I am assuming that we are defining the new variable as a linear combination of 2 "old" ones.
So let's say we have , and we define
where A,B are constants, and a,b are the indexes of two of the X's that you are going to combine.
So the correlation coef of this new one versus any old one is:
Let's reduce this to a function of the known variables.
Finally we know
And so we get:
Now you said you have estimates of all these parameters. I can't say how good your corr corf will be, if you stick your estimates into the formula. And looking back at this, maybe I should have started with the sample correlation coefficient formula, rather than with the population. So maybe you might want to rederive what I did using the same version. I think just trying to do the arithmetic in my head that you will get the same formula except that the population parameters will be replaced with the sample ones (the sample corr coeff just has an (n-1) instead of an nin the denominator which will cancel out ).