Portfolio Standard Deviation Question part, standard, because, average

sabre

Portfolio Standard Deviation Question:
On Page 178 of Secret Sauce it says that if the two assets in a two-asset portfolio are perfectly positively correlated (p = +1) then the standard deviation of the portfolio will just be the weighted standard deviations of each asset.  Isn’t this wrong?  My thinking is that if the correlation is ZERO then the portfolio would be the weighted average of the standard deviations, because a correlation of zero would elimanate the 3rd part of the portfolio standard deviation formula (it would make the 3rd part which is 2(w1)(w2)(SD1)(SD2)(P) equal to zero because P=0 and would leave behind the weighted SD)
Can somebody please explain this? or is this a type in the secret sauce?

pimpineasy

No… the statement is correct. Variance formula for two assets is (w1*sd1)^2 + (w2*sd2)^2 + 2*w1*w2*sd1*sd2*rho. If rho = 1, then this formula reduces to (w1*sd1 + w2*sd2)^2. Standard deviation is the square root, w1*sd1 + w2*sd2.

pimpineasy

Why does rho=1 eliminate the entire third section from the equation though.. wouldn’t rho=0 do this?

bulosehi

You are thinking that the formula for variance is (w1*sd1 + w2*sd2)^2 + 2*w1*w2*sd1*sd2*rho. This is wrong (see above).

andytrader

I appreciate your help but all I’m seeing is that applying a coefficient of zero to the end of the formula would eliminate the last section entirely whereas applying a coefficient of +1 would keep it as some positive value

kickthatcfa

I give up.

kmf229

(match+box)^2 = match^2 + box^2 + 2 match box
now match=w1*sd1
box=w2*sd2
and the 3rd part you are so kindly talking about refers to 2 * match * box * rho.
if rho = 1 - variance becomes (match + box)^2
if rho=0 - variance becomes match^2 + box^2
now figure…
and also closely see the brackets = the way you are thinking about it - and how ohai above has correctly explained it.

morebeans

Got it now.  Thank you both.