Introduction

Credit risk assessment is a critical aspect of financial analysis. In this chapter, we delve into the concept of the hazard rate and its significance in calculating the unconditional default probability of a credit asset. The hazard rate provides valuable insights into the likelihood of default, offering a robust tool for risk evaluation.

In the realm of credit risk evaluation, understanding the probability of default is pivotal. The hazard rate, often referred to as the instantaneous default rate, plays a key role in quantifying this probability. It captures the rate at which a credit asset transitions from a non-default state to a default state within an infinitesimally small time interval. By comprehending the hazard rate, we gain a deeper understanding of the dynamics of credit asset defaults.

Hazard Rate and Unconditional Default Probability

The hazard rate, denoted by λ(t), is a fundamental concept in survival analysis and credit risk modeling. It represents the conditional probability that a credit asset defaults within a very small time interval, given that it has survived up to that point in time. Mathematically, it can be expressed as follows:

$$\lambda (t)=\underset{\mathrm{\Delta }t\to 0}{lim}\frac{P(t\le T<t+\mathrm{\Delta }t\mid T\ge t)}{\mathrm{\Delta }t}$$

Here, $T$ represents the random variable denoting the time of default for the credit asset. The unconditional default probability $PD$ over a specific time horizon can be calculated using the integral of the hazard rate. This is expressed as:

$$PD(t)={\int }_0^t\lambda (u)e^{-{\int }_0^u\lambda (v)dv}du$$

In this equation, $t$ represents the time horizon for which we’re calculating the default probability.

Example: Let’s consider a credit asset with a hazard rate function $\lambda (t)=0.02$ and a time horizon of 3 years. Using the formula, we can calculate the unconditional default probability $PD(3)$:

$$PD(3)={\int }_0^{3}0.02e^{-{\int }_0^{u}0.02dv}du$$

After evaluating the integral, we find that $PD(3)$ is approximately 0.0586, or 5.86%.