# Yet Another Blog in Statistical Computing

I can calculate the motion of heavenly bodies but not the madness of people. -Isaac Newton

## A Two-Stage Approach in Loan-Level Consumer PD Models

In the traditional methodology of loan-level consumer probability of default (PD) models, a general functional form is taken as below

Logit(PD) = A * X + B * Y

where A * X is the linear combination of loan-level risk characteristics and B * Y is the linear combination of macro-economic indicators. While B * Y is considered a dynamic factor and might vary according to macro-economic conditions, A * X would usually remain static for each loan regardless of macro-economic scenarios during the forecast horizon, e.g. 9 quarters for CCAR exercises. However, based upon our observation in 2008 financial crisis, it is neither realistic nor conservative to assume the loan-level consumer risk characteristics, e.g. loan-to-value, credit score, and delinquent status, immune from the economic downturn.

In this article, a two-stage approach is proposed to address the above drawback in the legacy method by the means of injecting macroeconomic dynamics into loan-level risk characteristics that were assumed static in the PD model formulation. With this innovative two-stage method, it is assumed that the risk profile of each consumer loan, e.g. A * X, is jointly determined by both the historical profile and macro-economic dynamics such that

A * X = Z = Function(macroeconomic Indicator) offset by the historical Z for each consumer loan
==> Delta(Z) = C * M

where C * M represents the linear combination of macro-economic indicators or variants and scores the health of a general economy such that C * M is positive when the economy is deteriorating and is negative when the economy is improving. As a result, the loan-level risk characteristics, e.g. A * X, would be updated dynamically upon macroeconomic inputs for the whole forecast horizon across various macroeconomic scenarios.

It is worth mentioning that Fair Issac also announced the FICO score economic calibration service in 2013, which forecasts how FICO score would change under different macroeconomic scenarios and which shares the similar idea of the two-stage approach with the intent to address the same issue faced by CCAR practitioners in the banking industry.