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marginal PD


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David Harper CFA FRM

David Harper CFA FRM
Staff member
Hi ktm,

Per Saunders, marginal PD = conditional PD. Specifically, P [default during year X | survival until start of year X].

Unfortunately, this confused myself even in the April webinar due to Gujarati. In Gujarati, under more generalized probabilities (non credit), unconditional PDF/PMF = marginal PDF ... So when i said "marginal PD" on the webinar, Daniel noticed the discrepancy and asked something like "don't you mean conditional instead of unconditional?" ... because in probability distributions, marginal = unconditional.
for more on this, see http://www.bionicturtle.com/forum/viewthread/3051/
... ultimately, it depends on "conditional on what?"

(for this reason, I am thinking about only employing Hull's terminology; he refers to "conditional PD" and avoids this confusion... but the problem is: marginal PD is quite common in practice. But I am adding to my list for GARP's attention)

thanks, David


Active Member
Hi @David Harper CFA FRM CIPM ,
This whole marginal PD is giving me a headache :). Per the assigned Malz reading, marginal PD is the derivative of the cumulative PD with respect to time right ? This is not the same as conditional PD right (conditional PD = PD conditional on survival up to this point)?
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Active Member
I think I've got it now, with a little help from elsewhere and a lot of help from David's youtube video on the topic.
In Malz, marginal PD means the unconditional probability of default in a given year. Let's not confuse this with what me usually mean when we say unconditional PD.

Unconditional PD (it's also called cumulative)
The probability of defaulting between time T0 and Tn. We are making no assumptions here about previous defaults. For all we know the bond could have previously defaulted. For example the probability of defaulting between years 1 and 5

Marginal PD
Above, we had the probability of defaulting between years 1 and 5. Marginal PD will tell us the probability of defaulting in a specific year. For example the probability of defaulting in year 4.

Here is the Youtube video:


Active Member
Hi, If the Marginal PD is the confusing and seldom used is its formula = lambda*exp(-lambda*t) good for calculating any thing i.e. I used it and tried to match it with the value of conditional PD before going through the difference between Conditional PD and Marginal PD above and I am not sure where we will use "lambda*exp(-lambda*t)"


New Member

I am sorry to bother anyone by graving this old topic up of its grave! ; ) But I have a question regarding the "unconditional PD" (defined by Hull, the PD for every single period).

I understand the math and somehow ofcourse the concept, but I have difficulties to explain why the unconditional PDs decreases of the total period, e.g.:

2% Hazard

Year / PD unconditional
1 / 1.98
2 / 1.94
3 / 1.90
4 / 1.86
5 / 1.83

What is the the explanation (not mathematically!!) that the PD will decrease during the life of the bond, although there is more uncertainty for the future? It would make possibly sense to assume that - IF - the issuer survives the previous period, that the PD might decrease, but is this the reasen for this decease? How can we be sure that he will survive? Would it make not much more sense if the unPD increases instead of decreases? I hope that somebody can help me to get more insight of this topic.

With kind regards,

David Harper CFA FRM

David Harper CFA FRM
Staff member
Hi @pint0 I think it's a good question because I've definitely struggled with the non-mathematical interpretation of the unconditional PD. Despite not being currently assigned to the topic, I actual favor Hull's interpretation (emphasize mine): "We will refer to this as the unconditional default probability. It is the probability of default during the third year as seen today."

That's only three words ("as seen today") but I find them helpful. Why? Well, first we must acknowledge the key assumption of a constant hazard rate. Our assumption here is that the conditional default probability is constant; e.g., as the instantaneous PD, λ = 2.0% implies conditional PD is constant at 1 - exp(-2%) = 1.98%. This may not be realist but if we assume it then ...

The unconditional PD during a future Year X declines because it is from the perspective of today ("as we see it today, time zero") and, as a joint probability of (cumulatively) surviving to the beginning of year X and then (conditionally) defaulting during Year X, is must decline because the cumulative probability of survival declines as we extend the horizon forward. The conditional PD each year does not change, but we are pushing forward a declining survival rate. I literally visualize standing on a seashore and looking out to the horizon: if we were to imagine the constant conditional PD as a fixed-length piece of lumber that is constant length, as it drifts out to the horizon, it shrinks in appearance to us. The length of the lumber is unchanged (aka, conditional PD is unchanged) but the distance between us and the lumber is increasing (aka, the cumulative default probability is increasing such that cumulative survival is decreasing). By way of analogy, for me, the key here is that unconditional PD is from the perspective of today. I hope that's helpful,