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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?

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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.

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  • $\begingroup$ Thanks for the very detailed answer. Could you explain what "marginal probability is the density of the default time τ" means? Much appreciated. $\endgroup$ – AfterWorkGuinness Nov 20 '15 at 18:24
  • $\begingroup$ That is from your original text, as you said "the default time density function or marginal default probability ...". $\endgroup$ – Gordon Nov 20 '15 at 18:53
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    $\begingroup$ 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. $\endgroup$ – Gordon Nov 20 '15 at 19:08
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    $\begingroup$ 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 $P(\tau \le 4)$. $\endgroup$ – Gordon Nov 20 '15 at 19:21
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    $\begingroup$ 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. $\endgroup$ – Brian B Nov 20 '15 at 22:35

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