Forecasting Corporate Default Rates
Risk parameters are dynamic in nature, and understanding how these parameters change in time is a fundamental task for risk management.
In the first part of this example, we work with historical credit migrations data to construct some time series of interest, and to visualize default rates dynamics. In the second part of this example, we use some of the series constructed in the first part, and some additional data, to fit a forecasting model for corporate default rates, and to show some backtesting. A linear regression model for corporate default rates is presented, but the tools and concepts described can be used with other forecasting methodologies.
We work with historical transition probabilities for corporate issuers. This is yearly data for the period 1981-2005. The data includes, for each year, the number of issuers per rating at the beginning of the year and the number of new issuers per rating per year. There is also a corporate profits forecast, and a corporate spread. A variable indicating recession years, consistent with recession dates, is used mainly for visualizations.
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We start by performing some aggregations to get corporate default rates for Investment Grade (IG) and Speculative Grade (SG) issuers, and the overall corporate default rate.
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Aggregation and segmentation are relative terms. IG is an aggregate with respect to credit ratings, but a segment from the perspective of the overall corporate portfolio. Other segments are of interest in practice, for example, economic sectors, industries, or geographic regions. The data we use, however, is aggregated by credit ratings, so further segmentation is not possible.
Here is a visualization of the dynamics of IG, SG, and overall corporate default rates together. To emphasize their patterns, rather than their magnitudes, a log scale is used.
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The default rates obtained are examples of point-in-time (PIT) rates, only the most recent information is used to estimate them. On the other extreme, we can use all the migrations observed in the 25 years spanned by the dataset to estimate long-term, or through-the-cycle (TTC) default rates. Other rates of interest are the average default rates over recession or expansion years. The following figure shows the estimated PIT rates, TTC rates, and recession and expansion rates
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Using the credit data, you can build new time series of interest. We start with an age proxy that is used as predictor in the forecasting model in the second part of this example.
Age is known to be an important factor in predicting default rates. Age here means the number of years since a bond was issued. By extension, the age of a portfolio is the average age of its bonds. Certain patterns have been observed historically. Many low-quality borrowers default just a few years after issuing a bond. When troubled companies issue bonds, the amount borrowed helps them make payments for a year or two. Beyond that point, their only source of money is their cash flows, and if they are insufficient, default occurs.
We cannot calculate the exact age of the portfolio, because there is no information at issuer level in the dataset. However, and use the number of new issuers in year t-3 divided by the total number of issuers at the end of year t as an age proxy. Because of the lag, the age proxy starts in 1984. For the numerator, we have explicit information on the number of new issuers. For the denominator, the number of issuers at the end of a year equals the number of issuers at the beginning of next year. This is known for all years but the last one, which is set to the total transitions into a non-default rating plus the number of new issuers on that year.
We work with the following linear regression model for corporate default rates
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DefRate=β0+βageAGE+βcpfCPF+βsprSPR
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where
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AGE: Age proxy defined above
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CPF: Corporate profits forecast
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SPR: Corporate spread over treasuries
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As previously discussed, age is known to be an important factor regarding default rates. The corporate profits provide information on the economic environment. The corporate spread is a proxy for credit quality. Age, environment, and quality are three dimensions frequently found in credit analysis models. The in-sample fit, or how close the model predictions are from the sample points used to fit the model, is shown in the following figure.
To evaluate how this model performs out-of-sample, we set up a backtesting exercise. Starting at the end of 1995, we fit the linear regression model with the information available up to that date, and compare the model prediction to the actual default rate observed the following year. We repeat the same for all subsequent years until the end of the sample.
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For backtesting, relative performance of a model, when compared to alternatives, is easier to assess than the performance of a model in isolation. Here we include two alternatives to determine next year's default rate, both likely candidates in practice. One is the TTC default rate, estimated with data from the beginning of the sample to the current year, a very stable default rate estimate. The other is the PIT rate, estimated using data from the most recent year only, much more sensitive to recent events.
Here are the predictions of the three alternative approaches, compared to the actual default rates observed. Unsurprisingly, TTC shows a very poor predictive power. However, it is not obvious whether PIT or the linear regression model makes better predictions in this 10-year time span.
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