Bayes Theorem

[Deepak Chitnis]

Hi David,
Struggling with the bayes theorem question. Not Getting the correct answer please help.
Question: Suppose a manager for a fund of funds uses historical data to categorize managers as excellent or average. Based on historical performance, the probabilities of excellent and average managers outperforming the market are 80% and 50% respectively. Assume that the probabilities for managers outperforming the market is independent of their performance in prior years. In addition the fund of funds manager believes that only 15% of total fund managers are excellent managers. Assume that a new manager started three years ago and beat the market in each of the past three years.
1. Using the bayesian approach, what is the approximate probability that the new manager is an excellent manager today?
A.18.30%
B.27.5%
C. 32.1%
D.42.0%
2. What is the approximate probability that the new manager will outperform the market next year using the Baysian approach?
A.31.9%
B.51.2%
C.62.6%
D.80.0%
Please help me in these questions I have tries solving this using tree's approach. And I also have another query that is I have not found on the notes that steps that garp done with bayes theorem. These two question get the correct answer using GARP's terms of bayes. Like Getting the 80%^3 and 50%^3 and using bayes formula. How to solve this types of questions in exam using tree approach or metrics approach. Which step GARP supposed to be used by us. Please help me.(Also any advise for this kind of question if possible) Sorry for the trouble.
Thank you

 

[David Harper CFA FRM]

Hi @Deepak Chitnis Interesting question, what is the source? My instinct is that it is trickier than it looks, I'm not 100% confident in my answer due to the fact that the prior is three consecutive events, but my first reaction comports with two given answers. I would divide the outcomes into either:

• three consecutive years of outperformance
  • For excellent managers, that's 0.8^3
  • for average managers, that's 0.5^3
• then the other (binomial) outcome is simply NOT three years of outperformance
  • For excellent mangers, that's (1- 0.8^3)
  • for average, that's (1-0.5^3); i.e., so these are all other permutations that are NOT three consecutive years of outperformance

Then the "tree" is:

P(E) = 0.15

• P(O^3 | E) = 0.8^3; i.e., prob of three consecutive outperforms conditional on excellent manager
  
• P(O^3' | E) = 1 - 0.8^3

P(E') = 0.85

• P(O^3 | E') = 0.5^3; i.e., prob of three consecutive outperforms conditional on aveage (E' = not excellent) manager
  
• P(O^3' | E') = 1 - 0.5^3

Per Bayes, P(E | O^3) = P(E, O^3) / P(O^3) = (.15*.8^3) / [(.15*.8^3) + (0.85*.5^3) = .4192 or 42%. Then the next answer revised the probabilities from 15/85 to 42/58 per the Bayes assumption: (41.96% * 80%) + (1-41.96% * 50%) = 62.6%. I hope that helps,

 

[Deepak Chitnis]

Hi David,
Thank you for the answer. That helps a lot. I just wanted to ask you that ^3 why did we done this? And in the GARPs official book when there are two outcomes are available they did the same ^3 thing. But in our study notes or practice exam I didn't found anything like that. We simply solving the bayes theorem using drawing our tree but as you see we cannot solve these problem with simply using the tree. Correct me if I am wrong. Thank you for your help. And by the way I have many question sets of FRM part1 I found these type of question in that. Thanks againg.

 

[Deepak Chitnis]

Hi David,
I think you liked my question that's why you posed it on your fb page, just kidding. But do you think that GARP will ask that kind of question in the exam or do you have seen question like this in GARP practice question or in real exam?
Thank you,