Understanding Return
This section introduces risk and return, discusses how to understand return, and considers portfolio concerns such as diversification and weighting. It also discusses the expectations for expected returns, implications across portfolios, diversification, and understanding security lines. Risk considerations include the types of risk and measuring risk. Why are these topics of risk and return essential to consider?
Expected Return
In probability theory, the expected value of a random variable is the weighted average of all possible values.
LEARNING OBJECTIVE

Calculate the expected return of an investment portfolio
KEY POINTS
 In order to make investment decisions, investors often estimate the expected return of a potential investment.
 Expected value is a concept that the helps investors assess the value of a potential investment based on different future outcomes and a probability for each outcome.
 Once you have categories for different scenario's,
along with probabilities and returns in each scenario, you then
calculate your expected return by multiplying each probability by it's
respective outcome and adding these all together.
TERMS
 expected return
The expected return of a potential investment can be computed by computing the product of the probability of a given event and the return in that case and adding together the products in each discrete scenario.
 expected value
The expected value of a random variable is the weighted average of all possible values that this random variable can take on.
Imagine that your friend offered you a chance to play his game of dice. You have to pay $1 to play and he keeps your money if you roll anything other than a 6. Would you play it? If you answered "it depends", you are ready to learn about expected value.
Is a bet that doesn't always pay out, but pays out big when it does a good bet? Is playing the lottery a good bet? Would it be a good bet if tickets were only $0.10 instead of $1.00?
If your friend told you he would pay you $1 every time you roll a 6, you would be crazy to play. If your friend told you he would pay you $10 every time you roll a 6, you would be crazy not to play.
Expected value is calculated by multiplying the probability that something will happen by the resulting outcome if it happens. In the two cases described above, there are 5 ways to lose and only 1 to win. But in scenario 1, you can expect to lose $0.67 or 67% every time you bet, and in scenario two, you can expect to win $0.83 or 83% with each bet. This is a confusing topic in statistics and finance because on any given roll, there are only two outcomes  win or lose  and neither outcome involves $0.67 or $0.83 increments.
But to understand expected value, you have to imagine playing a particular game hundreds of times. If you were to sit at your friend's apartment and play the dice game 100 times, imagine what your bottom line would be. Even though you lose most of the time you roll in the 2nd scenario, when you win, you win big. After 100 games, you could expect to be up ($0.83 per roll) x (100 rolls) = $83. If you were foolish enough to play the game in the first scenario 100 times you would expect to be down $67 to your friend.
In finance, evaluating your expected return is important, but never as simple as evaluating a game of dice. Imagine that you are going to buy a house a year from today and you have $20,000 saved for that investment. Because you are smart enough to be using a free online text book, you are probably savvy enough to know that you can invest the money for a year and get some return on it. You are considering investing that money into stock of a Ski/Snowboard Mountain in Colorado, so you go talk to your snowboarding friend who lives on your floor of the dorm. He tells you that any given winter could be hella gnarly, totally chillax or wicked bogus, depending on how much it snows. You start doing some research, and you realize that how the stock has performed has everything to do with how much it snows. You create three different categories based on snowfall and label them hella gnarly (HG), totally chillax (TC) and wicked bogus (WB). In recent HG winters, those with over 20 feet of annual snowfall, the stock has averaged a 25% annual return. In recent TC winters, those with between 10 feet and 20 feet of snowdrop throughout the season, the stock has averaged 10%. In WB winters, those with less than 10 feet of total snowdrop, the stock has averaged 20%. Ok, so now what? Should you buy the stock?
If you said "it depends," that's a good answer. Being as smart as you are, you investigate recent weather patterns and you decide that there is a 25% chance of an HG year, a 60% chance of a TC year and only a 15% chance of a WB winter. You understand that past performance is never a guarantee of future results, but still you are happy with your research and you project an expected rate of return for your $20,000. How will it do?
Based on your research, you realize that the stock has an expected return that is calculated thus:
E[R]= (Probability_{HG})x(Return_{HG})+(Probability_{TC})x(Return_{TC})+(Probability_{WB})x(Return_{WB}) = (0.25)(25%) + (0.60)(10%)+(0.15)(20%) = 6.25% + 6% + 3% = 9.25%
If you were to invest the stock in the ski mountain, year after year, and your research proves accurate, you could expect to receive an average of 9.25% return each year. That is your expected return.
Source: Boundless
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