# 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?

## Expected Value

### Expected Value

# Expected Value

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

## Expected Value

### Expected Value

# Expected Value

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? ## Expected Value ### Expected Value # Expected Value One of the most important properties of expected value is the linearity of expectation. This means that $E(X+Y) = E(X)+E(Y)$ That is the expected value of the sum of two variables is equal to the sum of their individual expected values. Even when $$X$$ and $$Y$$ are dependent. The next two questions make use of this property. ## Expected Value ### Expected Value # Expected Value 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 (assuming these actions are all it contains) in one month (in dollars)? ## Expected Value ### Expected Value # Expected Value 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.

## Expected Value

### Expected Value

# Expected Value

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 $$7 \cdot 0.20 = 1.40.$$

## Expected Value

### Expected Value

# Expected Value

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?

## Expected Value

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# Expected Value

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 is the expected number of people who get their correct coat?

## Expected Value

### Expected Value

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