Estimates of genetic correlation from EBVs using Calo’s method

It is quite common to estimate genetic correlation from EBVs, for instance of proven bulls. Typically this is done when you have a small experiment with a few thousand records for a complex trait (say trait A), from which you derive mildly-accurate EBVs, and national evaluations for a well known trait (say trait B). It is essentially impossible to analyse both things together.
Correlating naively EBVs for trait A and B is wrong, because EBVs are not TBVs and they’re regressed towards 0. So the correlation is underestimated. Thus, Calo’s method is typically used. The references are:

• Calo, L. L., McDowell, R. E., VanVleck, L. D., & Miller, P. D. (1973). Genetic aspects of beef production among Holstein-Friesians pedigree selected for milk production. Journal of Animal Science, 37(3), 676-682.
• Blanchard, P. J., Everett, R. W., & Searle, S. R. (1983). Estimation of genetic trends and correlations for Jersey cattle. Journal of Dairy Science, 66(9), 1947-1954)

These are of course approximation that assume, for instance, unrelated animals or in other words, the reliability is so high that across-individual relationships can be ignored. So it possibly should be used only for individuals with high accuracy.
Equation [2] in Blanchard et al. to obtain the genetic correlation between traits A and B from the correlation of EBVs and respective reliabilities Rel is:

$latex r_{A,B} =\frac{\sqrt{(\sum(Rel_A) \sum(Rel_B))}}{\sum(Rel_A Rel_B)} r(EBV_A,EBV_B) $latex

Reading these papers is hard. In Blanchard’s paper b’s are reliabilities because defined as $latex b=\frac{var(EBV)}{var(TBV))} $latex. From Henderson we know that var(EBV) = cov(EBV, TBV) and

$latex Rel=\frac{cov(EBV,TBV)^2}{( Var(EBV)*Var(TBV))} $latex
which simplifies to

$latex Rel=\frac{cov(EBV,TBV)}{var(TBV)} = \frac{var(EBV)}{var(TBV)} $latex
which is equal to these b’s in Blanchard’s paper
For instance assume that all bulls have the same Rel for trait A of 0.3 and average Rel for trait B is 0.9, then r_A, B ≈ 1.9 × r(EBV_A, EBV_B).
so the correlation of EBVs can severely underestimate the genetic correlation.