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# Cumulative probability of default

#### ahnnecabiles

##### New Member
Hi David,

I have questions regarding the probability of default. First, regarding your screencast on the cumulative probability of default, why don't we use the 2-year spot rates for the treasury and corporate instead to compute for the 2-yr cumulative probability of default, i.e. 1-{1+(2-yr treasury)/1+(2-yr corporate)} = the 2-yr cumulative probability of default? Isn't it that the difference between the 2-yr treasury and 2-yr corporate includes the expected probability that the bond will default on the the two year period (its 2-yr cumulative probability)? Why do we have to take the long step of getting the forward rate to compute for the 2-yr cumulative probability?

Second, what is the difference between the cumulative probability of default and the conditional probability of default? According to Hull, the cumulative probability of default (the probability of the asset defaulting at the end of year 2 on the condition that it did not default on year 1) is just the sum of 2 marginal probabilities of default (i.e. marginal probability of default for year 1 and the marginal probability of default for year 2). Thus, if an asset has a 2-yr cumulative probability of default of .57 percent (as in his example in his book, Chapter 20 - credit risk), and has a marginal default probability of .20 percent in year 1, then it has .37 percent marginal default probability for year 2 (.57 - .20 = .37). Or, we will just add the two marginal probabilities to come up with the probability that it will default at the end of year 2. This is different from what Saunders says that it has to be 1-(probability of repayment year 1)(probability of repayment year 2). Whereas, conditional probability of default is the same as default intensity, in which the probability of default at time t is equal to 1-e^default intensity x t. I always thought that cumulative default probability and conditional default probability (the probability of default on the nth year on the condition that it did not default on the previous year/s) are the same. Please help.

Lastly David (sorry for the long queries but only your explanation on these could make me sleep :long, if we are asked, there are 10 independent bonds with the same marginal 1-yr edf of 5%, what is the probability that exactly one of the bonds will default at the end of the year? According to the solution (FRM Handbook p. 426), the probability is equal to 10 x .05 x (1-.05)^9. Why? Isn't it that it is about the probability of the union of x and y, such that p(x or y) = P(x) + P(y) where P(x) is the pd of one bond, such that the sum of their probabilities (10 x .05) is the probability that any one of them will default?

Hope my questions are clear, thanks so much. Will be waiting for your reply.

Thanks and more power.

#### David Harper CFA FRM

##### David Harper CFA FRM
Staff member
Subscriber
Hi Chinquee,

In regard to your first point, YES I AGREE that "shortcut" works. Note 2-year cumulative = 1 - (2 yr Probability of Repay)^2; i.e., we need to square to get the compounding thru both years.
I added a page behind the XLS on the member page; the red cell proves your point! I still like following Saunder's 'long way around' because it forces us to think of the spot rate curve as sequence of forward rates...

Regarding Hull's PDs:
I don't think Hull uses term marginal PD (please correct me if wrong?).
Hull's conditional PD is the same as Saunder's marginal PD. They both refer to, "what is probability in Year X given no prior defaults." Hull is saying his unconditional PDs sum to his cumulative PDs.

For your example, here is how they reconcile:

For the Baa, the 2-year marginal PD (i.e., PD in Year 2 conditional on no previous default) =
0.37%/(1-0.2%) = 0.37074 marginal PD in year 2
Such that 2-year cumulative PD = 1 - (1-p1)(1-p2) = 1 - (1-0.2%)*(1-0.37074%) = 0.57% (same as Hull)

So, note between them, we have:
* Hull's conditional PD is the same as Saunder's marginal PD
* Saunder's marginal PD (which is technically, yes, is a conditional PD): prob of default in year X conditional on no previous default
* Hull's "unconditional PD" is the PD for year X at the start (at time 0).
* Note Hull's unconditional PD must be < Saunder's marginal PD

To my thinking, I find Hull's unconditional PD to be the *least* useful/intuitive. Yes, we can add them, but really, we are just extracting them (in reverse) from his cumulative. Rather, I think Saunders gives the right focus: marginal (conditional) PD and cumulative PD.

(Please note this use of 'marginal' here in Saunders, as i think about it, is different than Gujarati. Gujarat has unconditional PDF = marginal PDF whereas Saunders marginal PD is really a conditional PD.)

In regard to: "I always thought that cumulative default probability and conditional default probability (the probability of default on the nth year on the condition that it did not default on the previous year/s) are the same," please note:

* Saunder's Marginal PD = Hull's Conditional PD = the probability the bond will default in the x-th year given it has not previously defaulted; e.g., probability bond defaults in 3rd year given (conditional on) no prior default
* Cumulative PD = probability that bond will default on any given year during an x-year horizon; e.g., probability bond defaults during five years (could be 1st year, 2nd year, 3rd, etc). Note we must be at the beginning of the x-year horizon otherwise we are in a marginal/conditional mood.

* Hull's un-conditional PD (in my humble opinion) is not particularly useful, or frankly relevant to the exam. Our focus is marginal vs. cumulative, but you have (impressively!) surfaced that marginal is a misnomer

On the basket CDS, the bonds are independent but the outcomes are not mutually exclusive. You could add them, per your union formula, if the outcomes where mutually exclusive. But if bond #1 defaults, this does not preclude bond #2 default. So, a union overstates because it includes the outcomes where bonds mutually default (e.g., the outcome were both bond #1 and bond #2 default). P (A or B) = P(A) + P(B) - overlaps. If it helps, a way to see the formula is:

* Okay, what is prob that the first SPECIFIC bond will default only. That is 5% * 95%^9 (i.e., and the rest do not default). That gives 3.15%. But it can be any of the bonds, not just the first, so how many combinations are there? There are 10 different "combinations" of this (i.e., i don't care which specific bond is first to default, so 10*5%*95%^9. So, if my bonds are ordered 1 thru 10, 3.15% is probability that only bond #1 defaults, and 10*3.15% is probability that only a single bond (#1 thru #10 defaults).

Hope that helps, great questions, thanks for engaging so thoughtfully!

David

#### ahnnecabiles

##### New Member
Hi David,

Got it! Really appreciate your efforts in clearly explaining and illustrating your points. Thanks so much!

God bless and more power.

#### sleepybird

##### Active Member
David,
Glad that I found this page. I was really confused with marginal PD vs. conditional PD. Can you help clarify my understanding below?
PD = 1 - (1+rf)/(1+y): This is a marginal PD in year 1?
PD = 1 - (1+rf)^2/(1+y)^2: This is a marginal PD over two years?
PD = 1 - (1+one year forward risk free)/(1+one year forward bond yield): This is a CONDITIONAL PD in year 2?
All the PDs above are risk-neutral PDs?
Finally, cumulative PD= 1 - (1-PD1)(1-PD2)(1-PD3).....: Only PD1 is a marginal PD, all other PD2, PD3,...PDn are conditional PD?

Here's a GARP FRM practice question: A corporate bond will mature in three years. The marginal probability of default in year one is 3%. The marginal probability of default in year two is 4%. The marginal probability of default in year three is 6%. What is the cumulative probability that default will occur during the three year period? Answer given: 1-(1-3%)(1-4%)(1-6%)=12.47%.
The 4% and 6% PDs for year 2 and year 3 are actually conditional PDs and not marginal PDs (just a messed-up of terminology between Saunder and Hull), am I correct?
I'm really struggling with this.

Thanks!