(see also post on linear combo of random variables…)
Develop quick intuitions — Quiz: consider A + B under independence assumption and then under 100% correlation assumption. When is variance additive, and when is stdev additive?
(First, recognize A+B is not a regular variable like “A=3, B=2, so A+B=5”. No, A and B are random variables, from 2 noisegens. A+B is a derived random variable that’s controlled from the same 2 noisegens.)
If you can’t remember which is which, remember independence means good diversification[intuitive], lower dispersion, lower spread-out around the expected return, thinner bell, lower variance and stdev.
Conversely, remember strong correlation means poor diversification [intuitive] , magnified variance/stdev.
–Case: 100% correlated, then A+B is exactly a multiple of A [intuitive], like 2*A or 2.4*A. If you think of a normal (bell) or uniform (rectangle) distribution, you realize 2.4*A is proportionally magnified horizontally by a factor of 2.4, so the width of the distribution increases by 2.4, so stdev increases by 2.4. In Conclusion, stdev is additive.
“variance is additive” applicable in the multi-period iid context.
simple rule — variance of independent A + B is the sum of the variances.
 0 correlation is sufficient
–Case: generalized — http://www.stat.ucla.edu/~hqxu/stat105/pdf/ch01.pdf P27 Eq5-36 is a good generalized formula.
V(A+B) = V(A) + V(B) + 2 Cov(A,B) …. easiest form
2*Cov(A,B) := 2ρ √V(A)V(B)
V( 7A ) = 7*7 V(A)