In this first session on defining and measuring risk, we're going to focus our attention on what we refer to as stand-alone risk. That is the risk that an investor faces when investing in a single asset on it's own. Now, before proceeding, it's important that we acknowledge that in the day-to-day world the term is risk bantered about to mean many different things. Common perceptions of risk include the chance of losing money, or the possibility that something will go wrong, or the likelihood that a project will fail. As you can see, risk is most often talked about with very negative connotations. But in finance, we take a broader view of risk, and begin by defining it as simply the chance that things might turn out other than expected. For better or worse. Now to illustrate this interpretation of risk. Let's work with an example. You are now the CFO of a very large listed company. Three of your division heads come to you and request funding for three alternative projects. You send them away and as your prerogative, you ask them to come back after collecting data relating to the past performance of similar projects that have previously been undertaken by the firm. Specifically you demand information on the rate of return generated each and every time that a similar project has been undertaken. They arrive back with the data and you collate the information into this table. So, working first with Project One, you see that on 20% of previous iterations in the project the project generated a rate of return of exactly 0% per annum. On 60% of occasions, the project generated a rate of 10% per annum. And on another 20% of occasions it generated a return of 20% per annum. Looking at Project Two, we see a greater variability in returns as compared with Project One, on both the upside, and the downside outcomes. With a 10% chance of the project generating a negative return of minus 10%, but on the upside, providing a 10% chance of a return of 30%. When we consider Project Three. We see that the variability of the returns have increased once again. With now the downside extending to a 2% chance of generating a return of -40%. Yet on the upside a 2% of a return of positive 60%. So, assuming that past return histories, or distributions, are a fair indication of what future distributions might look like, we can now estimate two very important measures from the previous data. Firstly, the expected return of each project is a measure of what we expect to get out of the project where that expectation is measured before the project begins. The second important measure is the Standard Deviation of Returns. Where we refer to these by the Greek letter sigma. Now this is a measure of the variability of the returns of the project, relative to the expected return of the project. Now, we're going to spend a fair bit of time working with these two measures in the next couple of modules, so it's important that we know exactly how to compute them. So firstly, let's draw up each of the three projects and, specifically, let's graph the returns that could occur for each project, which is on the horizontal axis of each graph, against the frequency with which each return occurred. Which is on the vertical axis of each graph. When you plot outcomes against frequency, the graph is known as a histogram. So, we can see from these that Project One has a range of outcomes centered much more closely on the return of 10% per annum. Whereas Project Three, while still centered on a return of 10% per annum, exhibits a greater number of occasions where the returns greatly exceeded or were far less than the 10% return. Now given our earlier definition of risk, referred to risk as suggesting that it involved the chances that things might turn out other than expected. Intuitively, it makes sense to think of Project Two as being riskier than Project One, and Project Three as being riskier than Project Two. So, let's calculate the expected return of Project One. To do so, we simply multiply each return that could occur by the likelihood of frequency of it occurring in the past. We then add up all those numbers we have just calculated. So there's a 20% chance that a return of zero occurs. 20% times zero equals zero. Then there's a 60% chance that a return of 10% occurs. 60% times 10% equals 6%. Finally, there's a 20% chance of a return of 20%. So 20% times 20% equals 4%. All of the other returns have a zero probability of occurring, and so they can be ignored. We then add each of the numbers we have calculated, 0%, 6% and 4%, and we end up with an expected return of 10% per annum for the project. We go through the same process for Projects Two and Three. And end up with expected returns for each of these projects, of 10% per annum as well. Now let me pause for a second, and take the opportunity to suggest to you that, even if you think you know what's going on, it probably makes sense at times like this, to pause the video and carefully reconstruct the numbers in the example yourself, so that you can assure yourself that you actually do understand what's going on. So all three assets have the same expected return, which isn't that surprising when we look at these three distributions. But what about risk? We definitely believe that the risk profile of the three projects is quite different. Let's see. The measure of risk that we will estimate here is what's referred to as the standard deviation. Which once again is denoted by the Greek letter sigma. Let me pause again and point out that standard deviation is itself derived from another measure of variability, that we will refer to as variance. Variance is denoted as sigma squared. That is, sigma to the power of 2. So standard deviation is simply the square root of variance. Don't worry, this will become very clear in a moment. The way that we calculate standard deviation is by firstly measuring the distance from expected return of each possible return. We then square that distance and multiply that number by the likelihood of the return occurring. We then add up all those numbers and that gives us the variance of returns. The square root of the variance is simply sigma and that's the standard deviation of returns. Let's take Project One once again. Starting with the return of 0%, the distance from the expected return of 10% is minus 10%. Step 2, we square that number and get 1%. Step 3, we multiply that number, 1%, by the likelihood that the initial return would occur, which is 20% and we get 0.2%. Doing this for each of the three possible outcomes of Project One yields a total sum of square differences of 0.4%, which is what we refer to as the project's variance, as denoted by sigma squared. The square root of this number is 6.32%, which is the project's standard deviation. So we go on and we repeat this for Projects Two and Three. And as our intuition told us earlier, we find that Project Two is riskier than Project One. And Project Three is riskier than Project Two. Project One has a standard deviation of returns of 6.32% per annum. Project Two a standard deviation of returns of 10.95% per annum, and Project Three 19.9% per annum standard deviation. Well that's well in good if you're the CEO of a large listed company, who can go order his or her minions, to go off and collect the intricate data required to build the histograms needed to then estimate project risk and expected return. But what if your out on your own? Lets say you're trying to get a hand on the risk of the shares of different companies. Well the good news is that's relatively straight forward. There are four simple steps to estimating the standard deviation of returns for a company's stock. Firstly, we download price information for the company. Secondly, we use those prices to calculate daily returns. Thirdly we utilize a spreadsheet program, like Excel, to calculate the standard deviation of returns. And finally, by convention, we scale up our daily standard deviation to a standard deviation as measured on an annual or per annum basis. So let's have a go at doing this using Kellogg's stock returns for the 2014 calendar year. Step one is to download the price file for this stock. Now I use the Yahoo Finance website to download Kellogg's prices, but there are other free databases around. One good thing about the Yahoo website is that it provides adjusted closed prices, which are useful, as they account for dividends. Otherwise, you will need to make sure you add dividends back in on the ex-dividend date to allow for the fact that shareholders have received part of their return on that day in the form of cash. Step two is to convert daily prices into daily returns. We do this by simply calculating the return for today by subtracting the closing price of yesterday from the closing price of today and then dividing that difference by the closing price yesterday. So when the price for Kellogg's fell from $58.14 on the 2nd of January to $57.92 on the 3rd of January 2014, we recorded daily return of -0.378%. So we'll repeat this using all of the daily closing prices for 2014. And observe the frequencies with which different returns occur. When we graph this we end up with a histogram, as shown by this graph. As you can see the most common daily return was around zero to half a percent. What about other companies? As you can see from these histograms, for the well known social media company Facebook, and the highly popular travel portal Trip Advisor, return distributions do vary remarkably between companies. With both of these companies exhibiting a greater spread in realized daily return than as experienced by Kellogg's. Interestingly, the S&P 500 index, which is a stock index that consists of 500 of the largest most frequently traded stocks in the US. That index exhibited a much tighter spread in returns, as compared with the individual companies documented here. The third step is to utilize a spreadsheet program, such as Microsoft Excel to calculate the standard deviation of returns using the STDEV function. As we have 252 trading days in 2014, we've downloaded 252 prices and calculated 251 returns. The Daily Standard Deviation and Returns for Kellogg's shares in 2014 is 1.0816%. The final step is to convert that Daily Standard Deviation, into a Yearly Standard Deviation. By simply multiplying the Daily Standard Deviation figure by the square root of the number of daily returns calculated over the full year. So in this case, by the square root of 251. This yields an annual standard deviation measure of 17.14% per annum. Now of course, if I had have used 12 months of monthly returns to estimate a monthly standard deviation, I would have multiplied this figure by the square root of 12. And if I had have calculated the weekly standard deviation by using 52 weekly returns, I would have scaled that figure by the square root of 52. Pretty straightforward, right? When I do this for each of the other return series. I end up with Facebook having a standard deviation of returns of 35.69% per annum. TripAdvisor's standard deviation of 40.8% per annum, and the stock market index, the S&P 500, a standard deviation of 11.33% per annum. Which gels neatly with our intuition that told us that Facebook looked riskier than Kellogg's, and TripAdvisor looked riskier than Facebook. Now an open question for us is, why might a stock market index exhibit less risk than the companies within the index? Well, we get to that point later in this module. In summary, in this session we've defined the concepts of expected return and standard deviation of return, which is a measure of risk that looks at the variability of returns, relative to an asset's expected return. We've also demonstrated how to measure standard deviation of returns, using either historical returns and probabilities or frequencies, and then historical returns on their own. Next up, we're going to consider how different investors might regard the trade-off between risk and return differently. And that might help explain why different investors hold different portfolios of assets with different risk profiles.