I was also having issues with the reduced form approaches. I assume q(t,T) is the probability at time t that the firm defaults before time T (i.e. analogous to _(T-t)q_(t) from contingencies).
When this is extended to the rating-transition approach the notes give qi(t,T) as "the probability that default occurs given that the debt has a rating i at time t."
"B(t,T) = P(t,T)-P(t,T)[(1-rr)qi(t,T)]"
is given as the formula for the price of the debt. To me this implies that qi(t,T) incorporates the rating transition matrix and risk-premium adjustment vector. So, qi(t,T) includes the probability that the bond changes credit rating during the period from t to T.
This does not correspond with the solution to Q&A Bank 5, question 2. That suggests that qi(t,T) "is the probability that the bond has credit rating i at time T," and that we must sum over all i to get the value of the bond.
Can anyone shed some light here? I'm inclined to go with the former approach. Also, the final answer does not help with evaluating credit derivatives whose payoff depends on changes in credit rating (although the rating transition matrix could be used to value that).