A person has an opportunity to bet on a coin. The coin is symmetric, it has a 0.5 probabilityof coming up heads, 0.5 probability of coming up tails. His initial wealth is w0 = $10, hisutility is u(w) = √w. If he bets $x, he gets w0 + 2x if the coin comes up heads, and w0 − x ifthe coin comes up tails.(a) Suppose you bet $1. What is the expected value of winnings?(b) What is the optimal bet amount?(c) Show that this person will not ever bet all of his money (all $10) even though the bethas a positive expected value.(d) How much would a risk-neutral person bet?(e) Consider a family of CRRA utility functions u(w) = w1−γ1−γ . Show that a person with anypositive degree of risk aversion γ > 0, no matter how small, would never bet all of hismoney.
Question
A person has an opportunity to bet on a coin. The coin is symmetric, it has a 0.5 probabilityof coming up heads, 0.5 probability of coming up tails. His initial wealth is w0 = x, he gets w0 + 2x if the coin comes up heads, and w0 − x ifthe coin comes up tails.(a) Suppose you bet 10) even though the bethas a positive expected value.(d) How much would a risk-neutral person bet?(e) Consider a family of CRRA utility functions u(w) = w1−γ1−γ . Show that a person with anypositive degree of risk aversion γ > 0, no matter how small, would never bet all of hismoney.
Solution
(a) The expected value of winnings can be calculated as follows:
E(Winnings) = 0.5*(w0 + 2x) + 0.5*(w0 - x) = 0.5*(10 + 21) + 0.5(10 - 1) = 0.512 + 0.59 = 6 + 4.5 = $10.5
(b) To find the optimal bet amount, we need to maximize the expected utility. The expected utility is given by:
E(U) = 0.5u(w0 + 2x) + 0.5u(w0 - x)
Taking the derivative of E(U) with respect to x and setting it equal to zero gives:
0 = 0.5*(1/√(w0 + 2x))2 - 0.5(1/√(w0 - x))
Solving this equation for x gives the optimal bet amount.
(c) Betting all of his money would result in a utility of zero if the coin comes up tails, since u(w0 - w0) = u(0) = 0. Since the utility function is increasing, this is less than the utility of not betting at all, u(w0) = √10. Therefore, he would not bet all of his money.
(d) A risk-neutral person would bet the amount that maximizes the expected value of winnings, which is $1 in this case.
(e) For a person with a CRRA utility function, the expected utility of betting all of his money is:
E(U) = 0.5u(w0 + 2w0) + 0.5u(0) = 0.5*(3w0)^(1-γ)/(1-γ) + 0
Since γ > 0, (3w0)^(1-γ)/(1-γ) < w0^(1-γ)/(1-γ) = u(w0). Therefore, E(U) < u(w0), so the person would not bet all of his money.
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