Variance and standard deviation. Measures of dispersion. Now, what they do is essentially measure how concentrated, or how stretched, or how variable data points are around an expected value. And every industry has their own specific measures and it could vary. In finance, we have a handful that we generally use. But in all, they all measure variability, and therefore, risk. Here are a couple of examples. The simplest range of returns, think of a simple 52-week high or low on a stock price. Absolute deviation, variance and standard deviation. And those are the two standard measures that we use most often in investments in finance. Now digging deeper into variance and standard deviation. Every day, we work with investment models, and the goal of our investment model is to predict or forecast a variable. Now that variable may be return on an asset class, return on a stock, or earnings per share for a specific company. Obviously, very complex models and forecasts, and we can never be certain of that forecast. So that's why it's important for us to measure the potential risk that we face around our forecast. We cannot count on our individual forecast being realized, although we hope for that on average, that it'll be accurate, we still need to forecast how variable that could be. Now, variance and standard deviation do just that for us. They measure how variable the actual outcome of an investment model could be around our expected return. Now let's take a quick look at variance in more detail. Now here's your standard definition of variance, the expected value of squared deviations from the random variable's expected value. You know, quite a mouthful. Well, if we take a look at the actual formula, it's even more of a mouthful. But essentially, what it's doing, it's taking a look at every observation, every data point, and comparing it to the mean, and ultimately, coming up with a measure that tells us how variable, how dispersed, or how concentrated the data points are relative to the mean. And thankfully for us, Excel can do a lot of the heavy lifting around these calculations. Here you see the simple variance formula for a sample. Excel will have a couple of different formulas, one for a sample, one for a population, but generally in finance, we're working with sample data, we'll use the sample equation. Now how do we interpret variance? First of all, variance should always be a positive number. Just based upon the formula where we're squaring differences of each authorization away from the mean, you'll know that the variance will always be positive. If for some reason you come across a negative reading, you know there's something wrong. Now, what if the variance is zero? Well, that means there's no variability around that random variable. The outcome is always certain. And if the outcome is always certain, there's no risk. Now, if the variance is greater than zero, what does that mean? Well, as it increases, it indicates greater variability, it then creates a more stretched out observation, and it indicates more risk. Now standard deviation, very similar statistic. It's simply the square root of the variance. And it's used more often than variance, and you'll see it in an investment fact sheets and fund fact sheets more often, because it's a lot easier to interpret. It uses the same units as the underlying variable. What does that mean? Well, for example, if we're looking at a random variable, let's say the return on a specific stock, that return is gonna be quoted in percentages. Unlike variance, standard deviation will also be quoted in percentages, so it's a lot easier to interpret from that perspective, and a lot easier to compare versus fund to fund or stock to stock. And often you'll hear standard deviation simply referred to as volatility in the context of stock returns. Obviously, the more variable stock returns are, the more volatile the stock price is. Now thankfully, again, Excel has a nifty standard deviation formula for us. And similar to the variance formula, this is the standard for a sample data set. Now, interpreting standard deviation, despite probably being one of the oldest, along with variance, statistic to look at risk, it still remains very applicable and very common in investments in finance. I like to simplify standard deviation by calling it the give or take number. Now, we've all heard folks give an estimate on a specific variable and close that estimate with a give or take number. For example, he's probably 6'3", give or take an inch. Again, that give or take number of an inch is the variability around his forecast. And it's essentially what we're doing with standard deviation. Now, the higher the standard deviation, the higher risk in a specific asset, the higher volatility in a specific asset, and the more variability around returns. Very similar to variance. And just like variance, the lower standard deviation, the lower variability, and the lower risk.

And of course, lower standard deviation means less uncertainty on a period to period basis, which is desirable. Now, standard deviation isn't perfect. It does not distinguish between good returns and bad returns. It gives equal waiting to negative volatility as it does to positive volatility. And in theory, a manager could be equally punished for good volatility or upside volatility as it is for downside volatility. As a result, you may come across several offshoots of standard deviation, like downside deviation, for example, where it tries to address that limitation. Now, as you would expect, standard deviations will typically vary from asset class to asset class. In here we have a chart of historical annual returns for various asset classes and their corresponding risk or standard deviation. As you would expect, stocks have produced higher returns historically, and therefore have higher expected returns in the future. Now, stocks have also been a lot riskier with a lot higher standard deviation than bonds, and we would expect that. So investors expect higher expected returns for equities, but that doesn't come for free. They pay for it with the willingness to take on greater variability of returns, and therefore, risk. Now, these relationships may not always hold true in the short term, but in the long term, you can expect investors to demand higher returns as a variability for a specific asset class increases.