Probabilities and expected returns. Now, first, let's look at the relationship between mean returns or average returns and expected returns. Average returns are simply historical returns, right? It's the return that an investor has actually received in the past. Conversely, an expected return is what an investor would expect to receive going forward. Now, generally over the long term, we would expect that the average return equal our expected return. Now, that's not always the case, but generally that's our initial assumptions in investment analysis. Now, how do we calculate expected returns? Very straightforward, it's just an average, simple average, arithmetic mean. We just add up each individual return and divide by N, N being the number of data points in our calculation. But let's look at that N a little closer. What does that actually mean? Well, when we divide by N, we're actually assuming that all occurrences are equally likely. Now, is that usually the case? Probably not. And that's where probabilities and weighted means come in. Now we'll use weighted means when we wanna combine not only the important information that we can glean from historical return data, but when we also want to incorporate our views on potential probabilities of various return streams. And this often will involve creating various cases, for example, a base case, a goods case, and a bad case around certain return streams so that we can put higher or lower weights on each occurrence. This often involves creating various cases, for example, a good case, a base case, and a bad case, where we can differentiate the different likelihoods of various occurrences within a return stream. And weighted means allow us to give weights to these various cases so that they match up with our views and incorporate both the historical data in our views into our forecast of an expected return. And here's your typical formula. It's simply a sum of all possible values and each are multiplied by a probability that's assigned to it. So instead of dividing by N, we're attaching a probability to each occurrence. And all these occurrences, all the probabilities must equal one. Now, let's go through an example. Given the two investments below, which investment has a higher expected return? Well, at first look, we would see that Security A and Security B have equal expected returns for three different cases, a base case, a bad case, and a good case. So they look pretty similar initially. But if we overlay probabilities around each case, we get a different story. So to calculate the expected return for Security A, we'll simply multiply each return by their corresponding probability and then sum them all together. And the resulting expected return for Security A is 9.9. If we do the same for Security B, we get a higher expected return of 16.5 and that's what we'd expect, right? 'Cause if you look at the probability distribution for Security B, you would see a higher probability attached to the base case and good case that come with higher expected returns. And Excel actually has a great formula for these type of calculations, and it's called sum product. And in sum product, you simply assign two different arrays. The first array would be the expected return in each case and the second array would be the probabilities and it would go ahead and multiply each and then sum them all together at the end to calculate your expected return.