A procedure for loss-optimising default definitions across simulated credit risk scenarios
AA procedure for loss-optimising default definitions acrosssimulated credit risk scenarios
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 new procedure is presented for the objective comparison and evaluation of default definitions. This allows thelender to find a default threshold at which the financial loss of a loan portfolio is minimised, in accordance withBasel II. Alternative delinquency measures, other than simply measuring payments in arrears, can also be evaluatedusing this optimisation procedure. Furthermore, a simulation study is performed in testing the procedure from‘first principles’ across a wide range of credit risk scenarios. Specifically, three probabilistic techniques are used togenerate cash flows, while the parameters of each are varied, as part of the simulation study. The results show thatloss minima can exist for a select range of credit risk profiles, which suggests that the loss optimisation of defaultthresholds can become a viable practice. The default decision is therefore framed anew as an optimisation problemin choosing a default threshold that is neither too early nor too late in loan life. These results also challenges currentpractices wherein default is pragmatically defined as ‘90 days past due’, with little objective evidence for its overallsuitability or financial impact, at least beyond flawed roll rate analyses or a regulator’s decree.
Keywords—
Decision Analysis; Credit Loss; Loan Delinquency; Default Definition
JEL:
C44, C63, G21.
Acknowledgements
This work 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.Word count (excluding front matter): 7643 ∗ ORC iD: 0000-0002-1708-0153 † Corresponding author: [email protected] a r X i v : . [ q -f i n . R M ] J u l procedure for loss-optimising default definitions across simulated credit risk scenarios 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). Retail credit is currently estimated at $13 trillion forthe US market, which largely consists of mortgages, credit cards, personal loans, vehicle financing, overdrafts andother revolving loans for the individual, as reported in The Board of Governors of the Federal Reserve System (US)(2018). For perspective, consumer debt in 2007 was 40% greater than total industry debt ($9.2 trillion) and morethan double total corporate debt ($5.8 trillion). Although greatest in the USA, consumer debt in other countries arenot far behind, e.g., the United Kingdom had debt levels in 2007 at £1.4 trillion – a staggering £400 billion growthwithin the span of a mere three years. While Canada’s consumer debt is estimated at $666 billion, this figure alsoconstituted 110% of total annual household income. In fact, this trend of debt levels exceeding household income istrue for quite a few countries for 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 application credit scorecards. These models rendered consistent approve/decline credit decisions that enabledgreater application volumes whilst keeping default risk aligned with a lender’s risk appetite. This is mainly achievedby only approving those applications with a predicted probability of default within a desired limit. Constructingthese scorecards involves finding a statistical relationship between a set of borrower-specific characteristics and thesuccessful (or failed) repayment outcome over time, using historical data. Naturally, the literature on credit scoringis considerable, e.g., Hand and Henley (1997), Hand (2001), Thomas, Edelman and Crook (2002, pp. 2–6, 41–86),Siddiqi (2005), Crook, Edelman and Thomas (2007), Hao, Alam and Carling (2010), and Louzada, Ara and2 procedure for loss-optimising default definitions across simulated credit risk scenarios
Fernandes (2016).The advent of these automated models did, however, call for a more methodical manner of measuring defaultbefore predicting the risk thereof. This includes capturing the development of loan delinquency over time, mostnotably using the accountancy-based number of payments in arrears, as calculated from the number of days pastdue. Specifically, the unpaid portion of an instalment is aged into several increasingly severe bins given the timeelapsed: 30 days, 60 days, 90 days, and so forth, as discussed in Cyert, Davidson and Thompson (1962). Usingthese resulting arrears categories , banks commonly specified three payments (or 90 days) in arrears as their point ofdefault, long before the introduction of the Basel II Capital Accords, which standardised default definitions to somedegree. This so-called ‘threshold’ is often pragmatically informed by managerial discretion, though supported bysome analysis, and generally ranges between 30–180 days, depending on data availability and the type of product, asdiscussed in Thomas et al. (2002, pp. 123–124). However, the direct financial implications of any chosen definitionare not readily known, nor accounted for when deciding the point of default. Therefore, the pursuit of scorecardmodelling excellence becomes questionable when the constructed response variable itself, i.e., the binary ‘good/bad’risk class that results from an applied default definition, is inherently arbitrary, as argued in Hand (2001).According to Finlay (2010, pp. 11–13), should an account accrue sufficient arrears (despite increased collectionefforts), then the lender rather pursues debt recovery, including selling off any assets underlying the original creditagreement. This is to say that every unpaid instalment erodes trust between bank and borrower but only up to acertain point, as characterised by the default definition. This study attempts to frame this point (and finding it)as a mathematical optimisation problem such that ‘default’ occurs neither too early nor too late during loan life.Too strict a threshold will marginalise accounts that would have resumed repayment, had the bank not been toobrash in its default decision. Conversely, too lenient a threshold may prove naive in tolerating increasing arrears atthe cost of liquidity risk. Moreover, profitability ought to be used as the basis for this optimisation, as argued anddiscussed in section 2 of this study. A procedure for this loss optimisation is then presented in section 3, along witha simulation framework for testing this procedure across various credit risk profiles. In particular, three probabilistictechniques are used to generate cash flows according to set parameters, which are then varied as part of the study.Finally, the simulation results are discussed in section 4, which demonstrate that the loss optimisation of defaultthresholds is a viable strategy for a select range of credit risk profiles. The estimation of the frequency of any event in a given sample fundamentally depends on the definition of the event.This is to say that while loan ‘default’ lies intrinsic to credit risk (and its estimation), the phenomenon thereofcertainly has many definitions, both historically and in modern times. These definitions typically vary by product,customer type, and bank, e.g., filing for bankruptcy, unfulfilled claims, negative net present values, overdrawingbeyond an agreed credit limit, as well as becoming three instalments in arrears, as discussed in Van Gestel andBaesens (2009, pp. 203–212) and Baesens, Rösch and Scheule (2016, pp. 137–138). The Basel II Capital Accordsalso standardised default definitions to some extent upon its introduction, while still leaving room for the lender’sdiscretion. Specifically, paragraph 452 of the Basel Committee on Banking Supervision (2006) defines ‘default’ For credit lines, e.g., credit cards and overdrafts, ‘payments in arrears’ are technically irrelevant since there are no amortisinginstalments. However, the number of days by which a facility is in excess of an agreed limit, can still be aged into these arrears categories.The ‘instalment’ is simply the amount required to recover from this overdrawn status. procedure for loss-optimising default definitions across simulated credit risk scenarios as one of the following two conditions. Firstly (and perhaps more commonly-known), the obligor has reached90 days past due (or three payments in arrears) on a material loan balance, or has been in excess of an advisedcredit limit for 90 days. Alternatively, the bank considers, in its opinion , that the obligor is unlikely to repay itsobligations in full, without the necessary intervention of the bank, e.g., liquidating any collateral. To help informthis opinion, Basel II also includes a few reasonable indicators of ‘default’, which are often promulgated verbatim by a particular country’s regulator, e.g., Regulation 67 of the Banks Act of South Africa (2012, pp. 1201–1202) thatdefines ‘default’ exactly the same way as in Basel II. At a minimum, these indicators include:1. The bank assigns a non-accrued status to the debt, thereby no longer charging interest;2. The bank writes down a portion of the debt, or raises a specific provision, since it believes credit quality hassignificantly deteriorated;3. The bank resolves to sell the debt at a material economic loss;4. The bank files for the obligor’s bankruptcy;5. The bank agrees to restructure the debt, which likely results in an overall reduced financial obligation;6. The obligor files for bankruptcy (or is placed therein), which will likely either delay or circumvent repayment.Most of these indicators (items 1–4) are retrospective in that they denote ‘default’ as a result of certain ex post actions taken by a bank. However, these actions are only reasonably pursued after a bank has already resolved thatcontinuing the credit agreement is of little financial benefit. In other words, the trust between bank and borrowerhas already eroded beyond a certain point, likely as a result of persistent non-payment. Consider that if reachingthis particular point already reflects ‘default’ in itself, then these specific default indicators do not signal ‘default’ asmuch as they merely reaffirm what a bank already considers to be obvious. This suggests the fallacy of circularreference, or petitio principii , on the premise of using these indicators in defining default when they themselvesare deduced by presumably the same default criteria. Lastly, items 5–6 ought to be considered more as possiblepredictors of default, rather than indicating definite default at a certain point in time – even though default isreasonably likely for those cases in practice.Apart from Basel II, the three main external rating agencies (Moody’s, Standard & Poor’s, and Fitch) use theirown but not too dissimilar default definitions in guiding investment decisions on a wide range of counterparts, asdiscussed in Van Gestel and Baesens (2009, pp. 115–117, 149–151, 208–209). While the original intent was toindicate investment-grade debt securities (mainly government bonds), modern ratings cover a much larger spectrumof companies and banks. In particular, Moody’s seeks to capture events that change the relationship betweenbondholder and issuer, as its philosophy of default. All three agencies signal ‘default’ when interest and/or capitalportions go unpaid, although a variable grace period apply: one day for Moody’s, 10–30 days for the others. Also,Moody’s does not consider technical defaults (e.g., covenant violations), while S&P does not consider the dividendsthat are due from preferred stock as ‘financial obligations’ and, as such, do not count missed dividend payments asdefaults. Despite these nuanced differences amongst the three agencies, the aim of specifying a default definition isthe same as that of Basel II, which is to find a certain ‘point of no return’ at which most delinquent accounts willremain delinquent and not recover.In addition to using their discretion, lenders also perform a statistical exercise called a roll rate analysis to helpchoose a default definition, as explained in Siddiqi (2005, pp. 33–42). This is best described as a cross-tabulation ofobserved transition rates amongst pre-binned arrears categories across a chosen outcome period (e.g., 12 months).4 procedure for loss-optimising default definitions across simulated credit risk scenarios The principle is to set ‘default’ to the category at which most accounts do not recover (or ‘cure’) to less severecategories, i.e., finding the ‘point of no return’. However, the choice of outcome period affects these roll ratessignificantly, which typically varies between 6–24 months in practice, as discussed in Thomas et al. (2002, pp. 91)and Van Gestel and Baesens (2009, pp. 101–102). In fact, the work of Kennedy, Mac Namee, Delany, O’Sullivanand Watson (2013) experimented with different outcome periods in predicting default (using a constant definitionthereof). Classifier accuracy degraded as the outcome period lengthened, though shorter periods also gave volatiledefault frequencies (due to seasonal effects). Moreover, too short a window may not adequately capture curing ratesdue to maturity, while an overly long window may become divorced from current market conditions, or may simplyrequire more data than available. Longer windows may also ignore oscillations between defaulting and curing, asdiscussed in Kelly and O’Malley (2016).Using different default definitions were first explored in Harris (2013b) and Harris (2013a), wherein the modelaccuracy of support vector machines predicting default are studied whilst employing various default definitions.However, while optimising accuracy is certainly worthwhile, the implications of variable default definitions foroverall profitability are less clear. Moreover, the work of Hand and Henley (1997) argues that a lender is mainlyconcerned with profitability when making a credit decision, and not as much with model accuracy. To that point,loan profitability also depends on factors other than delinquency, e.g., the market response to a lender’s risk-basedpricing (higher interest rates for riskier borrowers), as explored in Phillips (2013). Another factor is the amount ofloss provisions raised today in covering expected credit losses tomorrow, as explained in Van Gestel and Baesens(2009, pp. 38-74) and Finlay (2010, pp. 167–169). The recent introduction of IFRS 9, reviewed in Novotny-Farkas(2016), Xu (2016), Cohen, Edwards Jr et al. (2017) and Skoglund (2017), also aligns this loss estimation closer tocapital reservation, which are meant to absorb unexpected losses. In both cases, quantifying credit risk consists ofestimating default risk (probability of default, or PD), loss risk (loss given default, or LGD), and exposure risk(exposure at default, or EAD), as thoroughly discussed in Thomas (2009, pp. 289–293), Van Gestel and Baesens(2009, chap. 4–6), and Baesens et al. (2016, chap. 5–11). Naturally, these three components rely on a consistent andimmutable default definition, similar to loan pricing, with variable default definitions largely unstudied in literature.Although the exact reasons for retail credit defaults are innumerable (e.g., job loss, marital breakdown, financialnaivety, fraud), they are crudely grouped into either fraud (‘won’t pay’) or financial distress (‘can’t pay’), as exploredin Thomas (2009, pp. 282) and Bravo, Thomas and Weber (2015). Modelling the exact reason and its underlyingcauses is challenging in practice since lenders rarely keep record of defaulting reasons. Instead, a more tangibleapproach is to consider whether an impaired ability to repay is either persistent or temporary. Sufficient patienceon the lender’s part may allow certain financially-distressed borrowers enough time to recover and resume theirloan repayments, at the cost of accruing arrears and increased liquidity risk. On the other hand, too long a periodmay prove naive with the lender partially recovering monies (if at all) at an opportunity cost. Specifying a defaultthreshold therefore serves as a margin of tolerance towards accruing arrears before pursuing debt recovery instead,which aligns with the five-phase credit management model from Finlay (2010, pp. 11-13).In fact, an arrears-based default definition was intrinsic to the double hurdle PD model developed in Moffatt (2005).Both the payments in arrears (the ‘first’ hurdle) and the arrears amount itself (the ‘second’ hurdle) was used inclassifying overall ‘default’. This recognises that not all defaults are equal in their financial impact. The recent workof Kelly and McCann (2016) also supports this notion, using mortgage defaults from the Irish market to modelso-called ‘deep defaults’ (360+ days in arrears). A legal peculiarity during 2009-2013 made it extremely difficultfor Irish lenders to liquidate defaulted mortgages, which led to an artificially high level of arrears. Banks had little5 procedure for loss-optimising default definitions across simulated credit risk scenarios recourse but to relax their risk aversion and adapt to so-called ‘deeper’ defaults as the new norm. If nothing else,this particular instance casts doubt on both the meaning and supposed severity of the default decision itself as wellas using 90 days past due as the point of default.In summary, varying the outcome period in a roll rate analysis will give different transition rates amongst arrearscategories, simply due to sampling. This presents additional uncertainty as certain outcome periods may obscureidiosyncratic features of the loan portfolio. For example, it would be difficult to decide whether a particularly lowcuring rate (estimated across a certain outcome period) is attributable to the risk profile of borrowers, a shift inmarket conditions, or simply too short an outcome period – without conducting additional analysis. Furthermore,the mere possibility of curing from default injects uncertainty into a chosen default definition, which is supposedlythe point at which the relationship between bank and borrower ultimately crumbles away. Finally, there is littleobjective evidence for the presupposed profit-optimality of using Basel II’s 90 days past due as a sacrosanct defaultdefinition. For these reasons, a new approach to finding optimal default definitions is deemed necessary, based onprofitability instead of model accuracy or roll rate analyses.
The term ‘delinquency’ is interpreted as a measurable and variable quantity that signifies the severity of erodedtrust between bank and borrower. A ‘delinquency measure’ g should then reflect the extent of non-paymentfundamentally based on a borrower owing I t > (instalment) though only repaying R t ≥ (receipt) at a particulartime t . The function g then measures delinquency by quantifying the extent I t − R t by which the borrower chipsaway at the communal trust. For this study, three different delinquency measures (see appendix) are used:1. The popular accountancy-based number of payments in arrears, called the Contractual Delinquency (or CD -measure g );2. The Macaulay Duration index-based measure (or MD -measure g ) from Sah (2015), which is an index of theweighted average time to recover the capital portion of a loan;3. A modified version of g , called the Degree of Delinquency (or
DoD -measure g ), which incorporates thesizes of disrupted cash flows in assessing delinquency.Regardless of g , a procedure is developed in this section for finding the ‘best’ default threshold from a portfolioloss perspective. To test this procedure, a simulation framework is also described for generating various portfoliosacross several credit risk scenarios. 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 i with T i representing the contractual term of the i th account. Let v ( a ) t and v ( b ) t be standard actuarial discounting functions that respectively use an alternative risk-free interest rate and theclient interest rate in discounting back t periods. Let R it and I it be the receipt and expected instalment respectively6 procedure for loss-optimising default definitions across simulated credit risk scenarios at time t 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 i (cid:213) l = t + I il v ( b ) l − t , O ( i , t ) = for t = T 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. Having breached the default threshold (signifying broken trust), the lender’s objective changes tocollecting the maximum in the shortest time possible. However, changing the default threshold also implies avariable workout period between the default time and eventual resolution, i.e., curing from default or write-off. Asa simplifying assumption for this study, the loan is immediately written-off at some rate upon entering ‘default’,regardless of its definition. As such, let r E ∈ [ , ] be a loss rate applied on O ( i , t ) . Moreover, assume that A ( i , t ) is also partly written-off though at another loss rate r A ∈ [ , ] . Using two different rates recognises that therecovery success may differ between these two components (expected balance and arrears). Finally, let l ( i , t ) be thediscounted loss assessed at t and expressed as l ( i , t ) = O ( i , t ) r E + A ( i , t ) r A . (4)In optimising default definitions, let d ≥ be the default threshold such that the i th account is considered as ( g , d ) -defaulting if and only if g ( i , t ) ≥ d at any particular time t = , . . . , T i . Accordingly, let S D be the subset ofall ( g , d ) -defaulting accounts such that S D = (cid:8) i (cid:12)(cid:12) ∃ t ∈ [ , T i ] : g ( i , t ) ≥ d (cid:9) . (5)Since an account may enter and leave the ‘default’ state multiple times in reality, let t ( g , d ) i be the earliest moment ofdefault for the i th ( g , d ) -defaulting 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 i ] (cid:9) . (7)The difference in assessing losses using Eq. 4 between a ( g , d ) -defaulting and a ( g , d ) -performing account is simplythe time of assessment t , at either t = t ( g , d ) i or t = T i respectively. Finally, the discounted total loss L ( g , d ) of a7 procedure for loss-optimising default definitions across simulated credit risk scenarios given matured portfolio using the measure g and a default threshold d , is 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 ( i , T i ) . (8) Fig. 2.
Illustrating the loss optimisation of default thresholds across several delinquency measures. As a result,Measure 3 is chosen as the best measure with its minimum loss attained at point c .Losses can now be iteratively calculated across a range of thresholds d ∈ D g using a particular measure g ∈ (cid:8) g , g , g (cid:9) with the loss model (Eq. 8), thereby forming a loss curve across d for each g . To populate thesethresholds in D g , choose a sufficiently wide range of discrete thresholds d = , . . . , d N using g as a baseline. This d N is arbitrarily chosen as 60% of the contractual term in balancing computation time against the desired width ofthe eventual loss curve. Next, the ranges of real-valued measures g and g are binned into the same number ofthresholds using a combination of equal width discretisation and discretion. In summary, D g contains an equalnumber of thresholds for each measure g in the loss optimisation, which is accompanied by two preparatory steps:1. Delinquency must be measured for every account and across its history using g ∈ (cid:8) g , g , g (cid:9) ;2. A loss model L ( g , d ) must be applied at every relevant default threshold d ∈ D g .Finally, losses are aggregated twice by finding the minimum each time: first by threshold d for g and then by measure g . This forms the basis of the loss optimisation procedure. Specifically, each resulting loss curve – one for eachmeasure g – can first be inspected to find the lowest loss as well as the associated threshold for each minimum loss.Secondly, these minima m ( g ) , i.e., min L ( g , d ) = m ( g ) , coinciding at thresholds d ( g ) , i.e., arg d min L ( g , d ) = d ( g ) ,can then be compared to one another. The optimal measure g ∗ is then the one that yielded the lowest loss m ( g ) atcorresponding threshold d ( g ) , i.e., g ∗ = arg g min m ( g ) , as illustrated in Fig. 2. Note that this procedure can also beused for loss-optimising the threshold using a single measure g .8 procedure for loss-optimising default definitions across simulated credit risk scenarios 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 policies. Alternatively, a simulation-based approach is moreconducive to studying threshold optimisation from ‘first principles’ since a whole range of credit risk profiles andassociated assumptions can be simulated, contrasted by a real-world portfolio’s single profile. Moreover, somedelinquent accounts will simply never recover in reality, which implies a continuous stream of zeros in their 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 truncation parameter k ≥ , this effect is simulated at a certain starting point t (cid:48) = min (cid:0) j : g ( j ) ≥ k (cid:1) that only exists when delinquencyhas accrued sufficiently, i.e., the earliest period j ∈ [ , t c ] at which g ( j ) ≥ k is potentially triggered. A process,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 . (9)For the actual simulation study, consider N = , standard amortising loan accounts that are indexed by i = , . . . , N , with a fixed contractual term of t c = months, a fixed effective annual interest rate of 20%, and afixed principal amount such that the level instalment is I t = at every period t = , . . . , t c . An effective annualrisk-free rate of 7% is used in discounting, which is realistic for the South African market. Let the maximumloan size be L M = , and let r E = and r A = with the rationale that losses on arrears ought to bepenalised more than losses on expected balances. This is due to the latter being a decreasing quantity while theformer increases over time for a continuously delinquent loan.In simulating the receipt vector R of each loan account, three probabilistic techniques are now described. As a basictechnique (called random defaults ), let u t ∈ [ , ] be a randomly generated number at every period t = , . . . , t c andlet b be the probability of payment, i.e., P ( R t = I ) = b with I denoting the level instalment. Note that b = ischosen as a default value, though this is later varied. Each element R t within R is then populated with either I or 0,expressed as R t = I if u t < b otherwise . (10)Despite its simplicity, random defaults do not feasibly generate periods of consecutive non-payments followedby resumed payment, which frequently occurs in practice. Therefore, the so-called episodic defaults technique isalso used wherein p D = is the given probability of default, i.e., half the portfolio is bound to have a defaultepisode by design. Let l j be the number of consecutive non-payments to be simulated for the j th delinquent accountwithin the defaulting-segment. This episode length l j ∈ [ , k ] is sampled from the uniform distribution up to k ,coinciding with ( k , g ) -truncation. When applying ( k , g ) -truncation, accounts will only cure if they had less than k consecutive non-payments, as a limiting condition. Thereafter, the starting point o j ∈ [ , t c − l j ] of the episodeis also sampled from the uniform distribution up to t c − l j , which is to say the entire episode must fit within theremaining loan life. Finally, each element R t within R of the j th delinquent account is then simulated as R t = if o j ≤ t ≤ ( o j + l j ) I otherwise . (11)9 procedure for loss-optimising default definitions across simulated credit risk scenarios Realistically, an account may experience multiple default episodes during its life, though the previous episodictechnique only gives one such episode. Therefore, the
Markovian defaults technique is also defined wherein X t ∈ { P , D } denotes a random variable that can assume one of two states at each period t ; the state P : R t = I (paid) and the state D : R t = (delinquent). Then, let X , X , . . . be a sequence of random variables that form adiscrete-time first-order Markov chain. For simplicity, assume that every accounts starts off in the paying state,which implies that the initial state probabilities are P ( X = P ) = and P ( X = D ) = . Subsequently, the giventransition probabilities between states at future time t + , conditional on the current state at time t , are denoted bythe transition matrix as (cid:34) P PP P PD P DP P DD (cid:35) = (cid:34) P (cid:0) X t + = P (cid:12)(cid:12) X t = P (cid:1) P (cid:0) X t + = D (cid:12)(cid:12) X t = P (cid:1) P (cid:0) X t + = P (cid:12)(cid:12) X t = D (cid:1) P (cid:0) X t + = D (cid:12)(cid:12) X t = D (cid:1) (cid:35) = (cid:34) P PP − P PP − P DD P DD (cid:35) . (12) Following the aforementioned simulation approach, the parameters of each technique are now varied whileoptimising the default thresholds towards the lowest loss. The simulation results are grouped below by technique. (a)
Using ( , g ) -truncation (b) Using ( , g ) -truncation Fig. 3.
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), simulated loans are ( , g ) -truncated, while they are ( , g ) -truncated in (b). Thezoomed plots show that global minima occur at or near the truncation point, d = k , in both cases.In using this technique, ( k , g ) -truncation is applied to control the simulation and to serve as a sanity check. Intuitively,one expects that the lowest loss across default thresholds d to coincide wherever d = k , since simulated receipts arezeroed after having breached k . As an illustration, ( , g ) is first applied in Fig. 3a, which shows the lowest loss tooccur at d = for g . Note that k is arbitrarily set, though varied later. However, the choice of g ∈ (cid:8) g , g , g (cid:9) when applying ( k , g ) -truncation also introduces bias in the timing of these simulated non-payments. Specifically,10 procedure for loss-optimising default definitions across simulated credit risk scenarios the lowest loss (across all curves) is biased towards the curve of the same g used in truncation. As shown in Fig. 3b,when using ( , g ) -truncation instead, minimum loss now occurs approximately at d = k = for g .In general, minimum losses ought to occur wherever d = k when ( k , g ) -truncating simulated receipts. This is largelyconfirmed in Fig. 4 wherein various portfolios are generated in succession using different truncation parameters k = , . . . , . As a result, loss minima occur consistently at the truncation point d = k . Each increasing valueof k also yielded a smaller minimum loss as a result of the overall lessening truncation effect. Since receiptsare truncated less frequently as k increases, portfolios exhibit less delinquency, which explains both lower losscurves and lower loss minima. Although not shown, this behaviour is also true for g and g with associated ( k , g ) -truncation, though minima only occur approximately at d = k ± η with the discrepancy η becoming greateras k increases. Fig. 4.
Losses (as a proportion of summed principals) across thresholds d for the CD -measure g with ( k , g ) -truncation, using the random defaults technique. Several truncation points k = , . . . , are used, with the zoomedplot confirming that global minima in losses occur at each truncation point d = k .The effect of different credit risk profiles during loss optimisation is simulated by varying the parameter b andgenerating portfolios accordingly, as shown in Fig. 5. Keeping ( , g ) -truncation as a benchmark, loss minimaonly occur at d = k = for a certain range of . < b < . . This suggests that loss minima may only exist forcertain risk profiles in practice. Moreover, the two boundary cases of b = and b = in Fig. 5 also serve as areasonableness test in that the loss minimum should occur at d = for b = since all receipts will be zero bydesign. Conversely, if there is no credit risk, i.e., b = , then zero losses should occur across all thresholds d > since all receipts equal instalments. Credit risk scenarios can also be simulated by varying the loss rate r A andgenerating associated portfolios, as shown in Fig. 6 using g (though similar results hold for g and g ). Decreasingvalues of r A flattens the loss curve, which is sensible since arrears are penalised less. Conversely, increasing valuesof r A causes a greater ‘bend’ at the chosen truncation point, while loss minima materialises at the same point onlyfor a certain range of r A . This strengthens the previous result of loss optimisation across default thresholds being11 procedure for loss-optimising default definitions across simulated credit risk scenarios viable only for portfolios with particular risk profiles – not too much risk nor too little risk. Fig. 5.
Losses (as a proportion of summed principals) across thresholds d for the CD -measure g with ( , g ) -truncation, using the random defaults technique and several probabilities of payment b ∈ [ , ] . The zoomed plotshows a smaller range of . ≤ b ≤ . where loss minima occur at the chosen truncation point. Since this technique is tightly coupled with ( k , g ) -truncation by design, portfolios are generated accordingly for k = , . . . , , as shown in Fig. 7 for g (with similar results for g and g ). Clearly, the shapes of loss curves aredifferent, even though loss minima are still found at each successive truncation point. Also note that accountsresume payment provided that the length of their default episode is less than k . Then, longer default episodes (higher k ) seems to absorb the loss specifically introduced by truncation itself, which is signified by flattening loss curvesfor d ≥ k . Since higher k also implies that truncation occurs less frequently, a greater proportion of accounts withdefault episode length less than k will resume payment. In turn, arrears stabilise as truncation becomes less likely,which explains the flattening slopes of loss curves for greater k . In general then, small k implies shorter episodelengths but more truncated accounts, while large k means longer episode lengths but less truncated accounts. Itseems this technique generates portfolios with an interesting trade-off between default episode length and truncationfrequency, regarding their credit risk compositions. Since loss minima are still obtained at each truncation pointas hoped, it suggests that loss-optimising default thresholds will likely remain viable in practice when facingportfolios with more interesting characteristics. This includes portfolios where truly delinquent accounts (as proxyfor truncation) occur more frequently (e.g., unsecured lending), though accounts are also more prone to recoverfrom shorter bouts of delinquency, and vice versa . 12 procedure for loss-optimising default definitions across simulated credit risk scenarios Fig. 6.
Losses (as a proportion of summed principals) across thresholds d for the CD -measure g with ( , g ) -truncation, using the random defaults technique and several arrears loss rates r A ∈ [ , ] . The zoomed plot shows asmaller range of loss rates . ≤ r A ≤ where loss minima occur at the chosen truncation point. Fig. 7.
Losses (as a proportion of summed principals) across thresholds d for the CD -measure g with ( k , g ) -truncation, using the episodic defaults technique with p D = and several truncation points k = , . . . , . Thezoomed plot shows that loss minima occur at each successive truncation point.13 procedure for loss-optimising default definitions across simulated credit risk scenarios Using this technique, various credit risk contexts are simulated by substituting in a range of transition rates P PP ∈ [ , ] , while keeping P DD = constant at first, as shown in Fig. 8a for g (though similar results hold for g and g ). For P PP = , accounts are immediately absorbed into the delinquency state D since P DD = , therebyyielding a minimum loss at d = . Similarly, for P PP = , accounts will never leave the payment state P, whichexplains the zero loss curve across d > . As before, loss minima exist only for a limited range of . ≤ P PP ≤ . at d = , as a proxy for credit risk in practice. Curiously, loss minima at d = amounts to an unintended (andrather strict) ( , g ) -truncation effect due to specifying P DD = . In this case, any laxer truncation using k > willalso be indiscernible in effect due to the superseding absorption of P DD = . However, ( k , g ) -truncation ought tohave a more visible effect when using more transient rates P DD → later. Lastly, as another reasonableness check,the previous results in Fig. 3 can be emulated using Markovian defaults by specifying P PP = . and P DD = . ,whilst using the same ( , g ) and ( , g ) -truncation respectively.The range of rates P PP ∈ [ b L , b U ] wherein loss minima occur at d = depends on P DD , based on the simplepremise that it becomes less taxing to enter the delinquency state D, if one is also more prone to exit it again. Todemonstrate this, three reference values for P DD ∈ { . , . , . } are experimentally chosen as boundary cases.Fixing P DD = . , the range [ b L , b U ] shifts lower and narrows, eventually contracting to a single point as P DD is decreased, until it disappears when P DD < . . Moreover, loss minima at other thresholds d > (withoutapplying ( k , g ) -truncation) also appear for an upper non-overlapping range b U ≤ P PP < when P DD = . . As P DD is decreased, this range widens, e.g., becoming . ≤ P PP < when P DD = . . The Markov chain becomesincreasingly unstable for certain P PP as P DD is decreased since accounts start switching rapidly between states Pand D. Should this oscillation occur relatively early in loan life, then losses are higher owing to high expectedoutstanding balances. As a consequence, global loss maxima can start appearing when P DD ≤ . for certain P PP ,e.g., the range . ≤ P PP ≤ . when P DD = . and no truncation.By applying ( , g ) -truncation and fixing P DD = . , the aforementioned ideas are clearly manifesting in Fig. 8b.Firstly, it shows a diminishing range . < P PP < . wherein loss minima at d = are still found, owing to themajority being absorbed into delinquency. Secondly, an upper range . < P PP ≤ . exhibits loss minima at thechosen truncation point d = k , which is purposefully set high to distinguish from the previous loss minima at d = .For P PP > . , fewer accounts become (or stay) delinquent, which explains both the waning strength of truncationand the flattening loss curve at d ≥ k . Lastly, a small range . < P PP < . also has loss maxima, though therange expands significantly to P PP < . when removing truncation. Although believed to be a simulation artefact,the existence of loss maxima suggests an interesting dilemma wherein greater losses are attained when choosingsome middling d than what would have been the case when choosing either an incredibly conservative threshold(e.g., d = ) or remarkably naive threshold (e.g., d = ). Loss maxima can reasonably occur in practice, e.g.,real-world portfolios undergoing temporary macroeconomic stress, though further research is necessary.14 a) With transition rate P DD = (b) With transition rate P DD = . and ( , g ) -truncation Fig. 8.
Losses (as a proportion of summed principals) across thresholds d for the CD -measure g , using theMarkovian defaults technique with several transition rates P PP ∈ [ , ] . In (a) , P DD = is fixed with loss minimaoccurring for . ≤ P PP ≤ . at d = , highlighted in the zoomed plot. In (b) , P DD = . is fixed with ( , g ) -truncation wherein the zoomed plot shows an even smaller range of rates . < P PP < . where lossminima still occur at d = . Lighter encircled points show rates . < P PP ≤ . where loss minima occur at thechosen truncation point d = k . Darker encircled points show rates . < P PP < . where loss maxima occur atstaggered thresholds. procedure for loss-optimising default definitions across simulated credit risk scenarios Basel II, often promulgated verbatim by a country’s financial regulator, allows the lender reasonable freedom todefine their own default points. Yet many lenders pragmatically opt for the common ‘90 days past due’-definition anduse it accordingly across collection operations, analytics, pricing, and risk modelling, amongst others. Moreover,the financial implications of a particular definition are not readily considered when choosing a default point. Thereis often little objective evidence for a definition’s overall suitability, beyond the questionable results from roll rateanalyses or a regulator’s decree.Therefore, a procedure is presented in this study for optimising the default threshold using profitability (or loss) as itsbase. To facilitate this optimisation, three delinquency measures are formulated (see appendix). Each measure g isapplied on the historical cash flows of all loan accounts within a portfolio. Thereafter, the loss L ( g , d ) is calculatedfor each relevant default threshold d , resulting in a loss curve for each g . A loss curve can then be inspected to findthe threshold at which the lowest loss occurs, which concludes the loss optimisation. This procedure also allows forthe objective comparison and evaluation across multiple delinquency measures, should a lender wish to employ (ortest) alternative measures.In testing this procedure, a simulation study is conducted to generate loan portfolios across a wide range of creditrisk scenarios. Three probabilistic techniques are used to generate cash flows, with each technique aiming forincreased realism over the previous one. A simulation study allows for varying the parameters of each technique,thereby producing different risk profiles and portfolio characteristics. This is particularly useful for testing the lossoptimisation procedure from ‘first principles’. Indeed, the simulation results show that loss minima can exist for aselect range of credit risk profiles, which suggests that loss-optimising default thresholds can be a viable strategyin practice. These results also successfully frames the default decision as an optimisation problem in choosing adefault threshold that is neither too early nor too late in loan life.Future studies can focus on using real-world portfolio data in refining this procedure. Most real portfolios arecensored (or ‘incomplete’) since the majority of loan accounts have not yet reached maturity, excluding written-offand settled accounts. This is a non-trivial challenge since the procedure was developed with ‘completed’ accountsin mind. Furthermore, the particular loss model, as used in the procedure, can be refined and made estimablefrom real-world loss experiences. As an example, the LGD can perhaps be restructured in a way that allows forestimating loss risk conditioned on a particular default definition, amongst other factors. This will intersect with theexisting literature on credit loss modelling, which is currently enjoying greater research focus due to IFRS 9. References
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Three mathematical operations are now presented as delinquency measures and discussed. Firstly, the popularnumber of payments/months in arrears, called the
Contractual Delinquency (or CD -measure g ), is refined intoa more robust measure. Secondly, a more concise algorithm is contributed that creates the Macaulay Duration index-based measure (or MD -measure g ) from Sah (2015), which is an index of the weighted average time torecover the capital portion of a loan. Finally, a modified version of the MD -measure is introduced, called the Degreeof Delinquency (or
DoD -measure g ), which incorporates the sizes of disrupted cash flows in assessing delinquency. CD -measure g As a common measure, the unpaid portion of an amortising loan’s instalment is aged into a few increasingly severebins, given the time elapsed: 30 days, 60 days, 90 days, and so forth, as discussed in Cyert et al. (1962). This isoften converted to the number of payments in arrears (or arrears categories) simply by dividing the accumulatedarrears at a particular point in time with the level instalment, followed by rounding this ratio upwards to an integer.However, this is quite stringent in that even a small difference I t − R t = (cid:15) < will increase the payments in arrears,purely due to rounding. Should the ratio instead be rounded to the nearest integer, then a change in this measuredepends on whether the unrounded ratio is above or below 50%. This implied ‘threshold’ seems arbitrary andtoo fixed. Furthermore, this measure can potentially lag overall measurement when a significant overpayment isimmediately followed by a severe underpayment the following month. Lastly, its construction quickly becomescumbersome when the instalment is prime rate-linked and varies over time, which is common for secured lending.18 procedure for loss-optimising default definitions across simulated credit risk scenarios Therefore, a more comprehensive variant, called the CD -measure, is presented here that circumvents these challenges.Let the receipt vector be R = [ R , R , . . . , R T ] with its elements (or receipts) R t ≥ , and let the instalment vector be I = [ I , I , . . . , I T ] with its elements I t > . Both vectors are defined for a specific loan account across its discretetime periods t = , . . . , T , with t = representing the origination point and T denoting the tenure (or current loanage). Note that T may exceed the contractual term t c , especially in cases of extreme delinquency. The repaymentratio h t ∈ [ , ∞) is then defined as h t = (cid:18) R t I t (cid:19) ∀ t = , . . . , T and h = . (13)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 as an illustration. Next, aBoolean-valued decision function d ( t ) ∈ { , } is defined for t = , . . . , T , using Iverson brackets [ a ] that outputs1 if the enclosed statement a is true, and 0 if false, as d ( t ) = (cid:2) h t < z (cid:3) . (14)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) − . (15)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. 15 signifies the totalnumber of ‘payments’ by which accrued delinquency should be decreased. The floor is taken since g should reflectpayments in arrears and must therefore have a discrete scale. However, the currently owed instalment should berecognised first before reducing any accrued delinquency, by subtracting one instalment. For underpayment, i.e., R t < zI t , the delinquency is 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) . (16)Finally, the reduction in delinquency m ( t ) at time t is subtracted from delinquency measured at the previous period t − , thereby giving the net number of payments in arrears respective to z . The integer-valued CD -measure g ( t ) ≥ for t = , . . . , T is then recursively expressed 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) . (17)Note the necessary starting condition of g ( ) = , since a newly-disbursed loan account cannot yet be delinquent.19 procedure for loss-optimising default definitions across simulated credit risk scenarios MD -measure g Recently introduced in Sah (2015), the Macaulay Duration Index is based on bond duration, i.e., the weightedaverage time to recover the capital portion of a loan. By comparing actual duration to expected duration, itincorporates interest rates and the time value of money of arrears amounts in assessing delinquency. Naturally,its output cannot be compared directly to the previous g since both its scale and meaning differs. To ease theconstruction of g , a new algorithm is presented here.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 of aloan account, including at disbursement t = (to capture any account initiation fees). Considering the time valueof 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 annual compounding period p . Let L P denote the loan amount (or principal) that is to be amortised. While the Macaulay Duration is ordinarilycalculated at origination as the weighted average time to recover cash flows, here it is recursively calculated insteadat each subsequent period t = , . . . , T across the remaining m instalments. Naturally, this expected duration quantity, denoted as f ED ( t ) , tends towards 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 − tp (cid:19) (cid:21) ∀ t = , . . . , T . (18)However, Eq. 18 assumes that instalments I are free of uncertainty. A marked difference is reasonably expectedwhen substituting these instalments with the actual receipts R . Moreover, it becomes necessary to track the arrearsbalance as it develops (if it does) over the loan life. In line with Sah (2015), any arrears at any time are added to thelast expected (contractual) instalment at t = t c , since it represents the last contractual opportunity to repay any sucharrears, short of the lender intervening and restructuring the loan. This last instalment is then recursively updatedfor each subsequent period t , denoted by the vector I (cid:48) , which equals instalments I at first. Likewise, the actualduration f AD ( t ) is also recursively calculated for each subsequent period t . This is illustrated using pseudo-code inAlgorithm 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 the actualduration and the expected duration at time points t = , . . . , T − , which is expressed as g ( t ) = f AD ( t ) f ED ( t ) . (19)20 procedure for loss-optimising default definitions across simulated credit risk scenarios DoD -measure g 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. 19 can be altered such that the eventual g ( t ) is greater for larger loans than forsmaller loans by defining an appropriate multiplier function.Whilst refining g , note that it is only defined up to the contractual term t c . However, delinquency can continueeven past its contractual term T ≥ t c , likely due to persisting underpayment. Ignoring loan write-off policies for themoment, let d ( t ) ∈ { , } be a Boolean-valued decision function that returns 1 if the given time point t precedesthe contractual term t c , and 0 if otherwise. Using Iverson brackets, this is expressed as d ( t ) = (cid:2) t ≤ t c (cid:3) . (20)When t > t c , any arrears can clearly no longer be added to the last contractual instalment (since it has lapsed), as wasadded for I (cid:48) T at T = t when calculating g in Algorithm 1. Instead, at least one more payment, albeit out-of-contract,can reasonably be expected at every subsequent period t : t ≥ t c as long as collection efforts are actively pursued.Therefore, delinquency can now be computed up to time T instead of the previous T , with T either representing thecontractual term t c when t < t c , or becoming a moving target T = t when t ≥ t c . Note that both I and R willincrementally expand with additional elements for as long as collection efforts continues past the contractual term.A revised algorithm is given in Algorithm 2. 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 denote themaximum 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) . (21)21 procedure for loss-optimising default definitions across simulated credit risk scenarios 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) . (22)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) . (23)By including d ( t ) into ˜ f AD ( t ) in Eq. 23, accrued delinquency will not be inflated when overpaying at some period t . Finally, the real-valued Degree of Delinquency ( DoD ) measure g ( t ) ≥ is defined for t = , . . . , T − andexpressed 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) . (24)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