Tuesday, October 28, 2014

intuitive - stdev(A+B) when independent ^ 100% correlated

(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.

--Case: independent
"variance is additive" applicable in the multi-period iid context.

simple rule -- variance of independent[1] A + B is the sum of the variances.

[1] 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)


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