Beta

Create Beta model object for loss given default

Since R2022b

Description

Create and analyze a Beta model object to calculate loss given default (LGD) using this workflow:

1. Use fitLGDModel to create a Beta model object.

2. Use predict to predict the LGD.

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

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

Creation

Syntax

BetaLGDModel = fitLGDModel(data,ModelType)

BetaLGDModel = fitLGDModel(___,Name=Value)

Description

BetaLGDModel = fitLGDModel(data,ModelType) creates a Beta LGD model object.

BetaLGDModel = fitLGDModel(___,Name=Value) specifies options using one or more name-value arguments in addition to the input arguments in the previous syntax. The optional name-value arguments set the model object properties. For example, BetaLGDModel = fitLGDModel(data,'Beta',PredictorVars={'LTV' 'Age' 'Type'},ResponseVar="LGD",BoundaryTolerance=1e-4,WeightsVar="Weights") creates a BetaLGDModel object using a Beta model type.

Input Arguments

data — Data for loss given default

table

Data for loss given default, specified as a table.

Data Types: table

ModelType — Model type

string with value "Beta" | character vector with value 'Beta'

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.

Example: BetaLGDModel = fitLGDModel(data,'Beta',PredictorVars={'LTV' 'Age' 'Type'},ResponseVar="LGD",BoundaryTolerance=1e-4)

ModelID — User-defined model ID

"Beta" (default) | string | character vector

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

Description — User-defined description for model

"" (default) | string | character vector

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

Data Types: string | char

PredictorVars — Predictor variables

all columns of data except for ResponseVar (default) | string array | cell array of character vectors

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

ResponseVar — Response variable

last column of data (default) | string | character vector

Response variable, specified as ResponseVar and a string or character vector. The response variable contains the LGD data and must be a numeric variable. An LGD value of 0 indicates no loss (full recovery), 1 indicates total loss (no recovery), and values between 0 and 1 indicate a partial loss. By default, ResponseVar is set to the last column.

Data Types: string | char

BoundaryTolerance — Value to perturb LGD responses

1e-5 (default) | positive numeric

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

Data Types: double

SolverOptions — optimoptions object

object

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

WeightsVar — Column name containing weights

"" (default) | string scalar

Column name of the input table containing weights, specified as a string scalar.

Note

The default value ("") results in a weight of 1 for each row in data. All weight values in data must be nonnegative.

For an example using WeightsVar, see Create Weighted LGD Model.

Properties

ModelID — User-defined model ID

Beta (default) | string

User-defined model ID, returned as a string.

Data Types: string

Description — User-defined description

"" (default) | string

User-defined description, returned as a string.

Data Types: string

UnderlyingModel — Underlying statistical model

Read-only: compact linear model

This property is read-only.

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

PredictorVars — Predictor variables

all columns of data except for the ResponseVar (default) | string array

Predictor variables, returned as a string array.

Data Types: string

ResponseVar — Response variable

last column of data (default) | string

Response variable, returned as a string.

Data Types: string

BoundaryTolerance — Value to perturb LGD responses

1e-5 (default) | positive numeric

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

Data Types: double

WeightsVar — Column name containing weights

"" (default) | string scalar

Column name of the input table containing weights, returned as a string scalar. This property is also used to determine the weights variable for validation data when you use the modelDiscrimination or modelCalibration functions.

Object Functions

predict	Predict loss given default
modelDiscrimination	Compute AUROC and ROC data
modelDiscriminationPlot	Plot ROC curve
modelCalibration	Compute R-square, RMSE, correlation, and sample mean error of predicted and observed LGDs
modelCalibrationPlot	Scatter plot of predicted and observed LGDs

Examples

Create Beta LGD Model

This example shows how to use fitLGDModel to create a Beta model object for loss given default (LGD).

Load LGD Data

Load the LGD data.

load LGDData.mat
head(data)

      LTV        Age         Type           LGD   
    _______    _______    ___________    _________

    0.89101    0.39716    residential     0.032659
    0.70176     2.0939    residential      0.43564
    0.72078     2.7948    residential    0.0064766
    0.37013      1.237    residential     0.007947
    0.36492     2.5818    residential            0
      0.796     1.5957    residential      0.14572
    0.60203     1.1599    residential     0.025688
    0.92005    0.50253    investment      0.063182

rng('default');
NumObs = height(data);
c = cvpartition(NumObs,'HoldOut',0.4);
TrainingInd = training(c);
TestInd = test(c);

Create Beta LGD Model

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

BetaLGDModel = fitLGDModel(data(TrainingInd,:),'Beta',...
   ModelID='Example LGD Beta',...
   PredictorVars={'LTV' 'Age' 'Type'},...
   ResponseVar='LGD',...
   BoundaryTolerance=2e-05);
disp(BetaLGDModel)

  Beta with properties:

    BoundaryTolerance: 2.0000e-05
              ModelID: "Example LGD Beta"
          Description: ""
      UnderlyingModel: [1×1 risk.internal.credit.BetaModel]
        PredictorVars: ["LTV"    "Age"    "Type"]
          ResponseVar: "LGD"
           WeightsVar: ""

Display the underlying model.

disp(BetaLGDModel.UnderlyingModel)

Beta regression model:
     logit(LGD) ~ 1_mu + LTV_mu + Age_mu + Type_mu
     log(LGD) ~ 1_phi + LTV_phi + Age_phi + Type_phi

Estimated coefficients:
                           Estimate       SE        tStat       pValue  
                           ________    ________    _______    __________

    (Intercept)_mu          -1.3884     0.13117    -10.585             0
    LTV_mu                  0.61403     0.14989     4.0967    4.3513e-05
    Age_mu                 -0.47446    0.039808    -11.919             0
    Type_investment_mu       0.4516    0.084719     5.3305    1.0846e-07
    (Intercept)_phi          -0.113     0.12578    -0.8984       0.36908
    LTV_phi                0.030903     0.14692    0.21034       0.83343
    Age_phi                 0.23972    0.040152     5.9703     2.775e-09
    Type_investment_phi    -0.14724    0.078187    -1.8831       0.05982

Number of observations: 2093
Log-likelihood: -4976.51

Predict LGD

For Beta models, use predict to calculate the predicted LGD value, which is the unconditional expected value of the response, given the predictor values.

predictedLGD = predict(BetaLGDModel,data(TestInd,:))

predictedLGD = 1394×1

    0.0935
    0.1483
    0.3520
    0.0963
    0.1879
    0.2587
    0.2671
    0.0212
    0.1768
    0.2249
    0.1334
    0.1670
    0.2905
    0.1611
    0.1047
      ⋮

Validate LGD Model

Use modelDiscriminationPlot to plot the ROC curve.

modelDiscriminationPlot(BetaLGDModel,data(TestInd,:))

Use modelCalibrationPlot to show a scatter plot of the predictions.

modelCalibrationPlot(BetaLGDModel,data(TestInd,:))

More About

Beta Regression Models

Beta regression models model continuous variables that assume values in the open standard unit interval (0, 1).

The beta regression model is based on the assumption that the dependent variable is beta-distributed and that its mean is related to a set of regressors through a linear predictor with unknown coefficients and a link function. The beta regression model also includes a precision parameter that may be constant or depend on a potentially different set of regressors through a link function as well.

The beta regression model is based on an alternative parameterization of the beta density in terms of the mean and a precision parameter: μ = a / (a + b) and φ = a + b, 0 < μ < 1, φ > 0.

$$\begin{array}{l}f(y;\mu ,\varphi )=\frac{\Gamma (\varphi )}{\Gamma (\mu \varphi )\Gamma ((1-\mu )\varphi )}{y}^{\mu \varphi -1}{(}^1,\text{ 0<}y\text{<1,}\\ y\text{~B(}\mu \text{,}\phi \text{) with E(}y\text{)=}\mu ,\text{ VAR(}\phi \text{)=}\mu \text{(1-}\mu \text{)/(1+}\phi \text{)}\end{array}$$

The beta regression model is defined as

$$\begin{array}{l}{g}_1({\mu }_i)={\eta }_{1i}=x_i^T\beta ,\\ g_2({\varphi }_i)={\eta }_{2i}=z_i^T\gamma ,\end{array}$$

where β = (β₀,β₁,…,β_k)^T, ᵞ = (ᵞ₀,ᵞ₁,…,ᵞ_k)^T are the regression coefficients, η_1i, η_2i are the linear predictors, and x_i, z_i are the independent variables. g₁ and g₂ are the link functions where g₁ is logit and g₂ is log.

Parameter estimation is performed by maximum likelihood (ML).

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.

Version History

Introduced in R2022b

R2024a: Added WeightsVar name-value argument for Beta model

The Beta model supports a WeightsVar name-value argument for observation weights.

R2023a: modelAccuracy object function is renamed to modelCalibration function

The modelAccuracy object function is renamed to modelCalibration function. The use of modelAccuracy is discouraged, use modelCalibration instead.

R2023a: modelAccuracyPlot object function is renamed to modelCalibrationPlot function

The modelAccuracyPlot object function is renamed to modelCalibrationPlot function. The use of modelAccuracyPlot is discouraged, use modelCalibrationPlot instead.