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The Diversification Effect Metric
None of the diversification effect metrics above, although surely suitable for research purposes, seems as parsimonious for practical application as the direct measure of diversification effect used by Cheng and Roulac (2007) and De Wit (1997). The calculation directly taps the diversification effect without the need for inference or theoretical assumptions and without onerous matrices. In this calculation, the diversification effect lies in the relationship between two forms of the portfolio standard deviation equation. The first form is the generally accepted form, which weights each asset’s standard deviation by its allocation and its correlation coefficient with every other asset in the portfolio. The second form assigns only allocation weights, effectively assuming a perfect positive correlation between all portfolio assets. This form is the allocation-weighted mean standard deviation of each asset in the portfolio. The diversification effect resides in the difference between the results of these two equation forms, because the equations partition the effect of imperfect correlations. The generally accepted form incorporates the effect of correlation coefficients, and the allocation-weighted mean standard deviation does not. Cheng and Roulac expressed the relationship between the two standard deviation forms in an equally weighted portfolio as a ratio of portfolio standard deviation to the allocation-weighted mean standard deviation of individual assets:
Where:
B = Diversification effect
σp = Portfolio standard deviation weighted by allocations and the correlations between each asset in the portfolio
σj = Allocation-weighted mean portfolio standard deviation
This ratio divides the generally accepted form of the standard deviation, which incorporates both allocation and correlation weights, by the allocation-weighted mean standard deviation, which only weights asset standard deviations with allocations. It indicates the percent of risk remaining after quantifying the effect of correlation coefficients on portfolio risk. For the present article, we prefer:
Where:
DE = Diversification effect
By subtracting the Cheng and Roulac ratio from one, DE describes the percent of risk reduction that occurs as a function of correlation coefficients. A larger result indicates greater diversification effect.
Incremental diversification effect can also be useful, particularly in the process of evaluating expected effects of adding or exchanging portfolio assets. Incremental diversification effect is the change in diversification effect that occurs after assets are added or subtracted from a portfolio. We can express incremental diversification as:
Where:
IDE = Incremental diversification effect
DEj-1 = Diversification effect of the original portfolio (the portfolio before portfolio assets are changed)
We can simplify the equation as follows:
Where:
σp-1 = The original portfolio standard deviation weighted by allocations and the correlations between each asset in the portfolio
This ratio indicates the change in the percent of risk reduction that occurs as a function of correlation coefficients after adding or exchanging portfolio assets. Incremental diversification uses the relationship between the allocation-weighted mean standard deviation of the original portfolio (σj-1) and the generally accepted form of the standard deviation of the new portfolio (σp), effectively using the first standard deviation as the basis for the second and all subsequent comparisons.
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