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"Assume an investor with total wealth of $100 that has a constant relative risk aversion (CRRA) utility function. The functional formula for the CRRA utility function is given as

$\ U[W]=\frac{W^{1-θ}}{1−θ}$

Let the relative risk aversion coefficient θ = 3. (If necessary, scale the utility function by multiplying the numbers by the$\ 10^x$ for some x when you answer part b and c)

a) Show that the utility function is consistent with investors preferring higher wealth and lower variance.

b) Suppose there is a single risky asset. The risky asset’s payoff depends on the state of the world. The payoff is given as

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What is the expected utility of an investor whose portfolio consists of 100% in the risky portfolio? What is the expected utility of an investor whose portfolio consists of 100% in the risk-free asset?

c) What is the expected utility if the portfolio consists of 50% in the risky asset and 50% in the risk-free asset? Is it different from the average of the two answers given in part b)?

d) For an investor maximizing the expected utility function,$\ E(U[W])$, find the optimal weight invested in the risky portfolio."

I believe I got the wrong answer for part d) but I can't find the error. Would really appreciate it if someone could help me out!

a) \begin{eqnarray*} U'[W]&=&W^{-θ}\\ U'[W]&=&\frac{1}{W^3}\\\\ U''[W]&=&\frac{-θ}{W^{1-θ}}\\ U''[W]&=&-3W^2\\ \end{eqnarray*}

$\ U'[W]>0$ for$\ W>0$
$\ U''[W]<0$ for$\ W>0\ (QED)$

b) Risky Portfolio: \begin{eqnarray*} E(W) &=& p_1W_1 + p_2W_2\\ &=&0.5(1.2\cdot100)^2 + 0.5(0.92\cdot100)^2\\ &=&11432 \end{eqnarray*}

\begin{eqnarray*} E(U[W]) &=& -\frac{1}{2}\ \frac{1}{E(W^2)}\\\ &=&-4.37\cdot10^-5 \end{eqnarray*}

Risk-free Asset: \begin{eqnarray*} E(W) &=& p_1W_1 + p_2W_2\\ &=&0.5(1.02\cdot100)^2 + 0.5(1.02\cdot100)^2\\ &=&10865 \end{eqnarray*}

\begin{eqnarray*} E(U[W]) &=& -\frac{1}{2}\ \frac{1}{E(W^2)}\\\ &=&-4.81\cdot10^-5 \end{eqnarray*}

c)\begin{eqnarray*} E(W) &=& p_1W_1 + p_2W_2\\ &=&0.5(0.5\cdot1.2\cdot100+0.5\cdot1.02\cdot1.00)^2 + 0.5(0.5\cdot0.92\cdot100+0.5\cdot1.02\cdot100)^2\\ &=&10865 \end{eqnarray*} $\ E(U[W])=-4.60\cdot10^{-5}$

Average of the 2 answers:
$\ E(U[W]) = \frac{1}{2}\ E(U[W])_{risky} +E(U[W])_{risk-free}$
$\ E(U[W])=-4.59\cdot10^{-5}$

The expected utility is thus different from the average of the 2 answers in b)

d) Let y be the weight of the risky portfolio \begin{eqnarray*} E(W^2) &=& p_1W_1^2 + p_2W_2^2\\ &=&0.5(y\cdot1.2\cdot100 + (1-y)\cdot1.02\cdot100)^2 + 0.5(y\cdot0.92\cdot100 + (1-y)\cdot1.02\cdot100)^2\\ &=&0.5[(102+18y)^2 + (102-10y)^2] \end{eqnarray*} $\ E(U[W]) =-\frac{1}{2}\ \frac{1}{E(W^2)}\ $
$\ E(U[W]) =-\frac{1}{(102+18y)^2 + (102-10y)^2}\ $
$\ \frac{dEU}{dy}\ = 0$
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$\ y = -1.94$ (?)

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