Linear Combination

CFA Level I Quant
#1

Can someone point me to the explanation of this (I don’t understand what a linear combination of two or more normal random variables looks like/is - Not asking for proof/derivation, just trying to get my head around this statement):

A linear combination of two or more normal random variables is also normally distributed.

(Institute 534)

Institute, CFA. 2016 CFA Level I Volume 1 Ethical and Professional Standards and Quantitative Methods. CFA Institute, 07/2015. VitalBook file.

The citation provided is a guideline. Please check each citation for accuracy before use.

#2

You know the formula Rp = w1*R1 + w2*R2 + … wn*Rn, I.e. portfolio return is the weighted average of the returns of each security in the portfolio? That’s simply a linear combination.

#3

Ah I see thanks. Easier than thought. Now just need to wrap my head around the concept, to me it’s not intuitive. If you have assets with different means and standard deviations e.g. Asset1 has mean -500 and SD of 25 and Asset2 has a mean of 500 and SD of 25 for example, you’ll have a dual peaked distribution? Hardly normal… hmm… or does this have something to do with central limit theorem?

#4

Your example would be normally distributed around zero with a SD of ~ 35 (the sqrt of 25^2 + 25^2). Note, X & Y need to be independent.

https://en.wikipedia.org/wiki/Sum_of_normally_distributed_random_variables

#5

^ Slight correction: X and Y do not have to be independent. The general expression for the variance of aX + bY, where X and Y are random variables and a and b are constants, is as follows:

Var(aX+bY) = a2 * Var(X) +b2 * Var(Y) + 2*a*b*Cov(X,Y)

If X and Y are independent, then Cov(X,Y)=0 and the last term disappears.

#6

The distribution will still be normal with one peak. The mean and standard deviation will depend on weights in each asset class and the covariance between the asset classes.