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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.
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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.
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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.
Kobayashi and Nakajima Non-performing loans and debt restructuring 4 / 30
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.
Kobayashi and Nakajima Non-performing loans and debt restructuring 5 / 30
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).
Kobayashi and Nakajima Non-performing loans and debt restructuring 6 / 30
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.
Kobayashi and Nakajima Non-performing loans and debt restructuring 7 / 30
Benchmark model
.
. .
1 Introduction
.
. . 2 Benchmark model
.
. .
3 Model with non-performing loans
.
. .
4 Conclusion
Kobayashi and Nakajima Non-performing loans and debt restructuring 8 / 30
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.
Kobayashi and Nakajima Non-performing loans and debt restructuring 9 / 30
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
Kobayashi and Nakajima Non-performing loans and debt restructuring 10 / 30
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).
Kobayashi and Nakajima Non-performing loans and debt restructuring 11 / 30
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).
Kobayashi and Nakajima Non-performing loans and debt restructuring 12 / 30
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.
Kobayashi and Nakajima Non-performing loans and debt restructuring 13 / 30
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∗].
Kobayashi and Nakajima Non-performing loans and debt restructuring 14 / 30
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.
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Model with non-performing loans
.
. .
1 Introduction
.
. . 2 Benchmark model
.
. .
3 Model with non-performing loans
.
. .
4 Conclusion
Kobayashi and Nakajima Non-performing loans and debt restructuring 16 / 30
Model with non-performing loans
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.
Kobayashi and Nakajima Non-performing loans and debt restructuring 17 / 30
Model with non-performing loans
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.
Kobayashi and Nakajima Non-performing loans and debt restructuring 18 / 30
Model with non-performing loans
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.
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Model with non-performing loans
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
}.
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Model with non-performing loans
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?
Kobayashi and Nakajima Non-performing loans and debt restructuring 21 / 30
Model with non-performing loans
.
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.
Kobayashi and Nakajima Non-performing loans and debt restructuring 22 / 30
Model with non-performing loans
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).
Kobayashi and Nakajima Non-performing loans and debt restructuring 23 / 30
Model with non-performing loans
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.
Kobayashi and Nakajima Non-performing loans and debt restructuring 24 / 30
Model with non-performing loans
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 .
Kobayashi and Nakajima Non-performing loans and debt restructuring 25 / 30
Model with non-performing loans
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∗.
Kobayashi and Nakajima Non-performing loans and debt restructuring 26 / 30
Model with non-performing loans
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.
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Model with non-performing loans
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.
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Conclusion
.
. .
1 Introduction
.
. . 2 Benchmark model
.
. .
3 Model with non-performing loans
.
. .
4 Conclusion
Kobayashi and Nakajima Non-performing loans and debt restructuring 29 / 30
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.
Kobayashi and Nakajima Non-performing loans and debt restructuring 30 / 30