Conditional Transition Matrix
A Conditional Transition Matrix (also Stressed Migration Matrix) is a Transition Matrix whose transition probabilities between different states are conditional on (given the information from) the realization of certain random variables (the Stress Scenario).
Irrespective of the manner by which it is derived, a conditional matrix is a modelled outcome (depends on Model Assumptions), in contrast with the Empirical Transition Matrix that reflects more or less directly the observed transition rates
Model construction varies depending on the context. In Credit Risk Modelling context there are two broad categories
- Fitting Markov Chain models to market data that capture transition rates
- Fitting Markov Chain models to historical data (records) of state transitions
Models will typically invoke explanatory variables (covariates) which are linked to transition rates. We are then talking about "Dynamic Models of Transitions Rates"
Conditional migration probabilities can be derived for large collections of entities under the assumption of diversification of any idiosyncratic risk and the absence of unexplained variance. In such case specification of a set of common (systematic / macro) factors completely specifies the scenario. Conditional on systematic factor processes the random variables representing state transitions
are independent and identically distributed. Their limiting sum converges to their expected value (law of large numbers)
Given a dynamic model of transition rates, conditioning a transition matrix involves:
- specifying the total time period (time horizon)
- specifying the frequency (Temporal Grid)
- fixing the values of the model covariates (factors, drivers) at the desired frequency and for the required duration
Canonical (Gaussian) Model
This example is based on the canonical (Gaussian) model documented in Threshold Models. The transition thresholds for each period k are already known.
Transitions of the i-th entity are assumed to be governed by a model of the type
The conditional rating transitions probabilities are obtained by numerical integration. All entities are assumed to be members of a homogeneous ensemble (we drop the index i). Let denote the conditional probability density of the process that a given entity has not reached an absorbing state until period k-1.
The first period density is
Then for every subsequent period the conditional density is obtained by convolution
Conditional Transition Probabilities
The cumulative conditional probability for any desired transition is then given by
- A conditional transition matrix must satisfy the basic properties of a Transition Matrix
- When integrated over all possible scenarios the conditional transition matrix must reproduce the unconditional input. Symbolically, if F denotes the
(multivariate) distribution of the process Z: