Volatility for a portfolio may be calculated using the statistical formula for the variance of the sum of two or more random variables which is then square rooted. Alternatively, the volatility for a portfolio may be calculated based on the weighted average return series calculated for the portfolio. Both of these methodologies are discussed below.
1. The portfolio volatility formula
Consider a portfolio of three assets X, Y and Z with portfolio weights of a, b and c respectively. The portfolio volatility is:
Variance (X) = Variance in asset X’s returns, i.e. X’s returns volatility squared (σx2)
Variance (Y) = Variance in asset Y’s returns, i.e. Y’s returns volatility squared (σy2)
Variance (Z) = Variance in asset Z’s returns, i.e. Z’s returns volatility squared (σZ2)
ρxy = correlation coefficient of X and Y returns
ρxz = correlation coefficient of X and Z returns
ρyz = correlation coefficient of Y and Z returns
If the assets in the portfolio are independent of each other the correlation coefficient terms in the equation above would be zero. If the assets in the portfolio are perfectly correlated with each other then the volatility of the portfolio would simply be the weighted average sum of the asset return volatility.
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