What is the standard deviation of a portfolio that is invested 40% in stock Q and 60% in stock R? $$\begin{array} { l l c c c }
\text { State of } & \text { Probability of } & { \text { Returns if State Occurs } } \
\text { Economy } & \text { State of Economy } & \text { Stock } Q & \text { Stock R } \

\text { Boom }&25\% & 18 \% & 9 \% \ \
\text { Normal }& 75 \% & 9 \% & 5 \%\
\end{array}$$
A)0.7%
B)1.4%
C)2.6%
D)6.8%
E)8.1%

Question

What is the standard deviation of a portfolio that is invested 40% in stock Q and 60% in stock R? $$\begin{array} { l l c c c } 
\text { State of } & \text { Probability of } &  { \text { Returns if State Occurs } } \
\text { Economy } & \text { State of Economy } &  \text { Stock } Q & \text { Stock R } \

\text { Boom }&25\%  & 18 \% & 9 \% \ \
\text {  Normal }& 75 \% & 9 \% & 5 \%\
\end{array}$$
A)0.7%
B)1.4%
C)2.6%
D)6.8%
E)8.1%

Quizplus · Accepted Answer

First, we need to find the expected returns of stock Q and R:
$$	ext{Expected Return of Stock Q} = (0.25)(0.18) + (0.75)(0.09) = 0.105$$
$$	ext{Expected Return of Stock R} = (0.25)(0.09)+(0.75)(0.05) = 0.06$$
Next, we need to find the standard deviation of the portfolio:
$$\sigma_{p} = \sqrt{w_{q}^{2}\sigma_{q}^{2}+w_{r}^{2}\sigma_{r}^{2}+2w_{q}w_{r}\sigma_{q}\sigma_{r}ho_{q,r}}$$
where $w_{q}$ is the weight of stock Q in the portfolio, $w_{r}$ is the weight of stock R in the portfolio, $\sigma_{q}$ is the standard deviation of stock Q, $\sigma_{r}$ is the standard deviation of stock R, and $ho_{q,r}$ is the correlation coefficient between the two stocks.

We are not given the correlation coefficient in the problem, so we will assume that the two stocks are uncorrelated, meaning that $ho_{q,r} = 0$.

Substituting in the given values, we get:
$$\sigma_{p} = \sqrt{(0.4)^{2}(0.055)^{2}+(0.6)^{2}(0.027)^{2}} \approx 0.026 = 2.6\%$$

Therefore, the standard deviation of the portfolio is 2.6%. The answer is (C).

What Is the Standard Deviation of a Portfolio That Is