Question

Lottery - Let $1,000 be your current wealth. There are 100 people and each buys a lottery ticket at $5. The administrative cost of the lottery ticket per person is $5. If you win the lottery, you will get $500. There is only one person who can win the lottery. a. Define the gamble b. Calculate the expected value of the gamble c. Is this gamble favorable, fair, or unfavorable? d. Now, suppose your utility function is U = W5/2. Calculate the Certainty Equivalent and the risk premium for individual willing to buy the lottery (i.e., minimum amount of money he/she needs to receive to forgo the lottery)?. e. Now suppose that there are only 50 individuals participating in the lottery. Without calculating all the math again, what happens to risk premium? Does it increase, decrease, or stay the same?

Answer 1

Date: 1 a) consider the given Problem here there are 100 People each buys a lottery ticket at $5. the Probability that an individual will win the lottery is "1/100" The gamble is given by an individual having "$1000" then will have Choo0 +500-5) = $1,495 with Probability 1/100 will have (1.000 - 5) = $995 with Probability 99/100 (6) The expected value is EU = $1.495/100 + $ 995 * 99/100 - - $ 1000 c) Here an individual will get "$500" with Probability 1/100 if he win and will lose (-5) with Probability 99/100. so, the expected pay off is given by - Ex = $ 500/100 - 5* 99 (100 $ 5 - $4.95 = $0.05 zo so as the expected Pay off is Positive implied the gamble is favorable.

page No Date: / / d certainly equivalent is that amount of money at which the person is indifferent between gambling and taking the money, Probability of him winning the lottery is oool Probability of him losing the lottery is 0.99 The amount received by him after winning will be & (1000 +500) $ 1500 the amount lost by him after losing the lottery will be $ (1000 5) 6995 Uw5/2 He will be indifferent about taking the money and gambling only when - EUG = EU CIE Expected utility from gambling is equal to certainly equivalent is expected utility. EUG: 0.01 Cu >1500] + (0.99 [0-> 995] ROOL C1500) 512 +0.99 995) 5/2 : 0.01 X 87142125.3 + 0.99 x 31228972.92 - 8714211253 + 30916.6832 = 31788 104.5 EUGE = w 5/2 EUCE: EUG WS/2 : 31788 lou. 5 - W = (31788 104. 5) = 1002088

thos the certainly equivalent amount is $ 10021088 Now we need to calculate risk Premium Risk Premium it is the amount that the invesbor should expect to get to compensate for the loss he makes due to the risk. know that Expected utility from gambling = utility of the expected value - Risk Premium so Expected value of the lottery - 001 X1500 + 0,99 X995 = 15 + 985105 = 10oo.os so utility (Expected value of lottery - Risk Premium 0 - ( 1000 , 15 - 2)/2 = 0 Expected utility from gambling = =ool (1500) 8/2 +0.99 (995) 5/2 2 31788 lou, 5 — ☺ Equating ② and 31788 lou, 5 = C1ooo os -R / (10ooos -R) & (31788 104, 5) 2/5 1000.05 -R: 1002.088 R : 1000.05 - 2002.088 - 2.038

page No. Date: / Here negative sign shows that $2.038 is the amount that an individual is willing to spend to buy the risk Premium e) Now as less more of participants are there the probability of wining is more and Probability of losing is less than the Previous case. Thus the individual risk of losing money will be reduced comparatively. Thus the risk Premium that any person Participating in the gamble takes will be less than the previous one.