Question $4($Portfolio variance $1$pt$)$

Recall from class that the population variance and standard deviation of portfolio has the following

$1$

formulas. With two assets A and $B$ : If the portfolio weight on asset $A$ is $w^A$ and the the portfolio

weight on asset $B{w}^B,$ then

$r^p=w^A{r}^A+w^B{r}^B$

Var$(r^p)=(w^A{)}^2$Var$(r^A)+(w^B{)}^2$Var$(r^B)$

$+w^A{w}^B$Cov$(r^A,r^B)+w^B{w}^A$Cov$(r^A,r^B).$

With more than two assets: If the portfolio weight on asset $i$ is $w^i,$ and asset i has return $r^i,$ then

$r^p={\displaystyle \underset{i=1}{\overset{N}{}}}{w}^i{r}^i$

Var$(r^p)={\displaystyle \underset{i=1}{\overset{N}{}}}(w^i{)}^2$Var$(r^i)+{\displaystyle \underset{i}{\overset{N}{}}}{\displaystyle \underset{j}{\overset{N}{}}}{w}^i{w}^j$Cov$(r^i,r^j)$

Suppose that an investor holds a portfolio of bonds $x,Y$ and $Z.$ Portfolio weights of the three

assets are $20\%,50\%$ and $30\%,$ respectively. The table below shows the variance$-$covariance matrix

of the three assets. What is the standard deviation of the portfolio?