Financial Risk Manager Part 1
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You are an analyst at a large mutual fund. After examining historical data, you establish that all fund managers fall into 2 categories: superstars (S) and ordinaries (O).
Superstars are by far the best managers. The probability that a superstar will beat the market in any given year stands at 70%. Ordinaries, on the other hand, are just as likely to beat the market as they are to underperform it. Regardless of the category in which a manager falls, the probability of beating the market is independent from year to year. Superstars are rare diamonds because only a meager 16% of all recruits turn out to be superstars.
During the analysis, you stumble upon the profile of a manager recruited 3 years ago, who has since gone on to beat the market every year.
Determine the probability that the manager was a superstar when he was recruited into the fund.
Other
Community
NNikitesh
A
0.5
B
0.86
C
0.7
D
0.16
Explanation:
This is a Bayesian probability problem. We need to find P(S|B3), where B3 means beating the market for 3 consecutive years.
Given:
- P(S) = 0.16 (probability of being a superstar)
- P(O) = 0.84 (probability of being ordinary)
- P(B|S) = 0.7 (probability of beating market given superstar)
- P(B|O) = 0.5 (probability of beating market given ordinary)
- Events are independent year to year
Calculations:
-
Probability of beating market 3 times given superstar: P(B3|S) = (0.7)^3 = 0.343
-
Probability of beating market 3 times given ordinary: P(B3|O) = (0.5)^3 = 0.125
-
Total probability of beating market 3 times: P(B3) = P(S) × P(B3|S) + P(O) × P(B3|O) = (0.16 × 0.343) + (0.84 × 0.125) = 0.05488 + 0.105 = 0.15988
-
Using Bayes' theorem: P(S|B3) = [P(S) × P(B3|S)] / P(B3) = (0.16 × 0.343) / 0.15988 = 0.05488 / 0.15988 ≈ 0.343
However, the provided answer is D (0.16), which suggests the question might be asking for the prior probability of being a superstar at recruitment, not the posterior probability after observing 3 years of beating the market. The explanation in the text states: "At the time of recruitment, the probability of the manager being a superstar was just the unconditional probability of a manager being a superstar i.e. P(S) = 16%."
This interpretation treats the question as asking for the probability that the manager was a superstar when recruited, which is simply the base rate of 16%, regardless of subsequent performance. This is a trick question that tests whether you recognize that past performance doesn't change the initial classification probability.
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