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Survival Analysis Workflow assessing the impact of Macro-Economic Shocks on Credit Portfolios and predicting the Time of Default (By Anupam Saha, Software Specialist – Analytics, SAS R&D (India) Pvt. Ltd., and Naeem Siddiqi, Senior Solutions Specialist, SAS Institute Inc.)

Introduction

The recent global credit crisis has reiterated the need for bankers to focus on macro-economic factors (such as inflation, house price index, and so on) that severely impact the default behavior of credit portfolios. At the micro level, the response to these economic changes varies with individual capacity and tolerance. For example, although accounts within the same credit score or rating group show homogeneous behavior in default performance, the individual borrowers respond to macroeconomic shocks in a different manner than others in the same segment. This results in scattered default distributions within the predefined performance window. Because defaults are scattered across time, assessing the macro economic impact on a credit portfolio is a challenging task.

In credit scoring, the traditional probability-of-default (PD) model is used to estimate the individual likelihood to default. This involves generating a sample of new or existing accounts at one point in time and monitoring the performance of those accounts across a specified performance window to determine their default or non-default status. It is generally assumed that the probability of default is constant through the entire performance window, and the outcome of the model is a probability of default value (PD) that is valid for the full window. Because the defaults are scattered across time in the performance window, PD varies with respect to time within the same window. Along with the PD, the time of default can be a key piece of information for bankers, allowing them to create better account management strategies. The traditional methodology makes it difficult to calculate PD by different time points and the time of default within the specified performance window.

In this paper, we propose a survival analysis workflow for two purposes:

• to observe the impact of macro-economic shocks on default behavior of credit exposures

• to predict the time of default

We intend to use traditional PD models in conjunction with a survival analysis technique (explained later in this paper) for overcoming the challenges associated with traditional PD models. In this paper, we explain a survival model called “Cox Proportional Hazard Model” that can be used to establish a relationship between macro-economic variables and the hazard

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function (that is, the rate of change of probability of default at time t) at different time points in a observation window. The survival distribution of default can be estimated by this model. The survival function is used to determine the time of default. Further analysis is then performed to obtain the expected time-dependent default count distribution per segment. Finally, variances of macro-economic factors are used to stress-test the credit portfolio.

This paper also gives insights into the data requirements, SAS procedures for survival analysis, and the techniques to validate and monitor the workflow for accuracy and consistency.

Benefits of Using Survival Analysis in Credit Scoring

1. Modelers in banks are aware of the fact that macro-economic variables affect probabilities of default that are generated using models. These macro-economic variables can change at any time during the observation period. However, such variables, and in particular, the leading indicators, have not been included in the traditional logistic regression models used in banking. The impact of macro-economic variables on the PD value can be gauged by considering those variables as time-dependent covariates in the survival analysis model.

The survival function of default as well as the time of default can be derived from that survival model. Time of default here refers to the survival time of a particular account. Survival time is the time from when the account was opened to the date of default. If the account does not default by the end of the observation period, that observation can be considered as a censored event.

For analyzing the survival data, the hazard function gives the rate of change of probability of default at time t. ℎ( ) = lim∆ → ( ∆ | )∆

The probability of survival at time t can be given in terms of the hazard function: ( ) = exp{− ℎ( ) }

This is the probability of survival at time t. In credit scoring scenario, this is the probability that an account has not defaulted by time t after the account was opened (that is, 1-PD at time t). If the interval censoring is considered, then this indicates the probability that an account has not defaulted by time t after the start of the observation period.

There are several ways to generate the hazard function. We will use the Cox Proportional Hazard (Cox PH) model because it allows the inclusion of macro-economic variables as time-dependent covariates in the model. By Cox PH model,

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h (t) = λ (t) exp ( β X (t) + β X (t) + … β X (t))

where,

i is a subscript for observation i=1(1)n. h (t) = Hazard of the ith individual at time t.

λ0 (t) = Baseline hazard function. X (t) … … X (t) are the k time-dependent covariates of the ith individual at time t. β …..β are the regression parameters associated with the covariates, estimated by the Partial Likelihood method in Cox model.

For applying Cox PH model in the credit scoring environment, macro-economic variables should be captured at account default date, prior to default date (any lag period can be considered), at account open date, and at regular time intervals during the observation period. The change in macro-economic values between the account open date and the default date or prior to the default date (considering any lag period, for example, 6 months, 3 months, and so on) can also be considered as a time-dependent covariate in the model. The time of default and the default indicators in the observation window should also be captured. The survival probabilities for each of the individuals at failure time points can be estimated by using the baseline survival function and the parameter estimates β of β by maximum likelihood estimation and the macro-economic values at those time points. If a user changes the macro-economic values, the user can get the different survival probabilities of those accounts at default time points. In this way, the Cox PH model can help to see the impact of those variables on the probability of default.

2. Survival analysis can be used to determine the probability of default across time in an observation window. It is also possible to derive the survival function of default for future time points. This represents an advantage over the traditional PD-estimation methods, which limit the prediction to a predefined observation window. By generating PDs across time, a risk manager can adjust account treatment strategies according to an account’s risk level at that point in time, rather than rely on a single, long-term PD forecast that might not represent the account’s current risk levels.

Figure 1 shows an example of the probability of default across time in an observation window, as generated by using survival analysis. Figure 2 shows an example of the survival function of default across time in that observation window. Both the graphs are based on the same sample data. From Figure 1 and Figure 2, it is evident that as time passes, the probability of default increases, and the corresponding survival function decreases. At the early stages of the performance window, the change in the macro-economic values might not be significant to

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In addition to the macro-economic variables above, demographic (postal code, residential status, age, and so on) and behavioral variables (account-related variables) can also be considered. If the segmentation was based on these variables, then for the Survival model, only the macro-economic variables should be considered. The default time of each defaulted account should be captured in the modeling data. If the account does not default in the outcome period, then the entire outcome period should be considered as survival time for that account. The default indicator should also be considered in the modeling data. Macro-economic variables should be captured at account default date, prior to default date (any lag period can be considered), at account open date, and at regular time intervals during the observation period. The change in macro-economic values between the account open date and the default date or prior to the default date (considering any lag period, for example, 6 months, 3 months, and so on) can also be considered as time-dependent covariate in the model. For the survival model, interval censoring can be considered. If the defaults are captured for a specific time period, then the censoring technique can be considered as interval censoring. Table 2 shows the structure of a sample modeling analytical base table (ABT) for using survival analysis in a credit-scoring environment. Consider a sample data consisting of the following: 1. ID (account identification number) 2. Time (time of default of the account in a predefined observation period) 3. Default (censoring status where 1 = default and 0 = censored) 4. Consider that a time-dependent covariate is captured at default dates cov1 – cov6 (a covariate value at six default times in the observation period). covi denotes a covariate value at the ith time and i=1(1)6. For instance, the first account defaults at time3. Therefore, up to time 3, the covariate values have been captured. The second account defaults at time 4. Therefore, up to time 4, the covariate values have been captured. The third account does not default in the outcome period. Therefore, up to time 6, its covariate values can be captured. The third account will be considered a censored event.

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Table2:

Account id time default cov1 cov2 cov3 cov4 cov5 cov6

1 3 1 <value> <value> <value> 2 4 1 <value> <value> <value> <value> 3 6 0 <value> <value> <value> <value> <value> <value>

Study the impact of macro-economic variables on default behavior (using Cox model by PROC PHREG)

Many types of models have been used for survival analysis. In this paper, the Cox proportional hazard model has been mentioned. In SAS, the PHREG procedure performs regression analysis of the survival data based on the Cox proportional hazard model.

1. To handle time-dependent covariate in PHREG procedure, the counting process style of input is used. The advantage of this format that user can easily get the residual and influence statistics. In this format, for each record, multiple records are crated, one record for each distinct pattern of the time-dependent measurements. Each record contains a T1 value and a T2 value, representing the time interval T1and T2 during which the values of the explanatory variables remain unchanged. Each record also contains the censoring status at T2.

Following is an example of a modeling data set that is created in counting process style of input format. PHREG can be run on this type of data set.

Table 3:

Account id Time default T1 T2 status covariate

1 3 1 0 1 0 <value> 1 3 1 1 2 0 <value> 1 3 1 2 3 1 <value>

2. Consider that we have a scoring data set, and we want to find the predicted survival probability of default across time for the accounts in that data set.

Table 4:

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Account id time covariate

1 1 <value> 1 2 <value> 1 3 <value> 1 4 <value>

PROC PHREG can be run on the modeling data set (Table 3) and can generate survival probability for the accounts in the scoring data set (Table 4). Following is a sample SAS code for generating survival probabilities.

Data Zerocov; covariate= 0; Run;

proc sort data = scoredata; by time; run;

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retain lastS 1;

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16

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17

( ℎ ℎ − ℎ )

Where, n is the total number of accounts in the data.

The absolute difference of the actual and the predicted time of default can be considered. The discriminatory power of a time of default model gets better as the above measure gets closer to zero, indicating a closer match between the predicted and actual values.

Some other statistics (for example, Brier score) can also be generated in this scenario. A non-parametric test, such as median test, can also be performed to compare the median actual time of default with the median predicted time of default.

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Acknowledgements

I would like to thank Mrs. Billie Anderson, Mr. Ying So, Mr. David Park, Mr. Rajesh Khanwelkar, and the SAS Credit Scoring for Banking team.

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