How does Bayes' theorem work? I need a step-by-step CFA-style example.
Bayes' theorem is the hardest topic in CFA Level I Quant for me. The formula looks intimidating and I can't figure out when to apply it. Can someone show me how it works with a practical investment scenario?
Bayes' theorem is about updating your beliefs when you receive new information. It converts a prior probability into a posterior probability.
The Formula:
P(A | B) = [P(B | A) x P(A)] / P(B)
Investment Scenario:
An analyst at Crescent Capital classifies stocks as either "Growth" or "Value." Historical data shows:
- 40% of stocks are Growth: P(Growth) = 0.40
- 60% of stocks are Value: P(Value) = 0.60
- Among Growth stocks, 75% beat earnings: P(Beat | Growth) = 0.75
- Among Value stocks, 30% beat earnings: P(Beat | Value) = 0.30
Question: A stock just beat earnings. What's the probability it's a Growth stock?
Step 1: Find P(Beat) using the total probability rule:
P(Beat) = P(Beat | Growth) x P(Growth) + P(Beat | Value) x P(Value)
P(Beat) = 0.75 x 0.40 + 0.30 x 0.60 = 0.30 + 0.18 = 0.48
Step 2: Apply Bayes' theorem:
P(Growth | Beat) = [P(Beat | Growth) x P(Growth)] / P(Beat)
P(Growth | Beat) = (0.75 x 0.40) / 0.48 = 0.30 / 0.48 = 0.625
Interpretation:
Before the earnings announcement, there was a 40% chance the stock was Growth. After observing it beat earnings, the probability jumps to 62.5%. The earnings beat is evidence that updates our belief toward Growth.
The tree diagram approach (often easier on the exam):
| Path | Probability |
|---|---|
| Growth AND Beat | 0.40 x 0.75 = 0.300 |
| Growth AND Miss | 0.40 x 0.25 = 0.100 |
| Value AND Beat | 0.60 x 0.30 = 0.180 |
| Value AND Miss | 0.60 x 0.70 = 0.420 |
| Total | 1.000 |
P(Growth | Beat) = 0.300 / (0.300 + 0.180) = 0.300 / 0.480 = 62.5%
Exam tip: When you see phrases like "given new information," "after observing," or "updated probability" — that's Bayes. Draw the probability tree first, then use the formula.
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