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Subject:  CAGR/GSD Limits of Blends Date:  11/16/2007  12:46 AM
Author:  JuanBobsDad Number:  203944 of 262214

Recently there has been some revisiting the subject of blends with particular concern about what is possible and/or realistic (see Zeelot’s recent posts #203794 and #203916, for example). I only want to say something about what is possible of a blend in numeric terms (realism isn't my strong suit). I claim the following to be true.

1. The blend CAGR cannot be as large as the largest component CAGR nor can it be less than the smallest CAGR component of the blend.
2. The blend GSD cannot be any smaller than the smallest GSD component value.
a. The reduced GSD goal of a blend can be achieved if, and only if, at least two of the components are negatively correlated (i.e., the components vary “out of phase” so that when one tends to go up, another tends to go down).
b. If all the components are independent the GSD is a value in the neighborhood of the weighted sum of all the component GSD values.

As we all know, a blend is composed N simultaneous positions, each of which is customarily described by CAGR and GSD statistics (Note that periodic price ratios are also (1+%gain) values):

M = ln(1+CAGR) = m * average{ ln(price ratios) }
S = ln(1+GSD) = sqrt(m) * stdev{ ln(price ratios) }

where m is the number of rebalancing periods in a calendar year.

The only assumptions involved in these calculations are that investment returns are characterized by multiples of the original capital (as opposed to additive/incremental gains) and that these multipliers are independent and identically distributed random variables . Chebychev’s inequality gives meaning to the standard deviation and says that no more than 1/k2 of the values are more than k standard deviations away from the mean.

In real-dollar space, there is also an equal chance of getting an annual return greater than the CAGR or less than the CAGR. The inverse logarithmic transform only remaps the return values and does not affect the associated probabilities/frequencies. No more than 1/k2 of the years will deliver returns outside the range from exp(M-kS) to exp(M+kS).

exp(M - kS) = (1+CAGR) / (1+GSD)^k
exp(M + kS) = (1+CAGR) * (1+GSD)^k

It should be obvio