Assume we have equally invested in two different companies; ABC and XYZ. We anticipate that there is a 15% chance that next year's stock returns for ABC Corp will be 6%, a 60% probability that they will be 8% and a 25% probability that they will be 10%. In addition, we already know the expected value of returns is 8.2%, and the standard deviation is 1.249%. We also anticipate that the same probabilities and states are associated with a 4% return for XYZ Corp, a 5% return, and a 5.5% return. The expected value of returns is then 4.975%, and the standard deviation is 0.46%. Calculate the portfolio standard deviation:

Exam-Like

Community

TTanishq

Explanation:

Explanation

To calculate the portfolio standard deviation, we need to compute the portfolio variance first and then take its square root.

The covariance between ABC and XYZ is calculated using:

$\text{cov}(R_A, R_B) = \sum P(R_i) [R_i - E(R_i)][R_j - E(R_j)]$

Given:

$\text{cov(ABC, XYZ)} = 0.15(0.06 - 0.082)(0.04 - 0.04975) + 0.6(0.08 - 0.082)(0.05 - 0.04975) + 0.25(0.10 - 0.082)(0.055 - 0.04975)$

$= 0.15(-0.022)(-0.00975) + 0.6(-0.002)(0.00025) + 0.25(0.018)(0.00525)$

$= 0.15(0.0002145) + 0.6(-0.0000005) + 0.25(0.0000945)$

$= 0.000032175 + (-0.0000003) + 0.000023625$

$= 0.0000555$

Since the investment is equally weighted (50% in each):

$W_A = W_B = 0.5$

$\sigma_A = 0.01249, \quad \sigma_B = 0.0046$

Portfolio variance formula:

$\sigma_p^2 = W_A^2 \times \sigma_A^2 + W_B^2 \times \sigma_B^2 + 2 \times W_A \times W_B \times \text{cov}(R_A, R_B)$

$\sigma_p^2 = 0.5^2 \times (0.01249)^2 + 0.5^2 \times (0.0046)^2 + 2 \times 0.5 \times 0.5 \times 0.0000555$

$= 0.25 \times 0.000156 + 0.25 \times 0.00002116 + 0.5 \times 0.0000555$

$= 0.000039 + 0.00000529 + 0.00002775$

$= 0.00007204$

$\sigma_p = \sqrt{0.00007204} = 0.00849$

Therefore, the portfolio standard deviation is 0.00849 or 0.849%.

Powered ByGPT-5