4 Probability Trees and Conditional Expectations

Expected EPS of XYZ Incorporated

• Scenario: 60% chance economy is good, 40% chance it is bad
• If economy is good: 70% chance of $5.00 EPS, 30% chance of $3.50 EPS for XYZ
• If economy is bad: 80% chance of $1.50 EPS, 20% chance of $1.00 EPS for XYZ
• Expected EPS is the sum of (Joint Probability x EPS) = $3.29

Conditional Expectation

• An analyst increasing the discount rate next quarter doubling their earnings forecast is an example.
• Conditional expectation is how an expected value will change given another event.

Probability & the Technology Sector

• Economist estimates a 60% probability that the economy will expand next year
• The technology sector has a 70% probability of outperforming the market if the economy expands
• The technology sector has a 10% probability of outperforming the market if the economy does not expand
• Given the new information that the technology sector will not outperform the market, the probability that the economy will not expand is closest to 67%
• Bayes' formula can be used to update the probability given new information.
• P(economy does not expand | tech does not outperform) = P(economy does not expand and tech does not outperform) / P(tech does not outperform)

Bayes' Formula Example

• Probability of A is 0.4, probability of AC is 0.6
• Probability of (B | A) is 0.5, and the probability of (B | AC) is 0.2
• Using Bayes' formula, the probability of (A | B) is 0.625
• P(B): P(B) = [P(B | A) × P(A)] + [P(B | AC) × P(AC)], P(B) = [0.5 × 0.4] + [0.2 × 0.6] = 0.32
• P(A | B): P(A | B) = [ P(B | A) ÷ P(B) ] × P(A) = [0.5 ÷ 0.32] × 0.4 = 0.625

Investment in ACQ

• There is a 30% probability that ACQ will be acquired in the next two months
• If ACQ is acquired, there is a 40% probability of earning a 30% return and a 60% probability of earning 25%
• If ACQ is not acquired, the expected return is 12%
• The expected return on this investment is 16.5%
• E(r) = (0.70 × 0.12) + (0.30 × 0.40 × 0.30) + (0.30 × 0.60 × 0.25) = 0.165

Calculating Expected Value

• A two-sided, thick coin expected to land on its edge twice out of every 100 flips
• Equal probability of face up (heads) and face down (tails)
• Prize: $1 for heads, $2 for tails, $50 for edge
• The expected value of the prize on a single coin toss is $2.47
• Expected Payoff = ∑ Probability × Payoff

Bayes' Formula - Red Car Example

• In a parking lot with 100 red and blue cars: 40% of cars are red, 70% of red cars have radios, 80% of blue cars have radios
• Applying Bayes' formula, the probability that a car is red given that it has a radio is 37%

Bayes' Theorem - Earnings Decline Example

• An analyst expects that 20% of companies will decline in earnings next year
• If a company declines in earnings, there is a 90% probability that a specific ratio will be negative
• If a company does not decline in earnings, there is a 10% probability that the ratio will be negative
• If a randomly selected company has a negative ratio, based on Bayes' theorem, the updated probability that the company will experience a decline is 69%

Determining Probability of Default

• Bonds rated B have a 25% default chance in five years
• Bonds rated CCC have a 40% default chance in five years
• A portfolio is 30% B-rated and 70% CCC-rated bonds
• If a bond defaults in a five-year period, the probability it was a B-rated bond is 0.211
• P(B | default) = P(default and B) / P(default)

Calculating Expected Return and Standard Deviation

• An investment has a 40% probability of earning 10%, a 40% probability of earning 12.5%, and a 20% probability of earning 30%
• The mean expected return and the standard deviation of expected returns are 15.0% and 7.58%, respectively
• Mean = (0.4)(10) + (0.4)(12.5) + (0.2)(30) = 15%
• Standard deviation = √57.5 = 7.58

Portfolio Returns Example

• Tully Advisers, Inc. determined four possible economic scenarios and projected portfolio returns for two portfolios
• The expected return on Portfolio A is a probability-weighted average of 17%, 14%, 12%, and 8%
• Expected return = (0.15)(0.17) + (0.20)(0.14) + (0.25)(0.12) + (0.40)(0.08) = 0.1155 or 11.55%

Stock Price Expectations Sub Heading Text

• Tina O'Fahey, CFA, believes a stock's price in the next quarter relies on direction of the overall market and whether the company's next earnings report is good or bad
• The expected value if the overall market decreases is 0.4($60) + (1 – 0.4)($55) = $57

Stock Selection Probability

• John purchased 60% of stocks, with half considered good
• Andrew purchased 40% of stocks, with a fourth considered good
• The probability John selected a randomly chosen stock that turned out to be good is 0.75
• P(John | good) = P(good and John) / P(good)

Conditional Expectation

• Conditional expected values are contingent upon the occurrence of some other event
• The expectation changes as new information is revealed

Standard Deviation of Return

• Calculating the standard deviation of the return given:
  • 50% chance of a 12% return
  • 30% chance of a 10% return
  • 20% chance of a 15% return
• The standard deviation is 1.7%
• Standard deviation is the positive square root of the variance