Estimated study time: 45 minutes
Content:
Probability theory provides the mathematical framework for quantifying uncertainty in investment analysis. The probability of an event is a number between 0 and 1 representing the likelihood that the event occurs. Unconditional probability (marginal probability) is the probability of an event regardless of any other events: P(A). Conditional probability is the probability of an event given that another event has already occurred: P(A|B) = P(A and B) / P(B). This distinction is fundamental — the probability that a stock earns a positive return is different from the probability that it earns a positive return given that the market has declined. For independent events, P(A|B) = P(A): knowing that B occurred provides no information about A.
The multiplication and addition rules govern the calculation of joint and combined probabilities. The multiplication rule for joint probability states: P(A and B) = P(A|B) × P(B). For independent events, this simplifies to P(A and B) = P(A) × P(B). The addition rule states: P(A or B) = P(A) + P(B) – P(A and B). For mutually exclusive events (which cannot both occur), P(A and B) = 0, so P(A or B) = P(A) + P(B). Bayes' theorem is a critical extension: it allows updating of prior probabilities based on new information. The formula is: P(A|B) = [P(B|A) × P(A)] / P(B). In investment contexts, Bayes' theorem is used to update the probability of a scenario (e.g., recession) given observed economic data (e.g., yield curve inversion).
Expected value is the probability-weighted average of all possible outcomes: E(X) = Σ[P(xi) × xi]. For a portfolio, expected return is the weighted sum of expected returns of individual assets. Variance of a random variable X is E[(X – E(X))^2] = Σ[P(xi) × (xi – E(X))^2]. Covariance measures how two variables move together: Cov(X,Y) = E[(X–E(X))(Y–E(Y))]. Correlation normalizes covariance to the –1 to +1 range: Corr(X,Y) = Cov(X,Y) / [σ(X) × σ(Y)]. For a two-asset portfolio, variance = w1^2 × σ1^2 + w2^2 × σ2^2 + 2 × w1 × w2 × Cov(1,2). These formulas are the building blocks of portfolio theory.
Counting methods — combinations and permutations — are needed for probability problems involving selections from a group. Permutations count ordered arrangements: nPr = n! / (n–r)!. Combinations count unordered selections: nCr = n! / [r! × (n–r)!]. For example, if a committee of 3 is chosen from 10 candidates, the number of possible committees is 10C3 = 10! / (3! × 7!) = 120. Permutations matter when order counts — for example, ranking three candidates first, second, and third among ten gives 10P3 = 720 arrangements. The binomial distribution uses combinations to model the probability of exactly k successes in n independent trials where each trial has probability p of success: P(X=k) = nCk × p^k × (1–p)^(n–k). This is directly applicable to modeling investment outcomes across discrete scenarios.
Key Terms:
Quiz Questions:
Q1. The probability that a stock rises in a given month is 60%. The probability that interest rates rise in the same month is 40%. If these events are independent, what is the probability that both the stock rises AND interest rates rise?
A) 100% B) 76% C) 24% D) 20%
Answer: C — For independent events, P(A and B) = P(A) × P(B) = 0.60 × 0.40 = 0.24 = 24%. Independence means knowledge that interest rates rose does not affect the probability the stock rises. Option B uses the addition rule incorrectly (0.60 + 0.40 – 0.24 = 0.76 is P(A or B), not P(A and B)).
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Q2. An analyst believes there is a 30% probability of a recession next year. If there is a recession, the probability that a particular bond defaults is 40%. If there is no recession, the default probability is 5%. What is the unconditional probability that the bond defaults?
A) 45% B) 15.5% C) 12% D) 16.5%
Answer: B — Using total probability: P(default) = P(default|recession) × P(recession) + P(default|no recession) × P(no recession) = 0.40 × 0.30 + 0.05 × 0.70 = 0.12 + 0.035 = 0.155 = 15.5%. This is the law of total probability — weighting conditional probabilities by the probability of each conditioning scenario.
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Q3. A portfolio manager knows that if an earnings report beats expectations, the stock rises 80% of the time. If the report misses, the stock falls 70% of the time. The prior probability of beating earnings is 60%. After seeing the stock rise, what is the updated probability that the earnings report beat expectations? (Use Bayes' theorem.)
A) 80% B) 60% C) 72% D) 83%
Answer: D — P(beat) = 0.60, P(miss) = 0.40. P(rise|beat) = 0.80, P(rise|miss) = 0.30. P(rise) = 0.80 × 0.60 + 0.30 × 0.40 = 0.48 + 0.12 = 0.60. P(beat|rise) = [P(rise|beat) × P(beat)] / P(rise) = (0.80 × 0.60) / 0.60 = 0.48 / 0.60 = 0.80. Hmm, this gives 80%. Let me note that Bayes' theorem updates prior probability upward when the observed event (rise) is more likely given the hypothesis (beat). The answer is A (80%) if we use these numbers exactly.
Answer: A — Applying Bayes' theorem: P(beat|rise) = [P(rise|beat) × P(beat)] / P(rise) = (0.80 × 0.60) / [(0.80 × 0.60) + (0.30 × 0.40)] = 0.48 / (0.48 + 0.12) = 0.48 / 0.60 = 80%. The observation that the stock rose increases the probability of a beat from the prior 60% to the posterior 80%.
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Q4. An investment committee of 4 members is to be selected from 9 candidates. How many different committees are possible?
A) 3,024 B) 126 C) 36 D) 362,880
Answer: B — The number of combinations of 9 taken 4 at a time is: 9C4 = 9! / (4! × 5!) = (9 × 8 × 7 × 6) / (4 × 3 × 2 × 1) = 3,024 / 24 = 126. Since a committee has no order (member A is not different from member B on an unordered committee), we use combinations, not permutations. Option A (3,024) would be the answer for ordered arrangements (permutations, 9P4).
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Q5. A two-asset portfolio holds 40% in Asset X (standard deviation 10%) and 60% in Asset Y (standard deviation 15%). The correlation between X and Y is 0.5. What is the portfolio variance?
A) 0.0121 B) 0.0169 C) 0.0148 D) 0.0196
Answer: C — Portfolio variance = w_X^2 × σ_X^2 + w_Y^2 × σ_Y^2 + 2 × w_X × w_Y × Corr × σ_X × σ_Y = (0.4)^2 × (0.10)^2 + (0.6)^2 × (0.15)^2 + 2 × 0.4 × 0.6 × 0.5 × 0.10 × 0.15 = 0.16 × 0.01 + 0.36 × 0.0225 + 0.48 × 0.5 × 0.015 = 0.0016 + 0.0081 + 0.0036 = 0.0133. With a correlation below 1.0, diversification reduces variance below the weighted average of individual variances.
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