# Beta

Create Beta model object for exposure at default

## Description

Create and analyze a Beta model object to calculate the exposure at default (EAD) using this workflow:

1. Use fitEADModel to create a Beta model object.

2. Use predict to predict the EAD.

3. Use modelDiscrimination to return AUROC and ROC data. You can plot the results using modelDiscriminationPlot.

4. Use modelAccuracy to return the R-squared, RMSE, correlation, and sample mean error of predicted and observed EAD data. You can plot the results using modelAccuracyPlot.

## Creation

example

example

### Input Arguments

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Data for exposure at default, specified as a table.

Data Types: table

Model type, specified as a string with the value of "Beta" or a character vector with the value of 'Beta'.

Data Types: char | string

Name-Value Arguments

Specify optional pairs of arguments as Name1=Value1,...,NameN=ValueN, where Name is the argument name and Value is the corresponding value. Name-value arguments must appear after other arguments, but the order of the pairs does not matter.

User-defined model ID, specified as ModelID and a string or character vector. The software uses the ModelID text to format outputs and is expected to be short.

Data Types: string | char

User-defined description for model, specified as Description and a string or character vector.

Data Types: string | char

Predictor variables, specified as PredictorVars and a string array or cell array of character vectors. PredictorVars indicates which columns in the data input contain the predictor information. By default, PredictorVars is set to all the columns in the data input except for ResponseVar.

Data Types: string | cell

Response variable, specified as ResponseVar and a string or character vector. The response variable contains the EAD data and must be a numeric variable. By default, ResponseVar is set to the last column.

Data Types: string | char

Value to perturb EAD response values away from 0 to 1, specified as BoundaryTolerance and a positive scalar numeric.

Data Types: double

Limit variable, specified as LimitVar and a string or character vector. LimitVar indicates which column in data contains the limit amount. The limit amount value in the data must be a positive numeric value. The limit depends on the loan. If the loan is a credit card, the limit is the credit limit. If the loan is a mortgage, the limit is the initial loan amount. In general, LimitVar is the maximum amount that can be borrowed.

Note

LimitVar is required when ConversionMeasure is 'lcf'. For more information on LCF, see Conversion Measure Options.

Data Types: string | char

Drawn variable, specified as DrawnVar and a string or character vector. DrawnVar is the balance on the account at the time of observation, before default, and EAD is the balance at the time of default. DrawnVar indicates which column in data contains the drawn amount. The drawn variable value in the data can be a positive or negative numeric value.

Note

When the ConversionMeasure is 'lcf', DrawnVar is not required. In this case, DrawnVar is set to "".

Data Types: string | char

Response transform, specified as ConversionMeasure and a character vector or string. Limit conversion factor (LCF) is a fraction of the limit representing the total exposure. The EAD is then defined as the LCF times the limit (EAD = LCF*Limit).

Data Types: string | char

Options for fitting, specified as SolverOptions and an optimoptions object that is created using optimoptions from Optimization Toolbox™. The defaults for the optimoptions object are:

• "Display""none"

• "Algorithm""quasi-newton"

• "MaxFunctionEvaluations"500 ✕ Number of model coefficients

• "MaxIterations" — 1000

Note

When using optimoptions with a Beta model, specify the SolverName as fminunc.

The number of Beta model coefficients is determined at run time, depending on the number of predictors and the number of categories in the categorical predictors.

Data Types: object

## Properties

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User-defined model ID, returned as a string.

Data Types: string

User-defined description, returned as a string.

Data Types: string

Underlying statistical model, returned as a compact linear model object. The compact version of the underlying regression model is an instance of the risk.internal.credit.BetaModel class.

Data Types: object

Predictor variables, returned as a string array.

Data Types: string

Response variable, returned as a string.

Data Types: string

Limit variable, returned as a string.

Data Types: string

Drawn variable, returned as a string.

Data Types: string

Response transform, returned as a string.

Data Types: string

Value to perturb LGD response values away from 0 to 1, returned as a positive scalar numeric.

Data Types: double

## Object Functions

 predict Predict exposure at default modelDiscrimination Compute AUROC and ROC data modelDiscriminationPlot Plot ROC curve modelAccuracy Compute R-square, RMSE, correlation, and sample mean error of predicted and observed EADs modelAccuracyPlot Scatter plot of predicted and observed EADs

## Examples

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This example shows how to use fitEADModel to create a Beta model object for exposure at default (EAD).

UtilizationRate    Age     Marriage        Limit         Drawn          EAD
_______________    ___    ___________    __________    __________    __________

0.24359        25     not married         44776         10907         44740
0.96946        44     not married    2.1405e+05    2.0751e+05         40678
0        40     married        1.6581e+05             0    1.6567e+05
0.53242        38     not married    1.7375e+05         92506        1593.5
0.2583        30     not married         26258        6782.5        54.175
0.17039        54     married        1.7357e+05         29575        576.69
0.18586        27     not married         19590          3641        998.49
0.85372        42     not married    2.0712e+05    1.7682e+05    1.6454e+05
rng('default');
c = cvpartition(NumObs,'HoldOut',0.4);
TrainingInd = training(c);
TestInd = test(c);

Select Model Type

Select a model type for Beta.

ModelType = "Beta";

Select Conversion Measure

Select a conversion measure for the EAD response values.

ConversionMeasure = "LCF";

Use fitEADModel to create a Beta model object using the TrainingInd data.

Beta with properties:

BoundaryTolerance: 2.0000e-05
ModelID: "Beta"
Description: ""
UnderlyingModel: [1x1 risk.internal.credit.BetaModel]
PredictorVars: ["UtilizationRate"    "Age"    "Marriage"]
LimitVar: "Limit"
DrawnVar: ""
ConversionMeasure: "lcf"

Display the underlying model. The underlying model's response variable is the transformation of the EAD response data. Use the 'LimitVar' and 'DrawnVar' name-value arguments to modify the transformation.

Beta regression model:
logit(EAD_lcf) ~ 1_mu + UtilizationRate_mu + Age_mu + Marriage_mu
log(EAD_lcf) ~ 1_phi + UtilizationRate_phi + Age_phi + Marriage_phi

Estimated coefficients:
Estimate        SE          tStat        pValue
__________    _________    _________    __________

(Intercept)_mu                -0.68477       0.1145      -5.9807    2.5236e-09
UtilizationRate_mu              1.7029     0.077717       21.912             0
Age_mu                       -0.005633    0.0027489      -2.0492      0.040542
Marriage_not married_mu      -0.025614     0.051927     -0.49328       0.62186
(Intercept)_phi               -0.46429     0.095342      -4.8697    1.1837e-06
UtilizationRate_phi            0.41621      0.06701       6.2112    6.0944e-10
Age_phi                      -0.001282    0.0023261     -0.55112        0.5816
Marriage_not married_phi    0.00014903     0.042884    0.0034752       0.99723

Number of observations: 2627
Log-likelihood: -2931.19

EAD prediction operates on the underlying compact statistical model and then transforms the predicted values back to the EAD scale. You can specify the predict function with different options for the 'ModelLevel' name-vale argument.

105 ×

0.1758
0.1029
0.1528
0.0832
0.3261
0.5148
0.0648
0.0531
0.0712
0.3215
⋮

For model validation, use modelDiscrimination, modelDiscriminationPlot, modelAccuracy, and modelAccuracyPlot.

Use modelDiscrimination and then modelDiscriminationPlot to plot the ROC curve.

ModelLevel = "ConversionMeasure";

Use modelAccuracy and then modelAccuracyPlot to show a scatter plot of the predictions.

YData = "Observed";

Plot a histogram of observed EAD with respect to the predicted EAD.

figure;
histogram(AccData1.Observed);
hold on;
histogram(AccData1.(('Predicted_' + ModelType)));
legend('Observed','Predicted');

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## References

[1] Baesens, Bart, Daniel Roesch, and Harald Scheule. Credit Risk Analytics: Measurement Techniques, Applications, and Examples in SAS. Wiley, 2016.

[2] Bellini, Tiziano. IFRS 9 and CECL Credit Risk Modelling and Validation: A Practical Guide with Examples Worked in R and SAS. San Diego, CA: Elsevier, 2019.

[3] Brown, Iain. Developing Credit Risk Models Using SAS Enterprise Miner and SAS/STAT: Theory and Applications. SAS Institute, 2014.

[4] Roesch, Daniel and Harald Scheule. Deep Credit Risk. Independently published, 2020.

## Version History

Introduced in R2022b