Consider a risk-averse individual with a utility of wealth function U(w) = w^0.5 and initial wealth w = $500. He can invest all his wealth in a safe asset. The asset provides a guaranteed return r = 10% . He can also invest all his wealth in a risky Gamble with a potential capital gain of 60% of the asset's value and a potential capital loss of 30% of the asset's value.

The probability of a capital gain is 65% and the probability of a capital loss is 35%.

. Calculate the wealth in good state wt; and bad state w; for the risky gamble and the expected wealth from this gamble (ES) and compare this with the Expected wealth from safe investment . Given the individual's utility function, calculate the second derivative: U\"(w). What does the value of the derivative for positive values of W tell you about the risk preference of this individual? . For the risky gamble, calculate utility in the good state U(W;), utility in the bad state U(WL), the utility of expected wealth of the gamble MEG), and the expected utility from playing the gamble EUfG) . From the expected utility of the gamble, calculate the certainty equivalence CE of the gamble. . Using the certainty equivalence CE and expected gamble wealth EG, calculate the risk premium RP of the gamble for this individual. What does this risk premium tell you about this individual? [Risk Premium = EG - CE] . Sketch an appropriately labeled utility of wealth diagram with all the components of the gamble: utility of wealth function, wealth in the goodfbacl states, expected wealth from gamble, utility of wealth in the good/bad states, utility from expected wealth of gamble UlEG}, and expected utility of the gamble ElUlGll. . From the rst and second derivatives of U, find the coefficient of absolute risk aversion [CARA], and coefficient of relative risk aversion (CRRA): Raw) = 1353'; thw) = Racw) x w Your answers will be in terms of w . Are your results from the CARA and CRRA calculations consistent with the ArrowiPratt theory of risk averse behavior? Why