Today I would like to talk about risk and return, which are the fundamental concepts underlying all the theories of asset pricing. First of all, let me explain to you why those concepts are particularly important for us. We'll start with the simplest cases and step by step, we will proceed to more complicated examples. In the two previous sections, we have studied the fundamentals of debt and equity pricing based on the discounted cash flow approach. So far, we did not pay too much attention to where the expected return on a security comes from. We have mentioned though, that the expected return on the security used in the DCF based valuation can be treated as the opportunity cost of capital, which is the return of an alternative investment carrying the same level of risk. However, in practice, it's rather tricky to find such kind of investment. That's why we have a different way to approach the estimation of a security return. In this section, I would like to question myself, how can we estimate the return on a security? Indeed, talking about return, we need to distinguish between the two types of return. The return I have been talking about is called the required return and this is an example of an external estimation of the stock return. However, we can also find out what the expected return on a security is by observing its past performance at the market. In this case, the expected return will depend on historical prices of a security, which is in turn, the internal estimation of the stock required return. The most important thing is that in a perfect market, which acts as the theoretical background for most of the financial models, an expected return should be equal to the required return on a security. Meaning that indeed, in a perfect market, as we have previously discussed, the net present value of investing in any financial security will equal zero. Indeed, if an internal return generated by the security equals the external return required by the market, then indeed, we can conclude that the asset is priced correctly. In this case, by putting the equality sign between the expected return and required return, we may have a chance to determine the required return based on expected returns. How do we determine a single-period return on a security? In order to do that, let's consider a very simple example. Suppose yesterday you purchased one stock for $100, and today you sold it for 102. What is the return on this transaction? I believe it's easy to calculate that as your wealth has increased by 2 percent over just one day, then your return on this transaction will be 2 percent if the price goes up to 102. On the other hand, the price may decrease. If today you sold your security at the price of $95, then your return will be a loss equaling to minus 5 percent. In this case, we can see that a return on a single security is based on the price and derived from it. That's why we can say that the return on a single security is the function of the stock price. As we have seen on this example, it can be either positive or negative. In the previous example, we have calculated the daily return on a stock. However, we can use a different frequency. For example, we can compare the prices at the end of each week or at the end of each month and even at the end of each year. In these cases, we will be calculating weekly, monthly, and annual returns respectively. This may be pretty much different from daily returns, of course, and the frequency of our return calculation is generally determined by the sample that we are analyzing. As an example, I have considered the daily returns of Apple Incorporated Stock over the three years from 2018-2020. I started with collecting the closing prices for the stock and then calculating daily returns as a percentage change in the price compared to the closing of the previous day. In this example, I had about 750 observations of daily returns, about 250 each year. Now, I have to process this information to make a more general conclusion about the return on Apple Incorporated Stock. For example, taking a look at the daily returns of Apple Incorporated Stock in historical perspective, we can see that from day to day, the return on that stock was pretty much different. We could observe some days with high volatility when the exchange recorded a big increase, as well as the big decrease of the stock price. In other days, however, the change was not so dramatic, and in most cases, the fluctuations of the Apple Incorporated stock price measured by its daily returns was not very significant. In this case, taking this sample, including about 750 observations, we can make the distribution of these returns by using a histogram. Here you can see it on the picture. As we can see indeed, in most observations corresponding to most days, the daily change of Apple Incorporated Stock, which is the daily return of Apple Incorporated Stock, was not so big in almost 180 observations. It stayed within quite a narrow range between minus 0.5 and plus 0.5 percent. However, more dramatic changes in Apple Incorporated Stock price did not take place too often. As we can see, the tails of this distribution are rather thin. We can observe the same type of distribution for almost all types of financial and non-financial assets traded at the market. In statistics, such kind of distribution is often approximated by normal distribution, which is a very frequently used function in statistics. As we can see on this example, the returns of Apple Incorporated measured using daily returns also may follow this kind of normal distribution. How do we determine the expected return on a stock? An expected return, which is the same as the mathematical expectation, can be equal to the average historical return over the observation period. In our example, the observation period was three years. The expected return is the average historical return and can be calculated based on daily, weekly, monthly, or annual returns depending on our goals and objectives of our study. That means that indeed, talking about the future, we need to look into the past and we extrapolate the returns observed during the historical period for the forecastable future. This is one of the underlying assumptions of the portfolio theory and I will refer to it in the next section.