### Portfolio Standard Deviation

The standard deviation of a portfolio of assets, or portfolio risk, is not simply the sum of the risk of the underlying securities. Due to the correlation between securities, the computation of the portfolio risk must incorporate this correlation relationship.

## Computing Portfolio Standard Deviation

The portfolio standard deviation or variance, which is simply the square of the standard deviation, comprises of two key parts: the variance of the underlying assets plus the covariance of each underlying asset pair. Viewing a portfolio with two underlying assets, X and Y, we can compute the portfolio variance as follows:

$$\text{Portfolio variance} = w_X^2\sigma_X^2 + w_Y^2\sigma_Y^2 + 2 w_{X} w_{Y} \sigma_{X} \sigma_{Y} \rho_{XY}$$

Therefore,

$$\text{Portfolio standard deviaton} = \sqrt{w_X^2\sigma_X^2 + w_Y^2\sigma_Y^2 + 2 w_{X} w_{Y} \sigma_{X} \sigma_{Y} \rho_{XY}}$$

Where:

w = weight of the asset within the portfolio

σ = standard deviation

$$\rho$$ = correlation coefficient

Note that $$\sigma_{X} \sigma_{Y} \rho_{XY} = \text{Covariance}_{XY}$$

## Question

Given the following two-asset portfolio where asset A has an allocation of 80% and a standard deviation of 16% and asset B has an allocation of 20% and a standard deviation of 25% with a correlation coefficient between asset A and asset B of 0.6, the portfolio standard deviation is closest to:

A. 16.3%

B. 2.7%

C. 22%

Solution

We determine the portfolio variance as follows:

Portfolio variance = (0.8)² × (0.16)² + (0.2)² × (0.25)² + 2(0.8)(0.2)(0.16)(0.25)(0.6)

Then, we use the square root of the variance to get the standard deviation:

$$\text{Portfolio standard deviation} =\sqrt{2.66\%} = 16.3\%$$

Calculate and interpret portfolio standard deviation

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