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Calculating Expected Value of the Gamble for the Lottery

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You have just won the Wisconsin lottery! You have been offered (1) $0.5 million, or (2) a gamble in which you would receive a $1 million if a head were flipped and $ 0 if a tail came up.

a. What is the expected value of the gamble?
b. Would you take the sure $0.5 million or take the gamble? Why?
c. If you choose the sure $0.5 million are you a risk taker or risk averter?
d. Assume that you actually take the sure $0.5 million; you can invest it in either a US Treasury bond that will return $537,500 at the end of a year or a common stock that has a 50-50 chance of being either worthless or worth $1,150,000 at the end of the year.

(1) Calculate the expected dollar profit on the stock investment. (The expected profit on the US Treasury bond is $37,500.)

(2) Calculate the expected rate of return on the stock investment. (The expected rate of return on the US Treasury bond is 7.5%.)

(3) Would you invest in the bond or the stock? Why?

(4) Exactly how large would the expected profit (or expected rate of return) have to be on the stock investment to make you invest in the stock, given the 7.5% return on the bond?

(5) How might your decision be affected if, rather than buying one stock for $0.5 million, you could construct a portfolio of 100 stocks with $5,000 invested in each stock? Each of these 100 stocks has the characteristics as the one stock; that is, a 50-50 chance of being worth either $0 or $11,500 at year end. Would the correlation between these stocks matter? Explain.

Please show all work.

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This solution calculates the expected value of the gamble for the Wisconsin lottery.

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