Modeling Rating Transition Matrices forWholesale Loan Portfolios with Stata

Christopher F Baum Alper Corlu Soner Tunay

Boston College / DIW Berlin Risk Analytics, Citizens Financial Group

Stata Conference 2016, Chicago

Baum, Corlu, Tunay (BC / CFG) Modeling Rating Transitions StataConf 2016 1 / 33

Modeling asset quality rating transition matrices

Modeling asset quality rating transition matrices

Evaluation of the response of commercial banks’ loan portfolios tostressful economic conditions is mandated in the US byDodd–Frank and the Fed’s CCAR stress testing scenariosimposed on SIFIs.In this paper, we focus on how the asset quality ratings (AQRs) ofwholesale loans may vary in response to changes in themacroeconomic environment.A number of approaches have been developed to model the AQRtransition matrices of wholesale loans’ asset quality ratingmeasures.

Baum, Corlu, Tunay (BC / CFG) Modeling Rating Transitions StataConf 2016 2 / 33

Modeling asset quality rating transition matrices

Modeling asset quality rating transition matrices

Evaluation of the response of commercial banks’ loan portfolios tostressful economic conditions is mandated in the US byDodd–Frank and the Fed’s CCAR stress testing scenariosimposed on SIFIs.In this paper, we focus on how the asset quality ratings (AQRs) ofwholesale loans may vary in response to changes in themacroeconomic environment.A number of approaches have been developed to model the AQRtransition matrices of wholesale loans’ asset quality ratingmeasures.

Baum, Corlu, Tunay (BC / CFG) Modeling Rating Transitions StataConf 2016 2 / 33

Modeling asset quality rating transition matrices

Modeling asset quality rating transition matrices

Evaluation of the response of commercial banks’ loan portfolios tostressful economic conditions is mandated in the US byDodd–Frank and the Fed’s CCAR stress testing scenariosimposed on SIFIs.In this paper, we focus on how the asset quality ratings (AQRs) ofwholesale loans may vary in response to changes in themacroeconomic environment.A number of approaches have been developed to model the AQRtransition matrices of wholesale loans’ asset quality ratingmeasures.

Baum, Corlu, Tunay (BC / CFG) Modeling Rating Transitions StataConf 2016 2 / 33

Modeling asset quality rating transition matrices The single index approach

A commonly used technique in the financial industry is the singleindex approach, where creditworthiness is modeled as respondingto a systematic factor, Zt .Zt , meant to reflect the credit cycle, is expressed as a standardnormal variable, explaining the deviation of the transition matrixfrom the average transition matrix.The inherent symmetry in this approach is a weakness, as intimes of stress the distribution of transitions may become skewed,and larger transitions may be more frequent.The strongest objection to this approach: the default state istreated like any other column of the matrix, whereas it plays acrucial role in forecasting losses from the portfolio.We have recently implemented the single-index approach usingMata’s optimize functions.

Baum, Corlu, Tunay (BC / CFG) Modeling Rating Transitions StataConf 2016 3 / 33

Modeling asset quality rating transition matrices The single index approach

A commonly used technique in the financial industry is the singleindex approach, where creditworthiness is modeled as respondingto a systematic factor, Zt .Zt , meant to reflect the credit cycle, is expressed as a standardnormal variable, explaining the deviation of the transition matrixfrom the average transition matrix.The inherent symmetry in this approach is a weakness, as intimes of stress the distribution of transitions may become skewed,and larger transitions may be more frequent.The strongest objection to this approach: the default state istreated like any other column of the matrix, whereas it plays acrucial role in forecasting losses from the portfolio.We have recently implemented the single-index approach usingMata’s optimize functions.

Baum, Corlu, Tunay (BC / CFG) Modeling Rating Transitions StataConf 2016 3 / 33

Modeling asset quality rating transition matrices The single index approach

A commonly used technique in the financial industry is the singleindex approach, where creditworthiness is modeled as respondingto a systematic factor, Zt .Zt , meant to reflect the credit cycle, is expressed as a standardnormal variable, explaining the deviation of the transition matrixfrom the average transition matrix.The inherent symmetry in this approach is a weakness, as intimes of stress the distribution of transitions may become skewed,and larger transitions may be more frequent.The strongest objection to this approach: the default state istreated like any other column of the matrix, whereas it plays acrucial role in forecasting losses from the portfolio.We have recently implemented the single-index approach usingMata’s optimize functions.

Baum, Corlu, Tunay (BC / CFG) Modeling Rating Transitions StataConf 2016 3 / 33

Modeling asset quality rating transition matrices The single index approach

A commonly used technique in the financial industry is the singleindex approach, where creditworthiness is modeled as respondingto a systematic factor, Zt .Zt , meant to reflect the credit cycle, is expressed as a standardnormal variable, explaining the deviation of the transition matrixfrom the average transition matrix.The inherent symmetry in this approach is a weakness, as intimes of stress the distribution of transitions may become skewed,and larger transitions may be more frequent.The strongest objection to this approach: the default state istreated like any other column of the matrix, whereas it plays acrucial role in forecasting losses from the portfolio.We have recently implemented the single-index approach usingMata’s optimize functions.

Baum, Corlu, Tunay (BC / CFG) Modeling Rating Transitions StataConf 2016 3 / 33

Modeling asset quality rating transition matrices The single index approach

A commonly used technique in the financial industry is the singleindex approach, where creditworthiness is modeled as respondingto a systematic factor, Zt .Zt , meant to reflect the credit cycle, is expressed as a standardnormal variable, explaining the deviation of the transition matrixfrom the average transition matrix.The inherent symmetry in this approach is a weakness, as intimes of stress the distribution of transitions may become skewed,and larger transitions may be more frequent.The strongest objection to this approach: the default state istreated like any other column of the matrix, whereas it plays acrucial role in forecasting losses from the portfolio.We have recently implemented the single-index approach usingMata’s optimize functions.

Baum, Corlu, Tunay (BC / CFG) Modeling Rating Transitions StataConf 2016 3 / 33

Modeling asset quality rating transition matrices Advantages of Stata-based risk modeling

Advantages of Stata-based risk modeling

Stata’s facilities for data management and econometric modelinghave made it the tool of choice for many of the analytical tasksencountered in CFG’s retail and wholesale risk analysis.Our first approach to risk analysis of wholesale loans is based onthe fractional logit model, available for some time as a GLM, butnow explicitly supported by fracreg logit.Our fractional logit model estimates an entire set of models fordifferent rating classes in a single step.Our second approach to risk analysis of wholesale loans is basedon the SUR model(sureg), well supported in Stata for bothestimation and postestimation tasks.

Baum, Corlu, Tunay (BC / CFG) Modeling Rating Transitions StataConf 2016 4 / 33

Modeling asset quality rating transition matrices Advantages of Stata-based risk modeling

Advantages of Stata-based risk modeling

Stata’s facilities for data management and econometric modelinghave made it the tool of choice for many of the analytical tasksencountered in CFG’s retail and wholesale risk analysis.Our first approach to risk analysis of wholesale loans is based onthe fractional logit model, available for some time as a GLM, butnow explicitly supported by fracreg logit.Our fractional logit model estimates an entire set of models fordifferent rating classes in a single step.Our second approach to risk analysis of wholesale loans is basedon the SUR model(sureg), well supported in Stata for bothestimation and postestimation tasks.

Baum, Corlu, Tunay (BC / CFG) Modeling Rating Transitions StataConf 2016 4 / 33

Modeling asset quality rating transition matrices Advantages of Stata-based risk modeling

Advantages of Stata-based risk modeling

Stata’s facilities for data management and econometric modelinghave made it the tool of choice for many of the analytical tasksencountered in CFG’s retail and wholesale risk analysis.Our first approach to risk analysis of wholesale loans is based onthe fractional logit model, available for some time as a GLM, butnow explicitly supported by fracreg logit.Our fractional logit model estimates an entire set of models fordifferent rating classes in a single step.Our second approach to risk analysis of wholesale loans is basedon the SUR model(sureg), well supported in Stata for bothestimation and postestimation tasks.

Baum, Corlu, Tunay (BC / CFG) Modeling Rating Transitions StataConf 2016 4 / 33

Modeling asset quality rating transition matrices Advantages of Stata-based risk modeling

Advantages of Stata-based risk modeling

Stata’s facilities for data management and econometric modelinghave made it the tool of choice for many of the analytical tasksencountered in CFG’s retail and wholesale risk analysis.Our first approach to risk analysis of wholesale loans is based onthe fractional logit model, available for some time as a GLM, butnow explicitly supported by fracreg logit.Our fractional logit model estimates an entire set of models fordifferent rating classes in a single step.Our second approach to risk analysis of wholesale loans is basedon the SUR model(sureg), well supported in Stata for bothestimation and postestimation tasks.

Baum, Corlu, Tunay (BC / CFG) Modeling Rating Transitions StataConf 2016 4 / 33

Modeling asset quality rating transition matrices Our AQR modeling approach

Our AQR modeling approach

We depart from the single-index approach (or its multi-indexgeneralization) to propose a more flexible method of modelingAQR transitions.Our approach involves the modeling of the most likely events: (i)no change in asset quality, (ii) one-notch changes up or down onthe asset quality scale, (iii) transition to default.As historical transition matrices are dominated by the diagonaland adjacent diagonals, this approach should be able to capture alarge fraction of experienced ratings changes.

Baum, Corlu, Tunay (BC / CFG) Modeling Rating Transitions StataConf 2016 5 / 33

Modeling asset quality rating transition matrices Our AQR modeling approach

Our AQR modeling approach

We depart from the single-index approach (or its multi-indexgeneralization) to propose a more flexible method of modelingAQR transitions.Our approach involves the modeling of the most likely events: (i)no change in asset quality, (ii) one-notch changes up or down onthe asset quality scale, (iii) transition to default.As historical transition matrices are dominated by the diagonaland adjacent diagonals, this approach should be able to capture alarge fraction of experienced ratings changes.

Baum, Corlu, Tunay (BC / CFG) Modeling Rating Transitions StataConf 2016 5 / 33

Modeling asset quality rating transition matrices Our AQR modeling approach

Our AQR modeling approach

We depart from the single-index approach (or its multi-indexgeneralization) to propose a more flexible method of modelingAQR transitions.Our approach involves the modeling of the most likely events: (i)no change in asset quality, (ii) one-notch changes up or down onthe asset quality scale, (iii) transition to default.As historical transition matrices are dominated by the diagonaland adjacent diagonals, this approach should be able to capture alarge fraction of experienced ratings changes.

Baum, Corlu, Tunay (BC / CFG) Modeling Rating Transitions StataConf 2016 5 / 33

Modeling asset quality rating transition matrices Modeling the most likely transitions

Modeling the most likely transitions

This approach is implemented, using time series data on 16 assetquality ratings (AAA, AA+, AA, AA-...), with a set of fractional logitmodels of the probabilities of four events:

1 AQ unchanged from period t − 1 to t2 AQ increased by one rating from period t − 1 to t3 AQ decreased by one rating from period t − 1 to t4 AQ transition to default in period t

The remaining transition probabilities from the period t − 1 transitionmatrix are mechanically adjusted to meet the constraint that theprobabilities of events (1)–(4) plus remaining probabilities must sum toone for each AQ rating. This implies that the AQ transition matricesevolve dynamically in a relatively unconstrained manner.

Baum, Corlu, Tunay (BC / CFG) Modeling Rating Transitions StataConf 2016 6 / 33

Modeling asset quality rating transition matrices Modeling the most likely transitions

Modeling the most likely transitions

This approach is implemented, using time series data on 16 assetquality ratings (AAA, AA+, AA, AA-...), with a set of fractional logitmodels of the probabilities of four events:

1 AQ unchanged from period t − 1 to t2 AQ increased by one rating from period t − 1 to t3 AQ decreased by one rating from period t − 1 to t4 AQ transition to default in period t

The remaining transition probabilities from the period t − 1 transitionmatrix are mechanically adjusted to meet the constraint that theprobabilities of events (1)–(4) plus remaining probabilities must sum toone for each AQ rating. This implies that the AQ transition matricesevolve dynamically in a relatively unconstrained manner.

Baum, Corlu, Tunay (BC / CFG) Modeling Rating Transitions StataConf 2016 6 / 33

Modeling asset quality rating transition matrices Modeling the most likely transitions

Modeling the most likely transitions

This approach is implemented, using time series data on 16 assetquality ratings (AAA, AA+, AA, AA-...), with a set of fractional logitmodels of the probabilities of four events:

1 AQ unchanged from period t − 1 to t2 AQ increased by one rating from period t − 1 to t3 AQ decreased by one rating from period t − 1 to t4 AQ transition to default in period t

The remaining transition probabilities from the period t − 1 transitionmatrix are mechanically adjusted to meet the constraint that theprobabilities of events (1)–(4) plus remaining probabilities must sum toone for each AQ rating. This implies that the AQ transition matricesevolve dynamically in a relatively unconstrained manner.

Baum, Corlu, Tunay (BC / CFG) Modeling Rating Transitions StataConf 2016 6 / 33

Modeling asset quality rating transition matrices Modeling the most likely transitions

Modeling the most likely transitions

This approach is implemented, using time series data on 16 assetquality ratings (AAA, AA+, AA, AA-...), with a set of fractional logitmodels of the probabilities of four events:

1 AQ unchanged from period t − 1 to t2 AQ increased by one rating from period t − 1 to t3 AQ decreased by one rating from period t − 1 to t4 AQ transition to default in period t

The remaining transition probabilities from the period t − 1 transitionmatrix are mechanically adjusted to meet the constraint that theprobabilities of events (1)–(4) plus remaining probabilities must sum toone for each AQ rating. This implies that the AQ transition matricesevolve dynamically in a relatively unconstrained manner.

Baum, Corlu, Tunay (BC / CFG) Modeling Rating Transitions StataConf 2016 6 / 33

Modeling asset quality rating transition matrices Modeling the most likely transitions

Modeling the most likely transitions

This approach is implemented, using time series data on 16 assetquality ratings (AAA, AA+, AA, AA-...), with a set of fractional logitmodels of the probabilities of four events:

1 AQ unchanged from period t − 1 to t2 AQ increased by one rating from period t − 1 to t3 AQ decreased by one rating from period t − 1 to t4 AQ transition to default in period t

The remaining transition probabilities from the period t − 1 transitionmatrix are mechanically adjusted to meet the constraint that theprobabilities of events (1)–(4) plus remaining probabilities must sum toone for each AQ rating. This implies that the AQ transition matricesevolve dynamically in a relatively unconstrained manner.

Baum, Corlu, Tunay (BC / CFG) Modeling Rating Transitions StataConf 2016 6 / 33

Modeling asset quality rating transition matrices Modeling the most likely transitions

Modeling the most likely transitions

This approach is implemented, using time series data on 16 assetquality ratings (AAA, AA+, AA, AA-...), with a set of fractional logitmodels of the probabilities of four events:

1 AQ unchanged from period t − 1 to t2 AQ increased by one rating from period t − 1 to t3 AQ decreased by one rating from period t − 1 to t4 AQ transition to default in period t

The remaining transition probabilities from the period t − 1 transitionmatrix are mechanically adjusted to meet the constraint that theprobabilities of events (1)–(4) plus remaining probabilities must sum toone for each AQ rating. This implies that the AQ transition matricesevolve dynamically in a relatively unconstrained manner.

Baum, Corlu, Tunay (BC / CFG) Modeling Rating Transitions StataConf 2016 6 / 33

Modeling asset quality rating transition matrices Modeling the most likely transitions

Modeling the most likely transitions

This approach is implemented, using time series data on 16 assetquality ratings (AAA, AA+, AA, AA-...), with a set of fractional logitmodels of the probabilities of four events:

1 AQ unchanged from period t − 1 to t2 AQ increased by one rating from period t − 1 to t3 AQ decreased by one rating from period t − 1 to t4 AQ transition to default in period t

The remaining transition probabilities from the period t − 1 transitionmatrix are mechanically adjusted to meet the constraint that theprobabilities of events (1)–(4) plus remaining probabilities must sum toone for each AQ rating. This implies that the AQ transition matricesevolve dynamically in a relatively unconstrained manner.

Baum, Corlu, Tunay (BC / CFG) Modeling Rating Transitions StataConf 2016 6 / 33

Modeling asset quality rating transition matrices The fractional logit model

The fractional logit model

The fractional logit model of Papke and Wooldridge (J. AppliedEconometrics, 1996) is appropriate when the response variable isa proportion which may take on the values 0 and 1.Proportions data should not be modeled using linear regression,as it does not respect the bounds of 0 and 1, and thus canproduce predictions outside the (0,1) interval.In contrast to the Tobit model, which separately models thelikelihood that the response is 0 (or some other limiting value), thefractional logit produces a single set of marginal effects whichshow the effects of each explanatory variable on the proportion.The fractional logit model produces predictions on the same scaleas the original response variable, avoiding retransformation bias.

Baum, Corlu, Tunay (BC / CFG) Modeling Rating Transitions StataConf 2016 7 / 33

Modeling asset quality rating transition matrices The fractional logit model

The fractional logit model

The fractional logit model of Papke and Wooldridge (J. AppliedEconometrics, 1996) is appropriate when the response variable isa proportion which may take on the values 0 and 1.Proportions data should not be modeled using linear regression,as it does not respect the bounds of 0 and 1, and thus canproduce predictions outside the (0,1) interval.In contrast to the Tobit model, which separately models thelikelihood that the response is 0 (or some other limiting value), thefractional logit produces a single set of marginal effects whichshow the effects of each explanatory variable on the proportion.The fractional logit model produces predictions on the same scaleas the original response variable, avoiding retransformation bias.

Baum, Corlu, Tunay (BC / CFG) Modeling Rating Transitions StataConf 2016 7 / 33

Modeling asset quality rating transition matrices The fractional logit model

The fractional logit model

The fractional logit model of Papke and Wooldridge (J. AppliedEconometrics, 1996) is appropriate when the response variable isa proportion which may take on the values 0 and 1.Proportions data should not be modeled using linear regression,as it does not respect the bounds of 0 and 1, and thus canproduce predictions outside the (0,1) interval.In contrast to the Tobit model, which separately models thelikelihood that the response is 0 (or some other limiting value), thefractional logit produces a single set of marginal effects whichshow the effects of each explanatory variable on the proportion.The fractional logit model produces predictions on the same scaleas the original response variable, avoiding retransformation bias.

Baum, Corlu, Tunay (BC / CFG) Modeling Rating Transitions StataConf 2016 7 / 33

Modeling asset quality rating transition matrices The fractional logit model

The fractional logit model

The fractional logit model of Papke and Wooldridge (J. AppliedEconometrics, 1996) is appropriate when the response variable isa proportion which may take on the values 0 and 1.Proportions data should not be modeled using linear regression,as it does not respect the bounds of 0 and 1, and thus canproduce predictions outside the (0,1) interval.In contrast to the Tobit model, which separately models thelikelihood that the response is 0 (or some other limiting value), thefractional logit produces a single set of marginal effects whichshow the effects of each explanatory variable on the proportion.The fractional logit model produces predictions on the same scaleas the original response variable, avoiding retransformation bias.

Baum, Corlu, Tunay (BC / CFG) Modeling Rating Transitions StataConf 2016 7 / 33

Modeling asset quality rating transition matrices Modeling ‘staying put’

A stylized fact of transition matrix modeling is that the probabilityof an unchanged AQ rating (item (1) above) is substantial.Our fractional logit model of the probability of ‘staying put’ takesinto account the probabilities of transitions from each of thenon-default AQ ratings, augmented with relevant macroeconomicfactors.In the estimated model, we also found that interactions of themacro factors with transition probabilities, as well as with eachother, are very relevant.The model, fit to quarterly data for a portfolio of C&I loans from2000–2012 over 15 non-default AQ classes, yielded anR2 = 0.604.

Baum, Corlu, Tunay (BC / CFG) Modeling Rating Transitions StataConf 2016 8 / 33

Modeling asset quality rating transition matrices Modeling ‘staying put’

A stylized fact of transition matrix modeling is that the probabilityof an unchanged AQ rating (item (1) above) is substantial.Our fractional logit model of the probability of ‘staying put’ takesinto account the probabilities of transitions from each of thenon-default AQ ratings, augmented with relevant macroeconomicfactors.In the estimated model, we also found that interactions of themacro factors with transition probabilities, as well as with eachother, are very relevant.The model, fit to quarterly data for a portfolio of C&I loans from2000–2012 over 15 non-default AQ classes, yielded anR2 = 0.604.

Baum, Corlu, Tunay (BC / CFG) Modeling Rating Transitions StataConf 2016 8 / 33

Modeling asset quality rating transition matrices Modeling ‘staying put’

A stylized fact of transition matrix modeling is that the probabilityof an unchanged AQ rating (item (1) above) is substantial.Our fractional logit model of the probability of ‘staying put’ takesinto account the probabilities of transitions from each of thenon-default AQ ratings, augmented with relevant macroeconomicfactors.In the estimated model, we also found that interactions of themacro factors with transition probabilities, as well as with eachother, are very relevant.The model, fit to quarterly data for a portfolio of C&I loans from2000–2012 over 15 non-default AQ classes, yielded anR2 = 0.604.

Baum, Corlu, Tunay (BC / CFG) Modeling Rating Transitions StataConf 2016 8 / 33

Modeling asset quality rating transition matrices Modeling ‘staying put’

A stylized fact of transition matrix modeling is that the probabilityof an unchanged AQ rating (item (1) above) is substantial.Our fractional logit model of the probability of ‘staying put’ takesinto account the probabilities of transitions from each of thenon-default AQ ratings, augmented with relevant macroeconomicfactors.In the estimated model, we also found that interactions of themacro factors with transition probabilities, as well as with eachother, are very relevant.The model, fit to quarterly data for a portfolio of C&I loans from2000–2012 over 15 non-default AQ classes, yielded anR2 = 0.604.

Baum, Corlu, Tunay (BC / CFG) Modeling Rating Transitions StataConf 2016 8 / 33

Modeling asset quality rating transition matrices Modeling ‘staying put’

0.2.4.6.81

0.2.4.6.81

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2000q3 2004q3 2008q3 2012q3

2000q3 2004q3 2008q3 2012q3 2000q3 2004q3 2008q3 2012q3 2000q3 2004q3 2008q3 2012q3

AAA AA+ AA AA-

A+ A A- BBB+

BBB BBB- BB B

CCC CC C

Graphs by fr

blue: predicted series using gdpqgr and neg/pos unempDunemp Predicted Staying the Same Probability

Baum, Corlu, Tunay (BC / CFG) Modeling Rating Transitions StataConf 2016 9 / 33

Modeling asset quality rating transition matrices Modeling one-notch changes

After ‘staying put’, the most likely outcomes from one period to thenext are one-notch upgrades or downgrades in the AQ rating:items (2) and (3) above.These are modeled with separate fractional logits of the samespecification, taking transitions from each AQ rating asexplanatory factors.The same macroeconomic factors are included, interacted withthe AQ rating classes.The one-notch upgrade model yielded an R2 = 0.523 while theone-notch downgrade model yielded an R2 = 0.656.

Baum, Corlu, Tunay (BC / CFG) Modeling Rating Transitions StataConf 2016 10 / 33

Modeling asset quality rating transition matrices Modeling one-notch changes

After ‘staying put’, the most likely outcomes from one period to thenext are one-notch upgrades or downgrades in the AQ rating:items (2) and (3) above.These are modeled with separate fractional logits of the samespecification, taking transitions from each AQ rating asexplanatory factors.The same macroeconomic factors are included, interacted withthe AQ rating classes.The one-notch upgrade model yielded an R2 = 0.523 while theone-notch downgrade model yielded an R2 = 0.656.

Baum, Corlu, Tunay (BC / CFG) Modeling Rating Transitions StataConf 2016 10 / 33

Modeling asset quality rating transition matrices Modeling one-notch changes

After ‘staying put’, the most likely outcomes from one period to thenext are one-notch upgrades or downgrades in the AQ rating:items (2) and (3) above.These are modeled with separate fractional logits of the samespecification, taking transitions from each AQ rating asexplanatory factors.The same macroeconomic factors are included, interacted withthe AQ rating classes.The one-notch upgrade model yielded an R2 = 0.523 while theone-notch downgrade model yielded an R2 = 0.656.

Baum, Corlu, Tunay (BC / CFG) Modeling Rating Transitions StataConf 2016 10 / 33

Modeling asset quality rating transition matrices Modeling one-notch changes

After ‘staying put’, the most likely outcomes from one period to thenext are one-notch upgrades or downgrades in the AQ rating:items (2) and (3) above.These are modeled with separate fractional logits of the samespecification, taking transitions from each AQ rating asexplanatory factors.The same macroeconomic factors are included, interacted withthe AQ rating classes.The one-notch upgrade model yielded an R2 = 0.523 while theone-notch downgrade model yielded an R2 = 0.656.

Baum, Corlu, Tunay (BC / CFG) Modeling Rating Transitions StataConf 2016 10 / 33

Modeling asset quality rating transition matrices Modeling one-notch changes

0.2.4.6.81

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2000q3 2004q3 2008q3 2012q3 2000q3 2004q3 2008q3 2012q3

2000q3 2004q3 2008q3 2012q3 2000q3 2004q3 2008q3 2012q3

AA+ AA AA- A+

A A- BBB+ BBB

BBB- BB B CCC

CC C

Graphs by fr

blue: predicted series using unempDunemp and gdpqgr Predicted Upgrades

Baum, Corlu, Tunay (BC / CFG) Modeling Rating Transitions StataConf 2016 11 / 33

Modeling asset quality rating transition matrices Modeling one-notch changes

0.2.4.6.81

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2000q3 2004q3 2008q3 2012q3 2000q3 2004q3 2008q3 2012q3

2000q3 2004q3 2008q3 2012q3 2000q3 2004q3 2008q3 2012q3

AAA AA+ AA AA-

A+ A A- BBB+

BBB BBB- BB B

CCC CC

Graphs by fr

blue: predicted series using unempDunemp and gdpqgrPredicted Downgrades

Baum, Corlu, Tunay (BC / CFG) Modeling Rating Transitions StataConf 2016 12 / 33

Modeling asset quality rating transition matrices Modeling defaults

The last item to be explicitly modeled in this approach is thetransition to default, item (4) above.The fractional logit model of the probability of default takestransitions from each AQ ratio as explanatory factors.Macroeconomic factors are included, and interacted with the AQratio classes.The default model yielded an R2 = 0.987.

Baum, Corlu, Tunay (BC / CFG) Modeling Rating Transitions StataConf 2016 13 / 33

Modeling asset quality rating transition matrices Modeling defaults

The last item to be explicitly modeled in this approach is thetransition to default, item (4) above.The fractional logit model of the probability of default takestransitions from each AQ ratio as explanatory factors.Macroeconomic factors are included, and interacted with the AQratio classes.The default model yielded an R2 = 0.987.

Baum, Corlu, Tunay (BC / CFG) Modeling Rating Transitions StataConf 2016 13 / 33

Modeling asset quality rating transition matrices Modeling defaults

The last item to be explicitly modeled in this approach is thetransition to default, item (4) above.The fractional logit model of the probability of default takestransitions from each AQ ratio as explanatory factors.Macroeconomic factors are included, and interacted with the AQratio classes.The default model yielded an R2 = 0.987.

Baum, Corlu, Tunay (BC / CFG) Modeling Rating Transitions StataConf 2016 13 / 33

Modeling asset quality rating transition matrices Modeling defaults

The last item to be explicitly modeled in this approach is thetransition to default, item (4) above.The fractional logit model of the probability of default takestransitions from each AQ ratio as explanatory factors.Macroeconomic factors are included, and interacted with the AQratio classes.The default model yielded an R2 = 0.987.

Baum, Corlu, Tunay (BC / CFG) Modeling Rating Transitions StataConf 2016 13 / 33

Modeling asset quality rating transition matrices Modeling defaults

.00044

.00046

.00048.0005

.00052

.00054

.00056

.00058.0006

.0006.00065

.0007.00075

.0008.00085

.0009

.0008

.0009.001

.0011.0012.0013

.0011.0012.0013.0014.0015.0016.0017.0018

.0015

.002

.0025

.002.0022.0024.0026.0028

.003.0032.0034

.003

.0035

.004

.0045

.005

.0035.004

.0045.005

.0055.006

.0065

.0055.006

.0065.007

.0075.008

.0085.009

.007

.008

.009.01

.011.012.013

.01.011.012.013.014.015.016.017.018

.012

.014

.016

.018.02

.022

.024

.02.022.024.026.028

.03.032

.028.03

.032

.034

.036

.038.04

.042

.04.045

.05.055

.06.065

.07

2000q3 2004q3 2008q3 2012q3

2000q3 2004q3 2008q3 2012q3 2000q3 2004q3 2008q3 2012q3 2000q3 2004q3 2008q3 2012q3

AAA AA+ AA AA-

A+ A A- BBB+

BBB BBB- BB B

CCC CC C

Graphs by fr

blue: predicted series using gdpqgr and corpprofitPredicted Defaults

Baum, Corlu, Tunay (BC / CFG) Modeling Rating Transitions StataConf 2016 14 / 33

Modeling asset quality rating transition matrices Modeling defaults

We can also visualize the predictions of the model by asset qualityrating, in terms of the probability that a loan of a given rating is likely tostay in that rating, migrate to a different rating, or default. For example:

Baum, Corlu, Tunay (BC / CFG) Modeling Rating Transitions StataConf 2016 15 / 33

Modeling asset quality rating transition matrices Modeling defaults

0.2.4.6.81

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2000q1 2004q1 2008q1 2012q1 2000q1 2004q1 2008q1 2012q1 2000q1 2004q1 2008q1 2012q1 2000q1 2004q1 2008q1 2012q1

AAA AA+ AA AA-

A+ A A- BBB+

BBB BBB- BB B

CCC CC C SD

Graphs by tgt

blue: predicted series, red: historicalPredicted transitions from AA-

Baum, Corlu, Tunay (BC / CFG) Modeling Rating Transitions StataConf 2016 16 / 33

Modeling asset quality rating transition matrices Modeling defaults

0.2.4.6.81

0.2.4.6.81

0.2.4.6.81

0.2.4.6.81

2000q1 2004q1 2008q1 2012q1 2000q1 2004q1 2008q1 2012q1 2000q1 2004q1 2008q1 2012q1 2000q1 2004q1 2008q1 2012q1

AAA AA+ AA AA-

A+ A A- BBB+

BBB BBB- BB B

CCC CC C SD

Graphs by tgt

blue: predicted series, red: historicalPredicted transitions from BBB

Baum, Corlu, Tunay (BC / CFG) Modeling Rating Transitions StataConf 2016 17 / 33

Modeling asset quality rating transition matrices Modeling defaults

The fractional logit approach to modeling the most importantelements of the transition matrix is quite successful, and producespredictions in the probability metric without retransformation bias.As the diagonal, super- and sub-diagonals of the transition matrixcapture a sizable fraction of transition probabilities, the problem ofestimating a time-varying transition matrix is greatly simplified.This approach allows the introduction of relevant macroeconomicdrivers which affect the transition probabilities directly, as well asinfluencing the transition coefficients from each AQ rating class.

Baum, Corlu, Tunay (BC / CFG) Modeling Rating Transitions StataConf 2016 18 / 33

Modeling asset quality rating transition matrices Modeling defaults

The fractional logit approach to modeling the most importantelements of the transition matrix is quite successful, and producespredictions in the probability metric without retransformation bias.As the diagonal, super- and sub-diagonals of the transition matrixcapture a sizable fraction of transition probabilities, the problem ofestimating a time-varying transition matrix is greatly simplified.This approach allows the introduction of relevant macroeconomicdrivers which affect the transition probabilities directly, as well asinfluencing the transition coefficients from each AQ rating class.

Baum, Corlu, Tunay (BC / CFG) Modeling Rating Transitions StataConf 2016 18 / 33

Modeling asset quality rating transition matrices Modeling defaults

The fractional logit approach to modeling the most importantelements of the transition matrix is quite successful, and producespredictions in the probability metric without retransformation bias.As the diagonal, super- and sub-diagonals of the transition matrixcapture a sizable fraction of transition probabilities, the problem ofestimating a time-varying transition matrix is greatly simplified.This approach allows the introduction of relevant macroeconomicdrivers which affect the transition probabilities directly, as well asinfluencing the transition coefficients from each AQ rating class.

Baum, Corlu, Tunay (BC / CFG) Modeling Rating Transitions StataConf 2016 18 / 33

Modeling asset quality rating transition matrices An alternative approach using SUR

A SUR approach to modeling defaults

We now present an alternative empirical approach based onmodeling the observed default rates for each level of asset qualityin a portfolio of wholesale loans.This dynamic model expresses the default rate for asset quality i ,AQi,t , in terms of lagged values of AQi,t , AQi−1,t and AQi+1,t inaddition to relevant macroeconomic factors.The model is fit as a system of Seemingly Unrelated Regressions(SUR), a generalized least squares technique (Zellner, JASA1962) that allows contemporaneous correlation of errors to beused to gain efficiency in the estimation: in Stata, the suregestimation command.

Baum, Corlu, Tunay (BC / CFG) Modeling Rating Transitions StataConf 2016 19 / 33

Modeling asset quality rating transition matrices An alternative approach using SUR

A SUR approach to modeling defaults

We now present an alternative empirical approach based onmodeling the observed default rates for each level of asset qualityin a portfolio of wholesale loans.This dynamic model expresses the default rate for asset quality i ,AQi,t , in terms of lagged values of AQi,t , AQi−1,t and AQi+1,t inaddition to relevant macroeconomic factors.The model is fit as a system of Seemingly Unrelated Regressions(SUR), a generalized least squares technique (Zellner, JASA1962) that allows contemporaneous correlation of errors to beused to gain efficiency in the estimation: in Stata, the suregestimation command.

Baum, Corlu, Tunay (BC / CFG) Modeling Rating Transitions StataConf 2016 19 / 33

Modeling asset quality rating transition matrices An alternative approach using SUR

A SUR approach to modeling defaults

We now present an alternative empirical approach based onmodeling the observed default rates for each level of asset qualityin a portfolio of wholesale loans.This dynamic model expresses the default rate for asset quality i ,AQi,t , in terms of lagged values of AQi,t , AQi−1,t and AQi+1,t inaddition to relevant macroeconomic factors.The model is fit as a system of Seemingly Unrelated Regressions(SUR), a generalized least squares technique (Zellner, JASA1962) that allows contemporaneous correlation of errors to beused to gain efficiency in the estimation: in Stata, the suregestimation command.

Baum, Corlu, Tunay (BC / CFG) Modeling Rating Transitions StataConf 2016 19 / 33

Modeling asset quality rating transition matrices An alternative approach using SUR

In the empirical application, we have quarterly data for 1997–2012on default rates for 16 asset quality ratings from a wholesaleportfolio of C&I loans, where low ratings are of higher quality(AAA, AA+, AA...), with AQ16 denoting default (SD).Historical default rates vary considerably across asset qualityratings as well as over time.

Table: Selected descriptive statistics, 1997–2012

Rating mean s.d. min maxAAA .0018172 .0016586 0.0 .0086655A+ .0048467 .0025159 .0007106 .0095858BBB .0121260 .0063674 .0009095 .0227618CCC .0304619 .0171742 .0017301 .0758738

Baum, Corlu, Tunay (BC / CFG) Modeling Rating Transitions StataConf 2016 20 / 33

Modeling asset quality rating transition matrices An alternative approach using SUR

In the empirical application, we have quarterly data for 1997–2012on default rates for 16 asset quality ratings from a wholesaleportfolio of C&I loans, where low ratings are of higher quality(AAA, AA+, AA...), with AQ16 denoting default (SD).Historical default rates vary considerably across asset qualityratings as well as over time.

Table: Selected descriptive statistics, 1997–2012

Rating mean s.d. min maxAAA .0018172 .0016586 0.0 .0086655A+ .0048467 .0025159 .0007106 .0095858BBB .0121260 .0063674 .0009095 .0227618CCC .0304619 .0171742 .0017301 .0758738

Baum, Corlu, Tunay (BC / CFG) Modeling Rating Transitions StataConf 2016 20 / 33

Modeling asset quality rating transition matrices An alternative approach using SUR

In the empirical application, we have quarterly data for 1997–2012on default rates for 16 asset quality ratings from a wholesaleportfolio of C&I loans, where low ratings are of higher quality(AAA, AA+, AA...), with AQ16 denoting default (SD).Historical default rates vary considerably across asset qualityratings as well as over time.

Table: Selected descriptive statistics, 1997–2012

Rating mean s.d. min maxAAA .0018172 .0016586 0.0 .0086655A+ .0048467 .0025159 .0007106 .0095858BBB .0121260 .0063674 .0009095 .0227618CCC .0304619 .0171742 .0017301 .0758738

Baum, Corlu, Tunay (BC / CFG) Modeling Rating Transitions StataConf 2016 20 / 33

Modeling asset quality rating transition matrices An alternative approach using SUR

In the empirical application, we have quarterly data for 1997–2012on default rates for 16 asset quality ratings from a wholesaleportfolio of C&I loans, where low ratings are of higher quality(AAA, AA+, AA...), with AQ16 denoting default (SD).Historical default rates vary considerably across asset qualityratings as well as over time.

Table: Selected descriptive statistics, 1997–2012

Rating mean s.d. min maxAAA .0018172 .0016586 0.0 .0086655A+ .0048467 .0025159 .0007106 .0095858BBB .0121260 .0063674 .0009095 .0227618CCC .0304619 .0171742 .0017301 .0758738

Baum, Corlu, Tunay (BC / CFG) Modeling Rating Transitions StataConf 2016 20 / 33

Modeling asset quality rating transition matrices An alternative approach using SUR

0

.02

.04

.06

.08

1997q1 2001q1 2005q1 2009q1 2013q1

AAA A+BBB CCC

Historical default rates for selected asset quality ratings

Baum, Corlu, Tunay (BC / CFG) Modeling Rating Transitions StataConf 2016 21 / 33

Modeling asset quality rating transition matrices Evaluating persistence of default rates

The dynamic model we have implemented depends on thepersistence of default rates for each rating class.We can examine that persistence by computing theautocorrelation functions for each default rate.The autocorrelation functions for selected AQ ratings show thatbetween two and four quarterly lags of those ratings haveautocorrelation coefficients significantly differing from zero.These findings suggest that several lags should be included in theSUR estimation to capture the persistence of default rates.

Baum, Corlu, Tunay (BC / CFG) Modeling Rating Transitions StataConf 2016 22 / 33

Modeling asset quality rating transition matrices Evaluating persistence of default rates

The dynamic model we have implemented depends on thepersistence of default rates for each rating class.We can examine that persistence by computing theautocorrelation functions for each default rate.The autocorrelation functions for selected AQ ratings show thatbetween two and four quarterly lags of those ratings haveautocorrelation coefficients significantly differing from zero.These findings suggest that several lags should be included in theSUR estimation to capture the persistence of default rates.

Baum, Corlu, Tunay (BC / CFG) Modeling Rating Transitions StataConf 2016 22 / 33

Modeling asset quality rating transition matrices Evaluating persistence of default rates

The dynamic model we have implemented depends on thepersistence of default rates for each rating class.We can examine that persistence by computing theautocorrelation functions for each default rate.The autocorrelation functions for selected AQ ratings show thatbetween two and four quarterly lags of those ratings haveautocorrelation coefficients significantly differing from zero.These findings suggest that several lags should be included in theSUR estimation to capture the persistence of default rates.

Baum, Corlu, Tunay (BC / CFG) Modeling Rating Transitions StataConf 2016 22 / 33

Modeling asset quality rating transition matrices Evaluating persistence of default rates

The dynamic model we have implemented depends on thepersistence of default rates for each rating class.We can examine that persistence by computing theautocorrelation functions for each default rate.The autocorrelation functions for selected AQ ratings show thatbetween two and four quarterly lags of those ratings haveautocorrelation coefficients significantly differing from zero.These findings suggest that several lags should be included in theSUR estimation to capture the persistence of default rates.

Baum, Corlu, Tunay (BC / CFG) Modeling Rating Transitions StataConf 2016 22 / 33

Modeling asset quality rating transition matrices Evaluating persistence of default rates

-0.50

0.00

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Bartlett's formula for MA(q) 95% confidence bands

AAA

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Bartlett's formula for MA(q) 95% confidence bands

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Bartlett's formula for MA(q) 95% confidence bands

BBB

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Bartlett's formula for MA(q) 95% confidence bands

CCC

Baum, Corlu, Tunay (BC / CFG) Modeling Rating Transitions StataConf 2016 23 / 33

Modeling asset quality rating transition matrices Including macroeconomic factors

To capture common factors influencing all borrowers, we include a setof macroeconomic factors in each equation:

Real GDP growth, YoY, three-quarter centered moving averageUnemployment × the change in unemployment from t to t + 1Average weekly hours in manufacturing

Baum, Corlu, Tunay (BC / CFG) Modeling Rating Transitions StataConf 2016 24 / 33

Modeling asset quality rating transition matrices Including macroeconomic factors

To capture common factors influencing all borrowers, we include a setof macroeconomic factors in each equation:

Real GDP growth, YoY, three-quarter centered moving averageUnemployment × the change in unemployment from t to t + 1Average weekly hours in manufacturing

Baum, Corlu, Tunay (BC / CFG) Modeling Rating Transitions StataConf 2016 24 / 33

Modeling asset quality rating transition matrices Including macroeconomic factors

To capture common factors influencing all borrowers, we include a setof macroeconomic factors in each equation:

Real GDP growth, YoY, three-quarter centered moving averageUnemployment × the change in unemployment from t to t + 1Average weekly hours in manufacturing

Baum, Corlu, Tunay (BC / CFG) Modeling Rating Transitions StataConf 2016 24 / 33

Modeling asset quality rating transition matrices Including macroeconomic factors

To capture common factors influencing all borrowers, we include a setof macroeconomic factors in each equation:

Real GDP growth, YoY, three-quarter centered moving averageUnemployment × the change in unemployment from t to t + 1Average weekly hours in manufacturing

Baum, Corlu, Tunay (BC / CFG) Modeling Rating Transitions StataConf 2016 24 / 33

Modeling asset quality rating transition matrices Including macroeconomic factors

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Baum, Corlu, Tunay (BC / CFG) Modeling Rating Transitions StataConf 2016 25 / 33

Modeling asset quality rating transition matrices Evaluation of model performance

The explanatory power of the estimated model is very good, withindividual equations’ R2 values of 0.68 for AAA-rated borrowers,between 0.79 and 0.88 for AA+ through A rated borrowers, andgenerally above 0.92 for the remaining asset quality ratings.Tests for exclusion of the macroeconomic factors reject their nullhypothesis at the 90% or 95% level.The SUR technique improves upon equation-by-equation OLSwhen the residual correlation matrix contains sizable off-diagonalelements.The Breusch–Pagan test for a diagonal correlation matrix rejectsits null hypothesis at any level of confidence, implying that thereare important contemporaneous correlations among theequations’ errors: common shocks not .captured by the macrofactors.

Baum, Corlu, Tunay (BC / CFG) Modeling Rating Transitions StataConf 2016 26 / 33

Modeling asset quality rating transition matrices Evaluation of model performance

The explanatory power of the estimated model is very good, withindividual equations’ R2 values of 0.68 for AAA-rated borrowers,between 0.79 and 0.88 for AA+ through A rated borrowers, andgenerally above 0.92 for the remaining asset quality ratings.Tests for exclusion of the macroeconomic factors reject their nullhypothesis at the 90% or 95% level.The SUR technique improves upon equation-by-equation OLSwhen the residual correlation matrix contains sizable off-diagonalelements.The Breusch–Pagan test for a diagonal correlation matrix rejectsits null hypothesis at any level of confidence, implying that thereare important contemporaneous correlations among theequations’ errors: common shocks not .captured by the macrofactors.

Baum, Corlu, Tunay (BC / CFG) Modeling Rating Transitions StataConf 2016 26 / 33

Modeling asset quality rating transition matrices Evaluation of model performance

The explanatory power of the estimated model is very good, withindividual equations’ R2 values of 0.68 for AAA-rated borrowers,between 0.79 and 0.88 for AA+ through A rated borrowers, andgenerally above 0.92 for the remaining asset quality ratings.Tests for exclusion of the macroeconomic factors reject their nullhypothesis at the 90% or 95% level.The SUR technique improves upon equation-by-equation OLSwhen the residual correlation matrix contains sizable off-diagonalelements.The Breusch–Pagan test for a diagonal correlation matrix rejectsits null hypothesis at any level of confidence, implying that thereare important contemporaneous correlations among theequations’ errors: common shocks not .captured by the macrofactors.

Baum, Corlu, Tunay (BC / CFG) Modeling Rating Transitions StataConf 2016 26 / 33

Modeling asset quality rating transition matrices Evaluation of model performance

The explanatory power of the estimated model is very good, withindividual equations’ R2 values of 0.68 for AAA-rated borrowers,between 0.79 and 0.88 for AA+ through A rated borrowers, andgenerally above 0.92 for the remaining asset quality ratings.Tests for exclusion of the macroeconomic factors reject their nullhypothesis at the 90% or 95% level.The SUR technique improves upon equation-by-equation OLSwhen the residual correlation matrix contains sizable off-diagonalelements.The Breusch–Pagan test for a diagonal correlation matrix rejectsits null hypothesis at any level of confidence, implying that thereare important contemporaneous correlations among theequations’ errors: common shocks not .captured by the macrofactors.

Baum, Corlu, Tunay (BC / CFG) Modeling Rating Transitions StataConf 2016 26 / 33

Modeling asset quality rating transition matrices Evaluation of model performance

In-sample comparisons of the actual and predicted default rates foreach asset quality rating show that the model is able to capture thetrajectories of these series quite well through the period of the financialcrisis.

Baum, Corlu, Tunay (BC / CFG) Modeling Rating Transitions StataConf 2016 27 / 33

Modeling asset quality rating transition matrices Evaluation of model performance

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Default Rate, AAA Prediction

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AA-

Modeled Default Rate

Baum, Corlu, Tunay (BC / CFG) Modeling Rating Transitions StataConf 2016 28 / 33

Modeling asset quality rating transition matrices Evaluation of model performance

0.002.004.006.008

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Default Rate, A+ Prediction

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Default Rate, BBB+ Prediction

BBB+

Modeled Default Rate

Baum, Corlu, Tunay (BC / CFG) Modeling Rating Transitions StataConf 2016 29 / 33

Modeling asset quality rating transition matrices Evaluation of model performance

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.02.025

1998q3 2002q1 2005q3 2009q1 2012q3

Default Rate, BBB Prediction

BBB

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1998q3 2002q1 2005q3 2009q1 2012q3

Default Rate, B Prediction

B

Modeled Default Rate

Baum, Corlu, Tunay (BC / CFG) Modeling Rating Transitions StataConf 2016 30 / 33

Modeling asset quality rating transition matrices Evaluation of model performance

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Default Rate, CCC Prediction

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Default Rate, SD Prediction

SD

Modeled Default Rate

Baum, Corlu, Tunay (BC / CFG) Modeling Rating Transitions StataConf 2016 31 / 33

Modeling asset quality rating transition matrices Summary

The SUR model jointly captures the movements of default ratesacross asset quality ratings by exploiting the persistence in defaultrates for a given asset quality and its neighbors.The model can be augmented with macroeconomic factors tocapture common shocks affecting all borrowers.The SUR technique makes use of the contemporaneouscorrelation across equations’ errors to gain efficiency in estimationrelative to equation-by-equation OLS.

Baum, Corlu, Tunay (BC / CFG) Modeling Rating Transitions StataConf 2016 32 / 33

Modeling asset quality rating transition matrices Summary

The SUR model jointly captures the movements of default ratesacross asset quality ratings by exploiting the persistence in defaultrates for a given asset quality and its neighbors.The model can be augmented with macroeconomic factors tocapture common shocks affecting all borrowers.The SUR technique makes use of the contemporaneouscorrelation across equations’ errors to gain efficiency in estimationrelative to equation-by-equation OLS.

Baum, Corlu, Tunay (BC / CFG) Modeling Rating Transitions StataConf 2016 32 / 33

Modeling asset quality rating transition matrices Summary

The SUR model jointly captures the movements of default ratesacross asset quality ratings by exploiting the persistence in defaultrates for a given asset quality and its neighbors.The model can be augmented with macroeconomic factors tocapture common shocks affecting all borrowers.The SUR technique makes use of the contemporaneouscorrelation across equations’ errors to gain efficiency in estimationrelative to equation-by-equation OLS.

Baum, Corlu, Tunay (BC / CFG) Modeling Rating Transitions StataConf 2016 32 / 33

Modeling asset quality rating transition matrices The bottom line

The bottom line

In conclusion, Stata has become an essential element of the toolkit atCFG’s Risk Analytics group. Although other statistical tools anddatabases are also used for data management tasks, the group’seconometric modeling is heavily Stata-oriented due to the program’scapabilities, programmability, cost-effectiveness and overall ease ofuse.

Baum, Corlu, Tunay (BC / CFG) Modeling Rating Transitions StataConf 2016 33 / 33