Simulation-based optimisation of the timing of loan recovery across different portfolios
SSimulation-based optimisation of the timing of loanrecovery across different portfolios
Arno Botha ∗,a , Conrad Beyers †,a , and Pieter de Villiers ba Department of Actuarial Science, University of Pretoria, Private Bag X20, Hatfield, 0028, South Africa b Department of Electrical, Electronic, and Computer Engineering, University of Pretoria, Private Bag X20, Hatfield, 0028, South Africa
Abstract
A novel procedure is presented for the objective comparison and evaluation of a bank’s decision rules in optimisingthe timing of loan recovery. This procedure is based on finding a delinquency threshold at which the financial lossof a loan portfolio (or segment therein) is minimised. Our procedure is an expert system that incorporates the timevalue of money, costs, and the fundamental trade-off between accumulating arrears versus forsaking future interestrevenue. Moreover, the procedure can be used with different delinquency measures (other than payments in arrears),thereby allowing an indirect comparison of these measures. We demonstrate the system across a range of credit riskscenarios and portfolio compositions. The computational results show that threshold optima can exist across allreasonable values of both the payment probability (default risk) and the loss rate (loan collateral). In addition,the procedure reacts positively to portfolios afflicted by either systematic defaults (such as during an economicdownturn) or episodic delinquency (i.e., cycles of curing and re-defaulting). In optimising a portfolio’s recoverydecision, our procedure can better inform the quantitative aspects of a bank’s collection policy than relying onarbitrary discretion alone.
Keywords—
Optimisation; Credit Loss; Loan Delinquency; Collections; Expert systems
JEL:
C44, C63, G21.Word count (excluding front matter): 7514Figure count: 10 ∗ ORC iD: 0000-0002-1708-0153 † Corresponding author: [email protected] a r X i v : . [ q -f i n . R M ] S e p imulation-based optimisation of the timing of loan recovery across different portfolios Consumer credit has exponentially grown over the last few decades, largely spurred by the introduction of the creditcard during the 1950s, as discussed in Thomas (2009, pp. 2–3). At the time of writing, retail credit is estimated at$13 trillion for the US market, which largely consists of mortgages, credit cards, personal loans, vehicle financing,overdrafts and other revolving loans for the individual, as reported in The Board of Governors of the Federal ReserveSystem (US) (2018). For perspective, consumer debt in 2007 was 40% greater than total industry debt ($9.2 trillion)and more than double total corporate debt ($5.8 trillion). Although greatest in the USA, consumer debt in othercountries are not far behind, e.g., the United Kingdom had debts totalling £1.4 trillion in 2007 – a staggering £400billion growth within the span of but three years. While Canada’s consumer debt is estimated at $666 billion, thisfigure also constituted 110% of total annual household income. In fact, debt levels exceeded household income forborrowers from quite a few countries during the last twenty years, of which a few examples are shown in Fig. 1.
Fig. 1.
Consumer household debt-to-income over annual periods by country, including Australia (AUS), Canada(CAN), Finland (FIN), Greece (GRC), Netherlands (NLD), Norway (NOR), and the United States of America(USA). Reproduced from OECD (2018).This credit growth, as argued in Thomas (2009, pp. 1–6) and Thomas (2010), could not have been possiblewithout a degree of automation, historically facilitated by statistical decision-making models otherwise knownas credit scorecards. These models rendered consistent approve/decline credit decisions that enabled greaterapplication volumes whilst keeping default risk under control, i.e., the risk of the borrower reneging on repayments.This control is mainly achieved by only approving those applications with a predicted probability of default within adesired limit, which is usually aligned with a bank’s risk appetite. Constructing these scorecards involves finding astatistical relationship between a set of borrower-specific characteristics and the successful (or failed) repaymentoutcome over time, using historical data. Naturally, the literature on credit scoring is considerable, e.g., Hand andHenley (1997), Hand (2001), Thomas, Edelman and Crook (2002), Siddiqi (2005), Crook, Edelman and Thomas2 imulation-based optimisation of the timing of loan recovery across different portfolios (2007), Thomas (2009), Thomas (2010), Hao, Alam and Carling (2010), and Louzada, Ara and Fernandes (2016).The advent of these automated models did, however, call for a more methodical manner of "measuring default"before trying to predict the risk thereof. In most cases, the development of loan delinquency over time is capturedusing the number of payments in arrears from accountancy practices, which is constructed from days past due(DPD). Whilst practical and intuitive, this calculation (or the g delinquency measure as we will call it) has a fewflaws upon which alternative measures may improve, as discussed in the appendix. Nonetheless, banks commonlyspecified three payments (or 90 DPD) in arrears as a pragmatic point of ‘default’, long before the introduction of theBasel II Capital Accords. That said, this threshold can generally range between 30–180 days using managerialdiscretion and some analysis, as discussed in Thomas et al. (2002, pp. 123–124). However, the direct financialimplications of any chosen definition are not readily known, nor accounted for when deciding the point of default,especially when developing credit scoring models. Therefore and as originally argued in Hand (2001), pursuingmodelling excellence becomes questionable when the constructed outcome variable itself, as determined by thedefault definition, is inherently quite arbitrary. Fig. 2.
Illustrating the trade-off associated with two extreme arrears-based thresholds for two fictional loans.Threshold 1 is overly strict for loan 1 given that it cures later; but suitable for loan 2 since it never cures. Conversely,threshold 2 is overly naive for loan 2, though suitable for loan 1.Fundamentally, as an account continues to accrue arrears, the lender will respond by proportionately rampingup its collection efforts. Every unpaid instalment (or portion thereof) erodes the trust between bank and borrower,which is only tolerable up to a point. This ambiguous point may itself differ across portfolios and even banks,likely based on differences in risk appetites and market conditions. Regardless, having reached this point, the bankeffectively assumes that the troubled loan will helplessly fall into ever greater arrears if kept. Therefore, the lendershifts its focus to the immediate and maximal recovery of debt, including selling any collateral, as based on thefive-phase credit management model of Finlay (2010, pp. 11–13, 147–153). Presumably, this idea of reaching aso-called "point of no return" is the historical basis for a default definition, although most modern definitions also3 imulation-based optimisation of the timing of loan recovery across different portfolios contain more qualitative criteria. For Basel-compliant or IFRS 9-compliant types of credit risk modelling, anotherfactor to consider is that of competing regulatory requirements when defining ‘default’ across different jurisdictions.Furthermore, some lenders use multiple definitions for different purposes or across different portfolios – all ofwhich impedes the interpretation of ‘default’ in trying to cater for so many different contexts.Owing to the difficulties of defining ‘default’ precisely, we explore a more fundamental meaning of ‘default’ asthe portfolio-dependent, probabilistic, and risk-based "point of no return" beyond which loan collection becomessub-optimal if pursued. Our ‘default’ state is simply based on breaching a certain delinquency threshold, so that wemay assess the "net cost" of each candidate threshold. We want to find the best time at which the lender shouldforsake a loan and instead try to collect all it can. Furthermore, it is convenient to try and find this point from aloan delinquency-basis since the resulting measurements are scale-invariant and already incorporate behaviouralinformation on the borrower. Too strict a delinquency threshold will surely marginalise accounts that would haveresumed repayment (or cured from ‘default’), had the bank been more patient before initiating strict recovery.Conversely, too lenient a threshold may naively tolerate increasing arrears at the cost of greater liquidity risk andbigger capital buffers, which may introduce capital-inefficiencies. The goal now becomes to devise an expertsystem in which these two extremes (illustrated in Fig. 2) can be appropriately offset against each other. Doing socan theoretically form a proverbial ‘Goldilocks-zone’ in space that contains the ideal delinquency threshold for aportfolio, which translates to the ‘best’ time for loan recovery.
Fig. 3.
High-level steps of the contributed LROD-procedure.In this study, we develop such a system, called the
Loss-based Recovery Optimisation across Delinquency (LROD) procedure, as our main contribution. This procedure is summarised in three steps, shown in Fig. 3 andformally presented in subsection 3.1. Relevant literature is explored in section 2, including current practices onselecting default definitions as well as previous optimisation work on loan collection. Since we posit that differentportfolios will likely have different ‘ideal’ recovery thresholds, a simple simulation-based setup is described insubsection 3.2 as our testbed. This allows for examining recovery optimisation from "first principles" by randomlygenerating amortising loan portfolios with specifiable risk profiles, guided by expert judgement and industryexperience. Moreover, by tweaking this testbed’s simulation parameters appropriately, one can obtain quickmanagerial insight on the viability of recovery optimisation before embarking on any deep data work; a usefulsecondary contribution. We demonstrate the LROD-procedure in section 4 by conducting a broad computationalstudy using the aforementioned testbed. Threshold optima are found across most levels of default and loss risk, asmeasured by the probability of payment and loss rate respectively. Moreover, we test portfolios suffering fromsystematic defaults based on fixed patterns, as well as portfolios exhibiting episodic delinquency (cycles of curingand re-defaulting). Overall, our procedure delivers dynamic yet intuitive optimisation results for the timing of loanrecovery. It may therefore be used to improve a bank’s existing collection policies, with the accompanying sourcecode published in Botha (2020). Finally, we conclude the study in section 5 and outline areas of future research.4 imulation-based optimisation of the timing of loan recovery across different portfolios
The estimation of the frequency of any event in a given sample fundamentally depends on the definition of theevent. While loan ‘default’ lies intrinsic to credit risk (and its estimation), the phenomenon thereof certainly hasmany definitions, both historically and in modern times. These definitions typically vary by product, customertype, and bank, as discussed in Van Gestel and Baesens (2009, pp. 203–212) and Baesens, Rösch and Scheule(2016, pp. 137–138). Examples hereof include filing for bankruptcy, unfulfilled claims, negative net present values,overdrawing beyond an agreed credit limit, as well as becoming three instalments in arrears. Basel II standardiseddefault definitions to some extent upon its introduction, while still leaving room for the lender’s discretion, subjectto regulatory approval. Specifically, paragraph 452 of the Basel Committee on Banking Supervision (2006) defines‘default’ as one of the following two conditions. Firstly, the obligor has reached 90 DPD on a material loan balance,or has been in excess of an advised credit limit for 90 days. Alternatively, the bank considers, in its opinion , that theobligor is unlikely to repay its obligations in full, without the necessary intervention of the bank, e.g., liquidatingany collateral. To help inform this opinion, Basel II includes a few reasonable (but qualitative) indicators of "unlikeliness to pay" in paragraph 453. Examples include when debt restructuring leads to an overall reducedobligation, or partially selling off a debt at a loss.The requirements of Basel II are often promulgated verbatim by some regulators, e.g., Regulation 67 of theBanks Act of South Africa (2012, pp. 1201–1202). However, Basel II (and how it relates to ‘default’) is only trulyrelevant to estimating the amount of capital required for offsetting unexpected losses (UL). In turn, this requiresmodelling the expected losses (EL) for which a bank also holds provisions. EL is generally defined as the productof three specific risk parameters: 1) the Probability of Default (PD); 2) the Loss Given Default (LGD); and 3)the Exposure-At-Default (EAD). A comprehensive review of this topic is given in Thomas (2009, pp. 289–293),Van Gestel and Baesens (2009, chap. 4, 6), and Baesens et al. (2016, chap. 5–11). Relatedly, the new accountingstandard IFRS 9 (2014) firmly lodged the management of loss provisions as a deeply statistical exercise similar tothat of capital estimation, which is discussed in Novotny-Farkas (2016), Skoglund (2017) and Cohen, Edwards Jret al. (2017). However, even IFRS 9 does not impose a fixed default definition, instead requiring in paragraphB.5.5.37 that a particular definition simply be used consistently in a portfolio’s risk management. In addition, IFRS9 presumes 90 DPD as a default definition that may be superseded by alternatives if they are demonstrated as "reasonable" .Basel II and IFRS 9 regulate certain aspects of a default definition as it pertains to specific exercises, i.e.,modelling the UL and EL. However, the notion of ‘default’ extends to other areas in retail banking as well, mostnotably that of credit scoring, pricing, and collections. Selecting appropriate default definitions within these areasare often based on managerial discretion though supported by some analysis. In particular, the observed transitionsamongst increasingly severe arrears categories (30 days, 60 days, etc.) are cross-tabulated across a chosen length oftime in what is called a roll rate analysis . From Siddiqi (2005, pp. 33–42), the principle is to select a particularcategory as the default definition that is sufficiently stable in that accounts identified as ‘lost’ ought to remain lost atthe end of the outcome period. The chosen category should yield a minimum of accounts recovering from ‘default’on average. However, the direct loss implications associated with any definition may be a better criterion thanstability since the latter ignores any competing financial/opportunity costs that may actually exist when varyingthe default threshold. Furthermore, the ‘true’ transition rates can be obscured by the epoch of time from whichloan performance is sampled, which can certainly influence the chosen threshold. Perhaps the greatest source of5 imulation-based optimisation of the timing of loan recovery across different portfolios variation underlying these roll rates is the length of the outcome period, which can vary between 6–24 months inpractice, as discussed in Thomas et al. (2002, pp. 99) and Van Gestel and Baesens (2009, pp. 101–102).The work of Kennedy, Mac Namee, Delany, O’Sullivan and Watson (2013) and Mushava and Murray (2018)investigated the role of the outcome period using Irish and South African data respectively. The authors useddifferent time spans in predicting default risk and found that too short a window becomes insufficient in capturingthe transition rates due to seasonal effects and/or risk immaturity. Conversely, overly long windows may no longerrepresent the portfolio’s current risk composition, strategies, or even the current market conditions, in additionto yielding models with degrading accuracy. Furthermore, longer windows can ignore rapid transitions amongstdelinquency states, e.g., oscillating between defaulting and curing, as discussed in Kelly and O’Malley (2016),which is especially relevant for the accuracy of monthly EL estimates. The outcome length and the sample windoware clearly significant factors that complicate the choice of a default definition. As an example, a particularly lowcuring rate given a chosen definition cannot truly justify the latter (without conducting additional analysis) dueto these other confounding factors. Put differently, low curing rates may instead be attributed to an overly shortoutcome period or shifting market conditions – both of are reasons why a roll rate-based approach is deemed unfitfor dynamically finding the "point of no return" in this study.Varying the default threshold within a definition was first studied in Harris (2013b) and Harris (2013a) froma credit scoring perspective. The authors built default-classifiers using Support Vector Machines across variousthresholds and found that the model accuracy is affected by the chosen threshold. However, while optimisingaccuracy is certainly worthwhile, these results say little about the direct impact on profitability when varying thedefault threshold. As originally argued in Hand and Henley (1997) and Hand (2001), a lender is primarily interestedin the underlying profitability of a credit decision, with credit risk being but a facet thereof. Surely, borrowers withno arrears are likely to be profitable ventures for the bank, while accruing arrears up to a point can certainly lead toeventual losses. However, there is little objective evidence in literature for justifying the presumption of profitabilityunderlying 90 DPD as the ideal default threshold. Moreover, not all ‘defaults’ (or default thresholds) are equal, asdemonstrated in Kelly and McCann (2016) using Irish mortgage data. A legal peculiarity during 2009–2013 madeit extremely difficult for Irish lenders to liquidate troubled mortgages, which led to disproportionately deep levels ofarrears. The authors modelled so-called ‘deep defaults’ (e.g., 360+ DPD) across different arrears severities used asdefault thresholds, which yielded markedly different curing experiences. Amongst other things, these results castdoubt on the supposed finality of the classical 90 DPD threshold serving as the "point of no return".Selecting any DPD-based threshold will affect the associated probability of curing from the supposed ‘default’state. However, the chosen threshold’s suitability as a "point of no return" becomes questionable whenever thisprobability is nonzero. Furthermore, multi-period "episodes of delinquency" are more widespread in practice thanone would otherwise believe, based on anecdotal experience. The work of Thomas, Matuszyk, So, Mues andMoore (2016) demonstrated these patterns of periodic repayments using a four-state homogeneous Markov chain,in modelling the collections process of defaulted UK loans. The authors noted that these models may be used toevaluate write-off policies, even though this is not truly recommended. Instead, pursuing loan collection for anoptimal length of time was investigated in Mitchner and Peterson (1957), based on maximising net profit using USloans. The authors found that loan recovery should cease whenever the one-period expected repayment equalsthe collection cost itself. However, they assumed that a defaulted borrower is permanently absorbed into a payingregime once cured, which contrasts Thomas et al. (2016).A dynamic programming model was formulated in De Almeida Filho, Mues and Thomas (2010) in optimising6 imulation-based optimisation of the timing of loan recovery across different portfolios the collections process using unsecured European loans. The idea is to find the ideal recovery action and its optimalpursuit duration, which maximises the net recovery rate for an "average" debtor. These actions telephonic calls,demand letters, house visits, threats, legal steps, and write-off. However, cash flows from previous or future periodswere excluded from the state space formulation, which limits the approach’s tractability. This work was extended inSo, Mues, de Almeida Filho and Thomas (2019) by following a Bayesian approach on the individual debtor-level togive similarly optimised outputs. Within the same problem context, a Markov decision process was developed inLiu, He and Chen (2019) as an alternative approach using designed data. Similarly, an optimal collection action issought across both delinquency states and time. The authors calculated a schedule of optimised actions based onmaximising expected net present value, which superseded a static collection policy as an alternative. However,strong assumptions were made when designing both the data and elements within the authors’ method, which maynot be suitable in practice. Moreover, write-off was not structured as a candidate collection action, instead beingexogenously imposed within the Markov chain’s state space.In form, our study is closest to that of De Almeida Filho et al. (2010) and Liu et al. (2019), though a differentand arguably more general approach is followed. Specifically, we use delinquency measures instead of time andleverage the entire portfolio instead of using only ‘defaulted’ loans, which already imposes a particular "point of noreturn". We focus more fundamentally on if and when to abandon a loan based on accrued delinquency, instead ofpursuing various collection actions. From a literature perspective, our work attempts to bridge the branches ofcredit risk modelling and collection optimisation. We achieve this by framing the recovery decision’s timing as aloss-based optimisation problem wherein the ideal "point of no return" is sought.
We define delinquency as a time-dependent, varying, and measurable quantity that represents the extent of erodedtrust between bank and borrower. Relatedly, a delinquency measure g reflects the degree of non-payment based onthe fundamental idea of a borrower owing I t > (instalment) though only repaying R t ≥ (receipt) at a particulartime t . When R t < I t , the function g quantifies the extent I t − R t by which the bank incrementally loses confidencein the borrower honouring the original credit agreement. Three different delinquency measures g , g , and g are used in this study, with their construction detailed in the appendix. We develop the so-called Loss-basedRecovery Optimisation across Delinquency (LROD) procedure in subsection 3.1. This procedure attempts to findthe ‘best’ delinquency-based threshold for a given g at which the portfolio’s recovery decision is loss-optimised, asillustrated in Fig. 4. In addition, a simulation-based setup is described in subsection 3.2 by which portfolios can besystematically generated across various credit risk scenarios, in testing the LROD-procedure. Consider a portfolio of N loans, indexed by i = , . . . , N , and let g ( i , t ) denote the value of a particular measure g ∈ (cid:8) g , g , g (cid:9) at periods t = , . . . , t c i with t c i representing the contractual term of the i th account. Let v ( a ) t and v ( b ) t be standard discounting functions that use an alternative risk-free interest rate and the client interest raterespectively in discounting back t periods. Let R it and I it be the respective receipt and expected instalment at time t imulation-based optimisation of the timing of loan recovery across different portfolios Fig. 4.
Illustrating the loss optimisation of the recovery decision across several delinquency measures. As a result,Measure 3 is chosen as the best measure with its minimum loss attained at threshold c .for the i th account. Then, let R ( i , t ) be the summed historical receipts up to t , expressed as R ( i , t ) = t (cid:213) l = R il v ( a ) l . (1)For the remaining future instalments, let O ( i , t ) denote the expected outstanding balance at t , defined as O ( i , t ) = v ( a ) t t ci (cid:213) l = t + I il v ( b ) l − t , O ( i , t ) = for t = t c i . (2)To cater for arrears, let A ( i , t ) be the historical and cumulative shortfall up to t between instalments and receipts,given by A ( i , t ) = t (cid:213) l = (cid:16) I il − R il (cid:17) v ( a ) l . (3)Financial loss can only be realised when the lender disposes of the impaired asset, regardless of the extent ofimpairment. We purposefully define ‘default’ as a contrived state that carries a much more fundamental meaning,which becomes useful for optimising the eventual recovery decision. Having breached some threshold (signifyingbroken trust), the lender’s objective changes to collecting the maximum in the shortest time possible. As asimplifying assumption, a fixed portion of the loan is immediately written-off upon entering ‘default’. In reality,this portion will likely depend on many factors, including the workout period itself. This assumption can certainlybe relaxed in future research when refining this optimisation procedure and what is essentially its LGD-component.Accordingly, let r E ∈ [ , ] be a loss rate applied on O ( i , t ) . Moreover, assume that A ( i , t ) is partly written-off at adifferent loss rate r A ∈ [ , ] . Using two different rates recognises that the recovery success may differ between thesetwo components (expected balance and arrears). These static loss rates serve as placeholders for more sophisticated8 imulation-based optimisation of the timing of loan recovery across different portfolios loss models, presumably including all other costs. Finally, let l ( i , t ) be the discounted "blended loss" assessed at t and expressed as l ( i , t ) = O ( i , t ) r E + A ( i , t ) r A . (4)For optimising loan recovery, let d ≥ be a delinquency threshold such that the i th account is consideredas ( g , d ) -defaulting if and only if g ( i , t ) ≥ d at any particular time t = , . . . , t c i . Let S D be the subset of all ( g , d ) -defaulting accounts such that S D = (cid:8) i (cid:12)(cid:12) ∃ t ∈ [ , t c i ] : g ( i , t ) ≥ d (cid:9) . (5)Since an account may enter and leave the ( g , d ) -default state multiple times in reality, let t ( g , d ) i be the earliestmoment of ‘default’ for a qualifying account, defined as t ( g , d ) i = min (cid:0) t : g ( i , t ) ≥ d (cid:1) , ∀ i ∈ S D . (6)Similarly, let S P be the subset of all accounts considered as ( g , d ) -performing such that S P = (cid:8) i : g ( i , t ) < d ∀ t ∈ [ , t c i ] (cid:9) . (7)The difference in assessing losses between a ( g , d ) -defaulting and a ( g , d ) -performing account is simply the time ofassessment t , which is set at either t = t ( g , d ) i or t = t c i respectively within l ( i , t ) from Eq. 4. At each time t , thelender effectively decides an account’s membership between S D or S P , based on accrued delinquency g ( i , t ) and aparticular ( g , d ) -configuration. The latter is to be adopted as a portfolio-wide delinquency-based collection policyat the outset t = , as is common South African collection practice. In a sense, accrued delinquency forms thetime-invariant action space of a Markov decision process (MDP) in choosing d , whereas accrued delinquencyformed the state space in Liu et al. (2019). Accordingly, our state space is set membership itself, i.e., either S P or S D . However, we do not employ a classical MDP framework, instead opting for a simpler approach that facilitateschoosing a static ( g , d ) -policy. Both ‘payoff’ and the element of time is already accounted for in Eq. 4 by havingdiscounted the associated loss to t = for a given ( g , d ) -policy. As such, the objective function is simply the totalportfolio loss L g ( d ) for a particular ( g , d ) -configuration, defined as L g ( d ) = (cid:213) i ∈ S D l (cid:16) i , t ( g , d ) i (cid:17) + (cid:213) i ∈ S P l (cid:0) i , t c i (cid:1) . (8)Losses are iteratively calculated across a range of thresholds d ∈ D g using L g from Eq. 8 with a particularmeasure g ∈ (cid:8) g , g , g (cid:9) . In summary, three preparatory steps are necessary before conducting optimisation:1. Delinquency must be measured for every account and across its history using g ∈ (cid:8) g , g , g (cid:9) ;2. Select appropriate thresholds d ∈ D g on the domain of a particular g for optimisation;3. A portfolio loss model L g must be applied at every chosen threshold d ∈ D g of each g .The main recovery optimisation problem is effectively divided into smaller ( g , d ) -based sub-problems. The resulting L g ( d ) for each ( g , d ) -configuration is stored centrally, thereby forming a loss curve across d for each g . Theobjective then becomes a search for a threshold d (cid:48) ∈ D g such that L g ( d (cid:48) ) ≤ L g ( d ) for all chosen d ∈ D g . Moregenerally, if a global minimum m ( g ) exists on L g for a particular measure g , then L g is said to be minimised9 imulation-based optimisation of the timing of loan recovery across different portfolios at d ( g ) . Minimising again across the set formed by m ( g ) effectively allows indirect comparison of delinquencymeasures themselves at the portfolio-level. The optimal measure g ∗ is then the g that yielded the lowest loss at itscorresponding threshold, as illustrated in Fig. 4 and expressed as g ∗ = arg g min g ∈{ g , g , g } (cid:104) m ( g ) , m ( g ) , m ( g ) (cid:105) . (9)Alternatively, a single measure can be used, e.g., g , which simplifies the optimisation to finding a threshold d ( g ) ∈ D g that equals arg d min L g ( d ) if a minimum exists. Nonetheless, the optimisation’s feasibility reliesheavily on adequately populating the search space D g . Thresholds are trivially chosen as d = , . . . , d N forthe integer-valued g -measure since D g is a countable set, where d N is a reasonable (but admittedly arbitrary)proportion of the maximum contractual term, e.g., . However, this becomes more complicated for the real-valuedmeasures g and g since their search spaces contain infinite possible thresholds. As such, two competing interestsare balanced against each other when assembling D g : 1) too few thresholds that are inadequately spaced canobscure hidden optima and ruin the optimisation; 2) too many thresholds can become computationally burdensome.As a practical expedient, the output of g and g are binned into a discretionary range of thresholds by which D g ispopulated, followed by manual tweaks. A real-world portfolio inherently suffers from censoring insofar that delinquent loans are only kept on the balancesheet up to a certain point, as controlled by the bank’s write-off policies. Although eventually optimising therecovery decision of a real-world portfolio would be ideal, it is arguably prudent first to demonstrate the efficacyhereof from "first principles" on designed data. In this section, we describe a broad but simple simulation-basedsetup that is guided by expert judgement and industry experience. Using this setup as our testbed, replicableloan portfolios of varying risk levels and characteristics are iteratively generated, with which we can test ourLROD-procedure. This testbed is subsequently used to identify a certain range of credit risk profiles for whichoptima are found, simply by varying the simulation parameters.Some delinquent accounts will simply never recover in reality, which implies a continuous stream of zeros intheir receipts R = (cid:2) R , R , . . . , R t c (cid:3) after some point. Given a measure g ∈ (cid:8) g , g , g (cid:9) and a so-called truncationparameter k ≥ , this effect is simulated from a certain starting point t (cid:48) = min (cid:0) j : g ( j ) ≥ k (cid:1) that only exists whendelinquency has accrued sufficiently, i.e., the earliest period j ∈ [ , t c ] at which g ( j ) ≥ k is potentially triggered. Aprocess, called ( k , g ) -truncation, then changes R to R (cid:48) by R (cid:48) = (cid:2) R , R , . . . , R t (cid:48) , , . . . , (cid:3) if t (cid:48) exists R otherwise . (10)Consider N = , standard amortising loan accounts that are indexed by i = , . . . , N , with a fixedcontractual term of t c = months, a fixed effective annual interest rate of 20%, and a fixed principal amount suchthat the level instalment is I t = at every period t = , . . . , t c . Admittedly, these quantities are oversimplifiedand will typically vary in a real portfolio based on the level of credit risk and loan demand. However, samplingthem instead from stylised distributions (guided by expert judgement) did not have nearly the same effect as that of10 imulation-based optimisation of the timing of loan recovery across different portfolios credit risk in the optimisation itself. These simplifications are therefore justified for the time being. Furthermore,an effective annual risk-free rate of 7% is used in discounting, which is realistic for the South African market. Letthe maximum loan size be L M = , and let r E = and r A = with the rationale that losses on arrearsought to be penalised more than losses on expected balances. The latter is a decreasing quantity while the formerincreases over time for a persistently delinquent loan.In simulating the receipt vector R of each loan account, two probabilistic techniques are now described.As a basic technique (called random defaults ), let u t ∈ [ , ] be a randomly generated number at every period t = , . . . , t c and let b be the probability of payment, i.e., P ( R t = I ) = b with I denoting the level instalment. Notethat b = is chosen as a default value, though this is later varied. Each element R t within R is then populatedwith either I or 0, expressed as R t = I if u t < b otherwise . (11)Despite its simplicity, random defaults do not feasibly generate periods of consecutive non-payments followedby resumed payment, which frequently occurs in practice as "episodic delinquency". Therefore, the Markoviandefaults technique is defined where X t ∈ { P , D , W } denotes a random variable that can assume one of three statesat each period t ; the paying state P : R t = I , the delinquent state D : R t = , and the absorbing write-off stateW : R t ≥ t (cid:48) = from a certain point t (cid:48) onwards. Then, let X , X , . . . be a sequence of random variables that form adiscrete-time first-order Markov chain. We can reasonably assume that every account starts off as non-delinquent,i.e., P ( X = P ) = while P ( X ∈ { D , W }) = . Subsequently, the one-period transition probability from the currentstate i at t to the future state j at t + is denoted as P ij . However, let the write-off probabilities be sensibly setto . and respective to the starting states P and D. These values agree with general industry experience ofan unsecured portfolio, though can certainly be tweaked to the individual portfolio in practice. The remainingelements in the transition matrix can now be derived from but two probabilities, P PP and P DD . In turn, both of thesecan be systematically varied to generate a portfolio’s cash flows according to a certain level (or profile) of creditrisk. The transition matrix is accordingly expressed in Table 1.ToP D W F r o m P P PP − P PP − . − P DD − P DD Table 1:
A conceptual transition matrix for the Markovian defaults technique, wherein the rates P PP and P DD are tobe systematically varied. In this section, we demonstrate and test the LROD-procedure across a wide array of credit risk scenarios generatedusing the testbed described in subsection 3.2. The computational results are grouped below by technique, followedby suggestions for applying the LROD-procedure on real-world data.11 imulation-based optimisation of the timing of loan recovery across different portfolios
This technique leverages ( k , g ) -truncation to control the portfolio generation itself, thereby serving as a sanity checkwhen testing the optimisation results and its underlying logic. Intuitively, the lowest loss should be at threshold d = k , since receipts are zeroed after having breached k by design. As an illustration, ( , g ) -truncation is appliedin Fig. 5a, which shows the lowest loss occurs at d = for g as expected. However, the choice of g ∈ (cid:8) g , g , g (cid:9) when truncating introduces bias in the timing of cash flows, such that this g will likely contain the lowest loss aswell. This is demonstrated in Fig. 5b when using ( , g ) -truncation instead, where the minimum loss now occursapproximately at d = k = for g . Whilst seemingly artificial, truncation is merely used as a testing tool to helpbuild our intuition. However, the notion of truncation is plausibly similar to default contagion during a real-worldeconomic downturn, during which borrowers may default systematically at some level of accrued delinquency k onaverage. (a) Using ( , g ) -truncation (b) Using ( , g ) -truncation Fig. 5.
Losses (as a proportion of summed principals) across thresholds d by measure g ∈ (cid:8) g , g , g (cid:9) using therandom defaults technique. In (a), loans are ( , g ) -truncated, while they are ( , g ) -truncated in (b). In both cases,the zoomed plots show that global minima occur at or near the truncation point, d = k .Minimum losses ought to occur wherever d = k when ( k , g ) -truncating receipts. This intuition is largelyconfirmed in Fig. 6 wherein truncation parameters k = , . . . , are applied during portfolio generation. As a result,loss minima occur consistently at the truncation point d = k as expected, while holding other factors constant. Eachincreasing value of k also yielded a smaller minimum loss as a result of the overall lessening truncation effect. Sincereceipts are truncated less frequently as k increases, generated portfolios exhibit overall less delinquency (or creditrisk), which explains both lower loss curves and lower loss minima. Although not shown, this result holds similarlyfor g and g when used in truncation. Therefore, the optimisation is deemed sensitive to systematic defaults andcan react accordingly should the defaulting behaviour of borrowers converge, as simulated by truncation.12 imulation-based optimisation of the timing of loan recovery across different portfolios Fig. 6.
Losses across thresholds d for the g -measure with ( k , g ) -truncation, using the random defaults technique.Several truncation points k = , . . . , are used, with the zoomed plot confirming that global minima in lossesoccur at each truncation point d = k . Fig. 7.
Losses across thresholds d for the g -measure with ( , g ) -truncation, using the random defaults techniqueand several probabilities of payment b ∈ [ , ] . The zoomed plot shows a smaller range of . ≤ b ≤ . whereloss minima occur at the chosen truncation point. 13 imulation-based optimisation of the timing of loan recovery across different portfolios Besides truncation, this technique has another parameter that is arguably more relevant: that of the one-periodrepayment probability b . Each value of b corresponds to a particular level of credit risk during portfolio generation.By varying b , the effect of credit risk can be broadly tested when optimising loan recovery, as shown in Fig. 7.Applying ( , g ) -truncation as a benchmark, loss minima still occur at d = k = as expected, though only for acertain range of . < b < . . This suggests that optimising loan recovery in practice is infeasible for either veryrisky loan portfolios or near riskless portfolios. In particular, the two boundary cases of b = and b = in Fig. 7support this idea in that loans should be forsaken at the outset when b = , as evidenced by the loss minimum at d = , since all receipts will be zero-valued by design. Conversely, if there is no credit risk, i.e., b = , then noloss is made at any d > and loan recovery itself becomes a moot point. These computational results can directlytranslate into practical value when estimating the parameter b from a real-world portfolio, as well as estimating theextent of any underlying truncation effect. Fig. 8.
Losses across thresholds d for the g -measure with ( , g ) -truncation, using the random defaults techniqueand several arrears loss rates r A ∈ [ , ] . The zoomed plot shows a smaller range of loss rates . ≤ r A ≤ whereloss minima occur at the chosen truncation point.Intuitively, the loss experience (or LGD) of a particular portfolio ought to affect the results of recoveryoptimisation as well, especially when considering loan security in the event of default. In our context, this is testableby varying the loss rate r A during portfolio generation while holding other factors constant, as illustrated in Fig. 8using g (though similar results hold for g and g ). As a proxy for more secure portfolios, smaller values of r A lead to flatter loss curves, until reaching a point where recovery optimisation becomes infeasible. Conversely, largervalues of r A yield loss curves with a greater ‘bend’ at the chosen truncation point, which signifies the greater riskinvolved with more unsecured portfolios. Since b is held constant, we can conclude that once default does occur,the viability of recovery optimisation only increases with the risk of loss, which is intuitively sensible. This is tosay that unsecured portfolios will likely benefit even more from recovery optimisation than secured portfolios.14 imulation-based optimisation of the timing of loan recovery across different portfolios This technique affords greater flexibility in generating portfolios with more sporadic repayment histories. Accordingly,we demonstrate the LROD-procedure in Fig. 9 using some of the parametrisations of the underlying Markov chainthat yield optima across all delinquency measures. Evidently, the g -measure appears to outperform the othermeasures since it yields the lowest loss within each of these settings, including a number of other parametrisationsnot shown. However, summarily concluding the supremacy of g across all portfolios would be disingenuous.It is still possible that some real-world portfolios may be better served using measures other than g within theLROD-procedure (or more broadly in risk management). Our current objective is not to determine the best measureconclusively. Indeed, conducting such an empirical study would require expansive real-world data on all types ofportfolios across the risk spectrum, which is prohibitively impractical at this stage. That said, the g -measure ishenceforth used in this section given its supremacy in this instance. Fig. 9.
Losses across thresholds d by measure g ∈ (cid:8) g , g , g (cid:9) using the Markovian defaults technique to generatedifferent loan portfolios. Each panel explores a specific setting of the transition matrix, using the titular probabilitieswithin the matrix defined in Table 1. Encircled points indicate loss minima at associated thresholds d ( g ) .Using this technique, we devise a broad iterative scheme in systematically generating portfolios across theentire credit risk spectrum, as measured with g . In particular, P DD is held constant at a certain value whilevarying P PP , followed by fixing P DD to a different value and varying P PP again, and so on. This scheme allows forsuitably varying the transition matrix in Table 1 using fixed intervals, with some of the resulting loss curves andassociated loss minima presented in Fig. 10. The subplots in both panels (A) and (I) represent boundary cases thatconfirm intuition. Specifically, panel (A) demonstrates recovery optimisation for portfolios with highly transitivedelinquency states such that accounts immediately exit this state in the next period, once entered. Accordingly,the loss curves increasingly resemble a near risk-less case as the value of P PP tends towards 1, which is similar tosetting b = in Fig. 7 when using random defaults. In turn, recovery optimisation itself becomes progressivelyinfeasible in tandem with P PP approaching 1. Conversely, panel (I) showcases the loss curves of extremely riskyportfolios, which are again similar to setting b = in Fig. 7 as P PP approaches 0. More importantly, the fact thatloss minima occur at very small thresholds agrees intuitively with cutting losses sooner rather than later, especiallyfor extreme default risk.The remaining panels in Fig. 10 are perhaps the most instructive. As the delinquency state becomes more15 imulation-based optimisation of the timing of loan recovery across different portfolios Fig. 10.
Losses across thresholds d for the g -measure using the Markovian defaults technique with severaltransition rates P PP ∈ [ , ] and P DD ∈ [ , ] . Encircled points indicate loss minima at associated thresholds d ( g ) .absorbing (or less transient), i.e., moving from panel (B) to (F), we observe that the loss-optimal thresholds d ( g ) are increasingly staggered across both axes. This is to say that d ( g ) becomes progressively more sensitive to boththe threshold d and the value of P PP . Moreover, it is sensible that ever greater losses (at d ( g ) ) are associated withlower values of P PP since the latter implies less time being spent in the paying state, even as the delinquency statebecomes less transient. Furthermore, consider that d ( g ) increases in threshold-value when P PP decreases and P DD increases, i.e., moving from curve (i) down to curve (a) whilst moving across panels (B) to (F). This suggests thatgradually postponing loan recovery is the better strategy even as delinquency becomes more likely, at least up untila certain point, in this case, panel (G). However, this suggestion is counter-intuitive since one would rather cutlosses sooner than later when risk supposedly increases, which implies selecting lower thresholds instead. Twofactors help explain this phenomenon. Firstly, the relevant portfolios are increasingly turbulent by design when P PP changes from higher to lower values in each successive panel. The effect hereof is that loans start to oscillate quiterapidly between the paying and delinquent states as P PP decreases. The slightly increased rate of absorption into the16 imulation-based optimisation of the timing of loan recovery across different portfolios delinquent state (when moving across panels) is not sufficient to support earlier loan recovery as intuition wouldotherwise suggest, especially so when an account still frequently exits the delinquent state. This has the side-effectof muting the severity of ‘default’, which is plausible when curing from ‘default’ itself becomes increasingly likelydue to the same turbulence. Therefore, the associated opportunity cost of forsaking the loan earlier is too high whenfuture repayments are still likely to be received over the longer run, albeit sporadic. Accordingly, greater turbulencein a portfolio requires greater patience to collect upon these repayments, which is why postponing loan recovery (byvirtue of d ( g ) increasing) would be loss-optimal. Secondly, even if d ( g ) increases in value, the associated lossminimum reassuringly increases alongside P DD , as expected from more turbulent and riskier portfolios.There is little need for applying ( k , g ) -truncation on these results since the Markovian technique already hasa realistically-set write-off state that achieves the same effect. While additional truncation will surely confoundthe results, ( , g ) -truncation is experimentally applied in the interest of completeness. We obtain results (notshown) that are largely similar to that of random defaults in that loss minima still occur at or near k = acrossmost portfolios. The exceptions are the two boundary cases, i.e., at or close to panels (A) and (I). Furthermore, theMarkovian technique is especially geared towards generating "regime-switching" portfolios where accounts sufferfrom episodes of delinquency that vary in length, as controlled by the state probabilities. In this regard, episodicdelinquency is more common a phenomenon in practice than one would think, which is why investigating recoveryoptimisation for these cases is more valuable than exploring explicit truncation/write-off any further in this section. The steps in subsection 3.1 require data to be in a longitudinal-format, having measured delinquency in retrospectacross all accounts and time (usually monthly), based on expected instalments and actual receipts. Letting thecontractual term, client and risk-free rates, and even the loss rates vary across the portfolio ought not to impede thepractical use of the LROD-procedure. However, we have assumed the portfolio to be fully observed (or ’completed’)in this study, with little consideration given to any right-censoring and its effect on the receipt history of an account.This particular avenue is further explored in Botha, Beyers and De Villiers (2020). That said, simply excludingincomplete accounts from the dataset can sidestep this possible issue, though at the cost of a reduced sample size.The effect hereof will likely vary based on the typical tenure of the loan product.Our results, particularly those from subsection 4.2, can easily translate into practical value with relatively littleanalytical effort. For example, one can fit the same three-state Markov chain on a real portfolio’s delinquencyprogressions, just to obtain the associated transition rate estimates. In turn, these estimates can be used as a roughguide in finding a corresponding loss curve amongst all those presented in Fig. 10, i.e., a look-up exercise. Theassociated optimised threshold can provide a high-level idea of recovery optimisation, provided the assumptionsare reasonably met. That said, applying the LROD-procedure remains the imperative in order to capture allidiosyncrasies of a particular portfolio and the prevailing market conditions.
We explore a more fundamental meaning of loan ‘default’ by only using d as a variable threshold upon the domainof a delinquency measure g . Though different from current practices, this reinterpretation of ‘default’ better alignswith the rather probabilistic idea of breaching a certain "point of no return", having exceeded d on g . In principle,17 imulation-based optimisation of the timing of loan recovery across different portfolios keeping the loan any longer beyond this point becomes sub-optimal to abandoning it and recovering the maximuminstead. To this end, we contribute a novel optimisation procedure as an expert system to help find the ideal timefor debt recovery during loan life, based on accrued delinquency. This so-called LROD-procedure weighs twocompeting interests against each other: the prospect of reaping future revenue from troubled loans versus the costof retaining these loans any further. In principle, each ( g , d ) -configuration serves as a candidate collection policythat has a "net cost" if applied to a portfolio. The overall portfolio loss is then iteratively calculated across all suchpolicies using the procedure’s inner L g model. Doing so forms a loss curve for each g that can be inspected foran optimal threshold d ( g ) at which the lowest loss occurs, thereby concluding the optimisation. In addition, theLROD-procedure is formulated in such a way that it can be used with multiple loan delinquency measures. Thisfacilitates the objective testing of alternative measures, e.g., those provided in the appendix, that may better suit therecovery optimisation (or even broader risk management) of a portfolio. That said, our study objective is not toestablish the best measure conclusively, which would likely be a data-intensive and costly endeavour.Regarding results, we first describe a simple simulation-based setup in which the LROD-procedure (and itsgoal of recovery optimisation) is closely examined from "first principles". Using this setup as a testbed, we thenconduct a broad computational study wherein basic amortising loan portfolios are systematically generated byvarying the simulation parameters, though still constrained by expert judgement. Having spanned the entire creditrisk spectrum (as measured with the payment probability b ), our computational results show that optimising therecovery decision’s timing is viable across most risk levels, except at the extremes. We further demonstrate thatoptimised recovery times are sensitive to systematic defaults that may structurally affect a portfolio during aneconomic downturn, as approximated by the notion of ( k , g ) -truncation in our testbed. Another factor is that ofcollateral and the portfolio’s loss experience (or LGD), insofar that we successfully found optima across mostof the loss spectrum (as measured with the loss rate r A ). Moreover, recovery optimisation seems to become anincreasingly viable practice as the risk of loss increases.In addition, we test recovery optimisation on more turbulent portfolios wherein borrowers repay intermittently,thereby causing episodic delinquency. Once accounts oscillate rapidly between paying and nonpayment, ‘default’itself diminishes in severity, especially when curing also becomes more likely as a result of the very same turbulence.Accordingly, we found that optimised thresholds increased in value as turbulence develops, though only up to a point.Postponing loan recovery in tandem with greater turbulence is therefore strategically optimal since it allows greaterscope to collect upon these repayments, albeit sporadic. As a secondary contribution, the testbed itself can serve asa valuable tool in exploring the strategic viability of the LROD-procedure. Once appropriately parametrised, thetestbed can generate a wide variety of portfolios, which allows a bank to investigate (at least preliminarily) theprospects of recovery optimisation for a certain type of portfolio. Ultimately, the LROD-procedure can be used totweak existing collection policies and, perhaps in time, default definitions themselves.Future studies can focus on refining the LROD-procedure using real-world portfolio data. So-called ‘incomplete’portfolios, i.e., those wherein many loans have not yet reached contractual maturity, may prove a challenge forrecovery optimisation at this stage. The simplest solution would be to exclude the incomplete accounts, thoughunfortunately reducing the sample size as well. Alternatively, one can perhaps explore an appropriate forecastingapproach in future work. Furthermore, we currently assume homogeneity in that the optimised threshold is aportfolio-wide criterion. However, exploring segmentation schemes may be worthwhile such that the LROD-procedure yields an ideal threshold for each identified segment within the portfolio. Lastly, the current loss model L g can be refined by incorporating historical loss experiences and transforming it into a more dynamic component.18 imulation-based optimisation of the timing of loan recovery across different portfolios As an example, calculating the realised LGD generally requires a specific point of entering ‘default’. From this point,cash flows are observed during its workout up to the applicable write-off point. By introducing d as our ( g , d ) -defaultstate, the starting points of cash flows will naturally vary with d , thereby impacting the LGD calculation itself foreach ( g , d ) -configuration. Intuitively, longer or shorter workout periods will affect the loss experience, which willinfluence recovery optimisation based on our results in this study. This particular refinement will likely intersectwith the existing literature on credit loss modelling and IFRS 9, which as a field is currently quite in vogue. Acknowledgements
This study is financially supported by the Absa Chair in Actuarial Science, hosted at the University of Pretoria, withno known conflicts of interest that may have influenced the outcome of this work.
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Three mathematical operations are presented as delinquency measures in this appendix. Firstly, a variant of thewidely-used number of payments/months in arrears, called the g -measure (or CD -measure), is refined into a morerobust measure in subsection 6.1 using a weighting scheme. Secondly, a more concise algorithm is contributed insubsection 6.2 that creates the Macaulay Duration Index from Sah (2015), called the g -measure (or MD -measure),which is an index of the weighted average time to recover the capital portion of a loan. Lastly, a modified versionof g of our own invention is introduced in subsection 6.3, called the g -measure (or DoD -measure), whichincorporates the sizes of disrupted cash flows into delinquency assessment. CD ): the g -measure Days past due (DPD) from accountancy practices is commonly used in constructing a delinquency measure g ,whereby the unpaid portion of a loan’s instalment is binned into increasingly severe groups as each 30-day calendarmonth lapses: 30 days, 60 days, 90 days, and so forth, as discussed in Cyert, Davidson and Thompson (1962). Moreformally, g is defined as the function g ( t ) = f ( A t / I ) where A t denotes the accumulated arrears amount at discretetime t , I is the fixed instalment, and f is a chosen rounding function that maps the given input to the number ofpayments in arrears as the output. A common choice of f is the ceiling function, whereby the input is simplyrounded upwards to the nearest integer. 21 imulation-based optimisation of the timing of loan recovery across different portfolios However, this rounding scheme is quite stringent in that even a small difference I t − R t = (cid:15) < ZAR . willincrease the delinquency measurement, purely due to rounding. Depending on the volatility in R t over time, it ispunitive to penalise a borrower when (cid:15) is but a few cents. That said, a sensible boundary on (cid:15) must be applied,otherwise the idea of delinquency becomes meaningless. Should A t / I simply be rounded to the nearest integerinstead, then a change in g ( t ) over time [ t , t ] depends entirely on whether A t / I is above or below 50%. Thisimplied ‘threshold’ seems arbitrary, inflexible, and certainly at odds with the risk-based practices of a bank. Lastly,constructing g in practice quickly becomes cumbersome when the instalment is linked to an interest rate that variesover time, which is common for secured lending.There are two additional pitfalls to the g -measure. Firstly, overall measurement can become lagged by one (ormore) periods when a significant overpayment is immediately followed by a severe underpayment the followingmonth, purely due to the chosen f . Secondly, if A t accumulates interest on itself or attracts any fees, then g canbecome ‘corrupted’ due to its inherent reliance on A t . The potential exists for g to change in value, not due to afundamental breakdown in trust, but as a result of the lender’s own pricing structure or system constraints, whichmay artificially inflate the g -value. Moreover, the rounding scheme itself may exacerbate this effect. In both ofthese cases, the apparent "measurement error" in g can adversely affect the true accuracy of models predictingdefault risk.Therefore, a more comprehensive variant, called the CD -measure, is presented here that circumvents thesechallenges. Let the receipt vector be R = [ R , R , . . . , R T ] with its elements (or receipts) R t ≥ , and let theinstalment vector be I = [ I , I , . . . , I T ] with its elements I t > . Both vectors are defined for a specific loanaccount across its discrete time periods t = , . . . , T , with t = representing the origination point and T denotingthe tenure (or current loan age). Note that T may exceed the contractual term t c , especially in cases of extremedelinquency. The repayment ratio h t ∈ [ , ∞) is then defined as h t = (cid:18) R t I t (cid:19) ∀ t = , . . . , T and h = . (12)One can specify a certain threshold z ∈ [ , ] for h t , above which an account at time t is considered current andbeneath which it is considered delinquent. Note that z = is assumed in this study purely as an illustration,though the lender should certainly adjust this z accordingly. Next, a Boolean-valued decision function d ( t ) ∈ { , } is defined for t = , . . . , T , using Iverson brackets [ a ] that outputs 1 if the enclosed statement a is true, and 0 if false,as d ( t ) = (cid:2) h t < z (cid:3) . (13)Memory of past delinquency is introduced by defining another integer-valued function m ( t ) ∈ {− , , , . . . } for t = , . . . , T , which outputs the reduction in accrued delinquency (if any), as m ( t ) = (cid:18) (cid:22) h t z (cid:23) − (cid:19) (cid:16) − d ( t ) (cid:17) − d ( t ) = (cid:22) h t z (cid:23) (cid:16) − d ( t ) (cid:17) − . (14)This function m ( t ) gives the magnitude by which the measured delinquency at time t should be reduced (if at all) incatering for past delinquency. When overpaying, i.e., R t > I t , the ratio between h t and z in Eq. 14 signifies the totalnumber of ‘payments’ by which accrued delinquency should be decreased, as weighed by z . The rounding problem22 imulation-based optimisation of the timing of loan recovery across different portfolios from g is resolved in this measure when dividing by z since its specified value reflects the lender’s tolerancetowards underpayment by design. Accordingly, taking the floor of h t / z does not detract and merely enforces aninteger-valued scale in the eventual measure. Furthermore, the currently-owed instalment should be recognisedfirst before reducing any accrued delinquency, which is achieved by subtracting one instalment. For sufficientunderpayment, i.e., R t < zI t , the delinquency is sensibly increased by one payment, which resolves to m ( t ) = − when d ( t ) = .To indicate previous cases of delinquency using g at time t − , let d ( t ) ∈ { , } be another Boolean-valueddecision function for t = , . . . , T , which is defined using Iverson brackets again, as d ( t ) = (cid:2) g ( t − ) = (cid:3) . (15)The reduction in delinquency m ( t ) at time t is subtracted from delinquency as measured at the previous period t − , thereby giving net delinquency. The integer-valued CD -measure g ( t ) ≥ for t = , . . . , T is then recursivelyexpressed as g ( t ) = max (cid:34) , d ( t ) d ( t ) + (cid:0) − d ( t ) (cid:1) (cid:18) g ( t − ) − m ( t ) (cid:19) (cid:35) . (16)Note the necessary starting condition of g ( ) = , since a newly-disbursed loan cannot yet be delinquent. Theoutput for g is best interpreted as the z - weighted number of payments in arrears, weighed by the lender’s tolerance(or appetite) towards accrued arrears. Since delinquency only increases if h t < z by definition, a higher value of z effectively translates to greater risk-aversion, and vice versa for lower z -values. MD ): the g -measure The Macaulay Duration Index, recently introduced in Sah (2015), is based on bond duration, i.e., the weightedaverage time to recover the capital portion of a loan. This measure incorporates the loan’s interest rate as well as thearrears balance weighted by the time value of money. It is constructed as the ratio between the actual and expectedloan duration, reworked as the g -measure in this study. However, the values of g are incomparable to those of g since both the domains and meanings differ.Let ∆ t = I t − R t be the difference between the instalment I t and the receipt R t at every time point t = , . . . , T during the life of a loan, including at disbursement t = to capture any applicable initiation fees. Consideringthe time value of money, let v j = ( + r ) − j be a discounting function that uses a nominal monthly interest rate r .In addition, let δ be the continuously compounded rate with its nominal variant δ ( p ) = δ / p and with an annualcompounding period p = . Let L P denote the loan amount (or principal) that is to be amortised. Ordinarily, theMacaulay Duration is calculated (perhaps once) at origination as the weighted average time to recover sunk capitalfrom future cash flows. However, here it is recursively calculated instead at each subsequent period t = , . . . , T across the remaining m instalments as at each t . Naturally, this expected duration quantity, denoted as f ED , tendstowards zero over time as it nears the end of loan life, expressed as f ED ( t ) = T (cid:213) m = t (cid:20) (cid:18) I m v m − t L P (cid:19) (cid:18) ( m − t ) p (cid:19) (cid:21) ∀ t = , . . . , T . (17)However, Eq. 17 assumes that instalments I are free of uncertainty. When substituting these instalments with23 imulation-based optimisation of the timing of loan recovery across different portfolios the actual receipts R , a significant difference is intuitively expected. Moreover, it becomes necessary to track thearrears balance as it develops (if it does) over the loan life. In line with Sah (2015), any arrears at any time areadded to the last expected (contractual) instalment at t = t c , since it represents the last contractual opportunityto repay any such arrears, short of the lender intervening and restructuring the loan. This last instalment is thenrecursively updated for each subsequent period t , denoted by the vector I (cid:48) , which equals instalments I at first.Lastly, the actual duration f AD ( t ) is also recursively calculated for each subsequent period t . This is illustratedusing pseudo-code in Algorithm 1. Algorithm 1
Calculating g I (cid:48) : = I , where I = (cid:2) I , . . . , I T (cid:3) and T ≤ t c f AD ( ) : = f ED ( ) for t = , . . . , T do (cid:46) such that T ≤ t c I (cid:48)( T ) : = I (cid:48)( T ) + ∆ t (cid:16) + δ ( p ) p (cid:17) T − t , ∀ t = , . . . , T (cid:46) Add any arrears to I (cid:48)( T ) f AD ( t ) : = (cid:205) T | T ≤ t c m = t (cid:104) (cid:16) I (cid:48) m v ( m − t ) L P (cid:17) (cid:16) m − tp (cid:17) (cid:105) , ∀ t = , . . . , T end for Finally, the real-valued Macaulay Duration ( MD ) measure g ( t ) ≥ is then defined as the ratio between theactual duration and the expected duration at time points t = , . . . , T − , which is expressed as g ( t ) = f AD ( t ) f ED ( t ) . (18) DoD ): the g -measure From a cash flow perspective, an ideal delinquency measurement should penalise the non-payment of a larger loan’sinstalment to a greater degree than that of a smaller loan’s instalment, given the relatively larger impact on a bank’scash flow. Furthermore, the differences in risk concentration between a larger number of small loans versus a smallnumber of larger loans should also be incorporated by the ideal delinquency measure. As a possible solution, theactual duration f AD ( t ) from Eq. 18 can be altered such that the eventual g ( t ) is greater for larger loans than forsmaller loans by defining an appropriate multiplier.Note that g is only defined up to the contractual term t c . However, delinquency can continue even past itscontractual term T ≥ t c , likely due to persisting underpayment. Ignoring loan write-off policies for the moment, let d ( t ) ∈ { , } be a Boolean-valued decision function that returns 1 if the given time point t precedes the contractualterm t c , and 0 if otherwise. Using Iverson brackets, this is expressed as d ( t ) = (cid:2) t ≤ t c (cid:3) . (19)When t > t c , any arrears can clearly no longer be added to the last contractual instalment (since it has lapsed),as was added for I (cid:48) T at T = t when calculating g in Algorithm 1. Instead, at least one more payment, albeitout-of-contract, can reasonably be expected at every subsequent period t : t ≥ t c as long as collection efforts areactively pursued. Therefore, delinquency can now be computed up to time T instead of the previous T , with T either representing the contractual term t c when t < t c , or becoming a moving target T = t when t ≥ t c . Note thatboth I and R will incrementally expand with additional elements for as long as collection efforts continue past t c .A revised algorithm is given in Algorithm 2. 24 imulation-based optimisation of the timing of loan recovery across different portfolios Algorithm 2
Calculating g I (cid:48) : = I , where I = (cid:2) I , . . . , I T (cid:3) and < t c ≤ T T : = t c for t = , . . . , T do α : = I (cid:48)( T ) (cid:46) This refers to the element at the T th position of I (cid:48) T : = t c d ( t ) + t (cid:18) − d ( t ) (cid:19) (cid:46) T is either equal to t c or to t ≥ t c I (cid:48)( T ) : = I (cid:48)( T ) d ( t ) + ∆ t (cid:16) + δ ( p ) p (cid:17) T − t + α (cid:18) − d ( t ) (cid:19) (cid:16) + δ ( p ) p (cid:17) , ∀ t = , . . . , T β ( m ) : = m − t + − d ( t ) , ∀ t = , . . . , T (cid:46) Discounting periods, used in next two lines f ED ( t ) : = (cid:205) T m = t (cid:104) (cid:16) I m v β ( m ) L P (cid:17) (cid:16) β ( m ) p (cid:17) (cid:105) , ∀ t = , . . . , T f AD ( t ) : = f ED ( t ) , for t = f AD ( t ) : = (cid:205) T m = t (cid:104) (cid:16) I (cid:48) m v β ( m ) L P (cid:17) (cid:16) β ( m ) p (cid:17) (cid:105) , ∀ t = , . . . , T end for Afterwards, let λ ( L M , L P , s ) denote a multiplier function that inflates f AD ( t ) at the period t . Let L M denotethe maximum loan size and let s ∈ [ , ] be a real-valued sensitivity that represents the ‘strength’ at which to applythis inflationary effect. Let d ( t ) ∈ { , } be another Boolean-valued decision function that returns 1 if there iscurrently any accrued delinquency at t , and 0 otherwise, defined using Iverson brackets as d ( t ) = (cid:2) f AD ( t ) > f ED ( t ) (cid:3) . (20)As a simple example, this multiplier is defined as λ ( L M , L P , s ) = s (cid:18) − L M − L P L M (cid:19) . (21)The inflated variant of f AD ( t ) , denoted as ˜ f AD ( t ) , is given by ˜ f AD ( t ) = f AD ( t ) (cid:0) d ( t ) λ ( L M , L P , s ) + (cid:1) . (22)By including d ( t ) into ˜ f AD ( t ) in Eq. 22, accrued delinquency will not be inflated when overpaying at someperiod t . Finally, the real-valued Degree of Delinquency ( DoD ) measure g ( t ) ≥ is defined for t = , . . . , T − and expressed as g ( t ) = ˜ f AD ( t ) f ED ( t ) = (cid:18) g ( t ) f AD ( t ) (cid:19) ˜ f AD ( t ) = g ( t ) (cid:0) d ( t ) λ ( L M , L P , s ) + (cid:1) . (23)The sensitivity s , which is fixed in this study at s = (though should ideally be optimised), represents auniversal and intuitive lever at the lender’s disposal. Its adjustment can align with the lender’s particular riskappetite and tolerances. At s = , g collapses back into g , though it purposefully resembles a more risk-adverseform of g for s > . Delinquency values are more varied than those of g2