## bionomial distribution

pdwary
Posts: 3
Joined: Sun Aug 19, 2012 4:12 pm

### bionomial distribution

with ref to the slide 61 of quant 1. Question says

• Company ABC was incorporated on January 1, 2004. it has expected annual default rate of 10%.
Assuming a constant quarterly default rate, what is the probability that company ABC will not have
defaulted by April 1, 2004?
A. 2.4%
B. 2.5%
C. 97.4%
D. 97.5%
My question is why cant we equate pd=(.1)raise to the power 1/4? where pd be the quaterly probability of default.

what is the reason for starting the solution from non default that solution is the slide 62 ..starts with let nd be the probility of non default for a quater. then it was equated with 1-nd^4=.1

Tags:

praghavan039
Posts: 4
Joined: Tue Aug 14, 2012 3:59 pm

### Re: bionomial distribution

I agree.
As per your statement, pd = (0.1)^1/4 = 0.56234.. (This is the probability of default)
So, the probability that ABC would not have defaulted is 1 - 0.56234 = 0.43766 which is 43.7%

As per slide,

the probability of not defaulting for full year = (nd)^4
the probability of defaulting for full year = 1 - (nd)^4.

If the above statements can be said, can we also say that if

the probability of not defaulting in 1 quarter = nd
the probability of defaulting in 1 quarter = 1 - nd
the probability of defaulting in a full year = (1 - nd)^4.

If we equate this to 10%

(1 - nd)^4 = 0.1
1 - nd = 0.56234
nd = 0.43766 = 43.7%

So, is the formula supposed to be (1 - nd)^4 instead of 1 - (nd)^4 ??

praghavan039
Posts: 4
Joined: Tue Aug 14, 2012 3:59 pm

### Re: bionomial distribution

Looking further, I believe my above expln is wrong as i just consider one case of (1 - nd)^4. - Probability of defaulting in 1 full year. That is defaulting in all the four quarters.

But, for considering annual default rate, i failed to consider other possibilities that it can default in 1st quarter and not in 2nd, 3rd, 4th. It can default in 2nd qrtr and not in 1st, 3rd & 4th. It can default in 1st and 3rd quarter and not in 2nd and 4th, etc..

So, let nd be prob. of not defaulting in one quarter. There is one and only possibility that its not defaulting in all the four quarters(nd^4), that is annually. But, annual default rate could include various default possibilities as i explained in previous para.

So,

nd^4 - Only possibility that it will not default annually.
1 - (nd^4) - All the other possibilities it will default annually.
and hence it becomes

1 - (nd^4) = 0.1.

So, your assumption pd = (0.1)^ 1/4
pd^4 = 0.1 - It is just one of the possibility that it will default in all the four quarters. But, there could be many other possibilities as well. So, the above equation is not valid.