$($The Zero$-$beta CAPM.$)$ Consider a two$-$period economy with per$-$

fect financial markets, where three risky assets in positive net supply

are traded at date $0,$ and assets generate cash flows at date $1.$ There

are $I2$ investors in the date$-0$ financial markets. The investors only

want to consume at date $1,$ and investor $i$ is endowed with a mean

variance utility function $U_i(E[tilde(W{)}_i],$var$[tilde(W{)}_i]),$ where $U_i(*,*)$ is increasing

in its first argument and decreasing in its second argument, and tilde$(W{)}_i$ is

investor $i\text{'}$s date$-1($random$)$ terminal wealth. The date$-0$ equilibrium

rates of return on the three traded assets have the following first and

second moments $($notation follows from Lecture $4)$:

$e=\left[\begin{array}{c}\frac{1}{10}\\ \frac{1}{5}\\ \frac{3}{10}\end{array}\right],V=\left[\begin{array}{ccc}\frac{1}{20}& -\frac{2}{25}& \frac{1}{10}\\ -\frac{2}{25}& \frac{17}{100}& -\frac{6}{25}\\ \frac{1}{10}& -\frac{6}{25}& \frac{2}{5}\end{array}\right]$

$($i$)$ Recall the following parameters:

$a-=e^{\text{'}}{V}^{-1}e,b-=1^{\text{'}}{V}^{-1}e=e^{\text{'}}{V}^{-1}1,c-=1^{\text{'}}{V}^{-1}1,d=ac-b^2.$

Then we have $a=No.5_{,}b=No.6_{,}c=No.7_{?},$ and $d=$

No$.8.$

$($ii$)$ Recall the minimum variance portfolio $w_{mvp}.$ Let ${}_{mvp}$ denote the

expected rate of return on $w_{mvp}.$ Then there is a positive integer $n$

such that $\frac{n}{100}<{}_{mvp}<\frac{n+1}{100},$ where $n=No.9_{?}.$

$($iii$)$ Suppose that there are only three investors in this economy $(I=$

$3),$ and for all $i=1,2,3,$ investor $i\text{'}$s date$-0$ initial wealth is $W_i=i*W^{**},$

where $W^{**}>0$ is a constant. Let ${}_i$ denote the expected rate of return

on investor $i\text{'}$s equilibrium portfolio. You are told that ${}_1=0\mathrm{.}18,$

${}_2=0\mathrm{.}21$ and ${}_3=0\mathrm{.}2.$ Then the expected rate of return on themarket portfolio is ${}_m=$ No$.10.$ Moreover, one can show that the

market portfolio is

$w_m=\frac{1}{261}\left[\begin{array}{c}w\\ z\\ w\end{array}\right],$

where $w=No.{11}_{?}$ and $z=$ No$.12.$

$($iv$)$ Suppose that portfolio $k$ has

${}_k-=\frac{\text{c}\text{o}\text{v}[tilde(r{)}_k,\text{t}\text{i}\text{l}\text{d}\text{e}(r{)}_m]}{\text{v}\text{a}\text{r}[tilde(r{)}_m]}=\frac{3}{4}.$

Then the expected rate of return on portfolio $k,$ denoted $E[tilde(r{)}_k],$ is equal

to $\frac{301}{q},$ where $q=$ No$.13.$