### Portfolio Volatility – Market Risk Metrics

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**Sat Oct 29, 2022 6:48 am**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.

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 (σ

Variance (Y) = Variance in asset Y’s returns, i.e. Y’s returns volatility squared (σ

Variance (Z) = Variance in asset Z’s returns, i.e. Z’s returns volatility squared (σ

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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**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 (σ

_{x}^{2})Variance (Y) = Variance in asset Y’s returns, i.e. Y’s returns volatility squared (σ

_{y}^{2})Variance (Z) = Variance in asset Z’s returns, i.e. Z’s returns volatility squared (σ

_{Z}^{2})_{ρxy}= correlation coefficient of X and Y returns_{ρxz}= correlation coefficient of X and Z returns_{ρyz}= correlation coefficient of Y and Z returnsIf 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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