

Could u give a small hint?


E(Y+Z)=E(Y)+E(E(ZY))=E(Y)+0. so the expected values of both r equal. every risk averse individual would prefer Y to Y+Z coz they wont take additional risk if the expected value isnt increasing. risk neutral people would be indifferent. and risk loving people would prefer Y+Z. this means only statement 2 is correct.
i know this is wrong. in fact i think the first step is wrong.

Administrator

Sure. For understanding, lets do the problem for the finite state space or sample space i.e. S = {s(1), s(2), ..., s(n)}. Y takes only two values y(1) and y(2) i.e. Y: S > {y(1), y(2)}. And its given that Z is another random variable Z: S > R. It is also given that Pr(Y = y(1)) = pi(1) and Pr(Y = y(2)) = pi(2). And we know that E(ZY) = 0.
We can think of Y and Y + Z as two lotteries or gambles. The question asks you that if an individual has to choose between the two which one will he pick depending on his attitude towards risk?


Hmm..so is what I did completely wrong?


I am getting same answers as vasudha..
expected returns of both are same.
and risk in Y+Z is more..
So, just 2nd is true.

Administrator

This post was updated on .
Hi Vasudha and AJ,
Your conclusion is correct and I could also see that you have right reasons in mind. The only thing that is missing is the convincing expressions. Here is how you prove it:
Given that the person is risk averse, his utility function for money will be a concave function.
Eu(Y + Z)
= E(Eu(Y + Z)Y)
= pi(1) Eu((Y + Z)Y = y(1)) + pi(2) Eu((Y + Z)Y = y(2))
= pi(1) Eu((y(1) + Z)Y = y(1)) + pi(2) Eu((y(2) + Z)Y = y(2))
≤ pi(1) u(E(y(1) + Z)Y = y(1)) + pi(2) u(E(y(2) + Z)Y = y(2)) [By concavity of u(.)]
= pi(1) u(y(1)) + pi(2) u(y(2)) [E(ZY = y(1)) = 0 and E(ZY = y(2)) = 0]
= Eu(Y)
This automatically disproves (iii), (iv) and (v). For (i), just construct an example. As you said a risk loving person with appropriate choice of state space, pi, lotteries and utilities would do.


I see :) On Wed, Jun 20, 2012 at 5:53 AM, Amit Goyal [via Discussion forum] <[hidden email]> wrote:
Hi Vasudha,
Your conclusion is correct and I could also see that you have right reasons is mind. The only thing that is missing is the convincing expressions. Here is how you prove it:
Given that the person is risk averse, his utility function for money will be a concave function.
Eu(Y + Z)
= E(Eu(Y + Z)Y)
= pi(1) E(Eu(Y + Z)Y = y(1)) + pi(2) E(Eu(Y + Z)Y = y(2))
= pi(1) E(Eu(y(1) + Z)Y = y(1)) + pi(2) E(Eu(y(2) + Z)Y = y(2))
≤ pi(1) E(u(E(y(1) + Z)Y = y(1))) + pi(2) E(u(E(y(2) + Z)Y = y(2))) [By concavity of u(.)]
= pi(1) E(u(y(1))) + pi(2) E(u(y(2)))
= Eu(Y)
This automatically disproves (iii), (iv) and (v). For (i), just construct an example. As you said a risk loving person with appropriate choice of state space, pi, lotteries and utilities would do.

