Mastering Bayes’ Formula | CFA Level I Quantitative Methods

Hey there! In this article, we’ll wrap up probability concepts by discussing Bayes’ Formula and how it helps us learn from experience. We’ll also explore the total probability rule and how it’s used in investment problems. So, let’s dive in, shall we?

Quick Recap: The Total Probability Rule

Remember the total probability rule we learned earlier? Given mutually exclusive and exhaustive events “A” and “A^C,” we can find the unconditional probability of event B using this rule. This rule can also be applied to multiple events, like A1, A2, and A3.

We previously calculated the unconditional probability that the Dow Jones will go up in any given month (0.55), regardless of interest rate movements, using the total probability rule.

Reversing the Process: Bayes’ Formula

Bayes’ Formula: P(A|B) = P(B|A) * P(A) / P(B)

Now, what if we want to reverse the process and estimate the likelihood of a rate cut decision in the following month, given that the Dow Jones went up this month? That’s where Bayes’ Formula comes into play. We can use new information to update our probability model.

Here’s a step-by-step example:

1. Calculate the joint probability of B and A2 using the multiplication rule.
2. Rearrange the formula and apply the multiplication rule again.
3. Use known prior probabilities to update our model.
4. Plug in all the figures and calculate the updated probability (in this case, 0.07).

Bayes’ Formula helps us update the probability of an event based on new information. In this example, we updated the probability of a rate cut decision after learning that the Dow Jones went up for the month.

Practice Time: Applying Bayes’ Formula

EXAMPLE

Robert Holmes estimates that the probability of a buyout of Pinnacle is 70%. If it does, there is an 80% chance that the stock of Pinnacle will go up. If there is no buyout, the chances that the stock will go up is 55%. Given that the stock of Pinnacle went up, what is the updated probability of a buyout?

To solve this, we can use Bayes’ Formula:

1. Draw a tree diagram, if helpful.
2. Compute the joint probabilities using the multiplication rule.
3. Determine the event (B) and the new information (U).
4. Calculate the unconditional probability of U using the total probability rule.
5. Plug in the figures and calculate the updated probability (in this case, 0.77).

So, given that the stock went up, we now have a higher confidence (0.77) that the buyout will materialize.

And That’s a Wrap!

That’s it for our lesson on Bayes’ Formula! Now you know how to use this powerful tool to update probabilities based on new information. Next up, we’ll explore the principles of counting. Happy studying!