Learning outcomes. After completing this video you should be able to calculate the holding period return of a security, explain risk using standard deviation, calculate the expected return and risk of a security. Expected returns and risk. In this video, we will define what expected return and risk are and the relation between the two. Let's look at how you would calculate the return on an investment in a stock. Say you pay $50 to buy one share in a stock today. After a year, the stock price is $55. During the one year you hold the stock, it pays you a dividend of $1. So what return have you earned from holding the stock for a year? There are two parts. One is the income from the dividend, which is $1, and the other is a change in price from $50 to $55. You earned a profit of $6 on an initial investment of $50, which means your holding period return over the one year is 6 divided by 50, which is 12%. We can write a general formula for holding period returns as follows P sub 1- P sub 0 + D sub 1/P0. In this example the ability to calculate the holding period return depends on knowing how much dividend will be paid while you hold the stock, as well as what the stock price will be after a year. However, we almost never know what these future numbers will be. So all we can do is to estimate what the return over the next one year will be. The return hence is a random variable that is characterized by its possible outcomes and the probabilities associated with each outcome. We can then estimate an expected return of an investment using the possible outcomes and their probabilities. This expected return Is what we use as the discount rate when we calculate present and future values. Now let's turn to risk. What most people think of risk as being a bad thing, they relate it to the likelihood of losing money. Risk is actually the uncertainty of outcomes, both good as well as bad. In the context of investments, the risk is the uncertainty of future returns. This is usually measured using either the variance or standard deviation of returns. The larger the variance or standard deviation, the larger is the risk. Now let's see how expected return and risk are related. You have $100 and two investment choices to make. The first investment will give you $105 guaranteed after one year. Since there is no uncertainty or risk in this payoff, this is what is called a risk-free or riskless investment. The second investment will pay you either $150 after one year with a 60% probability or $80 after one year with a 40% probability. This investment is risky, as your future payoff is uncertain. The question is which investment should you invest in? To answer this question we need to compute the expected return and risk of the two investments. For the first investment the expected return is 105- 100, which is 5%. There is no uncertainty in the investment and hence its risk is 0. For the second investment, we have to compute the expected return for both possible payoffs. When the future return is $150, the expected return is 150- 100/100, which is 50%. When the future payoff is $80, the expected return is 80- 100/100, which is a -20%. With this investment, now there is a 60% chance of a 50% return, and a 40% chance of a -20% return. So its expected return is 0.6 x 50% + 0.4 x -20% = 22%. Clearly, the second investment seems more attractive, given its higher expected return. However, what is its risk? Its variance is 0.6 x 0.5- 0.2 to the whole squared + 0.4 x -0.2- 0.2 to the whole squared, which equals 0.1176. The standard deviation is simply the square root of variance. The square root of 0.1176 is 34.29%. The second investment has a return that is 17% higher but also has a risk that is 34.29% higher. As an investor, is the additional 17% return appropriate for the 34.29% risk? The answer is that it depends on two things. One, the investor's attitude towards risk, which is the domain of utility theory. Two, models that help compare the risk and return of different investment choices. These models are called asset pricing models. We will answer the question as to which investment is better after we've discussed utility theory next time.
