A Markov Chain Model for the Cure Rate of Non-Performing Loans
aa r X i v : . [ q -f i n . R M ] J un A Markov Chain Model for the Cure Rate ofNon-Performing Loans
Vilislav Boutchaktchiev1 May, 2018
Abstract
A Markov-chain model is developed for the purpose estimation ofthe cure rate of non-performing loans. The technique is performedcollectively, on portfolios and it can be applicable in the process ofcalculation of credit impairment. It is efficient in terms of data ma-nipulation costs which makes it accessible even to smaller financialinstitutions. In addition, several other applications to portfolio opti-mization are suggested.
Key words:
Cure Rate Estimation, Markov Chains, Survival Analy-sis, IFRS 9 Provisioning.
JEL Classification:
G21, M41
Introduction
In calculating the credit impairment the IFRS 9 standard permits,under certain conditions, the usage of cure rate in order to reducethe amount of bank provisions. The logic behind this allowance is,that if an impaired amount will eventually return to regular status,the bank need not calculate provisions on it. Several methodologicalmanuals are available to the banking community which (cf. e.g. [2]),often without stipulation on the assumptions on the model, providerecipes for calculation of a cure rate. Estimates made in this way turnout to be often overly conservative and, sometimes, dissatisfactory,because the basic assumptions of the model could not be verified. Thepresented technique allows to calculate with any desired accuracy andany desired frequency. (E.g., monthly, quarterly, etc.) The modeluses data only from the past 12 months in order to provide a mostrecent measurement of the cure rate. This is, sometimes, required byregulators for financial quantities measured collectively. This very factis the reason why in low-default portfolios, which are often those foundin small banks, the results of this type of method do not satisfy evenvery basic assumptions about the cure rate in general. For this reasonwe apply a smoothing method from Survival analysis. This method isthe topic of Section 3. he usage of Markov-chain models, in general, is a technique ac-cessible to the banking management and is part of their routine inaccessing credit risk and expected credit losses, several studies doc-ument and contribute to this practice, including [3], [4] and [5]. Itappears that Monte Carlo techniques similar to those demonstrated ine.g., [6] are applicable in the study of cure rate and it is my belief thatfuture interest in this subject would move in this direction.In Section 4 we produce two numerical examples using data formthree small Bulgarian banks .In addition, in Section 5 we show how this method provides severaltools to identify some portfolios where cure rate is inapplicable, butrather a different managerial approach will be more successful.I wish to thank my colleagues Jana Kostova and Alessandro Merlinifrom ASTOR BG Consulting for bringing this problem to my attentionand for providing the data for the actual computations. Cure rate is meant to measure the propensity of loans to return toregular status after they have been found delinquent. In a portfolio,collectively, the cure rate estimates what proportion of non-performingloans will be, in the end, repaid. Given the possibility of a loan whichis once cured to relapse or to move back and forth between categories itis not enough to simply measure the proportion over a certain horizonof time.For the purposes of this study we assume that a loan is considerednon-performing after it is found more than 90 days late. We make thefollowing assumptions1. The loan is finally cured after it becomes less than one monthpast due.2. Loans which are N or more months past due are considered lostand are written off.3. We should distinguish performing loans which have been grantedforbearance. According to [1] these would be loans to partiesexperiencing financial difficulties in meeting their obligations andthe bank has agreed to offer them special contractual terms. Ifsuch loan preserves its regular status for a year we consider itcured, otherwise we consider it lost.4. States are assigned to all loans in the portfolio, based on m , thewhole number of months past-due at time t = 0. The state where m = 0 is an absorbing state, as well as the one with m ≥ N .The forborne loans are assigned in a separate state. Hence, thenumber of states is N + 2.5. We assume the time periodicity of observation to the loan tapeis annual. We measure the probabilities of transition between Bulgarian banks are bound to report expected credit losses and to calculate provisionsbased on IFRS 9 since 01/01/2018. They are, in general, computing cure rates for theirretail loan portfolios and for other portfolios of standardized products. tates by observing the migration between states within a yearprior to time t = 0.6. We assume that the migration in the previous years is irrelevantto the further development of the portfolio. Moreover, we assumethat transition rates do not vary in time.Assumptions 1-3 are a question of bank policy and, although theysatisfy the requirement of IFRS 9, an alternative configuration may beset. Assumption 5 is inessential, although it should be noted that smallbanks often have shallow, low-default portfolios and high frequencyobservation leads to volatile cure rates. For a typical loan of the considered portfolio we have thus constructeda finite Markov chain of random variables { X t : t = 0 , , ... } , whichtake as value the state of the loan at year t . It has N + 2 states { S i : i = 0 , . . . , N + 1 } , describing the state the loan is. With anappropriate ordering we can assume that S denotes the state where m = 0, S denotes m ≥ N , S is the forborne state and, for any i > = 3, the state S i is characterized by M = i −
2. Hence, S is thefirst non-performing state, corresponding to M = 3. • Assumption 6 form Section 1 implies time homogeneity. • Assumptions 1 and 2 imply that S and S are absorbing statesand, hence, they form, each by itself two recurrent communica-tion classes. • If any part of the set of states T = { S i : i = 2 , . . . , N + 2 } formsa recurrent class this would imply that the loan contract for thisparticular portfolio can be optimized. We are giving an exampleto illustrate this in Section 4. For this reason we assume that T is the set of transitive states. • Assumption 3 implies that S is a transitive. In fact, P [ X n = S | X n − = S ] = p, P [ X n = S | X n − = S ] = q,P [ X n = S i | X n − = S ] = 0 , for i ≥ , where p and q , satisfying p + q = 1 are the probabilities to surviveand fail, respectively.We write the transition matrix A = ( p ( i, j ) = P [ X = S j | X = S i ],therefore as follows: A = . . . . . . T S . Furthermore, for the limit matrix A ∞ = lim n →∞ A n we have: A ∞ = . . . . . . T ∞ O , here O is the zero matrix. For the matrix T ∞ we have T ∞ = p qp q ... ... p N − q N − , where p i and q i satisfy p i + q i = 1 and are the probabilities of a loanshowing i months of payment delay to be cured or lost, respectively.Hence, in search for the cure rate, our goal is to study the vector( p , ...p N ). Proposition 1.
In the notation defined above, the probability to curefor a loan which is i months past due at time t = 0 can be found onthe i th row of the first column of the matrix T ∞ = ( I − S ) − T. Proof.
Since S is a substochastic matrix, representing the transitionrates of transitive states, we know that S n → O as n → ∞ . For thisreason the matrix I − S , with I — the identity matrix of size N − t ij the probability of a loan with initial state X = S i toreach eventually the state S j , j = 0 ,
1. (In the notation above, theseare the entries of the matrix T , t i = p i , and t i = q i .) We have t ij = P [ X n = S j for some n | X = S i ]= P [ X = S j | X = S i ]+ N X k =0 P [ X n = S j for some n | X = S k ] P [ X = S k | X = S i ]= p ( i, j ) + N X k =2 t kj p ( i, k ) . That is, T ∞ = T + T ∞ S Hence T ∞ = ( I − S ) − T. Let S ( x ) be probability of a loan to be cured if the initial state is atleast x months past due. For non-preforming states, i ≥ S ( x ) = P ( X n = S for some n | X = S i , i ≥ x + 2) . This function needs to satisfy the following conditions:1. S (0) = 1 . S ( x ) = 0 for x ≥ N S ( x ) is non-increasing.In addition, one would expect that chances of failure would increaseas a function of x , the months past-due. This is due, in part, to tworeasons. First, the longer delay signifies a more dire economic status.And second, portfolio manager would make more effort to increase theopportunities of survival of these loans which are less in delay, sincethey have better chance. This gives us an extra condition4 The logarithmic derivative S ( x ) dSdx ( x ) is decreasing.Generally the outcome of calculating the Markov chain need not satisfythese conditions. In order to smoothen the results we apply tools formthe Survival Analysis. (Cf. e.g. [7].) A common choice of survivalfunction is a best-fitting Weibull curve: S ( x ) = e − ( xλ ) k , corresponding to a Weibull distribution with CDF F ( x ) = 1 − S ( X ).Condition 4 simply means that the hazard rate is an increasing functionwhich would imply that the shape parameter k of the curve satisfies k >
1. The parameters k and λ must be chosen by fitting the CDF ofthis distribution to the points(0 , , (1 , p ) , (2 , p ) , . . . , ( N − , p N − ) , ( N, . After this, the cure rate of the portfolio is equal to S (3). Example . A Credit-Card Portfolio
We now consider the portfolio of select credit cards from a small bankat the end of March 2007. The total size of the portfolio is e N = 8. Thetransition matrix A , according to the notation above is: A = .
37 0 .
63 0 0 0 0 0 0 0 00 .
39 0 .
11 0 . .
157 0 .
008 0 .
015 0 .
11 0 .
06 0 .
02 0 . .
37 0 .
12 0 .
02 0 .
003 0 .
012 0 .
045 0 .
09 0 .
04 0 0 . .
05 0 .
32 0 .
09 0 .
004 0 .
107 0 .
113 0 .
141 0 .
102 0 .
073 00 0 .
45 0 0 0 0 .
19 0 .
119 0 .
149 0 .
012 0 .
080 0 . .
08 0 .
01 0 .
31 0 0 .
20 0 .
21 0 0 0 0 .
05 0 .
009 0 .
111 0 .
41 0 .
210 0 .
47 0 .
004 0 0 0 0 0 .
037 0 .
27 0 . Next, we compute the matrices ( I − S ) − : The point ( δ, p ) may be added to the sequence with a appropriately chosen small valueof the parameter δ . I − S ) − = .
127 1 .
187 0 .
018 0 .
08 0 .
166 0 .
179 0 .
121 0 . .
033 0 .
004 1 .
024 0 .
109 0 .
127 0 .
172 0 .
254 0 . .
114 0 .
006 0 .
132 1 .
221 0 .
216 0 .
288 0 .
254 0 . .
029 0 .
002 0 .
032 0 .
299 1 .
192 0 .
348 0 .
187 0 . .
017 0 .
001 0 .
018 0 .
164 0 .
048 1 .
549 0 .
237 0 . .
018 0 .
001 0 .
018 0 .
162 0 .
053 0 .
396 2 .
016 0 . .
012 0 0 .
007 0 .
064 0 .
021 0 .
21 0 .
708 1 . and T ∞ : T ∞ = .
37 0 . .
52 0 . .
398 0 . .
155 0 . .
038 0 . .
021 0 . .
021 0 . .
01 0 . . Next we fit a Weibull curve on ten points:(0 , , (0 . , . , (1 , . , (2 , . , (3 , . , (4 , . , (5 , . , (6 , . , (7 , . , (8 , , using a linear regression. The results are shown in Table 1.Table 1: The fitting of the Weibull curve for a Credit Card portfolio. In parenthesisare shown the t -statistic values from the linear regression. The ∗ denotes significanceto the 99% level. λ k R . ∗ . ∗ . .
91) (15 . k = 1 .
14 is different from 0 at the 99%level of significance. The one-sided hypothesis for k ≤ . The expected cure rate is computed as S (3) = 11 . Example . A State-Owned-Corporations Portfolio
A portfolio of select corporate loans to state-owned corporationswas tested for cure rate. The portfolio consists of 97 loans with totalvalue e The p -value of the one-sided Wald test comes to 0.049. . . . . Figure 1:
The effect of Weibull curve fitting. The curve labeled cr MC representsthe result of the Markov chain model; the final result is labeled cure rate . Both aredrawn as functions of months-past-due, mpd .The transition matrix A is as follows: A = .
37 0 .
63 0 0 0 0 0 0 0 00 0 0 0 .
25 0 0 . .
15 0 0 00 0 .
45 0 0 0 .
12 0 0 .
19 0 .
15 0 .
01 0 .
080 0 0 0 . .
25 0 .
45 0 0 00 0 0 0 . .
37 0 .
23 0 0 00 0 . .
01 0 0 .
08 0 .
31 0 0 .
20 0 .
21 0 0 0 .
01 0 0 .
05 0 .
11 0 .
41 0 .
210 0 .
47 0 .
01 0 0 0 0 0 .
03 0 .
27 0 . Notice, that states S , S and S form a separate recurrentcommunication class and, hence, S is not a transitive state. Thecurrent cure-rate model is inapplicable for this portfolio . In factwe conclude that there is a pattern of loans cycling between thesethree states without ever being lost or cured. As seen in Example 4.2, it is possible to find a recurrent class whichcontains both performing and non-performing states. This is an indica-tion that loans with this risk profile may oscillate between performance Most banks in Bulgaria are avoiding the use cure rate for corporate portfolios. nd non-performance without ever becoming cured or lost. Instead oftrying to remedy the situation using cure rate the management shouldseek to optimize the contract for this type of loans. Example 4.1 shows how the hazard rate might turn out constant. Infact, a large part of the tested portfolios of this type exhibited k sta-tistically indistinguishable from 1.This may be due simply to the shallowness of the portfolio, how-ever, it may indicate that young loans die unnecessarily quickly. There-fore, it may be worth for the bank management to consider contrac-tual change in order to better channel the behavioral patterns of theirclients. This model allows to compute expected time L i it takes for a loan instate S i to be resolved, i.e., to either fail, or cure. Proposition 2.
The time L i to absorption of a transitive state i , i ≥ , can be obtained by summing the entrees of the correspondingrow (i.e., the ( i − st row) of the matrix ( I − S ) − .Proof. One can see that that L i is the sum L i = N X j =2 L ( i, j ) , where L ( i, j ) is the expected time a state spends in state S j startinginitial state S i . Moreover, taking into account that( I − S ) − = I + S + S + · · · a simple computation shows, L ( i, j ) is the element of that matrix whichstays in ( i − st row and ( j − st column.Proposition 2 produces a tool for estimation of the expected pe-riod of uncertainty for loans. Furthermore, in a similar fashion onecan develop a early-warning system for prognosticating potential non-performing loans, by computing the times L (3 ,
5) and L (4 , I suggest a Markov-chain model which, together with a survival model,can be used to estimate the cure rate in a portfolio of loans withhomogeneous risk. Furthermore, we show that this technique produces the followinginstruments can be of use for making these portfolios more efficiently. The model was subsequently tested by computing cure rates for various select subport-folios of the retail product line over a period in years 2015-2016. The data was obtainedfrom three small Bulgarian banks. The analysis produced cure rates ranging between3-22%. These results appear in line with the guidance of [1]. . One can test if the portfolio demonstrates cyclical behavior, whichdefeats the purpose of computation of CR.2. One can compute the hazard-rate function of the portfolio andstudy it for further portfolio optimization, particularly in casesof unexpected hazard rate.3. Expected time-to-resolution together with probabilities of defaultmay be used for monitoring loans which are past-due over anextended period.4. Expected time-to-NPL can be computed to aid the developmentof an early warning system. References [1]
Asset Quality Review , Phase 2 Manual, ECB 2014.[2] Basel Committee on Banking Supervision
Prudential treatmentof problem assets — definitions of non-performing exposures andforbearance , Consultative Document. BIS, July 15, 2016.[3] Gaffney, E., R. Kelly, and F. McCann, (2014),
A transitions-basedframework for estimating expected credit losses , Research Techni-cal Papers 16/RT/14, Central Bank of Ireland.[4] Grimshaw, S. D. and Alexander, W. P. (2011),
Markov chainmodels for delinquency: Transition matrix estimation and fore-casting . Appl. Stochastic Models Bus. Ind., 27: 267-279.doi:10.1002/asmb.827[5] Robert A. Jarrow, David Lando, and Stuart M. Turnbull (2008)
A Markov Model for the Term Structure of Credit Risk Spreads .Financial Derivatives Pricing: pp. 411-453.[6] Todorov, V. and Dimov, I. T. (2016)
Monte Carlo methods formultidimensional integration for European option pricing , AIPConference Proceedings, 1773:1, 100009, doi:10.1063/1.4965003.[7] Weibull, W. (1951)
A Statistical Distribution Function of WideApplicability . Journal of Applied Mechanics, 18, 293-297.
Institute of Mathematics and Informatics, Bulgarian Academyof Sciences, Acad. G. Bonchev Str., Block 8, 1113 Sofia,Bulgaria
E-mail address: [email protected]@math.bas.bg