To calculate the Portfolio Variance (PV):
\[ PV = \sum_{i=1}^{n} w_i^2 \sigma_i^2 + \sum_{i=1}^{n-1} \sum_{j=i+1}^{n} 2 w_i w_j \rho_{ij} \sigma_i \sigma_j \]
Where:
Portfolio variance is a measure of the dispersion of returns of a portfolio. It is the aggregate of the actual returns of a given portfolio over a set period of time. Portfolio variance is a key concept in modern portfolio theory, which argues that the risk inherent in an investment should be measured by the variance of its return. It is calculated by multiplying the squared weight of each investment by its corresponding variance and adding two times the weighted average of the securities times their covariance. Portfolio variance is used to determine the volatility, or risk, of the portfolio. A higher portfolio variance indicates a higher degree of risk and volatility, and a lower portfolio variance indicates a lower degree of risk.
Let's assume the following values:
Using the formula:
\[ PV = 0.2^2 \times 0.1^2 + 0.3^2 \times 0.15^2 + 0.5^2 \times 0.2^2 + 2 \times 0.2 \times 0.3 \times 0.5 \times 0.1 \times 0.15+ 2 \times 0.2 \times 0.5 \times 0.5 \times 0.1 \times 0.2 + 2 \times 0.3 \times 0.5 \times 0.6 \times 0.15 \times 0.2 \]
\[ PV = 0.0004 + 0.002025 + 0.01 + 0.003 + 0.004 + 0.0054 = 0.024825 \]
The Portfolio Variance is 0.024825.
