Maximum risk reduction

[opcfkam]

Could anyone explain
1) if average correlation is 0.5, a portfolio of only 10 stocks get us to within 10% of that for the minimum variance portfolio (how to calc the result is 10 stocks and 10% of min variance portfolio???)
2) if average correlation is 0.1, a portfolio of only 90 stocks get us to within 10% of that for the minimum variance portfolio (how to calc the result is 10 stocks and 10% of min variance portfolio???)
Knowing that the above is derived from below 2 formula, but i don’t know how to interpret????
maximum risk reduction => variance protfolio approximate equal to average variance of asset * (correlation)
variance(p) = avg variance(asset)* [(1 - correlation)/n + correlation] )

 

[maratikus]

For simplicity consider equally weighted portfolios of stocks with the same variance (covariance of i = j: Var(i)=Covar(i,i)) and same covariance (for i j).
Var(x1+…xn)=sum(i,j): (1/n)*(1/n)*Covar(i,j)=sum(i,j=i): (1/n)*(1/n)*Covar(i,j)+sum(i,j not equal to i): (1/n)*(1/n)*Covar(i,j)=(1/n)*(1/n)*n*Var(i)+(1/n)*(1/n)*(n*(n-1))*Covar(i,j)=
Var(i)/n+((n-1)/n)*Covar(i,j)=Var(i)*(1/n+(1-1/n)*Cor(i,j))=Var(i)*((1-cor)/n+cor)
Maximum reduction is n-> inf -> min variance(p)=Var(i)*cor.
10% of min variance is reached at n: (1-cor)/n+cor = 1.1*cor -> n = (1-cor)/(0.1*cor)
does that help?