What is the deal with using the approximation of standard deviation of a portfolio of combined assets? When dealing with combined corner portfolios, Schweser uses an approximation of the weighted average of the standard deviations of the two corner portfolios being combined. However, in calculating the standard deviation of active risk they make you use the long method and basically truncate the equation by assuming the correlation between the assests is zero. Please tell me if I missing something, otherwise there appears to be no consistent application of the use of the approximation method. Short Approximation method: σ(ab) = Wa•σa + Wb•σb Long Method: σ(ab) = √ [(Waˆ2•σaˆ2) + (Wbˆ2•σbˆ2)] Both of these methods assume that correlation is zero so the 2WaWbρσaσb(a,b) goes to zero correct?
Son of a B#$ch! Ha. The greek letters turn to mush the second you hit post. Hey analyst forum… this is a finance and math forum. Time to upgrade the text software on this site.
Haven’t you learned yet? There’s the right way, the wrong way, and the CFAI way. Agree that this is a glaring inconsistency, but I’m just going to roll with it.
from the problems I have seen, if they want you using the long formula, they will give you SD and Covariance. If they don’t, they will give you expected returns and sharpe ratios.
active risk assumes managers have zero correlation. corner portfolios assume correlation of 1. That is the big difference.
I was just reading about this in the CFAI book.-approximation is acceptable when doing MVO procedures. Note however that the weighted method produces an “upper limit” on your standard deviation because it assume asset correlations are 1
L3BeatIt Wrote: ------------------------------------------------------- > I was just reading about this in the CFAI > book.-approximation is acceptable when doing MVO > procedures. Note however that the weighted method > produces an “upper limit” on your standard > deviation because it assume asset correlations are > zero. This is not correct. The upper limit of diversification would be a correlation of 1 (linear combination). Anything less than perfect correlation indicates diversification benefits and the efficient frontier begins to bend with that familiar shape we all know.
MWVT9, thanks for this clarification. I think I am understand the case in the longer equation (e.g. active risk) but can you elaborate on how the correlation of one affects the approximation method? Thanks! I would greatly appreciate that info.
for corner portfolios, you are assuming no diversification benefit (i.e. a straight line between the portfolios, not the curve from the efficient frontier due to sign constraint, etc.) because they’re distinct portfolio options, so it’s linear. when you have different portfolios that you’re mixing together (like trying to decompose total/active risk) you have to do the ‘long way’
Is the moral of this story, “On the exam, if you’re asked for the standard deviation of combined corner portfolios, just give them a weighted average”? That’s what I thought we’re supposed to do.
Thanks everyone. This helps.
LordJeffrey Wrote: ------------------------------------------------------- > MWVT9, thanks for this clarification. I think I am > understand the case in the longer equation (e.g. > active risk) but can you elaborate on how the > correlation of one affects the approximation > method? Thanks! I would greatly appreciate that > info. Both of them use the full equation for variance (standard deviation squared) that we are used to, but they have just simplified them. Variance=wi^2var,i+wj^2var,j+2*w,i*w,j*sd,i*sd,j*corrlation,i,j Square root of variance=standard deviation If you assume a zero correlation the third term is zero so you only need the first two terms If you assume correlation of 1, you can use the formula above (try it to convince yourself) or you can just use a weighted average of the two standard deviations. You will get the same answer.
ng30 Wrote: ------------------------------------------------------- > Is the moral of this story, “On the exam, if > you’re asked for the standard deviation of > combined corner portfolios, just give them a > weighted average”? That’s what I thought we’re > supposed to do. That is what I will do. Not that the reason is important, but I think the idea here is that the efficient frontier moves from a smooth curved line when there are no constraints (no corner ports) to a bunch of straight lines between corner portfolio when constrained (no longer smooth). I think this is why we assume corr of 1 here.
Good points.
McLeod81 Wrote: ------------------------------------------------------- > To the contrary, they are assuming that there are > no diversification benefits from holding two (or > more) asset classes. (corr = 1) > > stdev = (0.6)*(0.12) + (0.4)*(0.1) = 11.2% > > stdev = [(0.6^2)*(0.12^2) + (0.4^2)*(0.10^2) + 2*0.6*0.4*0.12*0.10*1] ^ (1/2) = 11.2% > > So, this assumption actually give you a more > conservative (higher) portfolio stdev. McLeod81 Wrote: ------------------------------------------------------- > Core-satellite: > Since we are dealing with ‘active risk’ and not > ‘total risk’ (stdev), we assume zero correlation > because the managers active risk (tracking error) > should be completely unrelated to one another. > > > Corner Portfolios: > In reality the correlation between asset classes > is less than one. We assume that they are > perfectly correlated (corr=1) w/no diversification > benefits in order to take a more conservative > route when determining whether the portfolio meets > risk guidelines.
Isn’t it the same? When the correlation is 1, the equation that includes the correlation givs the weighted averages of Std Dev.
sailawaycfa Wrote: ------------------------------------------------------- > Isn’t it the same? When the correlation is 1, > the equation that includes the correlation givs > the weighted averages of Std Dev. Yes.
But the active risk formula does not give you the weighted averages of stdev, because it assumes correlation among active returns = 0. StdevActive = [sum[(W^2)*(stdevA^2)] ] ^ (1/2)
Agree^