I have a question on the following exercise from chapter 9 of D. Luenberger, Investment Science, International Edition.

Exercise 2 (Wealth Independence)

 

Suppose an investor has exponential utility function $U(x) = -e^{-ax}$ and an initial wealth level of W. The investor is faced with an opportunity to invest an amount $w \le W$ and obtain a random payoff $x$.

 

Show that his evaluation of this incremental investment is independent of W.

I first considered that the utility after the investment is $-e^{-aW}*e^{-a(x-w)}$ and that this is a factor above the original utility of $-e^{-aW}$ which is independent of W, and then that's the solution.

However, this doesn't really take into consideration the randomness of x. For instance, I know of one possible example where if $x$ takes only 2 values with probability of $\frac{1}{2}$ each then it can be shown that $w$, the absolute value amount to be invested, is the exact same for any initial wealth $W$.

So if it's the case that this question requires a similar result for any random payoff $x$, how would I go about doing that if I don't know the probability function of $x$?

Perhaps I should consider $w$ as a proportion of $W$ and then show somehow that this is equal to some constant $\frac{k}{W}$ when $E(U(x))$ is maximized with respect to $x$.

If this is the approach, how do you differentiate $E(U(x))$ with respect to x?

I have a question on the following exercise from chapter 9 of D. Luenberger, Investment Science, International Edition.

Exercise 2 (Wealth Independence)

Suppose an investor has exponential utility function $U(x) = -e^{-ax}$ and an initial wealth level of W. The investor is faced with an opportunity to invest an amount $w \le W$ and obtain a random payoff $x$.

Show that his evaluation of this incremental investment is independent of W.

I first considered that the utility after the investment is $-e^{-aW}*e^{-a(x-w)}$ and that this is a factor above the original utility of $-e^{-aW}$ which is independent of W, and then that's the solution.

However, this doesn't really take into consideration the randomness of x. For instance, I know of one possible example where if $x$ takes only 2 values with probability of $\frac{1}{2}$ each then it can be shown that $w$, the absolute value amount to be invested, is the exact same for any initial wealth $W$.

So if it's the case that this question requires a similar result for any random payoff $x$, how would I go about doing that if I don't know the probability function of $x$?

Perhaps I should consider $w$ as a proportion of $W$ and then show somehow that this is equal to some constant $\frac{k}{W}$ when $E(U(x))$ is maximized with respect to $x$.

If this is the approach, how do you differentiate $E(U(x))$ with respect to x?

I have a question on the following exercise from chapter 9 of D. Luenberger, Investment Science, International Edition.

Exercise 2 (Wealth Independence)

Suppose an investor has exponential utility function $U(x) = -e^{ax}$ and an initial wealth level of W. The investor is faced with an opportunity to invest an amount $w \le W$ and obtain a random payoff $x$.

Show that his evaluation of this incremental investment is independent of W.

I first considered that the utility after the investment is $-e^{aW}*e^{-a(x-w)}$ and that this is a factor above the original utility of $-e^{aW}$ which is independent of W, and then that's the solution.

However, this doesn't really take into consideration the randomness of x. For instance, I know of one possible example where if $x$ takes only 2 values with probability of $\frac{1}{2}$ each then it can be shown that $w$, the absolute value amount to be invested, is the exact same for any initial wealth $W$.

So if it's the case that this question requires a similar result for any random payoff $x$, how would I go about doing that if I don't know the probability function of $x$?

Perhaps I should consider $w$ as a proportion of $W$ and then show somehow that this is equal to some constant $\frac{k}{W}$ when $E(U(x))$ is maximized with respect to $x$.

If this is the approach, how do you differentiate $E(U(x))$ with respect to x?

I have a question on the following exercise from chapter 9 of D. Luenberger, Investment Science, International Edition.

Exercise 2 (Wealth Independence)

Suppose an investor has exponential utility function $U(x) = -e^{-ax}$ and an initial wealth level of W. The investor is faced with an opportunity to invest an amount $w \le W$ and obtain a random payoff $x$.

Show that his evaluation of this incremental investment is independent of W.

I first considered that the utility after the investment is $-e^{-aW}*e^{-a(x-w)}$ and that this is a factor above the original utility of $-e^{-aW}$ which is independent of W, and then that's the solution.

However, this doesn't really take into consideration the randomness of x. For instance, I know of one possible example where if $x$ takes only 2 values with probability of $\frac{1}{2}$ each then it can be shown that $w$, the absolute value amount to be invested, is the exact same for any initial wealth $W$.

So if it's the case that this question requires a similar result for any random payoff $x$, how would I go about doing that if I don't know the probability function of $x$?

Perhaps I should consider $w$ as a proportion of $W$ and then show somehow that this is equal to some constant $\frac{k}{W}$ when $E(U(x))$ is maximized with respect to $x$.

If this is the approach, how do you differentiate $E(U(x))$ with respect to x?