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Standard deviation of a portfolio is measured by the formula -
(STDp) 2 = w12STD12 + w22STD22 + 2w1w2r1,2STD1STD2

where -
1. w1, w2 are weights of the respective investments Asset 1 and Asset 2
2. STDp is the standard deviation of the portfolio and STD1 and STD2 are the individual standard deviations
3. r1,2 is the correlation coefficient of a linear regression equation relating the returns of Asset 1 with the returns with Asset 2

Take a portfolio with two assets who have -
1. 10% potential return each
2. 20% standard deviation individually, and
3. a correlation coefficient of 0 between Asset 1 and Asset 2 (implying that the returns from Asset 1 are totally uncorrelated with (in layman’s terms, not dependent on) returns from Asset 2.

The standard deviation of a portfolio that comprises 50% investment each in Asset 1 and 2 is equal to -

(0.5)2 * (20) 2 + (0.5)2 * (20) 2 + 0 = +/- 14.1%

Now notice what happens when the rates of return on the assets are

1) perfectly correlated (r = 1) and

2) when they are perfectly negatively correlated (the returns on Asset 1 increase when the returns on Asset 2 decrease and vice versa; r = -1).

In the first scenario, substituting for r =1, the standard deviation of the portfolio is 20%, identical to that of the individual assets. Whereas, in the second scenario, substituting for r = -1, the standard deviation of the portfolio will be 0 i.e. the risk in the portfolio will be zero.

This highlights that diversification can be used to reduce the risk of a portfolio only if the diversification is across investments that are not perfectly correlated. Otherwise diversification will not result in reduction of the overall portfolio risk.