I am looking at the CFAi text and it is not clear if we can use linear aproximation for the standard deviation.
by that I mean if you need 50% portfolio A abd 50% portfolio B
is the Standard deviation 0.5sigmaA + 0.5sigmaB ???
yes
I was thinking that exact same thing yesterday. I looked at Schweser and they said to do linear approximation (i.e. weighted average) for the calculation of the standard deviation.
They said it will slightly overstate the standard devation – don’t know what CFAI officially says, hope this helps
yes thanks guys
^ the linear approx will always overstate the true stdev. this is a mathematical certainty. the real stdev of a ptf is:
[(WaSa)^2 + (WbSb)^2 + 2*rho*Wa*Wb*Sa*Sb]^1/2
assuming rho = 1 you can rearrange the above to be
[(WaSa + WbSb)^2]^1/2 = WaSa + WbSb, our linear approx.
rho must be less than or equal to 1 so our ptf stdev cannot be greater than the linear approx, and since rho is generally less than 1, it will be less than the linear approx. when we construct the ptf with the corner ptf we implicitly assume rho = 1 to get the stdev. remember when rho is 1 our frontier is a straight line connecting the two ptfs! which are doing
I think there’s an implicit assumption that the two corner portfolios are “close”. So, since the weights of the two porfolios are therefore also going to be close, the correlations will be close to one. So, the linear approximation of the std deviation should be a pretty close estimate of true s.d.