A Simple Portfolio Choice Problem.

This exercise helps you to get a better understanding of constrained optimization problems. We will deal with a simple portfolio choice problem. The household has an initial endowment of wealth $w_0$ and $w_1=w_0(1+r^p)$ is the terminal wealth for some portfolio return $r^p.$ This portfolio return depends on the investment in one risky and one risk$-$free asset and may be written as

$r^p={}^f{r}^f+r,$

where $r$ is the return on the risky asset. We assume that the share invested in the risky asset, $,$ is constrained:

${}_{?}\frac{?}{b}ar().$

Furthermore, we assume that there are only two possible realizations of the return on the risky asset, $r_{\text{l}\text{o}\text{w}}$ and $r_{\text{h}\text{i}\text{g}\text{h}},$ which are realized with probabilities $p$ and $1-p.$ We assume throughout that $r^f=0\mathrm{.}02,r_{\text{l}\text{o}\text{w}}=-0\mathrm{.}08,$ and $r_{\text{h}\text{i}\text{g}\text{h}}=0\mathrm{.}12.$ The objective function is given by $Eu(w_1).$ We will again assume a constant relative risk aversion $($CRRA$)$ utility function given by

$u(w_1)=\frac{1}{1-}{w}_1^{1-}$

The portfolio shares ${}^f$ and $$ must satisfy

${}^f+=1.$

It is now straightforward to write the portfolio return as

$r^p=r^f+(r-r^f).$

We substitute out $w_1=w_0(1+r^p).$ The maximization problem now writes as

$ma{x}_{}{E}_r[\frac{1}{}(w_0(1+r^f+(r-r^f)){)}^{}]$

$s.t.{}_{?}\frac{?}{b}ar()$

for $=1-.$

a$)$ Is the problem convex $($for reasonably small values of $)?$ If so$,$ what does it imply for the solution method?

Assume in the following an unconstrained problem, i$.$e$.,{}_{?}=-$ and $\frac{?}{\text{b}\text{a}\text{r}}()=.{?}^1$

b$)$ Show analytically that the optimal portfolio share is independent of initial wealth. This is an important result in portfolio theory due to Merton $(1969)$ and Samuelson $(1969).$

c$)$ Carefully define the objective function and solve the problem for $)=-3,p=(0\mathrm{.}1$ using a suitable algorithm from Exercise $1$ and denote the solution by ${}^{**}.$

d$)$ Plot $$ against the value of the objective function on the interval ${}^{***}-1,{}^{***}+1.$ Provide an economic interpretation.