Visual Representation of the Diversification Effect

The diversification effect is the gap between the sum of volatilities and the volatility of a sum. A simple image visualizes the formula of the standard deviation of a sum of two variables (Figure 30.1). The visualization shows the impact of correlation on the volatility of a sum. A vector whose length is the volatility represents each variable. The angle between the vectors varies in line with correlation. The vectors are parallel whenever the correlation is zero, and they are opposed when the correlation is -1 .With such conventions, the vector equal to the summation of the two vectors representing each variable represents the overall risk. The length of this vector is identical to the volatility of the sum of the two variables^{[1]}. The geometric visualization shows how the volatility of a sum changes when the correlation changes.

FIGURE 30.1 Geometric representation of the volatility of the sum of two random variables

FIGURE 30.2 Volatility of a sum when correlation changes

Figure 30.2 shows different cases. The volatilities of the two variables are set to 1 and the correlation only changes. The volatility of the sum is the length of the geometric summation of vectors 1 and 2. It varies between 0, when the correlation is -1, up to 2 when the correlation is +1. The intermediate case, when correlation is zero, shows that the volatility is "2. We extend now the formulas to the general case of a portfolio with many assets.

Extension to any Number of Variables

This formula extends the calculation of the variance and of the volatility to any number N of variables X.. This is the single most important formula for calculating the portfolio returns or value volatility, expanded later on using weights of assets within the portfolio. The formula also illustrates why the sum of individual risks is not the risk of the sum, due to the covariance terms. The general formula is:

In this formula, o2 is the variance of variable X, equal to the square of the standard deviation CT, and o is the covariance between variables X. and x. The sign X corresponds to a generalized summation over all couples of random variables X. and x. A similar, and more compact, notation writes that the variance of the random variables summing m random variables is the summation of all o with, whenever / — /', we obtain the variance o2 — a2.

[1] This result uses the formula for the variance of a sum, and by expressing the length of the diagonal as a function of the sides of the rectangle of which it is a diagonal.

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