What is the portfolio variance if 30% is invested in share S and 70% is invested in share T?
$\begin{array} { | l | l | l | l | }
\hline & & \multicolumn{3}{|c|} { \text { Returns if State Occurs } } \
\hline \text { State of Economy } & \text { Probability of State of Economy } & \text { Share S } & \text { Share T } \
\hline \text { Boom } & 40 \% & 12 \% & 20 \% \
\hline \text { Normal } & 60 \% & 6 \% & 4 \% \
\hline
\end{array}$
A).002220
B).004056
C).006224
D).008080
E).098000

Question

What is the portfolio variance if 30% is invested in share S and 70% is invested in share T?
$\begin{array} { | l | l | l | l | } 
\hline & & \multicolumn{3}{|c|} { \text { Returns if State Occurs } } \
\hline \text { State of Economy } & \text { Probability of State of Economy } & \text { Share S } & \text { Share T } \
\hline \text { Boom } & 40 \% & 12 \% & 20 \% \
\hline \text { Normal } & 60 \% & 6 \% & 4 \% \
\hline
\end{array}$
A).002220
B).004056
C).006224
D).008080
E).098000

Quizplus · Accepted Answer

To calculate the portfolio variance when investing in two assets, we use the formula:$$ \sigma_p^2 = w_S^2\sigma_S^2 + w_T^2\sigma_T^2 + 2w_Sw_T\sigma_S\sigma_T\rho_{S,T} $$where:- $w_S$ and $w_T$ are the weights of investments in share S and T, respectively.- $\sigma_S^2$ and $\sigma_T^2$ are the variances of returns for shares S and T, respectively.- $\rho_{S,T}$ is the correlation coefficient between the returns on S and T.First, we calculate the expected returns ($E(R)$) and variances ($\sigma^2$) for each share.For share S:- Expected return: $E(R_S) = 0.4(12\%) + 0.6(6\%) = 4.8\% + 3.6\% = 8.4\%$- Variance: $\sigma_S^2 = 0.4(12\% - 8.4\%)^2 + 0.6(6\% - 8.4\%)^2 = 0.4(3.6\%)^2 + 0.6(-2.4\%)^2 = 0.005184$For share T:- Expected return: $E(R_T) = 0.4(20\%) + 0.6(4\%) = 8\% + 2.4\% = 10.4\%$- Variance: $\sigma_T^2 = 0.4(20\% - 10.4\%)^2 + 0.6(4\% - 10.4\%)^2 = 0.4(9.6\%)^2 + 0.6(-6.4\%)^2 = 0.025344$To find the correlation coefficient ($\rho_{S,T}$), we need the covariance between S and T, which is not directly provided. However, we can calculate the portfolio variance directly if we assume the returns are independent (which is a simplification and may not reflect real-world conditions).The portfolio variance formula simplifies (without considering correlation) to:$$ \sigma_p^2 = w_S^2\sigma_S^2 + w_T^2\sigma_T^2 $$Given $w_S = 30\% = 0.3$ and $w_T = 70\% = 0.7$, we substitute the variances calculated:$$ \sigma_p^2 = (0.3)^2(0.005184) + (0.7)^2(0.025344) $$$$ \sigma_p^2 = 0.00046656 + 0.01239248 = 0.01285904 $$This calculation does not match any of the provided options directly because the correlation between S and T was not considered due to lack of data. However, the correct approach involves calculating expected returns, variances, and ideally, incorporating the covariance or correlation coefficient for a precise calculation. Given the options and the approach, there might have been a misunderstanding in the calculation process or an assumption made about the correlation. Without the exact correlation, we cannot accurately calculate the portfolio variance as per the options given. The correct answer should involve a detailed calculation considering all factors, but with the given data and typical approach, an exact match to the provided options is challenging without assuming or calculating the correlation coefficient.

What Is the Portfolio Variance If 30% Is Invested in Share