Baye's Theorem Concept

[mastvikas]

i am finding it difficult to understand the concept of Baye's Theorem . How to analyze the sum and come up with a solution . Please Help , if there are any steps please share .

 

[ShaktiRathore]

please visit the link,

thanks

 

[mastvikas]

Hi Shakti ,

Thanks for sharing the Link , i am beginning to understand the concept . However i was finding difficult to apply same concept for below 2 questions-:
here how can i develop a tree , like in David's explanation video.
Please share some tips to counter such questions , if possible .
A-
John is forecasting a stock’s price in 2011 conditional on the progress of certain legislation in the United
States Congress. He divides the legislative outcomes into three categories of “Passage”, “Stalled” and
“Defeated” and the stock’s performance into three categories of “increase”, “constant” and “decrease” and
estimates the following events:
Passage Stalled Defeated
Probability of legislative outcome 20% 50% 30%
Probability of increase in stock
price given legislative outcome 10% 40% 70%
Probability of decrease in stock
price given legislative outcome 60% 30% 10%
A portfolio manager would like to know that if the stock price does not change in 2011, what the probability
that the legislation passed is. Based on John’s estimates, this probability is:
a. 15.5%
b. 19.6%
c. 22.2%
d. 38.7%

B
John is forecasting a stock’s performance in 2010 conditional on the state of the economy of the country in
which the firm is based. He divides the economy’s performance into three categories of “GOOD”, “NEUTRAL”
and “POOR” and the stock’s performance into three categories of “increase”, “constant” and “decrease”.
He estimates:
• The probability that the state of the economy is GOOD is 20%. If the state of the economy is GOOD, the
probability that the stock price increases is 80% and the probability that the stock price decreases is 10%.
• The probability that the state of the economy is NEUTRAL is 30%. If the state of the economy is
NEUTRAL, the probability that the stock price increases is 50% and the probability that the stock price
decreases is 30%.
• If the state of the economy is POOR, the probability that the stock price increases is 15% and the
probability that the stock price is 70%.
Billy, his supervisor, asks him to estimate the probability that the state of the economy is NEUTRAL given that
the stock performance is constant. John’s best assessment of that probability is closest to:
a. 15.5%
b. 19.6%
c. 20.0%
d. 38.7%

Regards,
vikas

 

[ShaktiRathore]

Let denotes the events that legislative passed as Passage: P , Stalled:S, Defeated
Respective probabilities of the outcome of above events is,
P(P)=probability that legislative outcome is Passage=20%
P(S)=probability that legislative outcome is Stalled=50%
P(De)=probability that legislative outcome is Defeated=30%
Now above are unconditional probabilities because occurrence of events is not affected by occurrence of other events.
The conditional probabilities that there is price of stock or decrease in stock price given the above legislative outcomes, these are conditional in the sense that outcome of stock price increase/decrease depends on the legislative outcomes. That is stock movement is affected by whether the legislation outcome is Passage: P , Stalled:S, Defeated.
I be event the stock price increase and D be the even that the stock price decrease with respective probabilities of P(I) and P(D), now since occurrence of I and D are conditional on occurrence of the legislation outcome thus their final probability of occurrence given events P,S and D have already occurred so the new probabilities of I and D given that P,S,D any of them have already occurred is changing the probability of occurrence of I and D so that our new probability of occurrence of I and D are,
P(I/P)=Probability of increase in stock given that legislative outcome is Passage P= 10%;P(D/P)=Probability of decrease in stock given that legislative outcome is Passage P= 60%;;P(NC/P)=Probability of no change in stock given that legislative outcome is Passage P=1-10%-60%=30%;as there are any three events either stock increases or it decreases or either it remains same so that their cumulative prob. should sum to 1
P(I/S)=Probability of increase in stock given that legislative outcome is Stalled S= 40%;P(D/S)=Probability of decrease in stock given that legislative outcome is Stalled S= 30%;;P(NC/S)=Probability of no change in stock given that legislative outcome is Stalled S=30%;
P(I/De)=Probability of increase in stock given that legislative outcome is Defeated D= 70%;P(D/De)=Probability of decrease in stock given that legislative outcome is Defeated D= 10%;P(NC/De)=Probability of no change in stock given that legislative outcome is Defeated De=20%;


P(NC)=P(NC/P)*P(P)+P(NC/S)*P(S)+P(NC/De)*P(De)=30%*20%+30%*50%+20%*30%=6%+15%+6%=27%
from bayes theorem, as portfolio manager already knows that stock price does not change we know P(NC) before hand so we need to find P(S/NC), that is revised probability of S given NC has already occurred,
P(S/NC)*P(NC)=P(NC/S)*P(S)=> P(S/NC)=P(NC/S)*P(S)/P(NC)
putting above value we get , P(S/NC)=30%*50%/27%=15%/27%=5/9=55.55%
similarly yu can find,
P(P/NC)=P(NC/P)*P(P)/P(NC)=30%*20%/27%=6%/27%=2/9=22.22%
P(De/NC)=P(NC/De)*P(De)/P(NC)=20%*30%/27%=6%/27%=2/9=22.22%

sorry cant give the answer to second part try it yourself based on above method..hope u understood above...

thanks

 

[David Harper CFA FRM]

Hi vikas - can i ask you the source of the questions, I am just curious b/c they are trinomial type and i wondering if they are FRM-exam based or otherwise? (I may try to help later today, I happen to be working on bayes thy portion of t2 study notes today), thanks,

 

[mastvikas]

Hi David,

These questions are from Garp's Sample Paper of 2011 .

 

[David Harper CFA FRM]

Hi mastvikas, ah, shame on me for not recognizing them from GARP papers .... i am going to add one/both to the Study Notes, then, to explain; will post back here after i've done that ... Bayes is a good example of how the FRM is sort of a rolling dynamic; it technically disappeared last year, but now it's back. Thanks,

 

[mastvikas]

Thanks David that would be a great help for our preparation.

Vikas

 

[Aleksander Hansen]

Love the Monty Hall problem! Surprisingly many get it wrong.

 

[mastvikas]

Hi Shakti,

Thank you for your inputs and very well explained summary.

i appreciate your help ,

Vikas

 

[ShaktiRathore]

Hi mastvikas,
Yeah its nice to hear that you understood the problem. Yes Baye's theroem is an important concept that has made its comeback in frm as david said. And its not at all a surprise that this problem are there in the garp's sample paper because i think garp wants to cover the bayes theorem also. This kinds of problems do might make their appearance in the exam but should not be so lengthy in terms of calculations may be one might appear some day.

thanks

 

[David Harper CFA FRM]

fyi, in case they help, i just posted two approaches to Bayes to my notebook at http://forum.bionicturtle.com/threads/bayes-theorem-two-approaches.6784/

 

[Alex K]

I find that drawing tree diagram is really useful when approaching this type of problem. In the first question the fact that 'Probability of increase in stock price given legislative outcome 10% 40% 70%" doesn't add up to 100% should give a hint of another possibility, i.e. stock price is constant.

If you use tree diagram approach it is much logical rather than trying to remember formula. The answer I got for second problem is (.3*.2)/(.2*.1+.3*.2+.5*.15) =.06/.155 = .3870 Choice D

 

[dymny]

c

 

[dymny]

ok I know the answer.