Covariance and Correlation. Let's first dig into what covariance is. Now here's your standard definition for covariance. It's a measure of how returns on two assets move or do not move in the same direction. It's similar to variance in terms of how it's calculated, but instead of calculating variability of data points from a variables mean, we're comparing two variables to their means, and get an understanding of how these variations differ or are similar. Now, how do we interpret a covariance statistic? Well, if it's greater than zero, it means it's a positive relationship. Obviously, that would mean that assets to stocks to investments tend to move together. If it's less than zero, or negative, it's a negative relationship. They tend to move in opposite directions. The downside of covariance is that units make it very difficult to interpret further than that. The strength of the relationship and be able to compare covariances across assets is a bit difficult. Hence, the correlation statistic, also known as the correlation coefficient. And it's very similar to covariance in a sense that it allows us to get a sense of a positive or negative relationship between two assets. But unlike covariance, it's a lot easier to interpret and we're able to get a better understanding of how strong the positive or negative relationship really is. And it's easier to interpret, because it standardizes the values and places bounds of -1 and 1 on its possible values. So in addition to the degree or the strength of the relationship that we can measure, correlations also allow us easier comparisons between different pair of assets, which again, covariance does not allow us to do. And that's why you're more likely to see correlation data, because it's a lot more useful to us in finance and investments when we're comparing two assets. Now here's your typical correlation formula. As you can see, it's derived from the covariance of two assets, but then divided by each securities or each asset's standard deviation of returns. Luckily, Excel also has a formula to easily calculate correlation. And here, we're using the array functionality within Excel where the first array would be the returns of asset one and the second array would be the returns of asset two for a quick and easy calculation in Excel. Now, scatterplots. Scatterplots are formed by using data of two securities to plot along an x and y-axis. And here are a couple of examples. First, here's a scatterplot of the data being plotted from the lower-left to the upper-right. This graph shows a perfect correlation or correlation of +1. So as one asset moves, the other asset moves in the same direction, in the same proportion. Here's an example of just the mirror image. It goes from the top-left to the bottom-right. This is a correlation of -1 perfectly negative. So the two assets move in opposite directions by the same proportion. And this third scatterplot is a little bit more all over the place. And here is a correlation of zero. There's no relationship that's apparent. If one asset moves, it doesn't give us any information on how the other asset will potentially move. Now, visually inspecting a scatterplot like this is not typically sufficient to demonstrate a statistical relationship. It's just a a starting point. So for examining data, in order to assess whether there appears to be an underlying relationship. Now you'll rarely see measures in their perfect outer bounds as we see here. Rather, you'll typically see them between the -1 value and the +1 value. And then the strength of the relationship increases as the value approaches either of those outer bounds. Some other key points around correlation, and probably the most important use of the correlation statistic in finance, is that it helps us determine what asset to add to a portfolio to increase diversification. Couple of examples. The lower the correlation the more diversification benefits. That makes sense, right? Because the lower the correlation, the higher likelihood that assets move in different directions, increasing diversification. Of course, the complete opposite is true also. If we have two assets with a perfect correlation of +1, there are no diversification benefits by adding them together to a portfolio. That'll move together in the same direction by the same proportion, and therefore do not change the variability of returns in a portfolio. Another key point is that correlation does not equal causation. You may have heard that phrase before, correlation does not equal causation, but let's go through a quick example. Snow boots and car accidents have a correlation, but snow boots do not cause car accidents. You know, here the relationship is pretty obvious, right? You tend to wear boots when it snows outside, and the number of car accidents also increases when it snows outside. So it's obvious in this instance that the correlation between snow boots and car accidents is not from the relationship with themselves, but instead with a third variable, and in this case it being the snow. Let's take a quick look at a correlation matrix. It's a widely-used tool to get a sense of how assets or asset classes move together in their potential diversification benefits. Here we have a table showing correlation coefficients of asset classes, both on the x-axis and then in the y-axis, over the long-term. And in this case, it's 1926 to 2015. Now you can see that the diagonal on the table is always set to 1, because the correlation between one asset class to itself is always gonna be 1, obviously. As an example, let's take a look at the two data points highlighted in red for intermediate government bonds. Historically, intermediate government bonds have had the most negative correlation to equities, both small cap stocks and large cap stocks. This would show us that over the long-term, intermediate government bonds have been the best diversifier to include in an equity portfolio over the long-term.