CM2 - Utility Theorem (Q- Power Utility Pg -23)

[skharki1]

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 ?

 

[laura_mils]

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) )