# 2.2 Modelling Framework

Assessing the risk of default over multiple periods is crucial in credit risk analysis. We aim to estimate the time to default of an obligor, given its current credit factors , where each represents a credit factor. These factors include accounting ratios, market variables and macroeconomic variables. We seek to calculate the multi-period default probability, which represents the conditional probability of default given the observed credit factors at time 0.

The multi-period default probability is defined as the conditional probability:

The variable indicates the time elapsed since the observation of the explanatory variables, while denotes the vector of these explanatory variables, also known as credit factors or covariates, observed at time 0.

The conditional hazard function, denoted as represents the probability of default occurring between time and , given that default has not occurred before time and the credit factors at time 0 are :

where is a random default time. In our model, it is the age of the firm.

Once the conditional hazard function is known, the multi-period default probability can be calculated as the complement of the survival function:

where the survival function is given by:

.

The survival function , is derived from the baseline hazard function and the estimated coefficients of the Cox model. By fitting the Cox model to historical default data and covariates, we can estimate the hazard rate and subsequently the survival function. The survival function represents the probability that a borrower survives beyond time without defaulting.

The Hazard Rate Model involves expressing the conditional hazard rate as a function of time and covariates using a multiplicative form.

The multiplicative hazard rate model can be represented as:

where :

represents the baseline hazard function, which captures the hazard rate at time in the absence of covariates .

denotes the time-varying coefficient associated with covariate .

is the total number of covariates.

This model enables the hazard rate to vary over time and across different covariates, offering greater flexibility in capturing the dynamics of default risk. The time-varying coefficients enable the model to adapt to changes in credit factors and economic conditions over time.