Hello, welcome back, in this lecture, you're going to learn about the concept of risk, how do we define it and how can we measure it? First of all, what is risk? Well, risk means essentially we don't really know what's going to happen, right? There's uncertainty, we may have an expectation of what's going to happen, ex-ante, in other words, before investing. But the future return can rarely be precisely predicted ex-ante. We will only know the actual return when it's realized at a future date, in other words ex-post. So in other words risk means more things can happen than will happen. That is a range of possible outcomes and in advance we don't really know what the realize outcome is going to be. All right, let me start with a simple example, right. Suppose we toss a coin, and you win a dollar, if it comes up heads and you lose a dollar if it comes up tails. If it's a fair coin obviously you have a 50% of a chance of winning a dollar and 50% chance of losing a dollar. In other words, the possible outcomes are +1 or -1 with equal probability. Now if you play this game over and over and over and over again, what do you think you will end up with? Yes, I can hear you, saying zero of course, that's exactly right. On average you will break even and you will end up with a zero payoff. That is of course what we define as the expected outcome, last time, which is the probability weighted average 50% times +1 plus 50% times -1, which gives us 0, right. Note that, the actual outcome expose like the realize outcome is never 0, the expected outcome is 0. The ex-ante is payout 0 but the expose outcome will either be +1 or -1. All right, okay so that's just sort of a teaser for you, let's now look at a probable to distribution of returns, all right. So suppose you invest $100 in this XYZ stock, let's say at times zero and then there are three possible outcomes at t = 1. The share price will either end up going up to 140, or one year from now, or it will go up to 110, or it will go down to 80. Let's say that it will go up to 140 with the probability 25%. It will, most likely it will go up to 110, let's say that's 50% probability and then maybe it will go down to 80, let's say with 25% probability. In other words, the returns, these are the probabilities, the returns in each state is 40%, 10%, and -20%. Now how would you describe this distribution? Well, we can first find the mean, in other words, the central tendency, or what we might expect to have on average. What we call the expected return, which is the weighted average or the probability weighted average off the possible outcomes. So the expected return will be, 40% between 5% probability, it's going to be 40% with 50% probability we're going to get 10% return, and with again 25% probability we're going to lose 20%. So the expected return, it will be 10%. So our expected return, ex-ante is 10%, but note that in each state, the exposed return, the realized return will be different from what we expect. So we define risk as, how widely the actual returns can differ from central tendency, or the expected return. How much do they deviate? So we measure this dispersion as the variance or the standard deviation of the distribution of returns. Okay, so what is variance? Let me write it like this, what is the variance of returns? Well it is measured as the expectation of the expected deviations from. The squared deviations from the mean. So variance is the expected value of the squared deviations from the mean, so suppose you had m possible outcomes. Each with probability Pj, we can write the variance of returns, as the probability weighted average of each outcome's deviation, square deviation from the mean. Now we sometimes we typically denote the variance as sigma squared, and the standard deviation is the square root of that. So in our example then the variance is what? Well we can write is as, sigma squared or the variance of r return is going to be the probability weighted, square deviations from the mean. So its going to be 25% probability times the outcome in that state minus the mean squared plus 50% probability the outcome in that state minus the mean squared. Plus 25% probability minus 0.20 minus 0.10 squared. And if you do all that you're going to find that it is 0.045. So that's the variance, volatility or the standard deviation is the square of that, and that's going to be 0.21%. So the standard deviation of the one period returns on this XYZ stock given distribution is 21%. So the higher the volatility of these outcomes, anyway the higher the dispersion in these outcomes around the mean, the higher is the uncertainty and the higher is the risk. So variance or standard deviation for a given distribution of returns, gives a measure of how widely the possible outcomes are around this mean. So in this lecture, you learned the definition of risk. The variability of possible outcomes around the mean and variance and standard deviation provide us a measure of that uncertainty, or a measure of the risk.