Latest / VoraPrep Audio — Professional Licensure & Certification Exam Prep / [CFA Level I] CFAQM: Statistical Measures, Probability Trees, and Bayes' Formula
Transcript
- Hey, welcome back to VoraPrep Audio. Glad you’re here. Let's talk about a situation that comes up all the time on the exam, and definitely in the real world. Imagine you’re a portfolio manager. You’re looking at a new investment, and you have some historical data on how similar assets have performed. But the past isn’t the future, right? You also need to think about what might happen next. A recession, normal growth, maybe a boom. Then, a new economic report comes out. How do you take that new piece of information and update your forecast? How do you go from a gut feeling to a concrete number? That’s what this is all about. We’re going to walk through the statistical tools that let you do this systematically. You'll learn how to calculate expected returns, measure risk, and most importantly, use new data to intelligently update your beliefs. Let’s start with the basics of describing a set of data, like a stock's returns over the last few years. You have measures of central tendency, which are just fancy words for 'average' or 'typical'. The one everyone knows is the arithmetic mean. You just add up all the returns and divide by how many there are. Simple. This is your best guess for a single, upcoming period. It answers the question, "what do we think the return will be next year?" But what if you want to describe the average performance over multiple years? That’s different, because of compounding. For that, you need the geometric mean. Think of it this way. The arithmetic mean is like averaging your speed on different one-mile stretches of a road trip. The geometric mean is different. It's like finding the single, constant speed you would have needed to travel to cover the whole distance in the same amount of time. It gives you that true, compounded average growth rate. On the exam, if they ask for multi-period performance, they're pointing you toward the geometric mean. It's a huge trap people fall into. Of course, there's also the median, which is just the middle value if you line them all up. And there's the mode, which is the one that shows up most often. They're useful, but the two means are the heavy hitters. Now, knowing the average is only half the story. You also need to know how spread out the data is. That's where measures of dispersion come in. The big one here is standard deviation. It’s the single most important number for measuring risk. A low standard deviation means the returns are all clustered tightly around the average. That's predictable. A high standard deviation means the returns are all over the place. That's volatile and risky. Standard deviation is the square root of the variance, which is another measure of spread. Just remember: standard deviation equals risk. Okay, so that’s describing the past. What about predicting the future? That’s all about probability. There are a few flavors. You have a priori probability, which is based on pure logic, like the one-in-six chance of rolling a four on a die. Then there’s empirical probability, which comes from looking at historical data. For example, the probability of the market going up in a given month, based on the last 50 years. And finally, there’s subjective probability, which is your personal, expert judgment. When you start combining these, things get interesting. You have the unconditional probability. This is just the straight-up chance of something happening. We write it as P of A. For example, the probability of rain tomorrow is 30 percent. Period. But then you have conditional probability. This is huge. It’s the probability of A, given that B has already happened. We write it as P of A, given B. What's the probability of rain, given that the sky is full of dark clouds? It's probably a lot higher than 30 percent, right? The new information—the dark clouds—changed the probability. And that brings us to joint probability. This is the chance of two things both happening. We write it as P of A and B. For example, the probability that it's cloudy and it rains. You find this by multiplying the probability of the first event by the conditional probability of the second. So, P of A and B equals P of A given B, times P of B. A great way to keep all this straight, especially with a sequence of events, is a probability tree. Just picture a diagram that starts at one point and branches out. Each branch is a possible outcome, and you write the probability on that branch. To find the probability of a whole sequence of events, you just multiply your way down the branches. It turns a confusing paragraph of text into a clear visual map. This leads us to a crucial tool: Bayes' Formula. This sounds way more intimidating than it is. All it does is give you a mathematical way to update your beliefs when you get new information. It formalizes the process of going from a prior probability to a posterior probability. The formula states that your new, updated belief is your old belief, adjusted by how likely this new piece of information was. It’s the engine behind so much of modern finance and data analysis. If you start with a belief that there's a 20 percent chance of a recession, and then a terrible jobs report comes out, Bayes' Formula tells you exactly how to update that 20 percent to something higher. So, let's put this together with an example. Say you're calculating the expected value, or expected return, of an asset. This is just the weighted average of all the possible outcomes. Let’s say an asset could have a 20 percent return in a boom, and there's a 30 percent chance of a boom. In a normal growth period, it returns 10 percent, and that has a 50 percent probability. And in a recession, it loses 5 percent, with a 20 percent chance of that happening. To get the expected return, you multiply each possible return by its probability, and then add up all those results. Let's walk through it. First, the boom scenario. We have a 20 percent return with a 30 percent probability. So we multiply 0.20 by 0.30, which gives us 0.06. Next is the normal growth scenario. That's a 10 percent return with a 50 percent probability. So, 0.10 times 0.50 equals 0.05. Finally, the recession. The asset loses 5 percent, so that's a negative 0.05 return. The probability is 20 percent. Multiplying negative 0.05 by 0.20 gives us negative 0.01. Now we just add those three weighted outcomes together. We have 0.06, plus 0.05, minus 0.01. That sums to 0.10. So, the expected return for this asset is 10 percent. That 10 percent isn't a guarantee. It's the long-run average you'd expect if you could repeat this scenario over and over. Now, let's try a couple of questions you might see. I’ll give you a second to think before I walk through the answer. First one: An investment had annual returns of 10 percent, 20 percent, and then negative 15 percent over three years. What’s the geometric mean return? Okay, so remember, for geometric mean, we use the return factors. That’s one plus the return. So for year one, it’s 1.10. Year two is 1.20. And year three is 1 minus 0.15, which is 0.85. Now you multiply those together: 1.10 times 1.20 times 0.85. That gives you 1.122. Since it’s over three years, we take the cube root of that—or raise it to the power of one-third. That gets you about 1.039. Finally, you subtract one. So the geometric mean is 0.039, or 3.9 percent. That’s the actual compound annual growth rate. Here's another one, this time on probability. Let's say the probability a company’s earnings exceed expectations is 60 percent. If earnings do exceed expectations, the probability of the stock price going up more than 5 percent is 80 percent. But if earnings do not exceed expectations, the probability of the stock price going up more than 5 percent is only 30 percent. The question is: what is the total, unconditional probability that the stock price increases by more than 5 percent? Alright, this is a classic "law of total probability" problem, and a tree diagram would be perfect here. There are two paths to the stock going up. Path one is that earnings exceed expectations AND the stock goes up. The probability of that is 60 percent times 80 percent. So, 0.60 times 0.80, which is 0.48. Path two is that earnings do NOT exceed expectations AND the stock still goes up. The probability of earnings not exceeding is 1 minus 60 percent, so 40 percent. The chance of the stock going up in that case is 30 percent. So, 0.40 times 0.30, which is 0.12. To get the total probability, you just add the probabilities of those two paths together. So, 0.48 plus 0.12 gives us 0.60. That means there's a 60 percent chance overall that the stock price will increase by more than 5 percent. Before we wrap, a couple of key takeaways. First, know when to use the arithmetic versus the geometric mean. It's a guaranteed point if you get it right. Second, remember standard deviation is your go-to measure for risk. And third, really get comfortable with Bayes' Formula. Don't just memorize it; understand its purpose, which is updating your beliefs with new facts. VoraPrep is a full exam-prep app — lessons, practice questions, and an AI tutor in one place. Get started at Vora Prep dot com. Alright, that's it for today. You’re building a powerful toolkit here. Keep practicing, and I'll talk to you on the next one.