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LM04 Probability Trees and Conditional Expectations 2025 Level I Notes © IFT. All rights reserved 1 LM04 Probability Trees and Conditional Expectations 1. Introduction...........................................................................................................................................................2 2. Expected Value and Variance..........................................................................................................................4 3. Probability Trees and Conditional Expectations.....................................................................................6 4. Bayes' Formula and Updating Probability Estimates ............................................................................7 Summary......................................................................................................................................................................9 Required disclaimer: IFT is a CFA Institute Prep Provider. Only CFA Institute Prep Providers are permitted to make use of CFA Institute copyrighted materials which are the building blocks of the exam. We are also required to create / use updated materials every year and this is validated by CFA Institute. Our products and services substantially cover the relevant curriculum and exam and this is validated by CFA Institute. In our advertising, any statement about the numbers of questions in our products and services relates to unique, original, proprietary questions. CFA Institute Prep Providers are forbidden from including CFA Institute official mock exam questions or any questions other than the end of reading questions within their products and services. CFA Institute does not endorse, promote, review or warrant the accuracy or quality of the product and services offered by IFT. CFA Institute®, CFA® and “Chartered Financial Analyst®” are trademarks owned by CFA Institute. © Copyright CFA Institute Version 1.0
LM04 Probability Trees and Conditional Expectations 2025 Level I Notes © IFT. All rights reserved 2 1. Introduction Since many investment decisions are made in an environment of uncertainty, it is essential for portfolio managers and investment managers to have a fundamental grasp of probability concepts. In this learning module, we will focus on: • Calculation of the expected value, variance, and standard deviation for a random variable. • Using probability trees to visualize conditional expectations and the total probabilities for expected value. • Using Bayes’ formula to adjust probabilities with the arrival of new information. Instructor’s Note: Before we get into the actual concepts covered in the curriculum, some prerequisite fundamental concepts are presented below. Fundamental Concepts A random variable is an uncertain quantity/number. For example, when you roll a die, the result is a random variable. An outcome is the observed value of a random variable. For example, if you roll a 2, it is an outcome. An event can be a single outcome or a set of outcomes. For example, you can define an event as rolling a 2 or rolling an even number. Mutually exclusive events are events that cannot happen at the same time. For example, rolling a 2 and rolling a 3 are examples of mutually exclusive events. They cannot happen at the same time. Exhaustive events are those that cover all possible outcomes. For example, ‘rolling an even number’ or’ rolling an odd number’ are exhaustive events. They cover all possible outcomes. The two defining properties of probability are: • The probability of any event has to be between 0 and 1. • The sum of the probabilities of mutually exclusive and exhaustive events is equal to 1. Conditional v/s Unconditional probabilities Unconditional probability is the probability of an event occurring irrespective of the occurrence of other events. It is denoted as P(A). Unconditional probability is also called ‘marginal’ probability. Conditional probability is the probability of an event occurring given that another event has occurred. It is denoted as P(A|B), which is the probability of event A given that event B has occurred.
LM04 Probability Trees and Conditional Expectations 2025 Level I Notes © IFT. All rights reserved 3 Joint Probability and Multiplication Rule Multiplication rule is used to determine the joint probability of two events. It is expressed as: P(AB) = P(A|B) P(B) Rearranging the equation, we get the formula for computing conditional probabilities: P(A|B) = P(AB) / P(B) Example P(interest rates will decrease) = P(D) = 40% P(stock price increases) = P(S) P(stock price will increase given interest rates decrease) = P(S|D) = 70% Compute probability of a stock price increase and an interest rate decrease. Solution: P(SD) = P(S|D) x P(D) = 0.7 x 0.4 = 0.28 = 28% Addition Rule for Probabilities Addition rule is used to determine the probability that at least one of the events will occur. It is expressed as: P(A or B) = P(A) + P(B) – P(AB) P(AB) represents the joint probability that both A and B will occur. It is subtracted from the sum of the unconditional probabilities: P(A) + P(B), to avoid double counting. If the two events are mutually exclusive, the joint probability: P(AB) is zero and the probability that either A or B will occur is simply the sum of the unconditional probabilities for each event: P(A or B) = P(A) + P(B) Example P(price of A increases) = P(A) = 0.5 P(price of B increases) = P(B) = 0.7 P(price of A and B increases) = P(AB) = 0.3 Compute the probability that the price of stock A or the price of stock B increases. Solution P(A or B) = 0.5 + 0.7 – 0.3 = 0.9 Independent and Dependent Events If the occurrence of one event does not influence the occurrence of the other event, then the two events are called independent events.
LM04 Probability Trees and Conditional Expectations 2025 Level I Notes © IFT. All rights reserved 4 i.e. P(A|B) = P(A) or P(B|A) = P(B) Multiplication rule for independent events: P(AB) = P(A) P(B) Addition rule for independent events: P(A or B) = P(A) + P(B) – P(AB). (The addition rule does not change.) If the probability of an event is affected by the occurrence of another event, then it is called a dependent event. Total Probability Rule The total probability rule is used to calculate the unconditional probability of an event, given conditional probabilities. In investment analysis, we often formulate a set of mutually exclusive and exhaustive scenarios and then estimate the probability of a particular event. For example, let’s say that we have two scenarios S and non-S that are mutually exclusive and exhaustive. According to the total probability rule, the probability of any event P(A) can be expressed as: P(A) = P(AS) + P(ASC) Using the multiplication rule we get, P(A) = P(A|S) P(S) + P(A|SC) P(SC) If we have more than two scenarios, we can generalize this equation to: P(A) = P(AS1) + P(AS2) +... + P(ASn) = P(A|S1) P(S1) + P(A|S2) P(S2) + ... + P(A|Sn) P(Sn) 2. Expected Value and Variance Expected Value of a Random Variable The expected value of a random variable can be defined as the probability-weighted average of the possible outcomes of the random variable. For a random variable X, the expected value of X is denoted as E(X) and is calculated as: E(X) = ∑P(Xi ) Xi n i=1 where: Xi = One of n possible outcomes of the random variable X P(Xi) = Probability of Xi Variance of a Random Variable The expected value is our forecast, but we cannot count on the individual forecast being realized. This is why we need to measure the risk we face. Variance and standard deviation are examples of how we can measure this risk.