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# Deriving PD

#### Ali Ehsan Abbas

##### New Member
Dear @David Harper CFA FRM

Knowing default is characterized by a bernoulli distribution, can you please advise if an analytical solution exists to deriving PDs from sigma PD.

Let me be more precise..if sigma PD = 7%. What is PD? Would appreciate if you share the workout!

Thanks.

#### Matthew Graves

##### Member
Subscriber
Variance of a Bernouilli Distribution is given by p(1-p):

$\sigma&space;^2=p(1-p)&space;\therefore&space;p^2-p+\sigma^2=0$

$p=\frac{1\pm&space;\sqrt{1-4\sigma^2}}{2}$

Plugging in the SD of 0.07 gives p = 0.995 or 0.005

#### ShaktiRathore

##### Well-Known Member
Subscriber
Hi,
PD is a Bernoulli random variable therefore if 1 is the default state and 0 is the no default sate then E(PD)=PD.1+(1-PD).0=PD and
the variance(PD) = PD.(PD-1)^2+(1-PD).(PD-0)^2
variance(PD) = PD(1-PD)^2+PD^2.(1-PD) = PD.(1-PD)(1-PD+PD)=PD(1-PD).

We know that the variance of PD=sigma PD^2=PD*(1-PD)
=> PD-PD^2 = sigma PD^2 => PD^2-PD+sigma PD^2 =0 => PD = (1+sqrt(1-4* sigma PD^2))/2
for PD=0.07 => PD = (1+sqrt(1-4* 0.07^2))/2 = 0.9951 or PD = (1-sqrt(1-4* 0.07^2))/2=0.0049
thanks

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#### bpdulog

##### Active Member
So this was part of a question posted in the WhatsApp group. It says 7% is the SD of the "default indicator"? Are we supposed to assume default indicator = PD? Because I have not encountered that wording before

#### David Harper CFA FRM

##### David Harper CFA FRM
Staff member
Subscriber
@bpdulog "default indicator" does appear in De Servigny and Meissner (and probably elsewhere), it's a pretty common credit risk term. But I do not think that σ(indicator) implies σ(PD). Consider Altman's Z-score (https://en.wikipedia.org/wiki/Altman_Z-score) where default can be predicted based on the the Z-score which is the "indicator." In this case, the indicator itself is a linear function of several variables which can be mapped functionwise (e.g., via logistic transform) to a PD, but σ(indicator) will not equal σ(PD). So I would say that, at best, PD = F(indicator).

@Matthew Graves sweet solution, thank you!

#### bpdulog

##### Active Member
@bpdulog "default indicator" does appear in De Servigny and Meissner (and probably elsewhere), it's a pretty common credit risk term. But I do not think that σ(indicator) implies σ(PD). Consider Altman's Z-score (https://en.wikipedia.org/wiki/Altman_Z-score) where default can be predicted based on the the Z-score which is the "indicator." In this case, the indicator itself is a linear function of several variables which can be mapped functionwise (e.g., via logistic transform) to a PD, but σ(indicator) will not equal σ(PD). So I would say that, at best, PD = F(indicator).

@Matthew Graves sweet solution, thank you!
Thanks! @Ali Ehsan Abbas tagging for reference