Variance of Common Stocks?

Question

Explain why the variance of a portfolio of common stocks is not just the weighted average of the individual stock's variances. Under what ONE circumstance would it be?

Answer 1

Conceptually you can't use the weighted sum of the variances because each stock's return can vary either up *or down* by some amount, so in a portfolio of more than one stock, the variances of some stocks can offset that of others that moved in the opposite direction (especially if their returns are negatively correlated). For this reason, the variance of a portoflio of stocks usually goes *down* with diversification.

The only case where you *could* use the sum of all variances would be if the variance of all stocks were 0, but that never happens in the real world.