A theory of nonperforming loans and debt restructuring

Transcription

1 A theory of nonperforming loans and debt restructuring Keiichiro Kobayashi 1 Tomoyuki Nakajima 2 1 Keio University 2 University 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

2 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. Kobayashi and Nakajima Non-performing loans and debt restructuring 2 / 30

3 Introduction Non performing loans in Euro area and Japan Euro Area Japan Notes: Fraction of non-performing loans in total gross loans. Source: World Bank. Kobayashi and Nakajima Non-performing loans and debt restructuring 3 / 30

4 Introduction Non performing loans in some European countries 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

5 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

6 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

7 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

8 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

9 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

10 Benchmark model Short- and long-term loans Two types of loans to the firm: D t = value of long-term debt that the firm owes to the bank. k t = short-term (one-period) loans to the firm (working capital). b t+1 = repayment of the long-term debt in period t + 1: D t+1 = (1 + r)d t b t+1, Flow of funds: Firm b t+1 + (1 + r)k t b t+2 + (1 + r)k t+1 Bank k t k t+1 Kobayashi and Nakajima Non-performing loans and debt restructuring 10 / 30

11 Benchmark model Production and the firm owner s value F (k t ) = the firm s output in period t + 1. x t+1 = dividends to the owners of the firm: Limited liability: V t = value to the firm s owners: x t+1 = F (k t ) (1 + r)k t b t+1. x t+1 0. V t = r (x t+1 + V t+1 ). Kobayashi and Nakajima Non-performing loans and debt restructuring 11 / 30

12 Benchmark model One-sided lack of commitment The firm can choose to default in any period t, after receiving working capital k t. G(k t ) = the value of the outside opportunity of the firm; The bank would receive none when the firm defaults. Enforcement constraint: V t G(k t ). Kobayashi and Nakajima Non-performing loans and debt restructuring 12 / 30

13 Benchmark model Dynamic programming formulation The optimal contract can be obtained as: V (D) = max k,b, ˆD 1 [ F (k) (1 + r)k b + V ( ˆD) ], 1 + r [ b + ˆD ], s.t. D = r V (D) G(k), 0 F (k) (1 + r)k b. Kobayashi and Nakajima Non-performing loans and debt restructuring 13 / 30

14 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, Note: D = b r. V + D = 1 r [ F (k ) (1 + r)k ]. Kobayashi and Nakajima Non-performing loans and debt restructuring 14 / 30

15 Benchmark model Dynamics D max = largest level of long-term debt that can be credibly repaid: D max = arg V min is defined as V min = V (D max ). max D [0, ] {V (D) exists} For D 0 D max, let {Dt ce, Vt ce, kt ce, bt+1 ce } t=0 denote the solution to the optimal lending contract problem. Given D 0 (D, D max ], there exits a t such that k0 ce < k1 ce < < k t ce = k, and kt ce = k for all t t; V0 ce < V1 ce < < V t ce = V, and Vt ce = V for all t t; D0 ce > D1 ce > > D t ce = D, and Dt ce = D for all t t. Thus, the firm s output may be too small at first (debt overhang), but it converges to the efficient level in finite periods. Kobayashi and Nakajima Non-performing loans and debt restructuring 15 / 30

16 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

17 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 D 0 > D max. 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

18 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

19 Model with non-performing loans Feasible plans A plan {D t, V t, k t, b t+1 } t=0 is feasible if the following conditions are satisfied for all t 0: D t = (1 + r) (j+1) b t+j+1, j=0 V t = (1 + r) (j+1)[ ] F (k t+j ) (1 + r)k t+j b t+j+1, j=0 0 F (k t ) (1 + r)k t b t+1, V t G(k t ). Γ = the set of all feasible plans. Kobayashi and Nakajima Non-performing loans and debt restructuring 19 / 30

20 Model with non-performing loans Feasible repayment plans A repayment plan {b t+1 } t=0 is feasible if there exists {D t, V t, k t } t=0 such that {D t, V t, k t, b t+1 } t=0 Γ. Γ b = the set of all feasible repayment plans {b t+1 } t=0. d t ({b t+j+1 } j=0 ) = the PDV of a repayment plan {b t+j+1} j=0 evaluated in period t: d t ({b t+j+1 } j=0) = (1 + r) (1+j) b t+j+1. j=0 D max = maximum amount of repayable debt: { D max = max D R D = d 0 ({b t+1 } t=0) for some {b t+1 } t=0 Γ b}. Kobayashi and Nakajima Non-performing loans and debt restructuring 20 / 30

21 Model with non-performing loans Contractual values of debt D c 0 = contractual value of debt in period 0. Given a repayment plan {b t+1 } t=0, the contractual amount of debt, Dc t, evolves as Dt+1 c = (1 + r)dt c b t+1, t 0. If D0 c > D max, then D0 c > d 0 ({b t+1 } t=0) = (1 + r) (1+t) b t+1, {b t+1 } t=0 Γ b. t=0 Given D0 c > D max, the constrained efficiency would be achieved by officially reducing the amount of debt to D max 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

22 Model with non-performing loans Lemma Suppose that D0 c > D max. Then for any {b t+1 } t=0 Γb, Dt c > d t ({b t+j+1 } j=0), where {Dt c } are defined recursively as Dt c = (1 + r)dt 1 c b t for all t 1. In any period t, the bank has an incentive to void the existing plan {b t+j+1 } j=0 and make a new offer { b t+j+1 } j=0 Γb such that d t ({b t+j+1 }) < d t ({ b t+j+1 }) < Dt c. Thus, if D0 c > D max, the bank cannot make a commitment to any repayment plan {b t+1 } t=0 Γb. D c t is no longer a payoff-relevant state variable. Kobayashi and Nakajima Non-performing loans and debt restructuring 22 / 30

23 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 (k t, b t+1 ). The firm forms expectations about its future profits, Vt+1 e, and computes V e t = (1 + r) 1[ F (k t ) (1 + r)k t b t+1 + Vt+1 e ] The firm chooses to default in period t if and only if V e t < G(k t ). Kobayashi and Nakajima Non-performing loans and debt restructuring 23 / 30

24 Model with non-performing loans Game with non performing loans: Bank The bank also forms expectations about the future repayments from the firm, D e t+1, and chooses (k t, b t+1 ) by solving max (k t,b t+1 ) s.t. D e t = (1 + r) 1 (b t+1 + D e t+1), G(k t ) (1 + r) 1[ F (k t ) (1 + r)k t b t+1 + V e t+1], where Vt+1 e is taken as given by the bank. Equilibrium conditions: for all t 0, V e t = V t = D e t = D t = (1 + r) (j+1)[ ] F (k t+j ) (1 + r)k t+j b t+j+1, j=0 (1 + r) (j+1) b t+j+1. j=0 Kobayashi and Nakajima Non-performing loans and debt restructuring 24 / 30

25 Model with non-performing loans Constrained efficient allocation is not implementable Let {Dt ce, Vt ce, kt ce, bt+1 ce } t=0 denote the constrained efficient contract associated with the initial condition D 0 = D max. This is not an equilibrium in the game with non performing loans. Since V ce t = G(k ce t ), b ce t+1 = F (kt ce ) (1 + r)kt ce It can be shown that for all t 1, (1 + r)g(kt ce ) + Vt+1. ce F (kt ce ) (1 + r) (1 + r)g (kt ce ) < 0. Thus, given the firm s expectations, Vt+1 ce, the bank can collect more repayments by offering k t < kt ce. Kobayashi and Nakajima Non-performing loans and debt restructuring 25 / 30

26 Model with non-performing loans Markov equilibrium Restrict attention to Markov equilibrium: which reduces to: max (1 + (k,b) r) 1 (b + D e ), s.t. G(k) (1 + r) 1[ F (k) (1 + r)k b + V e]. max k F (k) (1 + r)k (1 + r)g(k) + V e Let k npl be the solution, which satisfies the FOC: Note: k npl < k. F (k npl ) (1 + r) (1 + r)g (k npl ) = 0. Kobayashi and Nakajima Non-performing loans and debt restructuring 26 / 30

27 Model with non-performing loans Persistence of inefficiency Equilibrium dynamics: k t = k npl < k, V t = V npl G(k npl ) < V, b t+1 = b npl F (k npl ) (1 + r)k npl rg(k npl ), D t = D npl bnpl r D max Whatever the cost of officially reducing debt is, if it exceeds D max D npl, the bank chooses not to do so. Kobayashi and Nakajima Non-performing loans and debt restructuring 27 / 30

28 Model with non-performing loans Summary Suppose that D0 c > D max. The constrained efficient allocation is obtained by officially reducing the amount of debt to D 0 = D max ; follow the Albuquerque-Hopenhayn type efficient contract starting from D 0 = D max. 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. Kobayashi and Nakajima Non-performing loans and debt restructuring 28 / 30

29 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

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