4. Monotonicity of Conditional PDs

The first mapping procedure described in the last part of Section 3 works under the assumption that the conditional PD given the score P[ D S = s] is a function that decreases in its argument s. As Figure 11.2 demonstrates this need not be the case. Are there any reasonable conditions such that monotonicity of the conditional PDs given the score is guaranteed?

An answer ^[4.] to this question, in particular, will provide us with a justification of the monotonicity assumption which underlies Equations 3.7a 7c. This assumption is needed to ensure that the proposed mapping procedure for having constant PDs over time really works.

We discuss the question in the context of a hypothetical decision problem. Assume that we consider a borrower chosen at random and have been informed about his or her score value. But, of course, we do not yet know whether the borrower will default. How could we infer the value of his or her default state variable Z? In formal terms: suppose that a realization ( s, z) of ( S, Z) has been sampled. s is observed, z is not yet visible. Is z = N or z = D?

One way to come to a decision would be to fix some set A of score values such that we infer default, D, as state if the borrower s score value...