# Diversification

Diversification is a measure of the commonality of a population. Greater diversification denotes a wider variety of elements within that population. Diversification is of central importance in investments. Diversification reduces the risk of a portfolio. It does not necessarily reduce the returns. This is why diversification is referred to as the only free lunch in finance.

Diversification can be quantified as the intra-portfolio correlation. This is a statistical measurement from negative one to one that measures the degree to which the various assets in a portfolio can be expected to perform in a similar fashion.

 Intra-portfolio correlation Percent of diversifiable risk eliminated 1 0% .75 12.5% .50 25% .25 37.5% 0 50% -.25 62.5% -.50 75% -.75 87.5% -1 100%

Portfolio balance yes yes occurs as the sum of all intra-portfolio correlations approaches negative one. Diversification is thus defined as the intra-portfolio correlation or, more specifically, the weighted average intra-portfolio correlation. Maximum diversification occurs when the intra-portfolio correlation is minimized. Intra-portfolio correlation may be an effective risk management measurement. The computation may be expressed as:

[itex]

Q = \frac{\sum_{i=1}^n\sum_{j=1}^n X_i X_j P_{ij}}{\sum_{i=1}^n\sum_{j=1}^n X_i X_j} [itex]

Where Q is the intra-portfolio correlation, [itex]X_i[itex] is the fraction invested in asset i, [itex]X_j[itex] is the fraction invested in asset j, [itex]P_{ij}[itex] is the correlation between assets i and j, and n is the number of different assets.

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