Is marginal probability of default the same as conditional probability of default?

Question

I'm thrown off by the term marginal probability of default. I've seen it defined by some authors as synonymous term for conditional probability of default

conditional probability of default: probability of defaulting given no default yet.

Which is solved for as such:

$PD_{conditional} = \frac{P(default\_anytime\_before\_period\_t1) - P(default\_in\_period\_t0)}{1-P(default\_in\_period\_t0)}$

I've also seen it defined as:

The default time density function or marginal default probability is the derivative of the default time distribution w.r.t. t:

$\frac{\partial}{\partial t}P[t^*<t]=F'(t) = \lambda e^{-\lambda t}$

Where

$t^*$ is the time of default

$t$ is the point in time we are observing from

$\lambda$ is the hazard rate

$F(t)$ = cumulative default time distribution = $P[t^* <t] = 1-e^{-\lambda}$

Question

Does this mean that $\lambda e^{-\lambda t}$ is an approximation of the conditional probability of default?

Answer 1

Based on your definition, they are certainly not the same. Generally, the marginal default probability is the probability that the default happens in a given time period, such as $[t, t+\Delta]$, that is, $P(t < \tau \le t+\Delta)$. Here, $\tau$ is the default time. See Chapter 10 of the book Counterparty Credit Risk and Credit Value Adjustment for definitions. Note that \begin{align*} P(t < \tau \le t+\Delta) &=P(\tau \le t+\Delta) - P(\tau \le t) \\ &\approx \Delta \frac{\partial P(\tau \le t)}{\partial t}. \end{align*} Then, people treat the marginal default probability, over a small time period, as the density $\frac{\partial P(\tau \le t)}{\partial t}$.

However, the conditional default probability is defined by \begin{align*} P(\tau \le t_1 \mid \tau > t) &= \frac{P\big((\tau \le t_1) \cap (\tau >t)\big) }{P(\tau >t)}\\ &=\frac{P(\tau \le t_1) - P(\tau \le t) }{1-P(\tau \le t)}, \end{align*} for $t_1 > t \ge 0$.

Let $t_1 = t + \Delta$, for $\Delta$ sufficiently small. Then \begin{align*} \frac{1}{\Delta} P(\tau \le t + \Delta \mid \tau > t) &= \frac{P(\tau \le t + \Delta) - P(\tau \le t) }{\Delta \big (1-P(\tau \le t)\big)}\\ &\approx \frac{1}{1-P(\tau \le t)} \frac{\partial P(\tau \le t)}{\partial t}\\ &=-\frac{\partial \ln \big[1-P(\tau \le t)\big]}{\partial t}\\ &=\lambda. \end{align*} In fact, the hazard rate is formally defined by \begin{align*} \lambda = \lim_{\Delta \rightarrow 0} \frac{1}{\Delta} P(\tau \le t + \Delta \mid \tau > t). \end{align*}

In literatures, the terms may be misused. Then, we need to pay attention to the specific definitions.