The structure of unrevealed Bads in Good/Bad risk scores

In calculating risk scores for making predictions and decisions about loan defaults, it is common practice to base assessments on a population of individuals whose loans have not yet attained a final status or trapped state of Good (G: paid in full) or Bad (B: default, bankrupt, written off, no response, etc). When active accounts are examined prior to end of loan term, we describe them as Contaminated Goods (CG) because they contain some Bads that default at a later time. In such cases, one can easily misestimate or misinterpret the eventual population odds and scores because the CG to B odds at any point in time is larger than G to B at the end of the loan. It is shown that if the risk score is a sufficient statistic and if the Information Odds score for Goods at the end-of-term is normal with variance σ2 in a population of terminated loan accounts, then so also is the conditional score distribution for Bads; surprisingly, the theoretical means are ±0.5σ2. When active accounts are contaminated by unrevealed Bads not yet classified as such, the conditional score distribution is a mixture of normal distributions with a variance larger than σ2; thus, variances of Active (CG) and Bad (B) accounts are unequal and the log of fitted odds versus score is convex, departing from the traditional assumption of a linear fit.