Transition Matrix Forecasting Model Validation

The purpose of this model was to project scenarios for Regulatory purposes. Federal Reserve provided forecasted values of various Macro Economic variables for Base, Adverse and Severely Adverse scenarios for 13 quarters.

  1. Quarterly Historical Transition Probability Matrix (TPM) from 2000Q1 to 2017Q2 from Moody’s were used. Thus, there were in total 70 TPMs.
  2. Moody’s also provided a Base Matrix that was long term average based on their business experience.
  3. A Stressed Matrix was calculated based on 2008-2009 recession.
  4. For each historical matrix a weight parameter ‘w’ was calculated which was based on the distance between Based Matrix and Stressed Matrix. Thus, a series of ‘w’ was calculated and this series was of length 70.
  5. This series was regressed with all the Macro Economic variables that were provided by the Fed and the best combination was chosen. The chosen combination was BBB Spread that was difference between treasury and corporate bond rates.
  6. Using the relationship and the Fed’s forecasts ‘w’ values were forecasted for Base, Adverse, and Severely Adverse scenarios.
  7. Using the projected values of ‘w’, the relationship of point 4 was reversed and forecasted TPM was calculated.

Key limitations of this methodology

Firstly, the optimization process limited the values of projected matrix.

Suppose the TPM is an 8×8 matrix, thus it is a 64 variable matrix which is approximated by one weight variable ‘w’. Primarily, ‘w’ averages the volatility of 64 variables and hence understates the volatility for some and overstates the volatility for other variables.

Secondly because of the optimization methodology, the values of projected upper half of the TPM gets capped by the value of Stressed Matrix and floored by the Base Matrix. Similarly, vice versa for the lower half of the TPM.

Thirdly, the historical quarterly TPM are sparse matrices so ‘w’ may not be able to capture the essence.


The methodology can be back tested by multiple approaches.

  1. Monte Carlo simulation

The projection was for 13 quarters, so from 66 TPMs random 13 TPMs were chosen multiple times with and without replacement. The cumulative TPMs were calculated and default values of various ratings were calculated. The project values were compared with various percentiles.

  1. The Severely Adverse projected TPM was compared with 13 quarter matrices starting from 2007Q4 to 2009Q3.
  2. The calculate 66 ‘w’ values were used to recalculated the respective historical TPM and they were compared with the actual historical TPM.

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