B. Assume there are only two assets in a portfolio. If this portfolio has a positive weight for each asset, can its (portfolio.s) variance be greater than the variance of returns on the asset in the portfolio that has the higher variance of the two? Explain. Can the variance of the portfolio be smaller than the variance of returns on the asset in the portfolio that has the smaller variance? Explain.
I'm looking for a mathematical proof along with theory to explain
Assuming the two assets are A and B, then
Asset A has a variance of Va, a standard deviation of SDa and a share of Wa;
Asset B has a variance of Vb, a standard deviation of SDb and a share of Wb;
Let Va > Vb,
and since SD = SQRT(V),
SDa > SDb
Also, by definition, Wa+Wb = 1
The variance of the portfolio is:
Variance (P) = Variance(A) * (Wa)^2 + Variance(B) * (Wb)^2 +2(Wa)*(Wb)*Covariance(A,B)
Variance (P)= Va* (Wa)^2 + Vb * (Wb)^2 +2 Wa * Wb* SDa * SDb) *Corr(A,B)
(* where "^2" means "squared", and Corr(A,B) is the Correlation ...
Assets/variance/portfolio question is solved.