Suppose that an investor has the opportunity (and funding ability) to pay $100,000 for a 50% chance to win $300,000 and a 50% chance of winning nothing.

Question

Suppose that an investor has the opportunity (and funding ability) to pay $100,000 for a 50% chance to win $300,000 and a 50% chance of winning nothing.
1. What is the expected value of the gamble?
2. What is the standard deviation of payoffs for this gamble?
3. Suppose that you have the opportunity (and funding ability) to repeat participation in this gamble for a total of five gambles. Each wager’s outcome is independent of the outcomes of all other wagers (the correlation coefficient between wager payoffs is zero). What is the expected value of this set of five wagers?
4. What is the standard deviation for this set of five wagers?
5. Which set of wagers has a higher expected payoff—that described in parts a and b of this question or that described in parts c and d of this question?
6. Which set of wagers has a lower risk as measured by standard deviation—that described in parts a and b of this question or that described in parts c and d of this question?
7. Which set of wagers seems to be preferable based on your answers to parts a through f—the single wager or the set of five wagers?
8. Devise an argument that if an individual finds the gamble described in parts a and b unacceptable, he will also find the gambles described in parts c and d unacceptable.

Prateek · Accepted Answer

1. The expected value of the gamble is the probability weighted value of the total amount to be won in each of the case, from which the investment of $100,000 has been subtracted thereon. Thus, the expected value of the gamble is $50,000.
The calculation is as follows:
Expected Value = [(Winning Probab. * $300,000) + (Losing Probab. * $0)] - $100,000
		= [(0.50 * $300,000) + (0.50 * $0)] - $100,000
		= $50,000
2. Standard Deviation of the payoff is computed using the expected payoff from the gamble which is $150,000. It is computed as follows:
Expected Payoff = [(Winning Probab. * $300,000) + (Losing Probab. * $0)] 
		= [(0.50 * $300,000) + (0.50 * $0)]
		= $150,000
Now, compute the standard deviation in the following manner:
S.D.= [[(Winning Probab. * ($300,000 - $150,000)] + [(Losing Probab. * ($0-$150,000)]]^1/2
       = [[0.50 * ($300,000 - $150,000)] + [0.50 * ($0 - $150,000)]]^1/2
       = $0
3. One can use the following formula to solve this problem.
Expected Value = 5 * Expected Value in 1 win
		= 5 * $50,000
		= $250,000
4. SD = 5 * $0
    = $0
5. Not enough information is provided by the student. Request them to provide (a), (b), (c), and so on parts of this question.
6. Not enough information is provided by the student. Request them to provide (a), (b), (c), and so on parts of this question.
7. Not enough information is provided by the student. Request them to provide (a), (b), (c), and so on parts of this question.
8. Not enough information is provided by the student.

Suppose that an investor has the opportunity (and funding ability) to pay $100,000 for a 50% chance to win $300,000 and a 50% chance of winning nothing. What is the expected value of the gamble? What...

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