Question

You are allocating money equally among 500 stocks. You believe all stocks have the same standard deviation of 35% and all stocks have the same pair wise correlation.

What is the standard deviation of the portfolio of 500 stocks if the correlation is 0?

What is the standard deviation of the portfolio of 500 stocks if the correlation is .8?

Much appreciated if I could get an explanation along with the solution! thank you

Answer 1

1) let X=X1+X2+X3+...+X₅₀₀

also Cov(Xi,Xj)=r*SD(Xi)*(Xj)=0*35*35=0

here as we know that Var(X)=Var(X1)+Var(X2)+Var(X3)+...Var(X500)+2*Cov(X1,X2)+2Cov(X1,X3)+...+2 Cov(X499,500)

=35²+35²+35²+..+35²+2*0+2*0+..+2*0 =35²*500=612500

hence standard deviation =sqrt(612500)=782.62

2) Cov(Xi,Xj)=r*SD(Xi)*(Xj)=0.8*35*35=980

Var(X)=Var(X1)+Var(X2)+Var(X3)+...Var(X500)+2*Cov(X1,X2)+2Cov(X1,X3)+...+2 Cov(X499,500)

=35²+35²+35²+..+35²+2*980+2*980+..+2*980 =35²*500+2*980*124750=245122500

hence standard deviation =sqrt(245122500)=15656.39