FRM 1 - Chapter 1 - Book 2 (Quantitative Analysis)

[TG2323]

Hi all, will we be expected to answer many state Bayesian problems in the FRM part 1 exam?
Example:
Suppose 3 types of managers: underperformers beat mkt 25% of the time. In-line performers beat mkt 50% of the time. Outperformers beat mkt 75% of the time. Prior belief is that manager has 60% chance of being in-line, 20% chance of being underperformer, and 20% chance of being outperformer. If manager beats mkt 2 years in a row, what should our updated beliefs be?

 

[David Harper CFA FRM]

Hi @TG2323 The FRM exam does like Bayesian problems, although the one you cite is a notch (or two notches) more difficult than you can expect on the exam. The exam has a time limit is one factor. But that question was inspired by previous author Miller, and it was among a set that was closely sampled for exam questions .... so the thing about that question, to your point, is that a many-state Bayes is unlikely, but GARP has tested a similar two-state question. Thanks,

 

[VPint6929]

Hi David,
Apologies for my foolishness if I have missed this. I am trying to understand the Bayesian problems but can't seem to understand how to arrive at the solution to this same problem, especially the "2 years in a row". This one is Sample Problem #5 in Chapter 1, Quantitative Analysis.

 

[VPint6929]

ChatGPT helped.

To find the probability that a manager beats the market 2 years in a row, we need to calculate the probability for each manager type and then use the principle of multiplication.

We are given the following probabilities: P(A) = 0.20 (20% chance of being an underperformer) P(B) = 0.60 (60% chance of being an in-line performer) P(C) = 0.20 (20% chance of being an outperformer) P(D|A) = 0.25 (25% chance of an underperformer beating the market) P(D|B) = 0.50 (50% chance of an in-line performer beating the market) P(D|C) = 0.75 (75% chance of an outperformer beating the market)
Now, using the principle of multiplication, we can calculate the probability of a manager beating the market 2 years in a row: P(D and D) = P(D) * P(D)

For the underperformers: P(D and D|A) = P(D|A) * P(D|A) = 0.25 * 0.25 = 0.0625
For the in-line performers: P(D and D|B) = P(D|B) * P(D|B) = 0.50 * 0.50 = 0.25
For the outperformers: P(D and D|C) = P(D|C) * P(D|C) = 0.75 * 0.75 = 0.5625

Now, we multiply these probabilities by the probability of each manager type: P(D and D) = P(D and D|A) * P(A) + P(D and D|B) * P(B) + P(D and D|C) * P(C) = (0.0625 * 0.20) + (0.25 * 0.60) + (0.5625 * 0.20) = 0.0125 + 0.15 + 0.1125 = 0.275

Therefore, the probability that a manager beats the market 2 years in a row is 0.275 or 27.5%.