank.


Introduction

Non performing loans in some European countries

0

5

10

15

20

25

30

35

1998 1999 2000 2001 2002 2003 2004 2005 2006 2007 2008 2009 2010 2011 2012 2013

Greece Ireland Italy Portugal Spain

Notes: Fraction of non-performing loans in total gross loans. Source: World Bank.


Introduction

Related evidence

Persistent effect of financial crises:

Reinhart and Reinhart (2010), Reinhart and Rogoff (2009): international

evidence that financial crises are followed by a decade-long slowdown of

output growth.

Evidence on evergreening and “zombie firms” in Japan:

Peek and Rosengren (2005), Caballero, Hoshi, and Kashyap (2008), etc.

It is important to note that zombie firms may recover.

Fukuda and Nakamura (2011): A majority of firms which are identified as

zombies by Caballero, Hoshi and Kashyap (2008) did recover substantially in

the first half of the 2000s.


Introduction

Outline of our framework

Existing theoretical analysis on non-performing loans is very limited.

We modify the model of Albuquerque and Hopenhayn (2004) and let the

firm’s debt non-state-contingent.

For simplicity, our benchmark model is deterministic.

Suppose that an unexpected shock hits the firm in period 0 so that the

contractual value of debt exceeds the maximum amount that the firm can

repay.

Such a shock may reflect an unexpected decline in the firm’s productivity, or in

the value of the collateral, etc.

The lender has two options:

reduce the amount of debt officially (debt restructuring);

retain the right to the original amount of debt (non-performing loans).


Introduction

Summary of the results

If the bank chooses to restructure debt officially, the levels of lending and

output converge to their first-best levels in finite periods.

If the bank chooses not to do so, the loans become non-performing.

The bank loses its ability to commit to a repayment plan.

The contract problem turns into that with two-sided lack of commitment.

The equilibrium level of output is permanently lower than their first-best levels

(zombie firms).

Our theory may help understand the experience of Japan in the 1990s and

2000s.


Benchmark model

.

. .

1 Introduction

.

. . 2 Benchmark model

.

. .

3 Model with non-performing loans

.

. .

4 Conclusion


Benchmark model

Benchmark model

a deterministic version of the model of Albuquerque and Hopenhayn (2004).

A bank lends to a firm.

One-sided lack of commitment:

the lender (bank) commits to long-term contracts; but

the borrower (firm) can choose to default.

r = common discount rate.


Benchmark model

Short- and long-term loans

Two types of loans to the firm:

Dt = value of long-term debt that the firm owes to the bank.

kt = short-term (one-period) loans to the firm (working capital).

bt+1 = repayment of the long-term debt in period t + 1:

Dt+1 = (1 + r)Dt − bt+1,

Flow of funds:

Firm bt+1 + (1 + r)kt bt+2 + (1 + r)kt+1

· · · ↑ ↓ ↑ ↓ · · ·Bank kt kt+1


Benchmark model

Production and the firm owner’s value

F (kt) = the firm’s output in period t + 1.

xt+1 = dividends to the owners of the firm:

xt+1 = F (kt) − (1 + r)kt − bt+1.

Limited liability:

xt+1 ≥ 0.

Vt = value to the firm’s owners:

Vt =1

1 + r(xt+1 + Vt+1).


Benchmark model

One-sided lack of commitment

The firm can choose to default in any period t, after receiving working

capital kt .

G(kt) = the value of the outside opportunity of the firm;

The bank would receive none when the firm defaults.

Enforcement constraint:

Vt ≥ G (kt).


Benchmark model

Dynamic programming formulation

The optimal contract can be obtained as:

V (D) = maxk,b,D̂

1

1 + r

[F (k) − (1 + r)k − b + V (D̂)

],

s.t. D =1

1 + r

[b + D̂

],

V (D) ≥ G (k),

0 ≤ F (k) − (1 + r)k − b.


Benchmark model

Efficient level of production

k∗ = (unconstrained) efficient level of production:

F ′(k∗) = 1 + r .

Define:

V ∗ = G (k∗),

x∗ = rV ∗,

b∗ = F (k∗) − (1 + r)k∗ − x∗,

D∗ =b∗

r.

Note:

V ∗ + D∗ =1

r

[F (k∗) − (1 + r)k∗].


Benchmark model

Dynamics

Dmax = largest level of long-term debt that can be credibly repaid:

Dmax = arg maxD∈[0,∞]

{V (D) exists}

Vmin is defined as Vmin = V (Dmax).

For D0 ≤ Dmax, let {Dcet ,V ce

t , kcet , bce

t+1}∞t=0 denote the solution to the

optimal lending contract problem.

Given D0 ∈ (D∗,Dmax], there exits a t̄ such that

kce0 < kce

1 < · · · < kcet̄ = k∗, and kce

t = k∗ for all t ≥ t̄;

V ce0 < V ce

1 < · · · < V cet̄ = V ∗, and V ce

t = V ∗ for all t ≥ t̄;

Dce0 > Dce

1 > · · · > Dcet̄ = D∗, and Dce

t = D∗ for all t ≥ t̄.

Thus, the firm’s output may be too small at first (debt overhang), but it

converges to the efficient level in finite periods.


Model with non-performing loans

.

. .

1 Introduction

.


.

. .


.

. .

4 Conclusion



Too much debt

To analyze non-performing loans, suppose that there is an unexpected shock

in period 0 so that

D0 > Dmax.

Such a situation may arise, for instance, when

there is a large negative shock to the productivity of the firm; or

a large decrease in the value of the collateral held by the firm.



Emergence of non-performing loans

Here, we assume that the bank decides not to change the contractual value

of debt.

Then the present discounted value of future repayments to the bank would

be less than the contractual value of the firm’s debt.

This might cause a serious problem because now the bank is no longer able

to commit to any repayment plan.

Thus, the lack of commitment becomes two-sided.



Feasible plans

A plan {Dt , Vt , kt , bt+1}∞t=0 is feasible if the following conditions are satisfied

for all t ≥ 0:

Dt =∞∑j=0

(1 + r)−(j+1)bt+j+1,

Vt =∞∑j=0

(1 + r)−(j+1)[F (kt+j) − (1 + r)kt+j − bt+j+1

],

0 ≤ F (kt) − (1 + r)kt − bt+1,

Vt ≥ G (kt).

Γ = the set of all feasible plans.



Feasible repayment plans

A repayment plan {bt+1}∞t=0 is feasible if there exists {Dt , Vt , kt}∞t=0 such

that {Dt , Vt , kt , bt+1}∞t=0 ∈ Γ.

Γb = the set of all feasible repayment plans {bt+1}∞t=0.

dt({bt+j+1}∞j=0) = the PDV of a repayment plan {bt+j+1}∞j=0 evaluated in

period t:

dt({bt+j+1}∞j=0) =∞∑j=0

(1 + r)−(1+j)bt+j+1.

Dmax = maximum amount of repayable debt:

Dmax = max{

D ∈ R∣∣∣ D = d0({bt+1}∞t=0) for some {bt+1}∞t=0 ∈ Γb

}.



Contractual values of debt

Dc0 = contractual value of debt in period 0.

Given a repayment plan {bt+1}∞t=0, the contractual amount of debt, Dct ,

evolves as

Dct+1 = (1 + r)Dc

t − bt+1, t ≥ 0.

If Dc0 > Dmax, then

Dc0 > d0({bt+1}∞t=0) =

∞∑t=0

(1 + r)−(1+t)bt+1, ∀{bt+1}∞t=0 ∈ Γb.

Given Dc0 > Dmax, the constrained efficiency would be achieved by officially

reducing the amount of debt to Dmax right away.

What would happen if the bank decides not to reduce the amount of debt?



.

Lemma

.

.

.

. ..

.

.

Suppose that Dc0 > Dmax. Then for any {bt+1}∞t=0 ∈ Γb,

Dct > dt({bt+j+1}∞j=0),

where {Dct } are defined recursively as Dc

t = (1 + r)Dct−1 − bt for all t ≥ 1.

In any period t, the bank has an incentive to void the existing plan

{bt+j+1}∞j=0 and make a new offer {b̃t+j+1}∞j=0 ∈ Γb such that

dt({bt+j+1}) < dt({b̃t+j+1}) < Dct .

Thus, if Dc0 > Dmax, the bank cannot make a commitment to any repayment

plan {bt+1}∞t=0 ∈ Γb.

Dct is no longer a payoff-relevant state variable.



Game with non performing loans: Firm

In each period t, the bank offers to the firm a pair of short-term loans and

repayment on the long-term debt (kt , bt+1).

The firm forms expectations about its future profits, V et+1, and computes

V et = (1 + r)−1

[F (kt) − (1 + r)kt − bt+1 + V e

t+1

]The firm chooses to default in period t if and only if

V et < G (kt).



Game with non performing loans: Bank

The bank also forms expectations about the future repayments from the firm,

Det+1, and chooses (kt , bt+1) by solving

max(kt ,bt+1)

Det = (1 + r)−1(bt+1 + De

t+1),

s.t. G (kt) ≤ (1 + r)−1[F (kt) − (1 + r)kt − bt+1 + V e

t+1

],

where V et+1 is taken as given by the bank.

Equilibrium conditions: for all t ≥ 0,

V et = Vt =

∞∑j=0

(1 + r)−(j+1)[F (kt+j) − (1 + r)kt+j − bt+j+1

],

Det = Dt =

∞∑j=0

(1 + r)−(j+1)bt+j+1.



Constrained efficient allocation is not implementable

Let {Dcet , V ce

t , kcet , bce

t+1}∞t=0 denote the constrained efficient contract

associated with the initial condition D0 = Dmax.

This is not an equilibrium in the game with non performing loans.

Since V cet = G (kce

t ),

bcet+1 = F (kce

t ) − (1 + r)kcet − (1 + r)G (kce

t ) + V cet+1.

It can be shown that for all t ≥ 1,

F ′(kcet ) − (1 + r) − (1 + r)G ′(kce

t ) < 0.

Thus, given the firm’s expectations, V cet+1, the bank can collect more

repayments by offering kt < kcet .



Markov equilibrium

Restrict attention to Markov equilibrium:

max(k,b)

(1 + r)−1(b + De),

s.t. G (k) ≤ (1 + r)−1[F (k) − (1 + r)k − b + V e

].

which reduces to:

maxk

F (k) − (1 + r)k − (1 + r)G (k) + V e

Let knpl be the solution, which satisfies the FOC:

F ′(knpl) − (1 + r) − (1 + r)G ′(knpl) = 0.

Note: knpl < k∗.



Persistence of inefficiency

Equilibrium dynamics:

kt = knpl < k∗,

Vt = V npl ≡ G (knpl) < V ∗,

bt+1 = bnpl ≡ F (knpl) − (1 + r)knpl − rG (knpl),

Dt = Dnpl ≡ bnpl

r≤ Dmax

Whatever the cost of officially reducing debt is, if it exceeds Dmax − Dnpl, the

bank chooses not to do so.



Summary

Suppose that Dc0 > Dmax.

The constrained efficient allocation is obtained by

officially reducing the amount of debt to D0 = Dmax;

follow the Albuquerque-Hopenhayn type efficient contract starting from

D0 = Dmax.

In this case, inefficiency will disappear in finite periods.

Without formal debt restructuring,

the bank holds non-performing loans;

the bank is no longer able to commit to a particular repayment plan;

the relationship between the bank and the firm exhibits two-sided lack of

commitment;

In this case, inefficiency will continue forever.


Conclusion

.

. .

1 Introduction

.


.

. .


.

. .

4 Conclusion


Conclusion

Conclusion

We develop a financial contracting problem with limited commitment, and

study what would happen when the firm’s debt exceeds the amount it can

repay.

The bank may or may not reduce the amount of debt officially.

If the bank chooses not to reduce it,

the loan becomes non performing;

the lack of commitment becomes two-sided;

inefficiency lasts forever.

Our theory may help interpret the experience of Japan’s lost decades.

Should be extended to a stochastic model with explicit costs of debt

restructuring.


Documents

A theory of nonperforming loans and debt restructuring · A theory of nonperforming loans and debt restructuring Keiichiro Kobayashi1 Tomoyuki Nakajima2 1Keio University 2University