as372w10a1 - choose Justify your choice 3 A decision...

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ActSc 372 WINTER 2010 Assignment 1 Due Date: January 25, 2010 (in class 12:30pm) 1. The certainty equivalent (CE) of a risk X is defined as u ( CE ) = E [ u ( X )] . Suppose now you are an expected utility maximizer with utility of wealth u ( w ) = w γ , w > 0 , 0 < γ < 1 and your loss exposure is the same as the coin tossing gamble as described in the St. Petersburg Paradox. Find an expression for the certainty equivalent of your loss exposure. What can you say about your risk aversion, as a function of γ ? 2. You are considering the following two projects to invest: Project A pays $99 with certainty, while project B pays $150 with probability 0.5, $100 with probability 0.3, and $20 with probability 0.2. (a) If you are an expected value maximizer, which project (A or B) would you choose? Justify your choice. (b) Now assume that your utility is of the type u ( x ) = x - αx 2 , where α > 0 and x < 1 / (2 α ). You know that you are indifferent between the choice of receiving $100 for sure, or $200 and $50 with a fifty-fifty chance. Which project would you
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Unformatted text preview: choose? Justify your choice. 3. A decision maker’s utility function is given by u ( w ) =-e-. 002 w ,w > 0. The decision maker has an initial wealth of w = 5 , 000 and faces a random loss X with a uniform distribution on (0 , 5000). What is the maximum premium this decision maker will pay for a coinsurance which reimburses 80% of the incurred loss? 4. An individual has the following utility function: u ( w ) = w 1-γ 1-γ , w > (a) State the additional condition(s) so that the above utility function can be used to describe a risk-averse agent. (b) Now assume that the X follows a lognormal distribution with mean μ and variance σ 2 ; i.e. log X ∼ N ( μ,σ 2 ) , where N ( μ,σ 2 ) denotes the normal distribution with mean μ and variance σ 2 . Show that the certainty equivalent of X is CE = e μ + σ 2 2 (1-γ ) . Hint: use moment generating function of a normally distributed random variable....
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