Hi everyone, Please find the below question Suppose Investor A has a power utility function with y = 1 , whilst Investor B has a power utility function with y = 0.5 . (i) Suppose that Investor B has an initial wealth of 100 and is offered the opportunity to buy Investment X for 100, which offers an equal chance of a payout of 110 or 92. Will the Investor B choose to buy Investment X? Solution provided : If Investor B buys X, then they will enjoy an expected utility of: 0.5 [ 2(sqrt(110) -1)] + 2(sqrt(92) -1)] = 18.08 If, however, they do not buy X, then their expected (and certain) utility is: 2 (sqrt(100) -1) = 18 Thus, as buying X yields a higher expected utility, the investor ought to buy it. My Solution : If Investor B buys X, then they will enjoy an expected utility of: 0.5*SQRT(110)+0.5*SQRT(92) = 10.03987576 Certainty Equivalent = (10.03987576)^2 = 100.7991 If, however, they do not buy X, then their expected (and certain) utility is: sqrt(100) = 10 Thus, as buying X yields a higher expected utility, the investor ought to buy it. my question is why are we multiplying by 2 and subtracting 1 from the equation. The bold part. 0.5 [ 2(sqrt(110) -1)] + 2(sqrt(92) -1)] How is it different from my solution ?
so look at the form of the power utility function: U(w) =( w^gamma -1) / gamma let gamma = 0.5 U(w) = (w^0.5 - 1) / 0.5 = 2 * (sqrt(w) - 1) (because 1/0.5 is the same as 2 and w^0.5 is the same as sqrt(w)) U(110) = 2*(sqrt(110) -1) and U(92) = 2*(sqrt(92) -1) it says in the question equal chance of a payout of 110 or 92, so that is 0.5 chance each. 0.5*(U(110) + U(92) ) 0.5*(2*(sqrt(110) -1) + 2*(sqrt(92) -1) )
