Probability of Default Model using SAS proc logistic

1. Loan Portfolio Manufacturing SME’s – Statistical Analysis
Manzar Ahmed
Advanced Credit Risk Management Coursework
1. Introduction
This paper will consider 310 manufacturing SMEs from a European country that constitutes a loan portfolio for
data collected in 2014.
The objective of this paper is to analyse the data in order to fit a probability of default (PD) model that
describes the relationship between the categorical response variable - default (0 = non-default and 1 = default)
and a set of predictor variables.
2. Statistical Analysis Tool
The loan portfolio data will be analysed using SAS University Edition, the tool is available for free to download
from the below link.
http://www.sas.com/en_gb/software/university-edition.html
The raw loans portfolio data has been uploaded into the SAS VM environment (via Jupyter Notebook) into the
/folders/myfolders/CreditRisk.
Please refer to appendix B of this document for the upload script as well as the scripts used for the analysis
and model in this paper.
3. Default Data
The loans portfolio data is contained in the SAS library CRDR called Default_Data.
Column Name
Type
Format
Description Example
Counterparty_Id Num 4. A key that uniquely identifies each counterparty in order of the data in the raw file.
Assigned using the automatic variable _N_ by SAS.
1
Default Num 1. Binary value 0 or 1 to indicate default. 1 = Default 0
Supplier_Target_Years Num 5.2 A temporal measure of financial sustainability expressed in years that considers all
short and medium term debts as well as other payables
0.36
Outside_Capital_Structure Num 5.2 This ratio evaluates a firm's capability to attract forms of financing other than
banks' loans. The higher the better
0.18
Cash_Ratio Num 5.2 The proportion of cash a company can generate in relation to its size 0.70
Capital_Tied_Up Num 5.2 Turnover of short term debts with respect to sales 0.12
Equity_Ratio Num 5.2 Financial leverage measure that divides equities by total assets 0.89
Cost_Income_Ratio Num 5.2 Efficiency measure that tells us how costs are changing compared to income 0.32
Trade_Payable_Ratio Num 5.2 How often does the company turn over during the year? A low ratio may be a sign
of chronic cash shortages
0.16
Liabities_Ratio Num 5.2 A debt ratio between long-term liabilities and total assets 0.83
Liquidity_Ratio Num 5.2 An index given an idea of how quickly a company can liquidate its assets to cover
short-term liabilities. The higher the better
0.70
Age Num 2 The age of the SME in years, by the end of 2014 3
Default_Status Char $15. Indicates if the counterparty has defaulted on their loan agreement. Will contain
either Default or Non-Default.
Non-
Default

2. 4. Data Analysis
The data will be analysed using proc univariate, proc means and proc corr to get a look and feel for the data.
4.1 Proc Univariate
The proc univariate procedure shows the distribution of the data, including the assessment of normality and
discovery of outliers. In the var statement of the procedure all the predictor variables have been listed in order
to get the feel for the data.
The following is available from the proc univariate analysis:
N, Mean, Standard Deviation, Skewness, Uncorrected Sum of Squares, Coefficient of Variation, Sum of
Weights, Sum of Observations, Variance, Kurtosis, Corrected Sum of Squares and Standard Error of the Mean
The UNIVARATE Procedure
SupplierTargetYears
OutsideCapitalStructure
CashRatio
CapitalTiedUp
EquityRatio
CostIncomeRatio
TradePayablesRatio
LiabilitiesRatio
LiquidityRatio
N 310 310 310 310 310 310 310 310 310
Mean 0.1974 0.5254 0.1265 0.0784 0.4342 0.2843 0.2407 0.6535 0.1211
Std Deviation 0.1601 0.2410 0.1018 0.0717 0.2599 0.1850 0.2083 0.2027 0.0982
Skewness 2.1444 -0.2262 1.8982 4.3089 0.3858 0.6971 1.0480 -0.5288 2.0616
Uncorrected SS 19.9969 103.4996 8.1618 3.4905 79.3137 35.6212 31.3595 145.0873 7.5213
Coeff Variation 81.1019 45.8652 80.4338 91.4701 59.8734 65.0720 86.5325 31.0113 81.0864
Sum
Observations
61.1900 162.8600 39.2200 24.2900 134.5900 88.1200 74.6100 202.5900 37.5300
Variance 0.0256 0.0581 0.0104 0.0051 0.0676 0.0342 0.0434 0.0411 0.0096
Kurtosis 5.5300 -1.0363 4.6882 30.4751 -0.9059 0.2093 0.1134 -0.3641 5.7568
Corrected SS 7.9188 17.9403 3.1998 1.5873 20.8799 10.5724 13.4026 12.6915 2.9777
Std Error Mean 0.0091 0.0137 0.0058 0.0041 0.0148 0.0105 0.0118 0.0115 0.0056

3. 4.2 Proc Means
The proc means procedure can be used analyse the mean, distribution and shape of the data by grouping the
data on the response variable default (using the class statement). The results show that there is a significant
difference in the mean and kurtosis of the predicator variables when grouped by counterparties which are
status default and non-default. The results are as expected as we would expect the healthier counterparties to
be different from the defaulted counterparties.
The MEANS Procedure
Default
Status
N
Obs Variable
Mean
StdDev
StdError
Kurtosis
Lower95%
CLforMean
Upper95%
CLforMean
Default 34 Supplier Target Years
Outside Capital Structure
Cash Ratio
Capital Tied Up
Equity Ratio
Cost Income Ratio
Trade Payable Ratio
Liabilities Ratio
Liquidity Ratio
0.294
0.655
0.094
0.118
0.298
0.331
0.270
0.768
0.093
0.150
0.165
0.088
0.076
0.223
0.171
0.179
0.159
0.086
0.026
0.028
0.015
0.013
0.038
0.029
0.031
0.027
0.015
-0.983
-0.207
2.164
-0.684
-0.083
-1.191
0.899
0.017
1.934
0.241
0.598
0.064
0.091
0.220
0.271
0.208
0.712
0.063
0.346
0.713
0.125
0.144
0.376
0.390
0.332
0.823
0.123
Non-Default 276 Supplier Target Years
Outside Capital Structure
Cash Ratio
Capital Tied Up
Equity Ratio
Cost Income Ratio
Trade Payable Ratio
Liabilities Ratio
Liquidity Ratio
0.186
0.509
0.130
0.074
0.451
0.279
0.237
0.639
0.124
0.157
0.244
0.103
0.070
0.260
0.186
0.212
0.203
0.099
0.009
0.015
0.006
0.004
0.016
0.011
0.013
0.012
0.006
7.475
-1.112
4.788
39.796
-0.912
0.443
0.101
-0.378
5.940
0.167
0.480
0.118
0.065
0.420
0.256
0.212
0.615
0.113
0.204
0.538
0.143
0.082
0.482
0.301
0.262
0.663
0.136

4. 4.3 Proc Corr
The proc corr procedure can be used to analyse the correlation (Pearson’s correlation coefficient) between
each predicator variables. The correlation cofficient will measure he lineasr dependence between each
predicator variables to give a value between +1 and -1 inclusive, where 1 is total positive linear correlation, 0 is
no linear correlation, and -1 is total negative linear correlation. The p-value tests if the correlation coefficient is
significant (highlighted in red).
4.4 Pearson Correlation Coefficients
The CORR Procedure
Pearson Correlation Coefficients, N = 310
Prob > |r| under H0: Rho=0
SupplierTargetYears
OutsideCapitalStructure
CashRatio
CapitalTiedUp
EquityRatio
CostIncomeRatio
TradePayableRatio
LiabilitiesRatio
LiquidityRatio
Supplier Target Years 1.00000
0.18792
0.0009
-0.18124
0.0014
0.34919
<.0001
-0.12503
0.0277
0.31932
<.0001
-0.16938
0.0028
0.13971
0.0138
-0.16088
0.0045
Outside Capital
Structure
0.18792
0.0009 1.00000
-0.16005
0.0047
0.28974
<.0001
-0.45627
<.0001
-0.43969
<.0001
0.65225
<.0001
0.47682
<.0001
-0.11401
0.0449
Cash Ratio
-0.18124
0.0014
-0.16005
0.0047 1.00000
-0.00434
0.9394
0.21627
0.0001
-0.07254
0.2028
0.13143
0.0206
-0.02776
0.6263
0.97047
<.0001
Capital Tied Up
0.34919
<.0001
0.28974
<.0001
-0.00434
0.9394 1.00000
-0.19404
0.0006
0.08843
0.1202
0.40803
<.0001
0.17815
0.0016
0.00605
0.9156
Equity Ratio
-0.12503
0.0277
-0.45627
<.0001
0.21627
0.0001
-0.19404
0.0006 1.00000
0.24874
<.0001
-0.45780
<.0001
-0.92874
<.0001
0.18564
0.0010
Cost Income Ratio
0.31932
<.0001
-0.43969
<.0001
-0.07254
0.2028
0.08843
0.1202
0.24874
<.0001 1.00000
-0.50407
<.0001
-0.25646
<.0001
-0.07783
0.1717
Trade Payable Ratio
-0.16938
0.0028
0.65225
<.0001
0.13143
0.0206
0.40803
<.0001
-0.45780
<.0001
-0.50407
<.0001 1.00000
0.47200
<.0001
0.15177
0.0074
Liabilities Ratio
0.13971
0.0138
0.47682
<.0001
-0.02776
0.6263
0.17815
0.0016
-0.92874
<.0001
-0.25646
<.0001
0.47200
<.0001 1.00000
0.02040
0.7205
Liquidity Ratio
-0.16088
0.0045
-0.11401
0.0449
0.97047
<.0001
0.00605
0.9156
0.18564
0.0010
-0.07783
0.1717
0.15177
0.0074
0.02040
0.7205 1.00000

5. 4.5 Scatter Plot Matrix

6. 5. Probability of Default Model (PD)
In this section the logistic regression function will be used to model the probability of default (PD) for the loans
portfolio data.
5.1 The Logit Link Function
The SAS proc logistic will be used to model the binary response variable default. The logit of the default will be
used as the response in the regression equation:
𝐿𝑛 (
𝑃
1 − 𝑃
) = 𝛽0 + ∑ 𝛽𝑖 𝑥𝑖
10
𝑖=1
P is defined as the probability that Default = 1 (Default_Status = ‘Default’). The Xs are the predicator variables
as follows:
𝑥1 Supplier Target Years
𝑥2 Outside Capital Structure
𝑥3 Cash Ratio
𝑥4 Capital Tied Up
𝑥5 Equity Ratio
𝑥6 Cost Income Ratio
𝑥7 Trader Payable Ratio
𝑥8 Liabilities Ratio
𝑥9 Liquidity Ratio
𝑥10 Age
The SAS proc logistic procedure will be used to estimate the beta’s.
5.2 PD Model - Logit
The full output for the proc logistic can be found in appendix 6.1.
The summary of the PD model output is as follows:
 The binary logit model was used with Fisher’s scoring optimisation technique
 The model used 310 observations (counterparties)
 The probability model was for Default_Status = ‘Default’
 The model convergence status shows it successfully converged to a solution
 Using a standard alpha criterion for significance of 0.05, the model is significant based on the
likelihood ratio and score, since chi-square is < .0001
Testing Global Null Hypothesis: BETA=0
Test
Chi-
Square DF Pr > ChiSq
Likelihood Ratio 39.3721 10 <.0001
Score 37.6438 10 <.0001
Wald 30.1195 10 0.0008

7.  The Analysis of Maximum Likelihood Estimate table shows the estimates for beta in the logistic
regression equation. The maximum likelihood estimate shows that the predicator variables are not
significant at the 0.05 alpha level
Analysis of Maximum Likelihood Estimates
Parameter DF Estimate
Standard
Error
Wald
Chi-
Square
Pr >
ChiSq
Intercept 1 -11.0281 3.2990 11.1746 0.0008
Supplier_Target_Year 1 0.0976 1.5079 0.0042 0.9484
Outside_Capital_Stru 1 3.2529 1.3918 5.4624 0.0194
Cash_Ratio 1 -7.6011 22.5496 0.1136 0.7361
Capital_Tied_Up 1 5.3388 2.7314 3.8205 0.0506
Equity_Ratio 1 2.6575 2.4262 1.1997 0.2734
Cost_Income_Ratio 1 2.6339 1.4102 3.4886 0.0618
Trade_Payable_Ratio 1 -2.2046 1.5860 1.9321 0.1645
Liabilities_Ratio 1 7.0681 3.3923 4.3411 0.0372
Liquidity_Ratio 1 4.6822 22.7943 0.0422 0.8373
Age 1 0.2205 0.1934 1.2998 0.2542
 The below table summarises the ability of the model to discriminate counterparties that will default
on their loan agreement. A typical value to report is the concordance statistic, labelled c. This value
indicates that 80% of the time the model is able to correctly predict counterparties that will default
on their loan agreement.
Association of Predicted Probabilities and Observed Responses
Percent Concordant 80 Somers' D 0.6
Percent Discordant 20 Gamma 0.6
Percent Tied 0 Tau-a 0.118
Pairs 9384 c 0.8
 The Hosmer-Lemeshow goodness of fit (GOF) test is a way to assess whether there is evidence for lack
of fit in a regression model. Results from the proc logistic is, HL chi-square of 8.5784 with 8 df,
yielding a p-value of 0.3791. As the p-value is greater than 0.05, it means we can reject the null
hypothesis i.e. evidence for lack of fit.

8. 5.3 Final Model
The beta estimates for the model are as follows:
Beta Parameter Estimate
0 Intercept -11.0281
1 Supplier Target Year 0.0976
2 Outside Capital Structure 3.2529
3 Cash Ratio -7.6011
4 Capital Tied Up 5.3388
5 Equity Ratio 2.6575
6 Cost Income Ratio 2.6339
7 Trade Payable Ratio -2.2046
8 Liabilities Ratio 7.0681
9 Liquidity Ratio 4.6822
10 Age 0.2205
5.4 PD Calculation
The SAS proc logistic procedure contains the outest option, which populates the beta’s into a sas dataset
called Default_Data_Est.
The below script generates the counterparty PD calculation into a new dataset called Counteryparty_PD.

9. 6. Appendix A
6.1 Proc Logistic
Model Information
Data Set CRDR.DEFAULT_DATA
Response Variable Default_Status Default Status
Number of Response Levels 2
Model binary logit
Optimization Technique Fisher's scoring
Number of Observations Read 310
Number of Observations Used 310
Response Profile
Ordered Value Default_Status Total Frequency
1 Default 34
2 Non-Default 276
Probability modelled is Default_Status='Default'.
Model Convergence Status
Convergence criterion (GCONV=1E-8) satisfied.
Model Fit Statistics
Criterion
Intercept
Only
Intercept and
Covariates
AIC 216.421 197.049
SC 220.158 238.151
-2 Log L 214.421 175.049
R-Square 0.1193 Max-rescaled R-Square 0.2389
Testing Global Null Hypothesis: BETA=0
Test Chi-Square DF Pr > ChiSq
Likelihood Ratio 39.3721 10 <.0001
Score 37.6438 10 <.0001
Wald 30.1195 10 0.0008

10. Analysis of Maximum Likelihood Estimates
Parameter DF Estimate
Standard
Error
Wald
Chi-Square Pr > ChiSq
Intercept 1 -11.0281 3.2990 11.1746 0.0008
Supplier_Target_Year 1 0.0976 1.5079 0.0042 0.9484
Outside_Capital_Stru 1 3.2529 1.3918 5.4624 0.0194
Cash_Ratio 1 -7.6011 22.5496 0.1136 0.7361
Capital_Tied_Up 1 5.3388 2.7314 3.8205 0.0506
Equity_Ratio 1 2.6575 2.4262 1.1997 0.2734
Cost_Income_Ratio 1 2.6339 1.4102 3.4886 0.0618
Trade_Payable_Ratio 1 -2.2046 1.5860 1.9321 0.1645
Liabilities_Ratio 1 7.0681 3.3923 4.3411 0.0372
Liquidity_Ratio 1 4.6822 22.7943 0.0422 0.8373
Age 1 0.2205 0.1934 1.2998 0.2542
Odds Ratio Estimates
Effect
Point
Estimate
95% Wald
Confidence Limits
Supplier_Target_Year 1.103 0.057 21.181
Outside_Capital_Stru 25.865 1.690 395.735
Cash_Ratio <0.001 <0.001 >999.999
Capital_Tied_Up 208.264 0.986 >999.999
Equity_Ratio 14.261 0.123 >999.999
Cost_Income_Ratio 13.928 0.878 220.913
Trade_Payable_Ratio 0.110 0.005 2.469
Liabilities_Ratio >999.999 1.521 >999.999
Liquidity_Ratio 108.003 <0.001 >999.999
Age 1.247 0.853 1.821
Association of Predicted Probabilities and Observed
Responses
Percent Concordant 80.0 Somers' D 0.600
Percent Discordant 20.0 Gamma 0.600
Percent Tied 0.0 Tau-a 0.118
Pairs 9384 c 0.800

11. Partition for the Hosmer and Lemeshow Test
Group Total
Default_Status = Default Default_Status = Non-Default
Observed Expected Observed Expected
1 31 0 0.22 31 30.78
2 31 2 0.60 29 30.40
3 31 1 0.97 30 30.03
4 31 0 1.35 31 29.65
5 31 1 1.74 30 29.26
6 31 4 2.31 27 28.69
7 31 1 2.96 30 28.04
8 31 5 4.24 26 26.76
9 31 6 6.83 25 24.17
10 31 14 12.79 17 18.21
Hosmer and Lemeshow Goodness-of-Fit
Test
Chi-Square DF Pr > ChiSq
8.5784 8 0.3791
0.00
0.25
0.50
0.75
1.00
Sensitivity
0.00 0.25 0.50 0.75 1.00
1 - Specificity
ROC Curve for M odel
AreaUnder theCurve= 0.8002

12. Influence Diagnostics
DefaultNon-DefaultDefault_Status
-2
-1
0
1
2
3
StdizedDevianceResidual
-2
0
2
4
6
StdizedPearsonResidual
-1
0
1
2
3
DevianceResidual
-2
0
2
4
6
PearsonResidual
0 100 200 300
CaseNumber
0 100 200 300
CaseNumber
Influence Diagnostics
DefaultNon-DefaultDefault_Status
0.0
0.2
0.4
0.6
0.8
CIDisplacementsCBar
0.00
0.25
0.50
0.75
1.00
1.25
CIDisplacementsC
0.0
0.1
0.2
0.3
0.4
0.5
Leverage
-2
-1
0
1
2
3
LikelihoodResidual
0 100 200 300
CaseNumber
0 100 200 300
CaseNumber

13. Influence Diagnostics
DefaultNon-DefaultDefault_Status
0
2
4
6
8
DevianceDeletionDifference
0
10
20
30
40
Chi-squareDeletionDifference
0 100 200 300
CaseNumber
0 100 200 300
CaseNumber
Influence Diagnostics
DefaultNon-DefaultDefault_Status
DfBetas
Cash_RatioOutside_Capital_Structure
Supplier_Target_YearsIntercept
-0.2
0.0
0.2
0.4
-0.2
0.0
0.2
0.4
0 100 200 300
CaseNumber
0 100 200 300
CaseNumber

14. Influence Diagnostics
DefaultNon-DefaultDefault_Status
DfBetas
Trade_Payable_RatioCost_Income_Ratio
Equity_RatioCapital_Tied_Up
-0.75
-0.50
-0.25
0.00
0.25
0.50
-0.75
-0.50
-0.25
0.00
0.25
0.50
0 100 200 300
CaseNumber
0 100 200 300
CaseNumber
Influence Diagnostics
DefaultNon-DefaultDefault_Status
DfBetas
AgeLiquidity_Ratio
Liabilities_Ratio
-0.4
-0.2
0.0
0.2
-0.4
-0.2
0.0
0.2
0 100 200 300
CaseNumber
0 100 200 300
CaseNumber

15. PredictedProbability Diagnostics
DefaultNon-DefaultDefault_Status
0.0
0.1
0.2
0.3
0.4
0.5
Leverage
0.00
0.25
0.50
0.75
1.00
1.25
CIDisplacementsC
0
2
4
6
8
DevianceDeletionDifference
0
10
20
30
40
Chi-squareDeletionDifference
0.0 0.2 0.4 0.6 0.8
Predicted Probability
0.0 0.2 0.4 0.6 0.8
Predicted Probability
Leverage Diagnostics
DefaultNon-DefaultDefault_Status
0.0
0.2
0.4
0.6
0.8
PredictedProbability
0.00
0.25
0.50
0.75
1.00
1.25
CIDisplacementsC
0
2
4
6
8
DevianceDeletionDifference
0
10
20
30
40
Chi-squareDeletionDifference
0.0 0.1 0.2 0.3 0.4 0.5
Leverage
0.0 0.1 0.2 0.3 0.4 0.5
Leverage

16. Influence onthe M odel Fit andParameter Estimates
DefaultNon-DefaultDefault_Status
0.25
0.50
0.75
1.00
1.25
CIDisplacementC
0.0 0.2 0.4 0.6 0.8
Predicted Probability
0
2
4
6
8
DevianceDifference
0.0 0.2 0.4 0.6 0.8
Predicted Probability
0
10
20
30
40
PearsonChi-SquareDifference
0.00
0.25
0.50
0.75
1.00
Probability
0 2.5E-1 5E-1 7.5E-1 1E0
Supplier Target Years
PredictedObserved
PredictedProbabilitiesfor Default_Status=Default with95% Confidence Limits
At Outside_Capital_Structure=0.525Cash_Ratio=0.127Capital_Tied_Up=0.078Equity_Ratio=0.434
Cost_Income_Ratio=0.284Trade_Payable_Ratio=0.241Liabilities_Ratio=0.654Liquidity_Ratio=0.121Age=2.758

17. 7. Appendix B
7.1 Data Step to Create Default Data

18. 7.2 Proc Univariate
7.3 Proc Means

19. 7.4 Proc Corr
7.5 Proc Logistic