The zero-inflated promotion cure rate model applied to financial data on time-to-default

Figures & data

Figure 1. Survival function of the zero-inflated cure rate model as presented in Louzada, Oliveira, and Moreira (2015).

Figure 2. Bias, square root of mean squared error and coverage probability (CP) of the maximum likelihood estimation $({\widehat{\mathit{\beta }}}_{10}$, ${\widehat{\mathit{\beta }}}_{11}$, ${\widehat{\mathit{\beta }}}_{20}$, ${\widehat{\mathit{\beta }}}_{21})$ of zero-inflated promotion cure rate regression model for simulated data under the three scenarios of parameters, obtained from Monte Carlo simulations with 1,000 replications and increasing sample size (n). 1 indicates scenario 1 with characteristic of having a low rate of STD and non-default. 2 indicates the scenario 2 with characteristic of having a moderate rate of STD and non-default. 3 indicates scenario 3 with a characteristic of having a high rate of STD and non-default.

Figure 3. Bias, square root of mean squared error and coverage probability (CP) of the maximum likelihood estimation $({\widehat{\mathit{\beta }}}_{30}$, ${\widehat{\mathit{\beta }}}_{31}$, ${\widehat{\mathit{\beta }}}_{40}$, ${\widehat{\mathit{\beta }}}_{41})$ of zero-inflated promotion cure rate regression model for simulated data under the three scenarios of parameters, obtained from Monte Carlo simulations with 1,000 replications and increasing sample size (n). 1 indicates the scenario 1 with characteristic of having a low rate of STD and non-default. 2 indicates the scenario 2 with characteristic of having a moderate rate of STD and non-default. 3 indicates scenario 3 with characteristic of having a high rate of STD and non-default.

Figure 4. MLEA, maximum likelihood estimation on average of the parameters $({\widehat{\mathit{\beta }}}_{10}$, ${\widehat{\mathit{\beta }}}_{11}$, ${\widehat{\mathit{\beta }}}_{20}$, ${\widehat{\mathit{\beta }}}_{21}$, ${\widehat{\mathit{\beta }}}_{30}$, ${\widehat{\mathit{\beta }}}_{31}$, ${\widehat{\mathit{\beta }}}_{40})$, ${\widehat{\mathit{\beta }}}_{41}$ of zero-inflated Promotion Cure rate regression model for simulated data under the three scenarios of parameters, obtained from Monte Carlo simulations with 1,000 replications and increasing sample size (n). 1 indicates the scenario 1 with characteristic of having a low rate of STD and non-default. 2 indicates scenario 2 with characteristic of having a moderate rate of STD and non-default. 3 indicates the scenario 3 with characteristic of having a high rate of STD and non-default.

Table 1. Frequency and percentage of the bank loan lifetime data

Number of customers	Number of STD ($T=0)$	Number of defaulters $(T>0)$	Number of censored $(T=\infty )$
5,733	321 (5.60%)	810 (14.13%)	4,602 (80.27%)

Table 2. Quantity of the available covariates

Covariate	Quantity of customers
Age group 1	503
Age group 2	3,088
Age group 3	1,220
Age group 4	922
Type of residence 1	629
Type of residence 2	4,056
Type of residence 3	998
Type of residence 4	50
Type of employment 1	956
Type of employment 2	4,777

Figure 5. Brazilian bank loan portfolio data.

Notes: Top panel, shows a histogram for the observed time-to-default variable of interest (left) and Kaplan–Meier survival curves stratified by age group (right). Bottom panel, Kaplan–Meier survival curves stratified by type of residence (left) and Kaplan–Meier survival curves stratified by type of employment (right).

Table 3. The zero-inflated promotion cure regression model for time to default on a Brazilian bank loan portfolio

Parameter	Dummy covariate (param${}^{(1)}$)	Estimate	S.E.${}^{(2)}$	p-value	Exp.${}^{(3)}$
$p_0$	Intercept $({\mathit{\beta }}_{10})$	$-0.6690$	0.1213	0.0000	0.5122
	Age group =4 $({\mathit{\beta }}_{11})$	$-0.8187$	0.2231	0.0002	0.4409
	Type of residence = 4 $({\mathit{\beta }}_{12})$	0.8653	0.4736	0.0677	2.3757
	Type of employment = 2 $({\mathit{\beta }}_{13})$	$-0.6473$	0.1434	0.0000	0.5234
$p_1$	Intercept $({\mathit{\beta }}_{20})$	0.9123	0.0957	0.0000	2.4901
	Type of residence = 1 $({\mathit{\beta }}_{21})$	$-0.2905$	0.1028	0.0047	0.7478
	Type of employment = 2 $({\mathit{\beta }}_{22})$	0.6331	0.0970	0.0000	1.8834
${\mathit{\alpha }}_1$	Intercept $({\mathit{\beta }}_{30})$	0.1730	0.0376	0.0000	1.1889
${\mathit{\alpha }}_2$	Intercept $({\mathit{\beta }}_{40})$	3.1855	0.0697	0.0000	24.1817
	Age group = 4 $({\mathit{\beta }}_{41})$	0.6895	0.1435	0.0000	1.9927

${}^1$ Related regression parameter to be estimated; ${}^2$ Standard error; ${}^3$ Exponentiation of the estimated parameter.

Figure 6. Brazilian bank loan portfolio. Kaplan–Meier survival curves stratified through the covariate selection given by the final promotion cure rate regression model presented in the Table 3.

Louzada, F., Oliveira, M. R., & Moreira, F. F. (2015). The zero-inflated cure rate regression model: Applications to fraud detection in bank loan portfolios. arXiv preprint arXiv:1509.05244.

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