# Portfolio Standard Deviation

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

## Computing Portfolio Standard Deviation

The portfolio standard deviation and variance are important. They involve the variance of the assets and the covariance between asset pairs. For a portfolio with assets X and Y, the portfolio variance can be calculated 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.

$$\sigma$$ = Standard deviation.

$$\rho$$ = Correlation coefficient.

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

## Question

Consider two assets in a portfolio. Asset A has an allocation of 80% and a standard deviation of 16%. Asset B has an allocation of 20% and a standard deviation of 25%. The correlation coefficient between asset A and asset B is 0.6. In this case, 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 \times (0.16)^2 + (0.2)^2 \times (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:

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

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