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Standard Deviation of Portfolio

By Yuriy Smirnov Ph.D.

Definition

Standard deviation of portfolio return measures the variability of the expected rate of return of a portfolio. Its value depends on three important determinants.

  1. Proportion of each asset in the portfolio.
  2. Standard deviation of return of each asset in the portfolio.
  3. The covariance of returns between each pair of assets in the portfolio.

The number of assets in the portfolio is important in diversification. The greater the number of assets (and the proportion of each asset is small), the importance of covariance is increasing. Thus, the standard deviation of return of a well-diversified portfolio depends more on covariances than on the standard deviation of return of individual assets.

Formula

The standard deviation of portfolio consisting of N assets can be calculated as follows:

Standard deviation of portfolio formula

where N is a number of assets in a portfolio, wi is a proportion of ith asset in a portfolio, wj is a proportion of jth asset in a portfolio, σ2 (ki) is variance of return of ith asset, and Cov(ki,kj) is covariance of returns of ith asset and jth asset.

In terms of correlation coefficient, the formula above can be transformed as follows:

Correlation coefficient formula

where R(ki,kj) is the correlation coefficient of returns of ith asset and jth asset, σ(ki) is the standard deviation of return of ith asset, and σ(kj) is the standard deviation of return of jth asset.

Formula of portfolio’s standard deviation in terms of correlation coefficient

Standard Deviation of Portfolio with 2 Assets

Consider the portfolio combining assets A and B. The formula above can be written as follows:

Formula of portfolio’s standard deviation with 2 assets

or

Formula of portfolio’s standard deviation with 2 assets

Standard Deviation of Portfolio with 3 Assets

The formula becomes more cumbersome when the portfolio combines 3 assets: A, B, and C.

Formula of portfolio’s standard deviation with 3 assets

or

Formula of portfolio’s standard deviation with 3 assets

As we can see, the more assets that are combined in a portfolio, the more cumbersome the formula of its standard deviation.

Calculation Examples

Example 1 – Portfolio of 2 Assets

A portfolio combines two assets: X and Y. The proportion of Asset X in the portfolio is 30%, and the

proportion of Asset Y is 70%. The standard deviation of return of Asset X is 21% and 8% for Asset Y.

Returns of Asset X and Asset Y are positively correlated as far as the correlation coefficient equals 0.347.

Let’s put these values into the formula above.

Standard deviation of portfolio with 2 assets calculation example

As we can see, the covariance between returns on assets is an important determinant of risk diversification.

Example 2 – Portfolio of 3 Assets

Consider the portfolio of three securities.

The correlation coefficients are shown in the correlation matrix below.

Now let’s put those values into the formula above.

Standard deviation of portfolio with 3 assets calculation example

As we can see, the standard deviation of portfolio of 8.809% is less than the standard deviation of any of its securities. This example demonstrates the action of diversification, namely, reducing risk by combining different assets in a portfolio. Reducing risk is achieved because the return on Security A and the return on Security B are negatively correlated.