Understanding Expected Values and Variance in Finance | CFA Level I Quantitative Methods
Expected Value: A Step Beyond Mean
Before we delve into expected value, let’s quickly recap the concept of mean. The mean is a historical statistic that represents the central value of a sample set. Now, let’s see how the expected value takes this concept a step further.
While the mean relies on past data, the expected value is a forward-looking inference about the true mean of a population. Calculating expected value involves a similar approach to weighted mean, but with some key differences:
- We use expected outcomes instead of actual observations.
- Each expected outcome is weighted by its probability of occurrence.
- The sum of all probabilities must equal 1.
- The expected value is the sum of the products of expected outcomes and their probabilities.
Working with Expected Values and Variance
Conditional Expected Values: Dealing with Uncertainty
Expected values can also be conditional, taking into account different possible scenarios. For example, different EPS values could be expected under different economic conditions, such as with or without a recession. A tree diagram can help us visualize these relationships and calculate expected values more efficiently.
Here’s how to create a tree diagram and calculate conditional expected values:
- Identify mutually exclusive and exhaustive events (e.g., recession or no recession).
- Calculate unconditional probabilities for each event (e.g., 0.6 for no recession, 0.4 for recession).
- Determine conditional probabilities for each outcome given the event in the previous branch.
- Use the conditional expected value formula to calculate the expected value under each scenario (e.g., $25.75 for EPS in case of a recession).
- Find the unconditional expected value using the total probability rule (e.g., $29.50).
Tree diagrams are a powerful tool for solving investment problems and are widely used throughout the CFA program. Make sure you understand this technique and practice it to sharpen your skills!