Library/CFA (Chartered Financial Analyst)/SchweserNotes 2022 Level I CFA Book 1: Quantitative Methods and Economics/Reading 3: Probability Concepts

Reading 3: Probability Concepts

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Probability Basics and Rules15 min

This section defines the core components of probability theory. A random variable represents an uncertain value, and an event is a collection of outcomes. Probabilities are measures between 0 and 1. The section covers the distinction between empirical, a priori, and subjective probabilities. It explains how to convert between probabilities and odds. Key rules include the multiplication rule for joint probabilities and the addition rule for the probability of either of two events occurring. The concept of independence is defined, distinguishing it from mutual exclusivity.

Key Points

Probability of an event is between 0 and 1.
Sum of mutually exclusive and exhaustive events is 1.
Odds for E = P(E) / (1 - P(E)).
Multiplication Rule: P(AB) = P(A|B) * P(B).
Addition Rule: P(A or B) = P(A) + P(B) - P(AB).
Independent events: P(A|B) = P(A).

Expected Value and Variance10 min

This section focuses on descriptive statistics for random variables. The expected value is the probability-weighted average of all possible outcomes. Variance measures the dispersion around the expected value, calculated as the probability-weighted sum of squared deviations. The total probability rule is introduced as a method to calculate unconditional probabilities by summing conditional probabilities weighted by their underlying event probabilities. This is often used in scenario analysis for investment returns.

Key Points

Expected Value E(X) = Sum(P(xi) * xi).
Variance = Sum(P(xi) * (xi - E(X))^2).
Total Probability Rule: P(A) = Sum(P(A|Si) * P(Si)).
Conditional expectations update views based on new info.

Portfolio Mathematics15 min

This section applies probability concepts to portfolios. It defines covariance as a measure of how two variables move together and correlation as a standardized measure of linear relationship bounded by -1 and +1. The expected return of a portfolio is a weighted average, but portfolio variance is non-linear, depending on the weights, variances, and covariances (or correlations) of the constituent assets. This non-linearity is the mathematical basis for diversification benefits.

Key Points

Portfolio Expected Return = w1*E(R1) + w2*E(R2).
Covariance Cov(1,2) = E[(R1 - E(R1))(R2 - E(R2))].
Correlation = Cov(1,2) / (StdDev1 * StdDev2).
Portfolio Variance (2 assets) = w1^2*Var1 + w2^2*Var2 + 2*w1*w2*Cov(1,2).

Bayes' Formula and Counting Rules10 min

The final section covers Bayes' formula and counting techniques. Bayes' formula provides a method to update a prior probability to a posterior probability given new information. Counting rules determine the number of ways outcomes can occur. These include the factorial function for arrangements, the labeling formula for grouping, the combination formula for selecting items without regard to order, and the permutation formula for selecting items where order matters.

Key Points

Bayes' Formula: P(Event|Info) = [P(Info|Event) / P(Info)] * P(Event).
n-Factorial (n!) counts ways to arrange n items.
Combination (nCr) selects r items from n, order irrelevant.
Permutation (nPr) selects r items from n, order matters.
Labeling formula handles assignments to multiple categories.

Questions

Question 1

Which of the following best describes a set of events that includes all possible outcomes?

Reading 3: Probability Concepts

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