# CRRA Utility Function Problem

"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

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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$$\ y = -1.94$$ (?)