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

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?

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.

• Thanks for the very detailed answer. Could you explain what "marginal probability is the density of the default time τ" means? Much appreciated. – AfterWorkGuinness Nov 20 '15 at 18:24
• That is from your original text, as you said "the default time density function or marginal default probability ...". – Gordon Nov 20 '15 at 18:53
• Generally, the marginal default probability is the probability that the default happens in a given time period, such as $[t, t+\Delta]$, that is, $P(t < \tau \le t+\Delta)$. As this is close to $\Delta \frac{\partial P(\tau \leq t)}{\partial t}$, people treat the marginal default probability, over a small time period, as the density. – Gordon Nov 20 '15 at 19:08
• Yes. We can say that the marginal default probability over year 4 is the probability that the default happens in year 4, that is, $P(3< \tau \le 4)$, while the cumulative default probability to year 4 is $P(\tau \le 4)$. – Gordon Nov 20 '15 at 19:21
• As a minor quibble, there is good reason to prefer a different definition of the hazard rate as the generalized derivative of the cumulative default probability, consistent with measure theory in general. – Brian B Nov 20 '15 at 22:35