Weighted sum of two normally distributed random variables incorporating correlation

Question

I am currently looking into a financial model that has two random variables which are normally distributed.

$X \sim N(0,1)$

$Y \sim N(0,1)$

The aim of the model is to create a third variable $Z$ from a weighted sum of the above by incorporating a measure of correlation $\rho$. In other words: $Z$ should be the result of $Y$'s own variance plus some correlation with a "market" variable $X$. The way the model incorporates this is by applying the following logic:

$$Z = X\sqrt{\rho} + Y\sqrt{1-\rho}$$

I would like to understand why $\sqrt{\rho}$ is used (instead for example just $\rho$) for the weighting and whether it is possible to apply this method to lognormal distributions as well.

Thanks.

Answer 1

The correlation between random variables is given as follows, $$\rho_{X,Y} = \frac{\mathbb{E}[(X - \mu_X)(Y - \mu_Y)]}{\sigma_X \sigma_Y} $$ Let's say the unit of X and Y is $U$. Then the unit of $\rho_{X,Y}$ is $\frac{U^2}{U.U}$ which is simply one, but the value of $\rho_{X,Y}$ is still a ratio of squares and if I want to use it for weighting my RV's I will use it's square root.