Topic 9_Risk&return - Topic 9 Risk and Return...

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Topic 9 Risk and Return Readings: Chap 9 Chap 10 (10.1-10.6, 10.9)
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Say hello to risk What is risk? The possibility that an actual return will differ from our expected return. Uncertainty in the distribution of possible outcomes.
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Why Does Risk Matter? For the simple reason that most people don’t like risk. What does this fact imply about the shape of people’s preferences (utility functions)? Utility of Wealth Wealth A concave utility function like this one implies that receiving $1 increases your utility less than losing $1 decreases your utility Utility increase with a $1 increase in wealth Utility decrease with a $1 decrease in wealth
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Example of Risky Business Example 1: Toss a coin. Heads you win, tails you lose. The shape of the utility curve on the previous slide says that the stakes matter. The stakes are $1, who wants to gamble? The stakes are $5, now who wants to gamble? What happens when the stakes are $10, $100, $1000, $1 million, or $10 million? The expected value of each gamble is the same, but virtually all people prefer the smaller gambles to the larger gambles.
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An Example of Risky Business Example: Instead of my current grading scale, you have the option to answer a single question on a date of my choosing. Consider two grading scales: (suppose that you have a 50-50% chance of getting it right) Scale 1 Scale 2 Correct Answer 70 100 Incorrect Answer 60 30 The expected grade in each case is 65. However, which of these scales do you prefer? Case 1 This grading scheme guarantees you pass the class Case 2 Pass and get the credit; fail and take it again
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The intuition of risk What mattered in our previous two examples? 1. The level of the expected outcome (actually we held this constant) 2. The dispersion of possible outcomes In the presence of risk one does not know beforehand what the return on any asset will be. However, sometimes we do have expectations about the possible outcomes and can assign probabilities to each outcome according to expectations.
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Measuring risk Say, we bet on coin flips. If it is a head, you win $1. If not, you lose $1. What are expected outcomes? What is the probability of each outcome? Head: +$1 50% Tail: -$1 50% What if you roll a die? What’re the expected outcomes and the probability of each outcome?
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Defining Expected Value The expected value (or mean) of a random variable X is given by E [ X ] X = p i X i i = 1 n where p i i = 1 n =1 Example: What is the expected value of the coin flip when the stakes are $1? E[X] = p 1 ($1) + p 2 (-$1) = .5 ($1) + .5 (-$1) = 0 What is the expected value of the coin flip when the stakes are $1000? E[X] = p 1 ($1000) + p 2 (-$1000) = .5 ($1000) + .5 (-$1000) = 0
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Example If you roll a die and get paid whatever number you get (i.e. you get $4 if you roll a 4), what’s your expected payoff?
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