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Probability Trees and Conditional Expectations

Quantitative Methods

Bayes' Formula and Updating Probability Estimates

Learning Outcome Statement:

calculate and interpret an updated probability in an investment setting using Bayes’ formula

Summary:

Bayes' formula is a statistical method used to update the probability of an event based on new evidence. It is particularly useful in investment contexts where decisions must be made with incomplete information. The formula helps in adjusting initial beliefs (prior probabilities) based on new data (likelihoods), resulting in updated beliefs (posterior probabilities).

Key Concepts:

Bayes' Formula

Bayes' formula is used to update the probability estimate for an event based on new information. It combines prior probability with new evidence to provide a posterior probability.

Prior Probability

The probability of an event based on existing knowledge before new evidence is presented.

Likelihood

The probability of observing the new evidence, given that the event has occurred.

Posterior Probability

The updated probability of an event occurring after taking into account the new evidence.

Total Probability Rule

A rule used to decompose the total probability of an event into a sum of probabilities conditional on scenarios or partitions of the sample space.

Formulas:

Bayes' Formula

P(A∣B)=P(B∣A)×P(A)P(B)P(A | B) = \frac{P(B | A) \times P(A)}{P(B)}

This formula is used to update the probability of event A given the occurrence of event B, incorporating new evidence into prior beliefs.

Variables:
P(A∣B)P(A | B):
Probability of event A given event B
P(B∣A)P(B | A):
Probability of event B given event A
P(A)P(A):
Prior probability of event A
P(B)P(B):
Total probability of event B
Units: dimensionless (probability)

calculate expected values, variances, and standard deviations and demonstrate their application to investment problems

Learning Outcome Statement:

Probability Trees and Conditional Expectations

Summary:

This LOS explores the use of probability trees and conditional expectations in calculating expected values and variances for financial metrics such as Earnings Per Share (EPS) under different scenarios. It introduces the total probability rule for expected value, which allows the calculation of unconditional expected values from conditional expected values across different scenarios. The content also covers the calculation of variance under different scenarios and its implications for assessing risk.

Key Concepts:

Expected Value

The expected value is a weighted average of all possible values that a random variable can take on, each value weighted by its probability of occurrence.

Conditional Expected Value

The expected value of a random variable given a particular scenario or condition. It is calculated by summing the products of the outcomes and their conditional probabilities.

Total Probability Rule for Expected Value

A principle that states the unconditional expected value can be expressed as the sum of conditional expected values, each weighted by the probability of its corresponding scenario.

Variance and Conditional Variance

Variance measures the dispersion of a set of values from their mean. Conditional variance is the variance calculated under a specific scenario, providing insights into risk given that scenario.

Formulas:

Expected Value of EPS

E(EPS)=0.15(2.60)+0.45(2.45)+0.24(2.20)+0.16(2.00)E(\text{EPS}) = 0.15(2.60) + 0.45(2.45) + 0.24(2.20) + 0.16(2.00)

Calculates the expected EPS by summing the products of each EPS outcome and its probability.

Variables:
E(EPS)E(\text{EPS}):
Expected value of EPS
0.15,0.45,0.24,0.160.15, 0.45, 0.24, 0.16:
Probabilities of respective EPS outcomes
2.60,2.45,2.20,2.002.60, 2.45, 2.20, 2.00:
EPS outcomes in USD
Units: USD

Total Probability Rule for Expected Value

E(X)=E(X∣S1)P(S1)+E(X∣S2)P(S2)+…+E(X∣Sn)P(Sn)E(X) = E(X | S_1)P(S_1) + E(X | S_2)P(S_2) + \ldots + E(X | S_n)P(S_n)

Expresses the unconditional expected value as a sum of conditional expected values, each multiplied by the probability of its scenario.

Variables:
E(X)E(X):
Unconditional expected value of X
E(X∣Si)E(X | S_i):
Expected value of X given scenario i
P(Si)P(S_i):
Probability of scenario i
S1,S2,…,SnS_1, S_2, \ldots, S_n:
Scenarios
Units: Unit of X

Variance of EPS

σ2(EPS)=0.15(2.60−2.34)2+0.45(2.45−2.34)2+0.24(2.20−2.34)2+0.16(2.00−2.34)2\sigma^2(\text{EPS}) = 0.15(2.60 - 2.34)^2 + 0.45(2.45 - 2.34)^2 + 0.24(2.20 - 2.34)^2 + 0.16(2.00 - 2.34)^2

Calculates the variance of EPS by summing the products of the squared deviations of EPS outcomes from their mean, each weighted by its probability.

Variables:
σ2(EPS)\sigma^2(\text{EPS}):
Variance of EPS
0.15,0.45,0.24,0.160.15, 0.45, 0.24, 0.16:
Probabilities of respective EPS outcomes
2.60,2.45,2.20,2.002.60, 2.45, 2.20, 2.00:
EPS outcomes in USD
2.342.34:
Expected EPS
Units: USD^2
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