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Author: JuanBobsDad One star, 50 posts Old School Fool Add to my Favorite Fools Ignore this person (you won't see their posts anymore) Number: of 254113  
Subject: CAGR/GSD Limits of Blends Date: 11/16/2007 12:46 AM
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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 obvious that if the log-transformed values are symmetrically distributed about the mean, M, the distribution of real-dollar returns will be positively skewed (simply as a consequence of the multiplicative process). Because GSD > 0 by definition, the skew must be positive (i.e., have a long tail of larger values).

From a statistical perspective, the real-dollar returns could just as well be described by distribution moments without the prestidigitation of the logarithmic data transform. There are fundamental reasons for mandating the transform approach, but it does present difficulties that tend to obfuscate the limits of blends. The distribution of annual returns, itself, can also be described by statistical moments. The first moment, of course, is the average which, because of the positive skew, the real-dollar average annual return will be a value larger than the CAGR.

Since the return of a blend is determined by adding the real-dollar values of the individual holdings, the annual average return of the blend is the weighted average of the component averages - a value between the largest and the smallest annual average returns of the blend components . Therefore, the CAGR of the blend must also be between the largest and smallest CAGRs in the blend.

As we all know, the second moment is the standard deviation or the square root of the variance. For any pair of holdings in the blend,

Var(aX + bY) = Var(X)*a^2 + Var(Y)*b^2 + Cov(X,Y)

where X is the set of (x-u) values,
with u being the average of the x values.

The covariance vanishes if the two screens are independent – the total variance of a blend of N holdings is then the weighted sum of the individual screen variances. The total variance is even larger if the (net) covariance is positive (meaning the individual holdings tend to vary up and down in unison). The value of a blend’s variance can be smaller than the sum of the variance values of the individual holdings if, and only if, the (net) covariance is negative (meaning the individual holdings tend to vary in opposition; when one moves up, another tends to move down). This is exactly what is desired of a blend. However, since the magnitude of the covariance can be no larger than the largest variance. Therefore, the blend GSD cannot be any smaller than the smallest GSD component.

From this it is clear that it is advantageous to choose low GSD screens for blending – at least one of the screens needs to have a small GSD because the blend cannot have a smaller GSD. It is also clear that it is advantageous to choose the higher CAGR screens or at least one higher CAGR screen because the blend CAGR cannot be larger. This is the easy part.

The success of a blend depends entirely on a negative net correlation between its components. Conventional wisdom is to “diversify across sectors” (negatively correlated components) but do the negative correlations remain relatively constant over time or can one be the victim of data mining errors with chance negative correlations (he asked rhetorically)?
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