ECONOMETRIC MODELLING OF DEFAULT AND MIGRATION RATES OF PORTFOLIO SEGMENTS (CPV)
Category: Risk Management in Banking
This section presents the macro-economic modelling of default and migration rates. This is the CPV (Wilson, 1997a, b) framework, which addresses explicitly the cyclical dynamics of these variables. The model focuses on portfolio segments, rather than individual obligors, and default or migration rates rather than probabilities. The underlying assumption is that of the homogeneous behaviour of credit standing of firms within a portfolio segment. This contrasts with the modelling of the credit standing of individual obligors. However, CPV allows us to define segments with more than one criterion, instead of using only risk classes.
The principle underlying CPV is to relate default and migration rates to external factors. Default and migration rates of portfolio segments relate to economic variables through a logit function, to obtain values in the 0 to 1 range. In addition, the model looks forward. Predicted default and migration rates are a function of the predicted values of these factors. Predicting economic variables uses standard time series modelling with Auto Regressive Integrated Moving Average (ARIMA) techniques. We review below the CPV building blocks, related to defaults and migrations, postponing the modelling of correlations to subsequent chapters (44, 45 and 48).
CPV Building Blocks
CPV relies on several modelling options:
• Default events depend on economic conditions materialized by an economic variable
summarizing the influences of country-industry factors Xk, where the index i designates a portfolio segment and k designates each specific factor.
• The default events modelled are default rates, ratios of default frequencies to a subpopulation of firms. Hence, CPV works with grouped data rather than individual data. These subpopulations are banking portfolio segments grouped by risk class for example.
• To look forward, it is necessary to use predicted values of the factors influencing the default rates. For this purpose, CPV needs to model the predicted values of factors using standard time series models.
• To convert the predicted values of factors into default rates, CPV uses the logit form making default rates a logit function of an index summarizing the predicted factors that influence default rates.
The economic factors driving the credit risk of obligors are economic, geographic or industry variables Xk that influence directly or indirectly the credit standing of obligors. This corresponds to the intuition that all firms are sensitive to common external conditions, with a different magnitude, in addition to firm-specific factors. The purpose is to capture the cyclical dynamics of credit risk of firms. The factors Xk represent the general risk of obligors, while the residual of the fit of default rates to represents the specific risk. The model therefore splits the general and specific components of credit risk explicitly. A major benefit of this modelling technique is to make explicit the cyclical dynamics of credit risk.
CPV uses three main building blocks for modelling default and migration rates of portfolio segments:
• The first block models default rates, at date t + 1, DR, t+1 by portfolio segment i as a function of the economic index ,t+1. It uses a logit model to ensure that the default rate remains in the 0 to 1 range of values, no matter what is the predicted ;t+1 value. The economic index is a linear combination of several industry-country factors Xk;t+1. The index i refers to a portfolio segment and the index k refers to economic factors. The date t is the current date, while t + 1 refers to a future date, one period ahead. The second block models the future values of the economic—industry factors Xk; t+i as a function of their past values and error terms using an ARIMA time series model. These economic factors are common to all portfolio segments, hence there is no index i . This allows the model to look forward using lagged values of economic factors.
• The third block models the migration probabilities MR,t+1 of transition matrices consistent with the variations of default rates, allowing the migration probabilities to shift upward or downward according to the values of the default rates.
These three building blocks use three main families of equations: the first family models default rates for each segment as a logit function of ;t+1; the second models predicted economic factors Xk;t+1 as an ARIMA function of lagged values and errors; the third relates the migration probabilities MR*;t+1 by segment to the modelled default rates. Values of variables are certain as of current date t and random as at future date t + 1.
This allows us to extend the model to generate random values at a future horizon, a critical modelling block to obtain a loss distribution and Value at Risk (VaR) for credit risk (see Chapter 48).
Econometric Migrations Modelling
Migrations are necessary to find the risk class at the horizon date in order to perform a risk-adjusted valuation of the facilities. The risk status at horizon determines the valuation of the facility using market spreads corresponding to this risk class. In CPV, migration rates are conditional on the state of the economy through the modelled default rates. Credit Portfolio View uses the econometric modelling of default rates to derive what migration rates should be for portfolio segments. Migration rates, just as default rates, depend on country-industry specifics, not risk class only. Details follow on CPV use and integrated econometric modelling of default rates and migration rates.
Existing migration matrices are historical. When using long-term migration statistics, it is possible to consider them as long-term averages, or unconditional migration probabilities. However, migrations are dependent on the state of the economy as much as default rates are. Hence, there is a need to model the migration rates between any pair of risk classes consistently with default rates. Since common factors influence both default and migration rates across risk classes, they correlate them. CPV extends the default rate modelling to migration rates, making them conditional on the state of the economy based on the same factors that influence default probabilities.
Migration rates apply to any pair of classes. The migration rates from segment i to segment j, MRj, are random. Consistency with default rates imposes that all migration rates, including the default state, should sum to 1. Therefore, when default rates vary, the sum of all other migration rates across a given risk class should also vary for the sum to remain 1. In order to preserve the consistency of the transition matrix, CPV uses a shift factor that modifies the migration rates in accordance with modelled default rate values.
CPV starts with the long-term average migration matrix, considered as an unconditional migration matrix. The default rates of portfolio segments result from the model. The model shifts the migration rates across risk classes according to their deviation from their long-term values. When the default rates get higher than average, all migration probabilities to non-default states diminish and shift towards lower values. When the default rates get lower than their long-term average, all migration probabilities to the non-default state increase to compensate the default rate decline. The shifts of migration rates are proportional to the deviation of the modelled default rate from its average long-term value.
In addition, CPV considers that investment grade assets are less sensitive to the cyclical dynamics of the economy. This is in line with intuition, suggesting that low risk firms have less predictable defaults than speculative grade firms, who tend to follow more closely the economic cycles. Accordingly, the specific risk, the risk unrelated to external factors, varies across risk classes and increases relative to the upper risk class. CPV provides an adjustment for this higher specific risk of the upper risk class.
Since we use predicted values of factors, the default and migration rates are also predicted variables. They are random as of current date since the factors driving their values are. This allows us to simulate future values complying with the model structure to generate distributions of segment default rates.