# Journal of Risk Model Validation

**ISSN:**

1753-9579 (print)

1753-9587 (online)

**Editor-in-chief:** Steve Satchell

# Beyond the contract: client behavior from origination to default as the new set of the loss given default risk drivers

####
Need to know

- This paper introduces a new set of client behavior based predictors.
- Parametric and non-parametric models are prepared for the recovery rate estimation.
- Adding client related variables reduces the errors and improve discrimination.
- The effect is more visible for fractional regression rather than regression tree.

####
Abstract

Modeling loss given default has increased in popularity as it has become a crucial parameter for establishing capital buffers under Basel II and III and for calculating the impairment of financial assets under the International Financial Reporting Standard 9. The most recent literature on this topic focuses mainly on estimation methods and less on the variables used to explain the variability in loss given default. In this paper, we expand this part of the modeling process by constructing a set of client-behavior-based predictors that can be used to construct more precise models, and we investigate the economic justifications empirically to examine their potential usage. The main novelty introduced in this paper is the connection between loss given default and the behavior of the contract owner, not just the contract itself. This approach results in the reduction of the values of selected error measures and progressively improves the forecasting ability. The effect is more visible in a parametric method (fractional regression) than in a nonparametric method (regression tree). Our findings support incorporating client-oriented information into loss given default models.

####
Introduction

## 1 Introduction and literature review

Loss given default (LGD) estimation causes many methodological and calculation problems, including the bimodal distribution (see Loterman et al 2012; Yao et al 2017), the need for a wide observation window,^{1}^{1} 1 Some recoveries can last eight years or more. the inclusion of partial recoveries, and difficulties in identifying proper predictors. Current studies mainly focus on finding new estimation methods, which can be more precise due to nonstandard statistical procedures, including the recently exploited two-stage modeling approach. In terms of applying new techniques, studies by Qi and Zhao (2011) and Brown (2012) may serve as an example: Brown (2012) compared methods such as ordinary least squares (OLS), beta regression, regression trees, least squares support vector machines (SVMs) and neural networks; Qi and Zhao (2011) focused on fractional response regression, inverse Gaussian regression, regression trees and neural networks. In addition, interesting approaches were adopted by Luo and Shevchenko (2013), who used the Markov chain Monte Carlo method, and Witzany et al (2012), who attempted to use survival time analysis techniques, in particular the proportional Cox model and its modifications, for LGD estimation. Alternatively, Loterman et al (2012) used two-stage modeling, which is becoming increasingly popular, by combining logistic regression with neural networks, OLS with regression trees and OLS with least squares SVMs, as well as many other methods.

The main issue in these types of models is the need to distinguish extreme LGD values from the middle range of the distribution. Yao et al (2017) investigated the problem in terms of classification (high and low values of LGD) and regression (values between high and low). They used the least squares support vector classifier and a set of regression methods (OLS, fractional response regression, etc). The same framework was also suggested by Gürtler and Hibbeln (2013), who tested a hypothesis about the superiority of a two-step model over direct LGD regression in terms of the coefficient of the determination measure. Based on these studies, as well as the work of Huang and Oosterlee (2011), Liu and Xin (2014) and Nazemi and Fabozzi (2018), we see a switch from canonical methods, such as OLS or historical averaging, to regression trees, SVM or beta regression. These new LGD practices can help with the bimodal LGD distribution either by being more flexible (eg, using beta regression or regression trees) or by dividing it into regions in which the estimation will give better results (two-stage models).

Less importance is attached to the issue of finding appropriate predictors that can better explain the LGD variance and improve the forecasting ability.^{2}^{2} 2 Corporate bonds were studied in Schuermann (2004), and small and medium-sized enterprise segments were studied in Chalupka and Kopecsni (2008). According to the Basel Committee on Banking Supervision (2017, p. 30), institutions can consider the factors related to transactions, obligors and institutions (in terms of an organization’s recovery processes or legal frameworks). Significant attention has been paid to choosing only the meaningful differentiating risk factors of transactions. Most of the latest studies focusing on retail banking consider contract-driven information as the predictors.^{3}^{3} 3 See Table 2 for a comparison.

There are three main sources of information when the set of predictors is to be established in LGD modeling (Ozdemir and Miu 2009, p. 17).

- (1)
Contract: loan-to-value (LTV), exposure at default (EAD), loan term, etc.

- (2)
Client: previous default indicator, employment status, monthly income, etc.

- (3)
Macroeconomics: house price index (HPI), consumer price index (CPI), etc.

The main contribution of our study is that it investigates the potential sources of risk driven by consumer behavior after credit is granted. Behavioral data is widely used in other areas of risk management, such as probability of default (PD) estimation (see Izzi et al 2012, p. 63; West 2000) or fraud detection (see Kovach and Ruggiero 2011; Fiore et al 2019). However, our study is, to the best of our knowledge, the first to add such wide-ranging data to the LGD model. We used transactional data, application data (about credit but also the full spectrum of banking services) and bank–client relation data to create new risk drivers, all with economic justification explicitly given and checked using out-of-sample data to confirm robustness. Our data set originated from a major Polish bank, where online communication (via a personal computer or a cell phone) between the bank and its customers is the primary contact channel. This allowed us to state that the fulfillment of the variables based on internet activity is sufficient^{4}^{4} 4 This is not always ensured in the case of LGD models, where at least a five-year time series is needed to perform an estimation. and covers the complete business cycle.^{5}^{5} 5 In line with the Basel II regulations (Basel Committee on Banking Supervision 2005, p. 65).

We checked whether the inclusion of the predictors describing the contract owner’s behavior leads to an increase in the precision and discrimination of LGD estimates. The research on estimation methods employs increasingly complicated structures, in which the appropriateness of, and consistency with, collection and recovery policies can be difficult to demonstrate (as requested in Basel Committee on Banking Supervision (2017, p. 74) and discussed in academia (see, for example, Martens et al 2011)). Thus, expanding the scope of information seems to be a reasonable choice to provide the best estimates possible. While a comprehensive approach to clients in LGD estimation is not widely recognized, it can be especially important in the nonmortgage loan (NML) segment, because recovery rate (RR) variability cannot be described by security possession. Assuming the new risk drivers have a significant impact on the LGD estimates, we evaluated whether parametric or nonparametric methods gave a larger increase in precision and discrimination.

The remainder of this paper is structured as follows. In Section 2, we propose the set of predictors with a calculation method and a link to LGD. In Section 3, we discuss champion and challenger approaches and goodness-of-fit measures. In Section 4, we estimate a model with standard explanatory variables as a champion approach. In Section 5, we discuss the challenger models with one or more new variables and goodness-of-fit measures to compare all variants. In Section 6, we present our conclusions.

## 2 Data and a description of the variables

Considering various ways of determining a dependent variable (Basel Committee on Banking Supervision 2005, p. 4), let LGD be defined as

$$\mathrm{LGD}=1-\mathrm{RR}=1-\sum _{t=1}^{n}\frac{{\mathrm{CF}}_{t}}{{(1+d)}^{t}}\left(\frac{1}{\mathrm{EAD}}\right).$$ |

This is a so-called workout approach, where the LGD is determined as 1 minus the sum of the discounted cashflows divided by the exposure at default (EAD) (Anolli et al 2013, p. 92).

A data set of NMLs used in the analysis (with the LGD meeting the definition above) was provided by a major Polish bank and consists of around 135 000 observations from May 2011 to December 2017. All the defaults come from nonsecured portfolios containing credit cards, cash loans and revolving loans given to private individuals or small and medium-sized enterprises (SMEs). Complete and incomplete cases were included in the reference data set, as the mean workout period was relatively long (30 months for complete defaults) (see Tanoue et al 2017). In such circumstances, the removal of incomplete defaults could lead to sample selection bias.

To estimate the partial RR for incomplete defaults from the moment of data origination until the end of the default, we used an approach presented in Starosta (2020). From the given methods, we chose the fractional regression model, which was the best approach reported for NMLs. In a second step, we divided the sample into 10 subsamples (based on the time-in-default characteristic) and calculated the partial RR from the start of the interval until the end of the default. Finally, we estimated the models and assigned predicted values to the actual values.

Variable | Minimum | Q1 | Mean | Q3 | Maximum |
---|---|---|---|---|---|

RR for closed cases | 0.0000 | 0.2372 | 0.6764 | 0.9980 | 1.0000 |

RR for full sample | 0.0000 | 0.2471 | 0.5527 | 0.9528 | 1.0000 |

The structure of the sample is different after including the incomplete cases, as presented in Table 1. This is mainly the result of the long-lasting defaults characterized by low recoveries, where the litigation process is still in progress. Removal of these cases could cause a serious underestimation of the LGD (Gürtler and Hibbeln 2013). The final distribution of the RR is presented in Figure 1.

As a benchmark, we estimated the model using the following standard set of predictors:

- •
type of product (credit card, cash loan, revolving loan);

- •
interest rate;

- •
requested amount;

- •
time on the books;

- •
principal amount;

- •
interest amount;

- •
EAD;

- •
tenor;

- •
decreasing installment indicator;

- •
second applicant indicator;

- •
length of relationship;

- •
client age.

The predictors in the above list generally coincide with the features from studies on retail banking in which LTV, EAD, time on the books and loan size are the most common risk drivers. Table 2 summarizes these studies. We mainly considered contract-driven indicators in our study, but we also included some client-based indicators. At this stage, macroeconomic variables were not included in any version of the model. This decision was driven mainly by the fact that there was no serious downturn period in the sample window for the Polish market.

Goodness-of-fit measures | ||||
---|---|---|---|---|

Study | Main risk drivers | ${?}^{\text{?}}$ | MAE | RMSE |

Leow (2010) |
LTV; Previous default indicator; Time on books; Security indicator; Age of property; Region of property |
0.233–0.268 | 0.101–0.121 | 0.158–0.161 |

Brown (2012) |
Age of exposure; Months in arrears; Loan amount; Application score; Joint application indicator; Time at bank; Residual status; Employment status; LTV; Time on books; Loan term; EAD; Utilization |
$-$0.695–0.497 | 0.034–0.431 | 0.122–0.693 |

Anolli et al (2013) |
Geographic area; Income; Age; Marital status; Vintage of contract; Duration; Installments; Delinquency; LTV; Gross Domestic Product; Unemployment rate; Saving ratio; HPI |
N/A | N/A | N/A |

Tong et al (2013) |
EAD; Loan amount; Property valuation; HPI; Time on books; Debt to value; Previous default indicator; Second applicant indicator; Loan term; Property age; Security type; Geographical region |
0.298–0.338 | N/A | N/A |

Zhang and Thomas (2012) |
Employment status;Mortgage indicator; Visa card indicator; Insurance indicator; Number of dependents; Personal loan account indicator; Residential status; Loan term; Loan purpose; Time at address; Time with bank; Time in occupation; Monthly expenditure; Monthly income |
0.029–0.107 | 0.352–0.408 | 0.406–0.493 |

Thomas et al (2010) |
Number of months in arrears; Application score; Loan amount; Time until default of the loan |
0.083–0.227 | N/A | N/A |

Yao et al (2017) |
Loan term; Time on books; Sum of transactions across all current accounts; Month in arrears; EAD; Delinquency status; Number of payments made; Most recent payment received; HPI; CPI |
0.069–0.647 | 0.154–0.318 | 0.211–0.356 |

As shown in the next section, we extended the basic set of predictors by including new factors that are not widely used in RR estimation but still have an economic link with a modeled phenomenon. We mainly focus on the client-based indicators, as there is almost nothing to add to the contract-driven indicators. We distinguished three main data sources, also mentioned in the credit scoring literature (see Anderson 2007, p. 275): transactional data, behavior on credit accounts and applications for other bank products.

### 2.1 Transactional data

The first new risk driver (cash_dep_acc) is the amount that the owner of the defaulted contract has in any other deposit accounts. Here, the link to the RR is straightforward. Having a “financial cushion” in the form of liquid assets such as cash makes it easier to return to a nondefault portfolio in a time of financial distress. In this scenario, the client’s motivation to use these funds plays a significant role; the client’s decision can be driven by the term of the deposit and the penalty for breaking it. Potentially, the expected benefits from longer deposits, which are more difficult to withdraw in the short term, could be greater than the benefits from repaying a loan. Although the size of the impact must be measured, the general rule is that the final RR increases with the amount of cash available on the deposit accounts. Variables can be determined for different time horizons. In the basic version, this is the sum of the balances of the deposit accounts at the time of the default. In a more complicated alternative, this could be the mean balance from the last $N$ months before the default.

The number of log-ins (n_login) is the second variable analyzed. If a financial institution has developed an effective internet banking system, at some point this becomes the easiest channel of communication for its clients. Nowadays, functionalities such as fully automatic credit analysis, chats/video chats or brokerage account management can be accessed via the internet without leaving home. Many clients use the internet as their main form of communication; they only go to a bank branch as a last resort. It is also useful for banks to know the best way to reach their clients. The hypothesis concerning this point is about higher RRs for clients who use internet banking more often, as they can be reached more easily by debt collection departments in the case of any financial difficulties. Similarly to cash_dep_acc, n_login can be checked at the time of default or during the last $N$ months.

### 2.2 Behavior on credit products

There is less information on NMLs than secured loans; thus, every piece of data should be considered as a potential factor. We therefore examined the connection between the contract owners and the financial institution. It should be a positive signal for the RR that an NML owner also has a mortgage loan (ML). The motivation related to repaying secured credit, where a house or a car is set as collateral, is significantly different from the case where a client has nothing tangible to lose. Empirical studies have confirmed that RR values are higher for secured credit than for nonsecured credit (Basel Committee on Banking Supervision 2005, p. 74), which leads to the hypothesis that possession of an ML contract by at least one owner leads to an increased RR when the NML contract is analyzed:

$$\text{ML\_indic}=\{\begin{array}{cc}1\hfill & \text{if at least one contract owner has ML},\hfill \\ 0\hfill & \text{otherwise}.\hfill \end{array}$$ |

In a more conservative version, in order to assign an $\text{ML\_indic}=1$ every owner has to have an ML contract.

Tenor and time on the books are widely known LGD predictors, and they are often used in studies. However, when a financial institution has many different tenors on offer, this could lead to a discrepancy in estimating the LGD for contracts with low tenors when most of a sample consists of long tenors, and vice versa. One way to deal with this situation is to use the credit life cycle phase instead of nominal tenor and time on the books:

$$\text{credit\_life\_cycle}=\frac{\text{time on the books}}{\text{tenor}}.$$ |

Time on the books should always lie in the interval $[0;1]$, and it should have a simple interpretation in almost all cases. If the value of a variable is larger, is should be possible to obtain a greater recovery. It is assumed that the clients’ motivation to repay credit is higher at the end of a schedule than at the beginning, when the perspective of getting rid of a financial burden is farther away. Thus, instead of taking the contract perspective expressed by the length of credit, the variable is presented from a client’s perspective and is expressed by how many months are left on the loan or how close the client is to full repayment.

Finally, we checked the information about delinquencies on any client contract in the selected historical period. Stating the past due amount on the analyzed contract as well as the other products owned can lead to lower RRs in comparison with clients with a clean delinquency history. However, if another contract is past due, then the collection department can take care of the rest of the client contract earlier. Thus, depending on the collection policy, a positive impact can also be valid. We define material delinquency as 30 days past due on an amount higher than 1% of EAD:

$$\text{n\_delinq}=\frac{\text{number of days with delinquency}}{\text{number of days possessing credit product within last year}}.$$ |

This variable can also be examined in the selected time period.

### 2.3 Requesting other bank products

During the life of a credit, the client’s cooperation with the bank can develop in different directions. Other credit products can be granted (or requested but not granted), an application for deposit products can be filled out or insurance products can be requested.^{6}^{6} 6 Cooperation between banks and insurance companies is getting closer, and insurance connected to repayment in the case of an unexpected event is now a standard add-on to any credit product. All these events can affect the LGD in various ways, so we examined each of them with the use of binary variables, defined as follows:

insurance | $=\{\begin{array}{cc}1\hfill & \text{if client bought insurance after initial credit was}\hfill \\ \hfill & \text{granted},\hfill \\ 0\hfill & \text{otherwise},\hfill \end{array}$ | ||

next_credit_granted | $=\{\begin{array}{cc}1\hfill & \text{if client got another loan after initial credit was}\hfill \\ \hfill & \text{granted},\hfill \\ 0\hfill & \text{otherwise},\hfill \end{array}$ | ||

next_credit_app | $=\{\begin{array}{cc}1\hfill & \text{if client applied for another loan after initial credit}\hfill \\ \hfill & \text{was granted},\hfill \\ 0\hfill & \text{otherwise},\hfill \end{array}$ | ||

deposit | $=\{\begin{array}{cc}1\hfill & \text{if client put a deposit down after initial credit was}\hfill \\ \hfill & \text{granted},\hfill \\ 0\hfill & \text{otherwise}.\hfill \end{array}$ |

The above features were selected to focus on expanding the potential determinants of LGD in the following directions:

- •
client behavior on a deposit account (managing inflows and outflows, saving propensity);

- •
degree of relationship with the bank (log-ons, but also channels of communication, possession of mobile applications, etc);

- •
product structure (requests for other products, both credit and insurance);

- •
seeking new relationships in core variables (inverting the perspective toward the client).

In our study, at least one new proposition from the list presented above is formulated to check the potential of each area. The results could suggest the most promising area of investigation for future research.

## 3 Methodology

The champion and challenger approaches are used to evaluate performance or to determine a set of benchmarking values (Apeh et al 2014). In our study, the champion method corresponds to the most effective model built with a standard set of explanatory variables. Estimation occurs via two methods.

The first method is fractional regression (see Belotti and Crook 2009; Bastos 2010), which is viewed as being a good benchmark for more complicated methods, but also as having the desired properties and giving interpretable results that are preferred from a regulatory point of view:

$$E({\mathrm{RR}}_{i}\mid {?}_{i})=\frac{1}{[1+\mathrm{exp}(-{?}_{i}?)]},$$ |

where $i=1,\mathrm{\dots},n$, $n$ is the sample size, $?$ are model parameters and ${?}_{i}$ is a vector of the explanatory variables for case $i$. In the estimation process, forward, backward and stepwise methods are used with a $p$-value of less than 0.05 as the criterion for adding a variable. The maximum value of the correlation between the final set of predictors is fixed at 80%. The variable with the higher loglikelihood is ultimately used, and the second one is removed.

A second method is the regression tree method (see Bastos 2010; Qi and Zhao 2011), a nonparametric approach that, in theory, better reflects nonlinear dependencies, such as in the LGD case. The interpretation is also straightforward, as this model results in a set of rules (binary splits) that can be shown as a combination of “if–else” statements. However, the method has some drawbacks, such as the potential instability connected to changes in the population or overfitting (Hastie et al 2008). It is necessary to take great care when tuning hyperparameters of a tree to limit these two issues. In this study, we implemented a standard set of tuning rules (see Qi and Zhao 2011; Nazemi and Fabozzi 2018).

- (1)
Select analysis of variance as the splitting selection method (applicable for continuous variables).

- (2)
Determine the complexity parameter based on tenfold cross-validation.

- (3)
Calculate the value of the minimum observations in a leaf in the manner proposed by Israel (1992):

$$n=\frac{{z}^{2}{\sigma}^{2}}{{\epsilon}^{2}},$$ where $z$ is the abscissa of the normal curve that cuts an area, $\epsilon $ is the desired level of precision at the tails and ${\sigma}^{2}$ is the variance of the LGD. The value obtained is rounded up to the nearest integer.

- (4)
Assign a value of $5$ to the hyperparameter to achieve stable and intuitive results. This parametrization should lead to the tree, where the maximum depth value should not be binding.

Each model is checked for precision and discrimination using the methods described in Table 3 on a holdout sample consisting of 30% of the total. The remaining 70% is used to train the model. We chose two methods – root mean square error (RMSE) and mean absolute error (MAE) – to evaluate precision, and two methods – the Gini index and the cumulative LGD accuracy ratio (CLAR) – to assess discrimination. Each of these proved useful in previous studies.

Measure | Calculation | Usage |
---|---|---|

RMSE | $\sqrt{{\displaystyle \frac{\text{1}}{n}}{\displaystyle \sum _{i=\text{1}}^{n}}{({\text{LGD}}_{i}-{\widehat{\text{LGD}}}_{i})}^{\text{2}}}$ | Bastos (2010) |

MAE | $\frac{\text{1}}{n}}{\displaystyle \sum _{i=\text{1}}^{n}}|{\text{LGD}}_{i}-{\widehat{\text{LGD}}}_{i}|$ | Tanoue et al (2017) |

Gini index${}^{*}$ | $\text{2AUROC}-\text{1}$ | Zhang and Thomas (2012) |

CLAR | Bucketing predicted and realized LGD | Ozdemir and Miu (2009) |

The precision values should be as low as possible, and the discrimination values should be as high as possible. The calculation of the RMSE, MAE and CLAR does not cause a major problem. However, in the case of the Gini coefficient we need to make a clarification: because a variable is expected to be binary, we divided each observation into two occurrences, and we assigned 1 to the first occurrence and 0 to the second. We then calculated the weighted Gini index, where the weight for the first case is the value of the $\mathrm{RR}$ and the weight for the second case is $1-\mathrm{RR}$. As there are no official benchmarks for any of the selected measures, we compared each model with all the others for all four measures. Additionally, we checked the increase in performance compared with the naive classifier expressed as the average value from the sample.

## 4 Champion models: empirical results

### 4.1 Fractional regression

In each approach, we treated the RR as a dependent variable for the estimations. The final form of the fractional regression model is presented in Table 4.

Standard | |||
---|---|---|---|

Parameter | Estimate | error | $?$ $\mathbf{(}\mathbf{>}{?}^{\text{?}}\mathbf{)}$ |

Intercept | 0.6254 | 0.0269 | $$0.0001 |

EAD (in thousands) | $-$0.0038 | 0.0012 | 0.0051 |

Length of relationship | 0.00296 | 0.0002 | $$0.0001 |

Client age | $-$0.0081 | 0.0006 | $$0.0001 |

Requested amount | 0.0099 | 0.0009 | $$0.0001 |

(in thousands) | |||

Tenor | $-$0.0077 | 0.0003 | $$0.0001 |

Interest amount | $-$0.0007 | 0.0001 | $$0.0001 |

(in thousands) |

From the table, we can draw some straightforward conclusions related to the estimates of the parameters. Both the length of relationship and the requested amount positively influenced RR. The longer a customer stays with the bank, the higher the proportion of debt that will be recovered after default, which is in line with the findings reported in Tong et al (2013) and Yao et al (2017). The same can be stated for the requested amount, where a positive sign was reported in Brown (2012) (in the same study, the sign was negative for another data set). This may be connected with the EAD estimate, which is negative, as reported by Tong et al (2013) and Tanoue et al (2017): high EAD values result in fewer recoveries. The model aims to differentiate cases with a high requested amount and a high EAD (there is still a high due amount, so there are lower recoveries) from a high requested amount and a low EAD (where the greater part of the outstanding credit has already been repaid or limit usage was low, so there is a chance to recover more, as the client could be more willing to complete the repayment). At this point, a new variable, such as the EAD divided by the requested amount, which can connect the two values mentioned above, may be useful. However, to compare the full specification, we did not consider this. Then, there is client age, which decreases the RR by 0.81% every year. To some extent, the same conclusion can be found in Belotti and Crook (2009), where the impact was estimated at the 0.346% level. The interest amount is the last variable included in the model; it has a negative influence on the RR.

### 4.2 Regression trees

The second model, based on regression trees, was tuned as follows:

- •
the minimum number of observations in a leaf was set equal to 181;

- •
the number of cross-validations was set equal to 10;

- •
the complexity parameter was set equal to 0.0006348369l;

- •
the maximum depth was set equal to 5.

In comparison with the fractional regression model, four additional variables are used to construct a tree; this is mainly due to the inclusion of nonlinear dependencies and no further assumptions being made about the correlation between the predictors. Table 5 shows the importance of each predictor; similar to the fractional regression method, the interest amount is one of the strongest predictors. Unsurprisingly, the variables connected to principal and interest, and those derived from these two variables, have the greatest impact on the RR prediction.

Variable | Importance |
---|---|

Interest amount | 659.3 |

Principal amount | 612.9 |

EAD | 584.7 |

Requested amount | 310.3 |

Interest rate | 271.2 |

Tenor | 242.1 |

Length of relationship | 134.8 |

Months on book | 49.4 |

Client age | 5.9 |

Decreasing installment | 2.5 |

indicator |

### 4.3 Comparison of the champion approaches

At this point, we analyzed the predictive accuracy to establish benchmarks that the challenger approaches need to beat.

Fractional | Regression | |
---|---|---|

Measure | regression | tree |

RMSE | 0.33242 | 0.32002 |

MAE | 0.29378 | 0.27700 |

Gini (%) | 20.08 | 26.01 |

CLAR | 0.7338 | 0.7233 |

The results suggest that both methods perform well on this particular data set (Table 6). The selected metrics are slightly in favor of the regression tree, but we are not neglecting any method at this point. Globally, the performance indicators are not very different from those reported in similar studies on nonmortgage products (see, for example, Yao et al 2017). As previously mentioned, there is a body of literature about other sophisticated methods, but we did not focus on finding the best statistical model. Thus, further research may be needed to obtain more insight about LGD modeling for unsecured loans, both here and when using client behavior variables.

## 5 Challenger approach: empirical results

In this section, we assess the performance of the newly created variables, first individually, then in terms of their interactions. In the first path, the data set is reestimated, and extended one variable at a time. This allows us to determine whether a specific risk driver is relevant in the RR modeling process. Second, the entire set of variables, both contract-level and client-level, is included in the estimation to verify the hypothesis of a potential relationship between the RR and the contract owner behavior.

(a) Fractional regression | ||||
---|---|---|---|---|

Variable | RMSE | MAE | Gini (%) | CLAR |

cash_dep_acc | 0.33142 | 0.29273 | 20.79 | 0.7354 |

n_login | 0.33074 | 0.29270 | 21.94 | 0.7372 |

ML_indic | 0.33194 | 0.29292 | 20.34 | 0.7315 |

credit_life_cycle | 0.32807 | 0.28905 | 23.73 | 0.7417 |

n_delinq | 0.33177 | 0.29295 | 20.61 | 0.7354 |

next_credit_granted | 0.33232 | 0.29353 | 20.42 | 0.7344 |

next_credit_app | 0.33219 | 0.29329 | 20.60 | 0.7349 |

deposit | 0.33234 | 0.29355 | 19.81 | 0.7319 |

insurance | 0.33171 | 0.29212 | 20.92 | 0.7361 |

(b) Regression tree | ||||

Variable | RMSE | MAE | Gini (%) | CLAR |

cash_dep_acc | 0.31982 | 0.27667 | 25.89 | 0.7189 |

n_login | 0.32002 | 0.27700 | 26.01 | 0.7233 |

ML_indic | 0.32002 | 0.27700 | 26.01 | 0.7233 |

credit_life_cycle | 0.31796 | 0.27499 | 27.87 | 0.7275 |

n_delinq | 0.32035 | 0.27758 | 25.69 | 0.7311 |

next_credit_granted | 0.32022 | 0.27512 | 26.12 | 0.7579 |

next_credit_app | 0.32033 | 0.27487 | 26.06 | 0.7416 |

deposit | 0.31978 | 0.27659 | 26.08 | 0.7241 |

insurance | 0.31978 | 0.27659 | 26.08 | 0.7241 |

The strongest influence, confirmed by six of the eight measures in both the parametric and nonparametric models, is exerted by credit_life_cycle. The precision of the model was increased by 0.00473 in the case of MAE, and by 3.65 percentage points in the case of Gini. This result suggests that inverting the perspective to a client view can elicit new insight from the raw contract data. The sign is strongly positive, which is compliant with the general assumption that being closer to full repayment has a positive effect on the RR. Another “changing perspective” variable is the share of EAD in a requested amount (or limit usage in the case of credit lines/credit cards), which determines the profile of the repayment pattern, such as decreasing installments, prepayments or high-volume usage. Each new variable achieves statistical significance at a 0.05 $p$-value level when analyzed separately. In the regression tree model, some of the variables (eg, the number of log-ins or the ML indicator) were not used in any division.

Standard | |||
---|---|---|---|

Parameter | Estimate | error | $?$ $\mathbf{(}\mathbf{>}{?}^{\text{?}}\mathbf{)}$ |

Intercept | $-$0.2587 | 0.0437 | $$0.0001 |

EAD (in thousands) | $-$0.0036 | 0.0004 | $$0.0001 |

Length of relationship | 0.0030 | 0.0002 | $$0.0001 |

Client age | $-$0.0094 | 0.0007 | $$0.0001 |

Tenor | 0.0033 | 0.0004 | $$0.0001 |

Months on book | $-$0.0045 | 0.0003 | $$0.0001 |

Sum of cash on deposit | 0.0041 | 0.0003 | $$0.0001 |

accounts (in thousands) | |||

ML indicator | 0.4546 | 0.0346 | $$0.0001 |

Credit life cycle | 1.5062 | 0.0426 | $$0.0001 |

Number of log-ins | $-$0.0206 | 0.0012 | $$0.0001 |

Delinquencies | 0.2042 | 0.0273 | $$0.0001 |

Insurance | $-$0.1617 | 0.0144 | $$0.0001 |

Deposit | $-$0.1588 | 0.0237 | $$0.0001 |

In the next step of the challenger approach, we examine the estimation of the entire training set using both base and new variables as predictors. In the fractional regression model, the base specification was used and all the new variables were added. Because the correlation coefficient between the indicators next_credit_granted and next_credit_approved was found to be 92%, the variable with the higher loglikelihood was used. The assumptions discussed in Section 3 hold for a regression tree.

Seven of the eight new variables were found to be statistically significant, which supports the hypothesis that the contract owner behavior is connected to the RR level (see Table 8). The same can be stated for the regression tree model (Table 9).

Considering the signs of the parameters, three of the four assumed directions hold; for the next three parameters, this direction is arguable. Being close to the end of the credit term, with all other factors unchanged, has a positive effect on the RR, which is a desirable property, as we expected it to work in the assumed direction. Taking into account the sum of cash on deposit accounts, we also found a positive coefficient sign, which confirms the hypothesis presented in Section 2. We used the average amount from the last three months before default in our study, but the analysis could definitely be widened to a six- or even a twelve-month window to select the best predictor. We should also consider that, in the selected portfolio, some clients may only have a credit product at a financial institution (with no debit account). It appears that the average RR for such clients can be found among clients with a positive amount of cash in their deposit accounts.

The third and fourth variables discussed are the number of log-ins and the ML indicator. Regarding the ML indicator, undoubtedly having an ML has a positive effect on the RR. However, the number of log-ins reveals a different behavior than expected. At the beginning, there is indeed a positive relationship, but a negative relationship develops as the number of log-ins increases. This suggests that, at some point, the client may try to stabilize their situation by logging in frequently, but the RR is not pushed in the desired direction. In this case, a change in the nonlinear specification can be considered, such as adding the squared version of a particular variable. Nevertheless, this behavior can be treated as unexpected, and it should be more fully investigated in further studies. Delinquencies on the other contracts possessed by the client have a positive effect on the RR, which could be due to the collection policy, as mentioned in Section 2.

The parameter sign for the last two variables is negative, but we did not make any initial assumptions about their influence, which implies the need to confirm the direction in further research.

Variable | Importance |
---|---|

Principal amount | 616.3 |

EAD | 614.5 |

Credit life cycle | 555.2 |

Interest amount | 526.9 |

Tenor | 282.2 |

Requested amount | 190.3 |

Indicator of next | 170.7 |

credit application | |

Indicator of next | 164.9 |

credit granted | |

Sum of cash on | 150.7 |

deposit accounts | |

Months on book | 148.9 |

Insurance | 94.7 |

Number of log-ins | 37.3 |

Length of relationship | 19.7 |

Delinquencies | 18.6 |

Decreasing | 1.1 |

installment | |

indicator | |

Client age | 0.1 |

In the regression tree model, the new variables have less influence, as the principal amount and EAD still play a major role. However, the credit life cycle is presented as one of the strongest risk drivers, and the indicator of the next credit application is in the top half of all the variables in Table 9. Finally, the information presented in Table 10 demonstrates the performance measures using behavioral characteristics.

The selected measures indicate that the new specification performs well. However, this could still be the result of incorporating more independent variables, so more research is required. To check the robustness of these metrics, a three-step validation was performed.

- (1)
Calculate the Akaike information criterion (AIC) and the Bayesian information criterion (BIC), which take the number of parameters into account and penalize it. As the regression tree is a nonparametric estimation method, we used the following equations to calculate the selected measures (Kuhn and Johnson 2013):

$\mathrm{AIC}$ $=n\mathrm{log}(\mathrm{RSS})+\alpha \text{number\_of\_leaves},$ $\mathrm{}\mathrm{BIC}$ $=n\mathrm{log}(\mathrm{RSS})+\mathrm{log}(n)\text{number\_of\_leaves},$ where RSS denotes the residual sum square.

- (2)
Estimate a naive classifier expressed as a mean and calculate the relative change from this classifier to the champion model and to the challenger model for error measures, where

$$\text{relative change}=\frac{\text{measure}-{\text{measure}}_{\text{reference}}}{{\text{measure}}_{\text{reference}}}.$$ - (3)
Perform the estimation only on new variables to check whether the relative change between this model and the naive classifier is at least as good as the change between the champion model and the naive classifier.

Fractional | Regression | |
---|---|---|

Measure | regression | tree |

RMSE | 0.32790 | 0.31746 |

MAE | 0.28845 | 0.27332 |

Gini (%) | 23.89 | 28.52 |

CLAR | 0.7426 | 0.7931 |

Measure | Champion | Challenger |
---|---|---|

AIC fractional regression | 128 203.74 | 126 198.68 |

BIC fractional regression | 128 274.18 | 126 329.42 |

AIC regression tree | $-$92 938.17 | $-$93 585.30 |

BIC regression tree | $-$95 852.01 | $-$93 456.06 |

Only | Only | |||||

Champion | Champion | Challenger | Challenger | new | new | |

Measure | FR | RT | FR | RT | FR | RT |

RMSE | $-$3.10 | $-$6.71 | $-$4.41 | $-$7.46 | $-$3.80 | $-$4.38 |

MAE | $-$3.39 | $-$8.91 | $-$5.14 | $-$10.12 | $-$4.20 | $-$5.09 |

CLAR | 22.14 | 20.39 | 23.60 | 32.01 | 22.79 | 23.44 |

The results in Tables 11 and 12 suggest that each approach performs better than a sample mean, but the degree of goodness-of-fit is wide. Taking fractional regression into consideration, the challenger model can be characterized by a material upgrade, and adding new variables significantly boosts the precision and discrimination. The robustness of this approach is confirmed by the AIC, the BIC and the out-of-sample precision. Moreover, only selecting the newly created variables seems to be better than using the champion approach (for example, a $-3.80$ versus a $-3.10$ gain on RMSE). This could stem from a greater linear dependency between the client-related variables than between the contract-related variables for changes in the RR, which is one of the assumptions of the fractional regression model. In addition, the AIC and the BIC confirm a better fit for the model with the new predictors, which allows us to say that the second model outperforms the base model, given the different specifications. These findings are slightly less obvious for the regression tree model. Even if our interpretation of the AIC and the BIC leads to the same conclusion as for the fractional regression, the relative change between the champion and challenger approaches is a little smaller, as is the influence of the new variables. In this situation, the gain in performance measures is clearly smaller for the “only new” approach than for the champion approach. We can argue that, taking into consideration this particular data set, the regression tree benefited from nonlinearity in the predictors, which reduced information gain from the client-oriented predictors. The same holds in terms of the discrimination measure, where “only new” is less effective than the champion approach but is still substantially better than a sample mean.

We believe that much more information, from the client, not just the contract, could be used for the LGD/RR estimation. Different recovery patterns can be seen for clients that are self-employed versus clients that are employed full time, or those that have already repaid some of their other obligations in their credit history versus new borrowers. More transactional data (such as the amount of inflows or payment patterns) or geolocation data (indicating changing jobs) can be adopted. Even information that is already available can be useful when used in a new manner, such as a share of EAD in the requested amount or the dynamics of log-ins, not just the mean. However, when creating new predictors, we should always consider their usefulness for the PD models, so the correlation between PD and LGD can be properly reflected in the capital requirements calculation.

## 6 Conclusion

The main aim of this study was to establish a connection between LGD and a new set of contract owner-oriented variables. We based the study on a large sample of NMLs from a Polish bank with an online communication system as a primary client contact channel. Evaluation of their performance shows that these new predictors are an effective supplement to the standard predictors, improving the LGD precision.

First, two techniques were applied to build champion models based on a standard set of predictors (fractional regression and regression tree), and then the performance of each new variable was checked and the estimation was performed on the entire set of variables. The regression tree method performed better than the fractional regression method in terms of the selected measures in both the champion and challenger approaches.

Second, adding client-based variables significantly reduced the error measures and increased discrimination in comparison with the champion approach; this upgrade is greater when using the fractional regression method than the regression tree method.

Third, in comparison with a naive classifier, such as the mean, client-based variables can have a significant influence on LGD precision. Moreover, in the fractional regression method, the impact of the client-based variables is the same as that of the contract variables. For the regression tree method, the impact of the client-based variables is half that of the contract variables.

We conclude that incorporating information about the contract owner’s behaviors plays a crucial role in the predictive accuracy of LGD modeling, and great care should be taken when choosing an appropriate estimation method.

## Declaration of interest

The author reports no conflicts of interest. The author alone is responsible for the content and writing of the paper.

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