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Expected Value

                 

A stock is currently trading at $50. Tomorrow it will announce either good news (and rise to $55) or bad news (and fall to $40). If you believe the probability of good news is 80%, should you buy or sell the stock for $50?

Whether you answered the last question intuitively or rigorously, you were computing an expected value. For a discrete random variable \(X,\) this is \[\sum_{x \in S} x \cdot P(X=x),\] where \(S\) is the set of values that \(X\) takes. For example, the expected value of the stock in the previous question was \[\sum_{x \in \{40,55\}} x \cdot P(X=x) = 40(0.2) + 55(0.8) = 52.\]

A stock is trading at $14. Tomorrow it will announce either good news (and rise to $20) or bad news (and fall to $10). What is the probability of good news, given that the stock is trading at its post-announcement expected value?

One of the most important properties of expected value is the linearity of expectation. This means that \[E(X+Y) = E(X)+E(Y),\] even when \(X\) and \(Y\) are dependent. The next two questions make use of this property.

The price of Stock A in one month is (approximately) normally distributed with mean $100 and standard deviation $25. The price of Stock B in one month is (approximately) normally distributed with mean $50 and standard deviation $5. They have a correlation of 90%. If you buy 3 shares of stock A and (short) sell two shares of stock B, what is the expected value of your portfolio in one month (in dollars)?

You are trading a weather contract that pays out $1 each day that it rains in a given 7-day week. If the probability of rain on any given day is 20%, what is the fair value of this 7-day contract (in dollars)?

Remark: These types of contracts can be used by agricultural companies and energy companies as hedges against unfavorable weather.

A useful construct for exploiting linearity of expectation is the indicator variable. It is 1 when some event occurs, and 0 otherwise. We can reframe the previous problem in terms of an indicator variable.

Take \(\mathbb{1}_{n}\) to be an indicator variable on the event of rain on day \(n.\) Then, the expected number of rainy days is \[E(\mathbb{1}_{1} + \mathbb{1}_2 + \cdots + \mathbb{1}_7),\] which by linearity of expectation is \[E(\mathbb{1}_{1}) + E(\mathbb{1}_2) + \cdots+ E(\mathbb{1}_7).\] Each of these variables has expected value \((0.20)(1) +(0.80)(0) = 0.20,\) so the expected value was \(7 \cdot 0.20 = 1.40.\)

You flip 10 coins and are paid $1 for each consecutive pair of heads. For example, if you flipped TTHHHTHHTT, you would be paid $3. What is the expected value of this game in dollars?

A work event gets very rowdy, and no one can remember which coat they brought, so each person takes a random coat as they leave. If each of the 100 employees brought exactly one coat, what the expected number of people who get their correct coat?

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