By Yuriy Smirnov Ph.D.

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

- Proportion of each asset in the portfolio.
- Standard deviation of return of each asset in the portfolio.
- 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.

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

where N is a number of assets in a portfolio, w_{i} is a proportion of *i*th asset in a portfolio, w_{j} is a proportion of *j*th asset in a portfolio, σ^{2} (k_{i}) is variance of return of *i*th asset, and Cov(k_{i},k_{j}) is covariance of returns of *i*th asset and *j*th asset.

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

where R(k_{i},k*j*) is the correlation coefficient of returns of *i*th asset and *j*th asset, σ(k_{i}) is the standard deviation of return of *i*th asset, and σ(k*j*) is the standard deviation of return of *j*th asset.

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

or

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

or

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

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.

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

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.

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.

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