AQR and Leverage

Damian over at Skill Analytics wrote a post on the AQR article from Allaboutalpha.

I agree with his sentiments regarding the way they determine their leverage. I would guess they don't use that formula to determine their leverage but it could be a simplification of something they do use. Nevertheless I would strongly advise not using it.

Let's use l as leverage. They have two portfolios A and B with correlation p and standard deviations stdev(A) and stdev(B). The standard deviation of the portfolio is
stdev(p)=[(stdev(A)/2)^2+(stdev(B)/2)^2+.5*stdev(A)*stdev(B)*p]^2

They set (stdev(A)+stdev(B))/2=l*stdev(p) or l=(stdev(A)+stdev(B))/(2*stdev(p))
Now, if I were to assume that stdev(B)=x*stdev(A) just for mathematical simplification
that would mean l=(1+x)*stdev(a)/(2*[(stdev(A)^2*(1+x^2))/4+.5*x*p*stdev(A)^2]^.5
and: l=(1+x)/[(1+2*x*p+x^2]^.5

So what we have from this little mathematical porn is that if there's no correlation then l=(1+x)/[1+x^2]^.5. In other words if the standard deviations of each asset are the same (x=1) and correlation is 0, then you'd use leverage l=2^.5 which is the maximum leverage you would use. Strangely, as the ratio of the two variances goes from something like x=.75 to 1.25, the peak is when x=1 and declines on either side. The same is generally true for other correlations except that the closer the correlation is to 1, the lower the leverage.

So why does this matter. Basically, if you were to use a system like this to determine your leverage, it is based on two things, the correlation between the two assets and the difference between the variances. In other words, the levels of variance do not matter in this framework, only the difference between the two assets' variances. The correlation part makes sense, but this seems a little too simplistic.